A comparison of discrete element simulations

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各向异性对软土力学特性影响的离散元模拟

DOI ：10.16030/ki.issn.1000-3665.202006027各向异性对软土力学特性影响的离散元模拟赵　洲1，宋　晶1,2,3，刘锐鸿1 ，杨守颖1 ，李志杰1（1. 中山大学地球科学与工程学院，广东广州　510275；2. 广东省地球动力作用与地质灾害重点实验室，广东广州　510275；3. 广东省地质过程与矿产资源探查重点实验室，广东广州　510275）摘要：软土预压工程中，初始和诱发各向异性对软土力学性质的影响十分显著，而现有研究缺乏对初始和诱发各向异性的统一研究方法。

采用离散单元法，以颗粒长宽比作为定量评价指标，构建真实形态的颗粒模型，生成5组不同沉积角的初始各向异性试样，并进行竖直和水平两方向加载的双轴模拟实验，研究了初始各向异性和诱发各向异性对软土力学特性影响；在细观层面，以颗粒为对象研究了颗粒接触形式和转动角度的变化规律，以接触为对象研究了配位数和接触法向各向异性的发展趋势，在此基础上探究抗剪强度指标与各向异性关系。

结果表明：初始和诱发各向异性共同影响试样力学性质，当加载方向和软土沉积方向垂直时，土体有最大的峰值强度。

颗粒接触形式中面面接触的比例随加载的进行逐渐增大，并影响着试样初始模量和抗剪强度，配位数和接触法向各向异性受颗粒接触形式的影响有不同的演化规律，并在加载后期趋于稳定；同时，初始各向异性试样相较各向同性试样有更大的黏聚力，诱发各向异性主要影响试样内摩擦角，进而影响试样抗剪强度。

关键词：软土；各向异性；峰值应力；抗剪强度指标；细宏观性质中图分类号：TU411.3 文献标志码：A 文章编号：1000-3665（2021）02-0070-08Discrete element simulation of the influence of anisotropy on themechanical properties of soft soilZHAO Zhou 1，SONG Jing1,2,3，LIU Ruihong 1 ，YANG Shouying 1 ，LI Zhijie1（1. School of Earth Sciences and Engineering , Sun Yat-Sen University , Guangzhou , Guangdong 510275, China ；2. Guangdong Provincial Key Lab of Geodynamics and Geohazards , Guangzhou , Guangdong 510275, China ；3. Guangdong Provincial Key Laboratory of Geological Processes and Mineral Resource Exploration ,Guangzhou , Guangdong 510275, China ）Abstract ：The initial and induced anisotropy has a significant effect on the mechanical properties of soft soil in preloading engineering. However, there is a lack of unified research methods for the initial and induced anisotropy. Discrete element method is adopted in this study, and the length-width ratio of particles is used as the quantitative evaluation index. Five types of initial anisotropy samples with different deposition angles are generated. The effects of initial anisotropy and induced anisotropy on the mechanical properties of soft soil are studied by vertical and horizontal loading. At the micro level, the contact form and rotation angle of particles are examined from the point of view of particles, and the development trend of coordination number and contact normal to anisotropy is studied from the point of view of contact. The relationship between shear strength index收稿日期：2020-06-12；修订日期：2020-08-04基金项目：国家自然科学基金项目（41877228；41402239；41877229）；广东省自然科学基金项目（2019A1515010554）；广州市科技计划项目（201904010136）第一作者：赵洲（1995-），男，硕士研究生，主要从事软土微观结构及数值模拟。

the mathematical theory of finite element method

the mathematical theory of finiteelement methodThe mathematical theory of the finite element method (FEM) is a branch of numerical analysis that provides a framework for approximating solutions to partial differential equations (PDEs) using discretization techniques. The finite element method is widely used in engineering and scientific disciplines to simulate and analyze physical phenomena.At the core of the FEM is the concept of dividing a domain into a finite number of elements, which are connected at nodes. The unknown solution within each element is approximated using a simple function, referred to as the basis function. These basis functions are usually polynomials of a certain degree, and their coefficients are determined by solving a set of linear equations.The mathematical theory of the FEM involves several key concepts and techniques. One of the fundamental principles is the variational formulation, which transforms the PDE into an equivalent variational problem. This variational problem is then discretized using the finite element approximation, resulting in a system of algebraic equations.Another important aspect is the assembly process, where the contributions from each element are combined to form the global stiffness matrix and right-hand side vector. This assembly is based on the integration of the basis functions and their derivatives over the element domains.Error estimation and convergence analysis are also essential components of the mathematical theory of the FEM. Various techniques, such as the energy method and the posteriori error estimators, are used to assess the accuracy of the finite element solution and to determine the appropriate mesh refinement for achieving convergence.Furthermore, the mathematical theory of the FEM includes the treatment ofboundary conditions, imposition of symmetries, and the development of efficient solvers for the resulting linear systems. It also addresses issues such as numerical stability,并行 computing, and adaptivity.In summary, the mathematical theory of the finite element method provides a comprehensive framework for numerically solving PDEs. It encompasses concepts such as element discretization, variational formulation, assembly, error estimation, and convergence analysis, which collectively enable the accurate and efficient simulation of a wide range of physical problems.。

Statistical mechanics of two-dimensional vortices and stellar systems

a r X iv:c o n d -ma t/212223v1[c ond-m at.stat-m ec h]1D ec22Statistical mechanics of two-dimensional vortices and stellar systems Pierre-Henri Chavanis Laboratoire de Physique Quantique,Universit´e Paul Sabatier,118,route de Narbonne 31062Toulouse,France Abstract.The formation of large-scale vortices is an intriguing phenomenon in two-dimensional turbulence.Such organization is observed in large-scale oceanic or atmo-spheric ﬂows,and can be reproduced in laboratory experiments and numerical simula-tions.A general explanation of this organization was ﬁrst proposed by Onsager (1949)by considering the statistical mechanics for a set of point vortices in two-dimensional hydrodynamics.Similarly,the structure and the organization of stellar systems (glob-ular clusters,elliptical galaxies,...)in astrophysics can be understood by developing a statistical mechanics for a system of particles in gravitational interaction as initiated by Chandrasekhar (1942).These statistical mechanics turn out to be relatively similar and present the same diﬃculties due to the unshielded long-range nature of the in-teraction.This analogy concerns not only the equilibrium states,i.e.the formation of large-scale structures,but also the relaxation towards equilibrium and the statistics of ﬂuctuations.We will discuss these analogies in detail and also point out the speciﬁcities of each system.1Introduction Two-dimensional ﬂows with high Reynolds numbers have the striking property of organizing spontaneously into coherent structures (the vortices)which dominate the dynamics [93](see Fig.1).The robustness of Jupiter’s Great Red Spot,a huge vortex persisting for more than three centuries in a turbulent shear between two zonal jets,is probably related to this general property.Some other coherent structures like dipoles (pairs of cyclone/anticyclone)and sometimes tripoles have been found in atmospheric or oceanic systems and can persist during several days or weeks responsible for atmospheric blocking.Some astrophysicists invoke the existence of organized vortices in the gaseous component of disk galaxies in relation with the emission of spiral density waves [99].It has also been proposed that planetary formation might have begun inside persistent gaseous vortices born out of the protoplanetary nebula [5,121,15,60,33](see Fig.2).As a result,hydrodynamical vortices occur in a wide of geophysical or astrophysical situations and their robustness demands a general understanding.Similarly,it is striking to observe that self-gravitating systems follow a kindof organization despite the diversity of their initial conditions and their en-vironement [9](see Fig.3).This organization is illustrated by morphological classiﬁcation schemes such as the Hubble sequence for galaxies and by simple2Pierre-Henri Chavanisrules which govern the structure of individual self-gravitating systems.For ex-ample,elliptical galaxies display a quasi-universal luminosity proﬁle described by de Vaucouleur’s R1/4law and most of globular clusters are well-ﬁtted by the Michie-King model.On the other hand,theﬂat rotation curves of spiral galaxies can be explained by the presence of a dark matter halo with a density proﬁle decreasing as r−2at large distances.The fractal nature of the interstellar medium and the large scale structures of the universe also display some form of organization.Fig.1.Self-organization of two-dimensional turbulentﬂows into large-scale vortices [93].These vortices are long-lived and dominate the dynamics.The question that naturally emerges is what determines the particular con-ﬁguration to which a self-gravitating system or a large-scale vortex settles.It is possible that their actual conﬁguration crucially depends on the conditions that prevail at their birth and on the details of their evolution.However,in view of their apparent regularity,it is tempting to investigate whether their organi-zation can be favoured by some fundamental physical principles like those of thermodynamics and statistical physics.We ask therefore if the actual states of self-gravitating systems in the universe and coherent vortices in two-dimensional2D vortices and stellar systems3Fig.2.A scenario of planet formation inside large-scale vortices presumably present in the Keplerian gaseous disk surrounding a star at its birth.Starting from a random vorticityﬁeld,a series of anticyclonic vortices appears spontaneously(upper panel). Due to the Coriolis force and to the friction with the gas,these vortices can eﬃciently trap dust particles passing nearby(lower pannel).The local increase of dust concen-tration inside the vortices can initiate the formation of planetesimals and planets by gravitational instability.This numerical simulation is taken from[15].turbulentﬂows are not simply more probable than any other possible conﬁgura-tion,i.e.if they cannot be considered as maximum entropy states.This statistical mechanics approach has been initiated by Onsager[101]for a system of point vortices and by Chandrasekhar[21]in the case of self-gravitating systems.It turns out that the statistical mechanics of two-dimensional vortices and self-gravitating systems present a deep analogy despite the very diﬀerent physical nature of these systems.This analogy was pointed out by Chavanis in[29,32,35] and further developed in[54,30,34,36,47,48].In the following,we will essentially discuss the statistical mechanics of2D vortices and refer to the review of Pad-manabhan[103](and his contribution in this book)for more details about the4Pierre-Henri Chavanisrge-scale structures in the universe as observed with the Hubble space tele-scope.The analogy with Fig.1is striking and will be discussed in detail in this paper. statistical mechanics of self-gravitating systems.We will see that the analogy be-tween two-dimensional vortices and(three-dimensional)self-gravitating systems concerns not only the prediction of the equilibrium state,i.e.the formation of large-scale structures,but also the statistics ofﬂuctuations and the relaxation towards equilibrium.This paper is organized as follows.In Sec.2,we discuss the statistical me-chanics of point vortices introduced by Onsager[101]and further developed by Joyce&Montgomery[70]and Pointin&Lundgren[107]among others(see a complete list of references in the book of Newton[98]).We discuss the existence of a thermodynamic limit in Sec.2.7and make the connexion withﬁeld theory. Statistical equilibrium states of axisymmetricﬂows are obtained analytically in Sec.2.8-2.9.The relation with equilibrium states of self-gravitating systems is shown in Sec.2.10.In Sec.3,we discuss the statistics of velocityﬂuctuations pro-duced by a random distribution of point vortices and use this stochastic approach to obtain an estimate of the diﬀusion coeﬃcient of point vortices.Application to2D decaying turbulence is considered in Sec.3.4.In Sec.4,we describe the relaxation of a point vortex in a thermal bath and analyze this relaxation in terms of a Fokker-Planck equation involving a diﬀusion and a drift.In Sec.5,we develop a more general kinetic theory of point vortices.A new kinetic equation is obtained which satisﬁes all conservation laws of the point vortex system and increases the Boltzmann entropy(H-theorem).We mention the connexion with2D vortices and stellar systems5 the kinetic theory of stars developed by Chandrasekhar[21].In Sec.6,we dis-cuss the violent relaxation of2D vortices and stellar systems.We mention the analogy between the Vlasov and the Euler equations and between the statistical approach developed by Lynden-Bell[90]for collisionless stellar systems and by Kuz’min[83],Miller[95]and Robert&Sommeria[111]for continuous vorticity ﬁelds.The concepts of“chaotic mixing”and“incomplete relaxation”are dis-cussed in the light of a relaxation theory in Sec.6.3.Application of statistical mechanics to geophysicalﬂows and Jupiter’s Great Red Spot are evocated in Sec.6.4.2Statistical mechanics of point vortices intwo-dimensional hydrodynamics2.1Two-dimensional perfectﬂowsThe equations governing the dynamics of an invisicidﬂow are the equation of continuity and the Euler equation:∂ρ∂t +(u∇)u=−16Pierre-Henri Chavaniswhere ∆=∂2xx +∂2yy is the Laplacian operator.In an unbounded domain,thisequation can be written in integral form asψ(r ,t )=−12πz × ω(r ′,t )r −r ′∂t +u ∇ω=0.(10)This corresponds to the transport of the vorticity ωby the velocity ﬁeld u .It is easy to show that the ﬂow conserves the kinetic energyE =u 22 (∇ψ)2d 2r =12 ωψd 2r ,(12)where the second equality is obtained by a part integration with the condition ψ=0on the boundary.Therefore,E can be interpreted either as the kinetic energy of the ﬂow (see Eq.(11))or as a potential energy of interaction between vortices (see Eq.(12)).2.2The point vortex gasWe shall consider the situation in which the velocity is created by a collection of N point vortices.In that case,the vorticity ﬁeld can be expressed as a sum of δ-functions in the formω(r ,t )=N i =1γi δ(r −r i (t )),(13)where r i (t )denotes the position of point vortex i at time t and γi is its circulation.According to Eqs.(9)(13),the velocity of a point vortex is equal to the sum of the velocities V (j →i )produced by the N −1other vortices,i.e.V i = j =i V (j →i )with V (j →i )=−γj |r j −r i |2.(14)2D vortices and stellar systems7 As emphasized by Kirchhoﬀ[79],the above dynamics can be cast in a Hamil-tonian formγi dx i∂y i,γidy i∂x i,(15) H=−12m).Thisis related to the particular circumstance that a point vortex is not a material particle.Indeed,an isolated vortex remains at rest contrary to a material particle which has a rectilinear motion due to its inertia.Point vortices form therefore a very peculiar Hamiltonian system.Note also that the Hamiltonian of point vortices can be either positive or negative(in the case of vortices of diﬀerent signs)whereas the kinetic energy of theﬂow is necessarily positive.This is clearlya drawback of the point vortex model.2.3The microcanonical approach of Onsager(1949)The statistical mechanics of point vortices wasﬁrst considered by Onsager[101] who showed the existence of negative temperature states at which point vortices of the same sign cluster into“supervortices”.He could therefore explain the formation of large,isolated vortices in nature.This was a remarkable anticipation since observations were very scarce at that time.Let us consider a liquid enclosed by a boundary,so that the vortices are conﬁned to an area A.Since the coordinates(x,y)of the point vortices are canonically conjugate,the phase space coincides with the conﬁguration space and isﬁnite:dx1dy1...dx N dy N= dxdy N=A N.(17)This striking property contrasts with most classical Hamiltonian systems con-sidered in statistical mechanics which have unbounded phase spaces due to the presence of a kinetic term in the Hamiltonian.As is usual in the microcanonical description of a system of N particles,we introduce the density of statesg(E)= dx1dy1...dx N dy Nδ E−H(x1,y1,...,x N,y N) ,(18)8Pierre-Henri Chavaniswhich gives the phase space volume per unit interaction energy E.The equilib-rium N-body distribution of the system,satisfying the normalization condition µ(r1,...,r N)d2r1...d2r N=1,is given byµ(r1,...,r N)=1T =dS2π i<jγiγj ln|r i−r j|.(22)2D vortices and stellar systems9 Making the change of variable x=r/R,weﬁnd that g(E,V)=V N g(E′,1) with E′=E+1βV 1+ββV 1+γ2(N−1)γ2.(25)We shall see in Sec.2.8that this negative critical inverse temperature is the minimum inverse temperature that the system can achieve.If,on the other hand,we consider a neutral system consisting of N/2vortices of circulationγand N/2vortices of circulation−γ,weﬁndP=N8π .(26)This result is well-known is plasma physics[113].The critical temperature at which the pressure vanishes is now positiveβc=8π∂t =−Ni=1γ∇δ(r−r i(t))V i.(28)Since V i=u(r i(t),t),we can rewrite the foregoing equation in the form∂ω10Pierre-Henri ChavanisSince the velocity is divergenceless,we obtain∂ω4π i=jγ2 ln|r i−r j|1=−2D vortices and stellar systems 11whereg (r 1,r 2,t )=µ(r 1,...,r N ,t )d 3r 2...d 2r N ,(36)is the two-body distribution function.In the mean-ﬁeld approximation,whichis exact in a properly deﬁned thermodynamic limit with N →+∞(see Sec.2.7),we haveg (r 1,r 2,t )=P (r 1,t )P (r 2,t ).(37)Accounting that N (N −1)≃N 2for large N ,the average energy takes the formE =−12ω ψd 2r =u 2n i !.(40)The logarithm of this number deﬁnes the Boltzmann ing Stirlingformula and considering the continuum limit in which ∆,ν→0,we get the classical formula S =−NP (r )ln P (r )d 2r ,(41)where P (r )is the density probability that a point vortex be found in the sur-face element centered on r .At equilibrium,the system is in the most probablemacroscopic state,i.e.the state that is the most represented at the microscopic level.This optimal state is obtained by maximizing the Boltzmann entropy (41)at ﬁxed energy (39)and vortex number N ,or total circulationΓ=Nγ=ω d 2r .(42)12Pierre-Henri ChavanisWriting the variational principle in the formδS−βδE−αδΓ=0,(43) whereβandαare Lagrange multipliers,it is readily found that the maximum entropy state corresponds to the Boltzmann distributionω =Ae−βγψ,(44) with inverse temperatureβ.We can account for the conservation of angular momentum L= ω r2d2r(in a circular domain)and impulse P= ω yd2r (in a channel)by introducing appropriate Lagrange multipliersΩand U foreach of these constraints.In that case,Eq.(44)remains valid provided that we replace the streamfunctionψby the relative streamfunctionψ′=ψ+Ω2π i<j ln|r i−r j|2D vortices and stellar systems13 where the potential of interaction has been normalized by R.For simplicity, we have ignored the contribution of the images but this shall not aﬀect the ﬁnal results.We now introduce the change of variables x=r/R and deﬁne the functionu(x1,...,x N)=1N2γ2.(49) In terms of these quantities,the density of states can be rewritteng(E)=2πV NNu(x1,...,x N) .(50)The proper thermodynamic limit for a system of point vortices with equal cir-culation in the microcanonical ensemble is such that N→+∞withﬁxedΛ.Wesee that the box size R does not enter in the normalized energyΛ.Therefore,the thermodynamic limit corresponds to N→+∞withγ∼N−1→0and E∼1. This is a very unusual thermodynamic limit due to the non-extensivity of thesystem.Note that the total circulationΓ=Nγremainsﬁxed in this process.For suﬃciently large N,the density of states can be writteng(E)≃ Dρe NS[ρ]δ Λ−E[ρ]) δ 1− ρ(r)d2r ,(51) withS[ρ]=− ρ(r)lnρ(r)d2r,(52) E[ρ]=−114Pierre-Henri Chavaniswhich is the normalization factor in the Gibbs measure1µ(r1,...,r N)=ln Z.(57)βUsing the notations introduced previously,we can rewrite the integral(55)in the formZ(β)=V N N0... N0N i=1d2x i eηu(x1,...,x N),(58)whereβNγ2η=2D vortices and stellar systems15 and phase transitions occur.This is the case,in particular,for the gravitational problem(see Sec.2.10).Finally,the grand canonical partition function is deﬁned byZ GC=+∞ N=0z N2(∇ξ)2−ρ(r)ξ(r)}d2r=e−12(∇ξ)2d2r+√−βγξ(r),we can easily carry out the summation on N to obtainZ GC= Dφe−12(∇φ)2−µ2eφ(r)},(67)T eff=−βγ2,µ2=−zβγ2.(68) Therefore,the grand partition function of the point vortex gas corresponds to a Liouvilleﬁeld theory with an actionS[φ]=12(∇φ)2−µ2eφ(r)}.(69)While the previous description is formally correct if we deﬁne Z and Z GC by Eqs.(55)and(64),it must be noted however that the canonical and grand canon-ical ensembles may not have a physical meaning for point vortices.In particular, it is not clear how one can impose a thermal bath at negative temperature. On the other hand,the usual procedure to derive the canonical ensemble from the microcanonical ensemble rests on a condition of additivity which is clearly lacking in the present case.2.8Axisymmetric equilibrium states in a diskLet us consider a collection of N point vortices with circulationγconﬁned within a disk of radius R.At statistical equilibrium,the streamfunctionψis solution of the Boltzmann-Poisson equation(45).If we work in a circular domain,we must in principle account for the conservation of angular momentum.This can lead to bifurcations between axisymmetric and oﬀ-axis solutions[117].We shall,16Pierre-Henri Chavanishowever,ignore this constraint for the moment in order to obtain analytical ex-pressions for the thermodynamical parameters.This is a suﬃcient approximation to illustrate the structure of the problem,which is our main concern here.If we conﬁne our attention to axisymmetric solutions,the Boltzmann-Poisson equation(45)can be written1dr r dψξddξ =λe−φ,(72)φ(0)=φ′(0)=0,(73)withλ=1ifβ<0andλ=−1ifβ>0.It turns out that this equation can be solved analytically as noticed by a number of authors.With the change of variables t=lnξandφ=2lnξ−z,Eq.(72)can be rewrittend2zdz(λe z).(74)This corresponds to the motion of aﬁcticious particle in a potential V(z)=λe z. This equation is readily integrated and,returning to original variables,weﬁnally obtaine−φ=18ξ2)2.(75)From the circulation theorem(6)applied to an axisymmetricﬂow,we have−dψ2πr.(76) whereΓ(r)= r0ω(r′)2πr′dr′is the circulation within r.Taking r=R and introducing the dimensionless variables previously deﬁned,we obtainη≡βγΓπR2(η+4)1η+4r22D vortices and stellar systems 17At positive temperatures (η>0),the vorticity is an increasing function of the distance and the vortices tend to accumulate on the boundary of the domain (Fig.4).On the contrary at negative temperatures (η<0),the vorticity is a decreasing function of the distance and the vortices tend to group themselves in the core of the domain to form a “supervortex”(Fig.5).These results are consistent with Onsager’s prediction [101].We also conﬁrm that statistical equilibrium states only exist for η>ηc =−4,as previously discussed.At this critical temperature,the central vorticity becomes inﬁnite and the solution tends to a Dirac peak:ω (r )→Γδ(r ),for η→ηc =−4.(79)00.20.40.60.81r/R1234567<ω>/ω∗η>0η=0η=20Fig.4.Statistical equilibrium states of point vortices at positive temperatures (ω∗=Γ/πR 2).The vortices are preferentially localized near the wall.The energy deﬁned by Eq.(39)can be written in the dimensionless formΛ≡2πE2η2 α0φ′(ξ)2ξdξ.(80)The integral can be carried out explicitly using Eq.(75).Eliminating αbetweenEqs.(80)and (77),we ﬁnd that the temperature is related to the energy by the equation of state Λ=1ηln418Pierre-Henri Chavanis00.20.40.60.81r/R246810<ω>/ω∗η<0η=−2η=−3η=−3.5Fig.5.Statistical equilibrium states of point vortices at negative temperatures showing a clustering.For η=−4,the vortices collapse at the center of the domain and the vorticity proﬁle is a Dirac peak.0.250.50.751Λ=2πE/Γ2−10−8−6−4−20246810η=βγΓ/2πΛ0=1/8ηc =−4Fig.6.Equilibrium phase diagram (caloric curve)for point vortices with equal circula-tion conﬁned within a disk.For simplicity,the angular momentum has not been taken into account (Ω=0).2D vortices and stellar systems1900.250.50.751Λ=2πE/Γ2−7−6−5−4−3S /NΛ0=1/8η=0η>0η<0ηc =−4Fig.7.Entropy vs energy plot for a system of point vortices with equal circulation conﬁned within a disk.The entropy (41)can also be calculated easily from the above results.Within an unimportant additive constant,it is given byS ηln 4−1+820Pierre-Henri Chavanisaccount,the density of states is given byg (E,L )=δ E −H (r 1,...,r N ) δ L −N i =1γr 2i N i =1d 2r i ,(83)and the angular velocity of the ﬂow byΩ=2T∂SL ,oneﬁnds the exact resultΩ=2N8π.(85)Therefore,the vorticity ﬁeld is determined by the Boltzmann distributionω =Ae −βγψ′,(86)where ψ′is the relative streamfunctionψ′≡ψ+Ω4βL(4+η)r 2.(87)For η=0,one hasω =γNΓLr 2.(88)For large r ,the asymptotic behavior of Eq.(86)isω ∼14L(4+η)r 2(r →+∞),(89)where we have used ψ∼−(Γ/2π)ln r at large distances.From Eq.(89),one sees that η≥−4is required for the existence of an integrable solution.Inserting the relation (86)in the Poisson equation (7),we get−∆ψ′=Ae −2πη2πLη(4+η).(90)With the change of variablesξ=γNN 2γ2−2πηdξ2+1dξ=2πηe φ−(4+η),(92)2D vortices and stellar systems21 and the vorticity(86)becomesω =N2γ22πL,if r≤(2L/Γ)1/2,(94)and ω =0otherwise.This vortex patch is the state of minimum energy at ﬁxed circulation and angular momentum.Forη→−4,one has approximatelyω =N2γ2(1−AπηγN4L(4+η)r2.(95)Theﬁrst factor is an exact solution of Eq.(92)with the second term on the right hand side neglected(see Sec.2.8).The second factor is a correction for large r, in agreement with the asymptotic result expressed by Eq.(89).The parameter A tends to inﬁnity asη→−4and is determined from the condition ω d2r=Γby the formulaπA+ln(πA)=−C−ln 1+η|r j−r i|3,(97)where F(j→i)is the force created by star j on star i.The force can be written as the gradient F=−∇Φof a gravitational potentialΦwhich is related to the stellar densityρ(r,t)=Ni=1mδ(r−r i),(98)by the Poisson equation∆Φ=4πGρ.(99)22Pierre-Henri ChavanisFurthermore,the equations of motion(Newton’s equations)can be put in the Hamiltonian formm d r i∂v i,md v i∂r i,H=1|r i−r j|.(100)In the analogy between stellar systems and two-dimensional vortices,the star densityρplays the role of the vorticityω,the force F the role of the velocity V and the gravitational potentialΦthe role of the streamfunctionψ.The crucial point to realize is that,for the two systems,the interaction is a long-range un-shielded Coulombian interaction(in D=3or D=2dimensions).This makes the connexion between point vortices and stellar systems deeper than between point vortices and electric charges for example.In particular,point vortices can organize into large scale clusters,like stars in galaxies,while the distribution of electric charges in a neutral plasma is uniform.There are,on the other hand, fundamental diﬀerences between stars and vortices.In particular,a star creates an acceleration while a vortex creates a velocity.On the other hand,the gravi-tational interaction is attractive and directed along the line joining the particles while the interaction between vortices is rotational and perpendicular to the line joining the vortices.Despite these important diﬀerences,the statistical mechanics of2D vortices and stellar systems are relatively similar.Like the point vortex gas,the self-gravitating gas is described at statistical equilibrium by the Boltzmann distri-butionρ =Ae−βΦ,(101) obtained by maximizing the Boltzmann entropy atﬁxed mass M and energy E. Its structure is therefore determined by solving the Boltzmann-Poisson equation∆Φ=4πGAe−βΦ,(102) where A andβ>0have to be related to M and E.This statistical mechanics approach has been developed principally for globular clusters relaxing towards equilibrium via two-body encounters[9].It is clear that the Boltzmann-Poisson equation(102)is similar to the Boltzmann-Poisson equation(45)for point vor-tices at negative temperatures.The density proﬁle determined by these equations is a decreasing function of the distance,which corresponds to a situation of clus-tering(see Figs.5and8).The similarity of the maximum entropy problem for stars and vortices,and the Boltzmann-Poisson equations(102)(45),is aﬁrst manifestation of the formal analogy existing between these two systems.However,due to the diﬀerent dimension of space(D=3for stars instead of D=2for vortices),the mathematical problems diﬀer in the details.First of all,the density proﬁle determined by the Boltzmann-Poisson equation(102)in D=3decreases like r−2at large distances leading to the so-called inﬁnite mass2D vortices and stellar systems 23−50510ln(ξ)−20−15−10−50ln (e −ψ)singularsphereξ−2Fig.8.Density proﬁle of the self-gravitating gas at statistical equilibrium.The dashed line corresponds to the singular solution ρ=1/2πGβr 2.problem since M = +∞0ρ4πr 2dr →+∞[19].There is no such problem forpoint vortices in two dimensions:the vorticity decreases like r −4,or even more rapidly if the conservation of angular momentum is accounted for,and the totalcirculation Γ= +∞0ω2πrdr is ﬁnite.The inﬁnite mass problem implies thatno statistical equilibrium state exists for open star clusters,even in theory.A system of particles in gravitational interaction tends to evaporate so that the ﬁnal state is just two stars in Keplerian orbit.This evaporation process has been clearly identiﬁed in the case of globular clusters which gradually lose stars to the beneﬁt of a neighboring galaxy.In fact,the evaporation is so slow that we can consider in a ﬁrst approximation that the system passes by a succession of quasiequilibrium states described by a truncated isothermal distribution function (Michie-King model)[9].This justiﬁes the statistical mechanics approach in that sense.Another way of solving the inﬁnite mass problem is to conﬁne the system within a box of radius R .However,even in that case,the notion of equilibrium poses problem regarding what now happens at the center of the conﬁguration.The equilibrium phase diagram (E,T )for bounded self-gravitating systems is represented in Fig.9.The caloric curve has a striking spiral behavior parametrized by the density contrast R =ρ(0)/ρ(R )going from 1(homogeneous system)to +∞(singular sphere)as we proceed along the spiral.There is no equilibrium state below E c =−0.335GM 2/R or T c =GMm24Pierre-Henri ChavanisΛ=−ER/GM 20.51.52.5η=βG M /R Fig.9.Equilibrium phase diagram for self-gravitating systems conﬁned within a box.For suﬃciently low energy or temperature,there is no equilibrium state and the system undergoes gravitational collapse.−1−0.500.51 1.5Λ=−ER/GM 200.511.522.533.5η=βG M /R µ=105µ=104µ=103µ=102µ=10Fig.10.Equilibrium phase diagram for self-gravitating fermions [41].The degeneracy parameter µplays the role of a small-scale cut-oﬀǫ∼1/µ.For ǫ→0,the classical spiral of Fig.9is recovered.2D vortices and stellar systems25 mass goes to zero.Therefore,the singularity contains no mass and this process alone cannot lead to a black hole.Since the T(E)curve has turning points,this implies that the microcanon-ical and canonical ensembles are not equivalent and that phase transitions will occur[103].In the microcanonical ensemble,the series of equilibria becomes un-stable after theﬁrst turning point of energy(MCE)corresponding to a density contrast of709.At that point,the solutions pass from local entropy maxima to saddle points.In the canonical ensemble,the series of equilibria becomes unsta-ble after theﬁrst turning point of temperature(CE)corresponding to a density contrast of32.1.At that point,the solutions pass from minima of free energy (F=E−T S)to saddle points.It can be noted that the region of negative speciﬁc heats between(CE)and(MCE)is stable in the microcanonical en-semble but unstable in the canonical ensemble,as expected on general physical grounds.The thermodynamical stability of isothermal spheres can be deduced from the topology of theβ−E curve by using the turning point criterion of Katz[75]who has extended Poincar´e’s theory of linear series of equilibria.The stability problem can also be reduced to the study of an eigenvalue equation associated with the second order variations of entropy or free energy as studied by Padmanabhan[102]in the microcanonical ensemble and by Chavanis[37]in the canonical ensemble.This study has been recently extended to other statisti-cal ensembles[44]:grand canonical,grand microcanonical,isobaric....The same stability limits as Katz are obtained but this method provides in addition the form of the density perturbation proﬁles that trigger the instability at the critical points.It also enables one to show a clear equivalence between thermodynamical stability in the canonical ensemble and dynamical stability with respect to the Navier-Stokes equations(Jeans problem)[37,44].These analytical methods can be extended to general relativity[38].It must be stressed,however,that the statistical equilibrium states of self-gravitating systems are at most metastable: there is no global maximum of entropy or free energy for a classical system of point masses in gravitational interaction[2].Phase transitions in self-gravitating systems can be studied in detail by in-troducing a small-scale cut-oﬀǫin order to regularize the potential.This can be achieved for example by considering a system of self-gravitating fermions (in which case an eﬀective repulsion is played by the Pauli exclusion principle) [62,8,52,41,43]or a hard spheres gas[3,103,120].Other forms of regularization are possible[59,128,45].For these systems,there can still be gravitational collapse but the core will cease to shrink when it feels the inﬂuence of the cut-oﬀ.The result is the formation of a compact object with a large mass:a“fermion ball”or a hard spheres“condensate”.The equilibrium phase diagram of self-gravitating fermions is represented in Fig.10and has been discussed at length by Chavanis [41]in the light of an analytical model.The introduction of a small-scale cut-oﬀhas the eﬀect of unwinding the classical spiral of Fig.9.For a small cut-oﬀǫ≪1,the trace of the spiral is still visible and the T(E)curve is multivalued (Fig.11).This can lead to a gravitationalﬁrst order phase transition between a gaseous phase with an almost homogeneous density proﬁle(upper branch)and a。

UDEC中文指导说明

通用离散元用户指导（U D E C 3.1）2004.9目录1 引言 (1)1.1 总论 (1)1.2 与其他方法的比较 (2)1.3 一般特性 (2)1.4 应用领域 (3)2 开始启动 (4)2.1 安装和启动程序 (4)2.1.7 内存赋值 (4)2.1.9 运行UDEC (5)2.1.10 安装测试程序 (5)2.2 简单演示－通用命令的应用 (5)2.3 概念与术语 (6)2.4 UDEC模型：初始块体的划分 (8)2.5 命令语法 (9)2.6 UDEC应用基础 (10)2.6.1 块体划分 (10)2.6.2 指定材料模型 (16)2.6.2.1 块体模型 (16)2.6.2.2 节理模型 (17)2.6.3 施加边界条件和初始条件 (19)2.6.4 迭代为初始平衡 (21)2.6.5 进行改变和分析 (24)2.6.6 保存或恢复计算状态 (25)2.6.7 简单分析的总结 (25)2.8 系统单位 (26)3 用UDEC求解问题 (27)3.1 一般性研究 (27)3.1.1 第1步：定义分析模型的对象 (28)3.1.2 第2步：产生物理系统的概念图形 (28)3.1.3 第3步：建造和运行简单的理想模型 (28)3.1.4 第4步：综合特定问题的数据 (29)3.1.5 第5步：准备一系列详细的运行模型 (29)3.1.6 第6步：进行模型计算 (29)3.1.7 第7步：提供结果和解释 (30)3.2 产生模型 (30)3.2.1 确定UDEC模型合适的计算范围 (30)3.2.2 产生节理 (32)3.2.2.1 统计节理组生成器 (32)3.2.2.2 VORONOI多边形生成器 (34)3.2.2.3 例子 (34)3.2.3 产生内部边界形状 (35)3.3 变形块体和刚体的选择 (38)3.4 边界条件 (42)3.4.1 应力边界 (42)3.4.1.1 施加应力梯度 (43)3.4.1.2 改变边界应力 (44)3.4.1.3 打印和绘图 (44)3.4.1.4 提示和建议 (45)3.4.2 位移边界 (46)3.4.3 真实边界－选择合理类型 (46)3.4.4 人工边界 (46)3.4.4.1 对称轴 (46)3.4.4.2 截取边界 (46)3.4.4.3 边界元边界 (49)3.5 初始条件 (50)3.5.1 在均匀介质中的均匀应力：无重力 (50)3.5.2 无节理介质中具有梯度变化的应力：均匀材料 (51)3.5.3 无节理介质中具有梯度变化的应力：非均匀材料 (51)3.5.4 具有非均匀单元的密实模型 (52)3.5.5 随模型变化的初始应力 (53)3.5.6 节理化介质的应力 (54)3.5.7 绘制应力等值线图 (55)3.6 加载与施工模拟 (57)3.7 选择本构模型 (62)3.7.1 变形块体材料模型 (63)3.7.2 节理材料模型 (64)3.7.3 合理模型的选择 (65)3.8 材料性质 (71)3.8.1 岩块性质 (71)3.8.1.1 质量密度 (71)3.8.1.2 基本变形性质 (71)3.8.1.3 基本强度性质 (72)3.8.1.4 峰后效应 (73)3.8.1.5 现场性质参数的外延 (77)3.8.2 节理性质 (80)3.9 提示和建议 (81)3.9.1 节理几何形状的选择 (81)3.9.2 设计模型 (81)3.9.3 检查模型运行时间 (82)3.9.4 对允许时间的影响 (82)3.9.5 单元密度的考虑 (83)3.9.6 检查模型响应 (83)3.9.7 检查块体接触 (83)3.9.8 应用体积模量和剪切模量 (83)3.9.9 选择阻尼 (84)3.9.10 给块体和节理模型指定模型和赋值 (84)3.9.11 避免圆角误差 (85)3.9.12 接触嵌入 (85)3.9.13 非联结块体 (86)3.9.14 初始化变量 (86)3.9.15 确定坍塌荷载 (86)3.9.16 确定安全系数 (86)3.10 解释 (88)3.10.1 不平衡力 (88)3.10.2 块体/网格结点的速度 (88)3.10.3 块体破坏的塑性指标 (89)3.11 模拟方法 (89)3.11.1 有限数据系统模拟 (89)3.11.2 混沌系统的模拟 (90)3.11.3 局部化、物理的不稳定性和应力路径 (91)1 引言1.1 总论通用离散元程序（UDEC，Universal Distinct Element Code）是一个处理不连续介质的二维离散元程序。

Advances in discrete element modelling of underground

REVIEW ARTICLEAdvances in discrete element modelling of underground excavationsCarlos Labra ÆJerzy Rojek ÆEugenio On˜ate ÆFrancisco ZarateReceived:5November 2007/Accepted:6May 2008/Published online:17July 2008ÓSpringer-Verlag 2008Abstract The paper presents advances in the discrete element modelling of underground excavation processes extending modelling possibilities as well as increasing computational efﬁciency.Efﬁcient numerical models have been obtained using techniques of parallel computing and coupling the discrete element method with ﬁnite element method.The discrete element algorithm has been applied to simulation of different excavation processes,using dif-ferent tools,TBMs and roadheaders.Numerical examples of tunnelling process are included in the paper,showing results in the form of rock failure,damage in the material,cutting forces and tool wear.Efﬁciency of the code for solving large scale geomechanical problems is also shown.Keywords Coupling ÁDiscrete element method ÁFinite element method ÁParallel computation ÁTunnelling1IntroductionA discrete element algorithm is a numerical technique which solves engineering problems that are modelled as a large system of distinct interacting bodies or particles that are subject to gross motion.The discrete element method (DEM)is widely recognized as a suitable tool to model geomaterials [1,2,4,8].The method presents important advantages in simulation of strong discontinuities such as rock fracturing during an underground excavation or rock failure induced by a tunnel excavation.It is difﬁcult to solve such problems using conventional continuum-based procedures such as the ﬁnite element method (FEM).The DEM makes possible the simulation of different excavation processes [5,7]allowing the determination of the damage of the rock or soil,or evaluation of cutting forces in rock excavation with roadheaders or TBMs.Different possibil-ities of DEM applications in simulation of tunnelling process are shown in the paper.Examples include new developments like evaluation of tool wear in rock cutting processes.The main problem in a wider use of this method is the high computational cost required by the simulations ﬁrst of all due to large number of discrete elements usually required.Different strategies are possible in addressing this problem.This paper will present two approaches:parall-elization and coupling the DEM and FEM.Parallelization techniques are useful for the simulation of large-scale problems,where the number of particles involved does not allow the use of a single processor,or where the single processor calculation would require an extremely long time.A shared memory parallelization of the DEM algorithm is presented in the paper.A high per-formance code for the simulation of tunnel construction problems is described and examples of the efﬁciency of thebra ÁE.On˜ate ÁF.Zarate International Center for Numerical Methods in Engineering,Technical University of Catalonia,Gran Capitan s/n,08034Barcelona,Spaine-mail:clabra@ E.On˜ate e-mail:onate@ F.Zaratee-mail:zarate@J.Rojek (&)Institute of Fundamental Technological Research,PolishAcademy of Sciences,Swietokrzyska 21,00049Warsaw,Poland e-mail:jrojek@.plActa Geotechnica (2008)3:317–322DOI 10.1007/s11440-008-0071-2code for solving large-scale geomechanical problems are shown in the paper.In many cases discontinuous material failure is localized in a portion of the domain,the rest of it can be treated as continuum.Continuous material is usually modelled more efﬁciently using the FEM.In such problems coupling of the discrete element method with the FEM can provide an optimum solution.Discrete elements are used only in a portion of the analysed domain where material fracture occurs,while outside the DEM subdomainﬁnite elements can be bining these two methods in one model of rock cutting allows us to take advantages of each method. The paper presents a coupled discrete/ﬁnite element tech-nique to model underground excavation employing the theoretical formulation initiated in[5]and further devel-oped in[6].2Discrete element method formulationThe discrete element model assumes that material can be represented by an assembly of distinct particles or bodies interacting among themselves.Generally,discrete elements can have arbitrary shape.In this work the formulation employing cylindrical(in2D)or spherical(in3D)rigid particles is used.Basic formulation of the discrete element formulation using spherical or cylindrical particles wasﬁrst proposed by Cundall and Strack[1].Similar formulation has been developed by the authors[5,7]and implemented in the explicit dynamic code Simpact.The code has a lot of original features like modelling of tool wear in rock cut-ting,thermomechanical coupling and other capabilities not present in commercial discrete element codes.Translational and rotational motion of rigid spherical or cylindrical elements is described by means of the Newton–Euler equations of rigid body dynamics:M D€r D¼F D;J D_X D¼T Dð1Þwhere r D is the position vector of the element centroid in a ﬁxed(inertial)coordinate frame,X D is the angular veloc-ity,M D is the diagonal matrix with the element mass on the diagonal,J D is the diagonal matrix with the element moment of inertia on the diagonal,F D is the vector of resultant forces,and T D is the vector of resultant moments about the element central axes.Vectors F D and T D are sums of all forces and moments applied to the element due to external load,contact interactions with neighbouring spheres and other obstacles,as well as forces resulting from damping in the system.Equations of motion(1)are inte-grated in time using the central difference scheme.The overall behaviour of the system is determined by the cohesive/frictional contact laws assumed for the inter-action between contacting rigid spheres(or discs in2D).The contact law can be seen as the formulation of the material model on the microscopic level.Modelling of rock or cohesive zones requires contact models with cohesion allowing tensile interaction force between particle.In the present work the simplest of the cohesive models,the elastic perfectly brittle model is used.This model is char-acterized by linear elastic behaviour when cohesive bonds are active:r¼k n u n;s¼k t u tð2Þwhere r and s are the normal and tangential contact force, respectively,k n and k t are the interface stiffness in the normal and tangential directions and u n and u t the normal and tangential relative displacements,respectively. Cohesive bonds are broken instantaneously when the interface strength is exceeded in the tangential direction by the tangential contact force or in the normal direction by the tensile contact force.The failure(decohesion)criterion is written as:r R n;j s j R t;ð3Þwhere R n and R t are the interface strengths in the normal and tangential directions,respectively.Breakage of cohe-sive bonds allows us to simulate fracture of material and its propagation.In the absence of cohesion the frictional contact is assumed with the Coulomb friction model.3Coupling the DEM and FEMIn the present work the so-called explicit dynamic formu-lation of the FEM is used.The explicit FEM is based on the solution of discretized equations of motion written in the current conﬁguration in the following form:M F€r F¼F ext FÀF int Fð4Þwhere M F is the mass matrix,r F is the vector of nodal displacements,F F ext and F F int are the vectors of external loads and internal forces,respectively.Similarly to the DEM algorithm,the central difference scheme is used for time integration of(4).It is assumed that the DEM and FEM can be applied in different subdomains of the same body.The DEM and FEM subdomains,however,do not need to be disjoint—they can overlap each other.The common part of the subdomains is the part where both discretization types are used with gradually varying contribution of each modelling method.This idea follows that used for molecular dynamics coupling with a continuous model in[9].The coupling of DEM and FEM subdomains is provided by additional kinematical constraints.Interface discrete elements are constrained by the displacementﬁeld of overlapping interfaceﬁnite elements.Making use of thesplit of the global vector of displacements of discrete ele-ments,r D ,into the unconstrained part,r DU ,and the constrained one,r DC ,r D ={r DU ,r DC }T ,additional kine-matic relationships can be written jointly in the matrix notation as follows:v ¼r DC ÀNr F ¼0;ð5Þwhere N is the matrix containing adequate shape functions.Additional kinematic constraints (5)can be imposed by the Lagrange multiplier or penalty method.The set of equations of motion for the coupled DEM/FEM system with the penalty coupling is as follows"M F 0000"M DU 0000"M DC 0000"J D26643775€r F €r DU €r DC _X D 8>><>>:9>>=>>;¼"F ext F À"F int F þN T k DF v "F DU"F DC Àk DF v "T D 8>><>>:9>>=>>;ð6Þwhere k DF is the diagonal matrix containing on its diagonal the values of the discrete penalty function,and globalmatrices "M F ;"M DU ;"M DC and "J D ;and global vectors"F int F ;"F ext F ;"F DU ;"F DC and "T D are obtained by aggregation of adequate elemental matrices and vectors taking into account appropriate contributions from the discrete and ﬁnite element parts.Equation (6)can be integrated in time using the standard central difference scheme.4Application of DEM to simulation of tunnelling process Fracture of rock or soil as well as interaction between a tunnelling machine and rock during an excavation process can be simulated by means of the DEM.This kind of analysis enables the comparison of the excavation process under different conditions.4.1Simulation of tunnelling with a TBMSimpliﬁed models of a tunnelling process must be used due to a high computational cost of a full-scale simulation in this case.We assume that the TBM is modelled as a cylinder with a special contact model for the tunnel face is adopted.Figure 1presents a simpliﬁed tunnelling process.The rock sample,with a diameter of 10m and a length of 7m,is discretized with randomly generated and densely com-pacted 40,988spheres.Discretization of the TBM geometry employs 1,193rigid triangular elements.Tunnelling pro-cess has been carried out with prescribed horizontal velocity 5m/h and rotational velocity of 10rev/min.Rock properties of granite are used,and the microscopic DEM parameters corresponding to the macroscopic granite properties are obtained using the methodology described in [10].A special condition is adopted to eliminate the spherical particles in the face of the tunnel.Each particle,which is in contact with the TBM and lacks cohesive contacts with other particles,is removed from the model.Thus,the advance of the TBM and the absorption of the material in the shield of the TBM is modelled.Figure 1a,c presents the displacement of the TBM and the elimination of the rock material.The area affected by the loss of cohesive contacts,resulting in material failure is shown in Fig.2.This loss of cohesion can be considered as damage ,because it produces the change of the equivalent Young modulus.4.2Simulation of linear cutting test of single disccutter Simulation of the linear cutting test was performed.A rock sample with dimensions of 13591095cm is repre-sented by an assembly of randomly generated and densely compacted 40,449spherical elements of radii ranging from 0.08to 0.60cm.The granite properties are assumed in the simulation and appropriate DEM parameters are evaluated.The disc cutter is treated as a rigid body and the parameters describing its interaction with the rock are as follows:contact stiffness modulus k n =10GPa,Coulomb friction coefﬁcient l =0.8.The velocity of the disc cutter is assumed to be 10m/s.Fig.1Simulation of TBM excavation:Evolution and elimination ofmaterialFig.2Simulation of TBM excavation:Damage over tunnel surfaceFigure 3a shows the discretization of the disc cutter.Only the area of the cutter ring in direct interaction with the rock is discretized with discrete elements due to the com-putational cost reasons.The whole model is presented in Fig.3b.The evolution of the normal cutting force during the process is depicted in Fig.4a.The values of the forces should be validated,because the boundary condition can affect the results.The evolution of the wear,using the for-mulation presented in [5],can be seen in Fig.4b.The elimination of the discrete elements,where the wear exceed the prescribed limit,permit the modiﬁcation of the disc cutter shape,which leads to a change of the interaction forces.In the present case,a low value of the wear constant is considered,in order to maintain the initial tool shape.Accumulated wear indicates the areas where the removal of the tool material is most intensive.An acceleration of the wear process using higher values of the wear constant is required in order to obtain in a short time considered in the analysis the amount of wear equivalent to real working time.5High performance simulationsOne of the main problems with the DEM simulation is the computational cost.The contact search,the force calcula-tion for each contact,and the large number of elements necessary to resolve a real life problem requires a high computational effort.High performance computation,and parallel implementation could be necessary to run simu-lations with large number of time steps.The advances of the computer capabilities during last years and the use of multiprocessors techniques enable the use of parallel computing methods for the discrete element analysis of large scale real problems.A shared memory parallel version of the code is tested.The main idea is to make a partition of the mesh of particles and use each processor for the contact calculation at different parts of the mesh.The partition process is performed using a special-ized library [3].The calculation of the cohesive contacts requires most of the computational cost.A special structure for the database,and the dynamic load balance is used in order to obtain a good performance for the simulations.Two different structures for the contact data are used in order to have a good management of the information.The ﬁrst data structure is created for the initial cohesive contacts,where a static array can be used.The other data structureisFig.3Linear cutting test simulation:a cutter ring with partial discretization;b full discretized model0.014Table 1Times for different number of processors Time (s)versus processors 124Total404.31272.93156.85Static contacts (per step)0.12790.06920.0351Dynamic contacts (per step)0.00590.00570.0055Time integration (per step)0.04260.03570.0344Speed up1.001.842.58designed for the dynamic contacts,occurring in the process of rock fragmentation,and the interaction between differ-ent bodies.The management of this kind of contact is completely dynamic,and it is not necessary to store vari-ables with the history information.Table 1presents the times of parallel simulations of a tunnelling process,which was described earlier.The main computational cost is due to the cohesive contacts evalu-ation.The results shown in the table conﬁrm that a good speed-up has been achieved.6DEM and DEM/FEM simulation of rock cutting A process of rock cutting with a single pick of a roadheader cutter-head has been simulated using discrete and hybrid discrete/ﬁnite element models.In the hybrid DEM/FEM model discrete elements have been used in the part of rock mass subjected to fracture,while the other part have been discretized with ﬁnite elements.In both models the tool is considered rigid,assuming the elasticity of the tool is irrelevant for the purpose of modelling of rock fracture.Figure 5presents results of DEM and DEM/FEM sim-ulation.Both models produce similar failures of rock during cutting.Cutting forces obtained using these two models are compared in Fig.6.Both curves show oscilla-tions typical for cutting of brittle rock.In both cases similar values of amplitudes are observed.Mean values of cutting forces agree very well.This shows that combined DEM/FEM simulation gives similar results to a DEM analysis,while being more efﬁcient numerically—computation time has been reduced by half.7Conclusions •Discrete element method using spherical or cylindrical rigid particles is a suitable tool in modelling of underground excavation processes.•Use of the model in a particular case requires calibra-tion of the discrete element model using available experimental results.•Discrete element simulations of real engineering prob-lems require large computation time and memory resources.•Efﬁciency of discrete element computation can be improved using technique of parallel computations.Parallelization makes possible the simulation of large problems.•The combination of discrete and ﬁnite elements is an effective approach for simulation of underground rock excavation.Acknowledgments The work has been sponsored by the EU project TUNCONSTRUCT (contract no.IP 011817-2)coordinated by Prof.G.Beer (TU Graz,Austria).References1.Cundall PA,Strack ODL (1979)A discrete numerical method for granular assemblies.Geotechnique29:47–65Fig.5Simulation of rock cutting:a DEM model,b DEM/FEM model2.Campbell CS(1990)Rapid granularﬂows.Annu Rev Fluid Mech2:57–923.Karypis G,Kumar V(1998)A fast and high quality multilevelscheme for partitioning irregular graphs.SIAM J Sci Comput 20:359–3924.Mustoe G(ed)(1992)Eng Comput9(2).Special issue5.On˜ate E,Rojek J(2004)Combination of discrete element andﬁnite element methods for dynamic analysis of geomechanics put Methods Appl Mech Eng193:3087–3128 6.Rojek J(2007)Modelling and simulation of complex problems ofnonlinear mechanics using theﬁnite and discrete element meth-ods(in Polish).Habilitiation Thesis,Institute of Fundamental Technological Research Polish Academy of Sciences,Warsaw7.Rojek J,On˜ate E,Zarate F,Miquel J(2001)Modelling of rock,soil and granular materials using spherical elements.In:2nd European conference on computational mechanics ECCM-2001, Cracow,26–29June8.Williams JR,O’Connor R(1999)Discrete element simulationand the contact problem.Arch Comp Meth Eng6(4):279–304 9.Xiao SP,Belytschko T(2004)A bridging domain method forcoupling continua with molecular put Methods Appl Mech Eng193:1645–166910.Zarate F,Rojek J,On˜ate E,Labra C(2007)A methodology todetermine the particle properties in2d and3d dem simulations.In:ECCOMAS thematic conference on computational methods in tunnelling EURO:TUN-2007,Vienna,Austria,27–29August。

弹性力学第4章

5Formulation and Solution StrategiesThe previous chapters have now developed the basicﬁeld equations of elasticity theory.Theseresults comprise a system of differential and algebraic relations among the stresses,strains,anddisplacements that express particular physics at all points within the body under investigation.In this chapter we now wish to complete the general formulation byﬁrst developing boundaryconditions appropriate for use with theﬁeld equations.These conditions specify the physicsthat occur on the boundary of body,and generally provide the loading inputs that physicallycreate the interior stress,strain,and displacementﬁelds.Although theﬁeld equations are thesame for all problems,boundary conditions are different for each problem.Therefore,properdevelopment of boundary conditions is essential for problem solution,and thus it is importantto acquire a good understanding of such development bining theﬁeldequations with boundary conditions then establishes the fundamental boundary value problemsof the theory.This eventually leads us into two different formulations,one in terms ofdisplacements and the other in terms of stresses.Because these boundary value problems aredifﬁcult to solve,many different strategies have been developed to aid in problem solution.Wereview in a general way several of these strategies,and later chapters incorporate many ofthese into the solution of speciﬁc problems.5.1Review of Field EquationsBefore beginning our discussion on boundary conditions we list here the basicﬁeld equationsfor linear isotropic elasticity.Appendix A includes a more comprehensive listing of allﬁeldequations in Cartesian,cylindrical,and spherical coordinate systems.Because of its ease of useand compact properties,our formulation uses index notation.Strain-displacement relations:e ij¼12(u i,jþu j,i)(5:1:1)Compatibility relations:e ij,klþe kl,ijÀe ik,jlÀe jl,ik¼0(5:1:2)83Equilibrium equations:s ij,jþF i¼0(5:1:3) Elastic constitutive law(Hooke’s law):s ij¼l e kk d ijþ2m e ije ij¼1þnEs ijÀnEs kk d ij(5:1:4)As mentioned in Section2.6,the compatibility relations ensure that the displacements arecontinuous and single-valued and are necessary only when the strains are arbitrarily speciﬁed.If,however,the displacements are included in the problem formulation,the solution normallygenerates single-valued displacements and strain compatibility is automatically satisﬁed.Thus,in discussing the general system of equations of elasticity,the compatibility relations(5.1.2)are normally set aside,to be used only with the stress formulation that we discuss shortly.Therefore,the general system of elasticityﬁeld equations refers to the15relations(5.1.1),(5.1.3),and(5.1.4).It is convenient to deﬁne this entire system using a generalized operatornotation asJ{u i,e ij,s ij;l,m,F i}¼0(5:1:5) This system involves15unknowns including3displacements u i,6strains e ij,and6stresses s ij.The terms after the semicolon indicate that the system is also dependent on two elastic materialconstants(for isotropic materials)and on the body force density,and these are to be given apriori with the problem formulation.It is reassuring that the number of equations matches thenumber of unknowns to be determined.However,this general system of equations is of suchcomplexity that solutions by using analytical methods are essentially impossible and furthersimpliﬁcation is required to solve problems of interest.Before proceeding with development ofsuch simpliﬁcations,it is usefulﬁrst to discuss typical boundary conditions connected with theelasticity model,and this leads us to the classiﬁcation of the fundamental problems.5.2Boundary Conditions and Fundamental ProblemClassiﬁcationsSimilar to otherﬁeld problems in engineering science(e.g.,ﬂuid mechanics,heat conduction,diffusion,electromagnetics),the solution of system(5.1.5)requires appropriate boundaryconditions on the body under study.The common types of boundary conditions for elasticityapplications normally include speciﬁcation of how the body is being supported or loaded.Thisconcept is mathematically formulated by specifying either the displacements or tractions atboundary points.Figure5-1illustrates this general idea for three typical cases includingtractions,displacements,and a mixed case for which tractions are speciﬁed on boundary S tand displacements are given on the remaining portion S u such that the total boundary is givenby S¼S tþS u.Another type of mixed boundary condition can also occur.Although it is generally not possible to specify completely both the displacements and tractions at the same boundarypoint,it is possible to prescribe part of the displacement and part of the traction.Typically,this 84FOUNDATIONS AND ELEMENTARY APPLICATIONStype of mixed condition involves the speciﬁcation of a traction and displacement in two different orthogonal directions.A common example of this situation is shown in Figure 5-2for a case involving a surface of problem symmetry where the condition is one of a rigid-smooth boundary with zero normal displacement and zero tangential traction.Notice that in this example the body under study was subdivided along the symmetry line,thus creating a new boundary surface and resulting in a smaller region to analyze.Because boundary conditions play a very essential role in properly formulating and solving elasticity problems,it is important to acquire a clear understanding of their speciﬁcation and use.Improper speciﬁcation results in either no solution or a solution to a different problem than what was originally sought.Boundary conditions are normally speciﬁed using the coordinate system describing the problem,and thus particular components of the displacements and tractions are set equal to prescribed values.For displacement-type conditions,such a speciﬁ-cation is straightforward,and a common example includes ﬁxed boundaries where the dis-placements are to be zero.For traction boundary conditions,the speciﬁcation can be a bit more complex.Figure 5-3illustrates particular cases in which the boundaries coincide with Cartesian or polar coordinate surfaces.By using results from Section 3.2,the traction speciﬁcation can be reduced to a stress speciﬁcation.For the Cartesian example in which y ¼constant ,Displacement Conditions Mixed ConditionsTraction ConditionsFIGURE 5-1Typical boundary conditions.T (n)y u Rigid-SmoothFIGURE 5-2Line of symmetry boundary condition.Formulation and Solution Strategies 85the normal traction becomes simply the stress component s y ,while the tangential traction reduces to t xy .For this case,s x exists only inside the region,and thus this component of stress cannot be speciﬁed on the boundary surface y ¼constant .A similar situation exists on the vertical boundary x ¼constant ,where the normal traction is now s x ,the tangential traction is t xy and the stress component s y exists inside the domain.Similar arguments can be made for polar coordinate boundary surfaces as shown.Drawing the appropriate element along the boundary as illustrated allows a clear visualization of the particular stress components that act on the surface in question.Such a sketch also allows determination of the positive directions of these boundary stresses,and this is useful to properly match with boundary loadings that might be prescribed.It is recommended that sketches similar to Figure 5-3be used to aid in the proper development of boundary conditions during problem formulation.Consider the pair of two-dimensional example problems with mixed conditions as shown in Figure 5-4.For the rectangular plate problem,all four boundaries are coordinate surfaces,andr(Cartesian Coordinate Boundaries)(Polar Coordinate Boundaries)xFIGURE 5-3Boundary stress components on coordinatesurfaces.(n)(n)(n)= σy = 0= t xy = 0(n)T x (n)T y = t xy =0,Traction Condition (Non-Coordinate Surface Boundary)(Coordinate Surface Boundaries)FIGURE 5-4Example boundary conditions.86FOUNDATIONS AND ELEMENTARY APPLICATIONSthis simpliﬁes speciﬁcation of particular boundary conditions.Theﬁxed conditions on the left edge simply require that x and y displacement components vanish on x¼0,and this speciﬁca-tion does not change even if this were not a coordinate surface.However,as per our previous discussion,the traction conditions on the other three boundaries simplify because they are coordinate surfaces.These simpliﬁcations are shown in theﬁgure for each of the traction speciﬁed surfaces.The second problem of a tapered cantilever beam has an inclined face that is not a coordinate surface.For this problem,theﬁxed end and top surface follow similar procedures as in theﬁrst example and are speciﬁed in theﬁgure.However,on the inclined face,the traction is to be zero and this does not reduce to a simple speciﬁcation of the vanishing of individual stress components.On this face each traction component is set to zero,giving the resultT(n)x¼s x n xþt xy n y¼0T(n)y¼t xy n xþs y n y¼0where n x and n y are the components of the unit normal vector to the inclined face.This is the more general type of speciﬁcation,and it should be clearly noted that none of the individual stress components in the x,y system will vanish along this surface.It should also be pointed out for this problem that the unit normal vector components are constants for all points on the inclined face.However,for curved boundaries the normal vector changes with surface position.Although these examples provide some background on typical boundary conditions,many other types will be encountered throughout the text.Several exercises at the end of this chapter provide additional examples that will prove to be useful for students new to the elasticity formulation.We are now in the position to formulate and classify the three fundamental boundary-value problems in the theory of elasticity that are related to solving the general system ofﬁeld equations(5.1.5).Our presentation is limited to the static case.Problem1:Traction problemDetermine the distribution of displacements,strains,and stresses in the interior of an elastic body in equilibrium when body forces are given and the distribution of the tractions are prescribed over the surface of the body,T(n)i(x(s)i)¼f i(x(s)i)(5:2:1) where x(s)i denotes boundary points and f i(x(s)i)are the prescribed traction values. Problem2:Displacement problemDetermine the distribution of displacements,strains,and stresses in the interior of an elastic body in equilibrium when body forces are given and the distribution of the displacements are prescribed over the surface of the body,u i(x(s)i)¼g i(x(s)i)(5:2:2) where x(s)i denotes boundary points and g i(x(s)i)are the prescribed displacement values.Formulation and Solution Strategies87Problem3:Mixed problemDetermine the distribution of displacements,strains,and stresses in the interior of an elasticbody in equilibrium when body forces are given and the distribution of the tractions areprescribed as per(5.2.1)over the surface S t and the distribution of the displacementsare prescribed as per(5.2.2)over the surface S u of the body(see Figure5-1).As mentioned previously,the solution to any of these types of problems is formidable,andfurther reduction and simpliﬁcation of(5.1.5)is required for the development of analyticalsolution methods.Based on the description of Problem1with only traction boundary condi-tions,it would appear to be desirable to express the fundamental system solely in terms ofstress,that is,J(t){s ij;l,m,F i}thereby reducing the number of unknowns in the system.Likewise for Problem2,a displacement-only formulation of the form J(u){u i;l,m,F i}wouldappear to simplify the problem.We now pursue such specialized formulations and explicitlydetermine these reducedﬁeld equation systems.5.3Stress FormulationFor theﬁrst fundamental problem in elasticity,the boundary conditions are to be given only interms of the tractions or stress components.In order to develop solution methods for this case,it is very helpful to reformulate the general system(5.1.5)by eliminating the displacementsand strains and thereby cast a new system solely in terms of the stresses.We now develop thisreformulated system.By eliminating the displacements,we must now include the compatibil-ity equations in the fundamental system ofﬁeld equations.We therefore start by using Hooke’slaw(5.1.4)2and eliminate the strains in the compatibility relations(5.1.2)to gets ij,kkþs kk,ijÀs ik,jkÀs jk,ik¼n 1þn (s mm,kk d ijþs mm,ij d kkÀs mm,jk d ikÀs mm,ik d jk)(5:3:1)where we have used the arguments of Section2.6,that the six meaningful compatibility relations are found by setting k¼l in(5.1.2).Although equations(5.3.1)represent the compatibility in terms of stress,a more useful result is found by incorporating the equilibrium equations into the system.Recall that from(5.1.3),s ij,j¼ÀF i,and also note that d kk¼3. Substituting these results into(5.3.1)givess ij,kkþ11þns kk,ij¼n1þns mm,kk d ijÀF i,jÀF j,i(5:3:2)For the case i¼j,relation(5.3.2)reduces to s ii,kk¼À1þn1ÀnF i,i.Substituting this result backinto(5.3.2)gives the desired relations ij,kkþ11þns kk,ij¼Àn1Ànd ij F k,kÀF i,jÀF j,i(5:3:3)This result is the compatibility relations in terms of the stress and is commonly called theBeltrami-Michell compatibility equations.For the case with no body forces,these relations canbe expressed as the following six scalar equations:88FOUNDATIONS AND ELEMENTARY APPLICATIONS(1þn)r2s xþ@2@x(s xþs yþs z)¼0(1þn)r2s yþ@2@y(s xþs yþs z)¼0(1þn)r2s zþ@2@z(s xþs yþs z)¼0(1þn)r2t xyþ@2@x@y(s xþs yþs z)¼0(1þn)r2t yzþ@2@y@z(s xþs yþs z)¼0(1þn)r2t zxþ@2(s xþs yþs z)¼0(5:3:4)Recall that the six developed relations(5.3.3)or(5.3.4)actually represent three independentresults as per our discussion in Section2.6.Thus,combining these results with the threeequilibrium equations(5.1.3)provides the necessary six relations to solve for the six unknownstress components for the general three-dimensional case.This system constitutes the stressformulation for elasticity theory and is appropriate for use with traction boundary conditionproblems.Once the stresses have been determined,the strains may be found from Hooke’s law(5.1.4)z,and the displacements can be then be computed through integration of(5.1.1).As perour previous discussion in Section2.2,such an integration process determines the displace-ments only up to an arbitrary rigid-body motion,and the displacements obtained are single-valued only if the region under study is simply connected.The system of equations for the stress formulation is still rather complex,and analytical solutions are commonly determined for this case by making use of stress functions.Thisconcept establishes a representation for the stresses that automatically satisﬁes the equilibriumequations.For the two-dimensional case,this concept represents the in-plane stresses in termsof a single function.The representation satisﬁes equilibrium,and the remaining compatibilityequations yield a single partial differential equation(biharmonic equation)in terms of thestress function.Having reduced the system to a single equation then allows us to employ manyanalytical methods toﬁnd solutions of interest.Further discussion on these techniques ispresented in subsequent chapters.5.4Displacement FormulationWe now wish to develop the reduced set ofﬁeld equations solely in terms of the displacements.This system is referred to as the displacement formulation and is most useful when combinedwith displacement-only boundary conditions found in the Problem2statement.This develop-ment is somewhat more straightforward than our previous discussion for the stress formulation.For this case,we wish to eliminate the strains and stresses from the fundamental system(5.1.5).This is easily accomplished by using the strain-displacement relations in Hooke’s lawto gives ij¼l u k,k d ijþm(u i,jþu j,i)(5:4:1) which can be expressed as six scalar equationsFormulation and Solution Strategies89s x¼l@u@xþ@n@yþ@w@zþ2m@u@xs y¼l@u@xþ@v@yþ@w@zþ2m@v@ys z¼l@u@xþ@v@yþ@w@zþ2m@w@zt xy¼m @u@yþ@v@x,t yz¼m @v@zþ@w@y,t zx¼m @w@xþ@u@z(5:4:2)Using these relations in the equilibrium equations gives the resultm u i,kkþ(lþm)u k,kiþF i¼0(5:4:3)which are the equilibrium equations in terms of the displacements and are referred to as Navier’s or Lame´’s equations.This system can be expressed in vector form asm r2uþ(lþm),(,Áu)þF¼0(5:4:4) or written out in terms of the three scalar equationsm r2uþ(lþm)@@x@u@xþ@v@yþ@w@zþF x¼0m r2vþ(lþm)@@y@u@xþ@v@yþ@w@zþF y¼0m r2wþ(lþm)@@z@u@xþ@v@yþ@w@zþF z¼0(5:4:5)where the Laplacian is given by r2¼(@2=@x2)þ(@2=@y2)þ(@2=@z2).Navier’s equations arethe desired formulation for the displacement problem,and the system represents three equa-tions for the three unknown displacement components.Similar to the stress formulation,thissystem is still difﬁcult to solve,and additional mathematical techniques have been developedto further simplify these equations for problem mon methods normally employthe use of displacement potential functions.It is shown in Chapter13that several suchschemes can be developed that allow the displacement vector to be expressed in terms ofparticular potentials.These schemes generally simplify the problem by yielding uncoupledgoverning equations in terms of the displacement potentials.This then allows several analyt-ical methods to be employed to solve problems of interest.Several of these techniques arediscussed in later sections of the text.To help acquire a general understanding of these results,a summaryﬂow chart of the developed stress and displacement formulations is shown in Figure5-5.Note that for thestress formulation,the resulting system J(t){s ij;l,m,F i}is actually dependent onlyon the single material constant Poisson’s ratio,and thus it could be expressed asJ(t){s ij;n,F i}.90FOUNDATIONS AND ELEMENTARY APPLICATIONS5.5Principle of SuperpositionA very useful tool for the solution to many problems in engineering science is the principle ofsuperposition.This technique applies to any problem that is governed by linear equations.Under the assumption of small deformations and linear elastic constitutive behavior,allelasticityﬁeld equations(see Figure5-5)are linear.Furthermore,the usual boundary condi-tions speciﬁed by relations(5.2.1)and(5.2.2)are also linear.Thus,under these conditions allgoverning equations are linear,and the superposition concept can be applied.It can be easilyproved(see Chou and Pagano1967)that the general statement of the principle can beexpressed as follows:Principle of Superposition:For a given problem domain,if the state{s(1)ij,e(1)ij,u(1)i}is asolution to the fundamental elasticity equations with prescribed body forces F(1)i andsurface tractions T(1)i,and the state{s(2)ij,e(2)ij,u(2)i}is a solution to the fundamentalequations with prescribed body forces F(2)i and surface tractions T(2)i,then the sta-te{s(1)ijþs(2)ij,e(1)ijþe(2)ij,u(1)iþu(2)i}will be a solution to the problem with body forcesF(1)iþF(2)i and surface tractions T(1)iþT(2)i.In order to see a more direct application of this principle,consider a simple two-dimensionalcase with no body forces as shown in Figure5-6.It can be observed that the solution to themore complicated biaxial loading case(1)þ(2)is thus found by adding the two simplerproblems.This is a common use of the superposition principle,and we make repeated use ofthis application throughout the text.Thus,once the solutions to some simple problems aregenerated,we can combine these results to generate a solution to a more complicated case withsimilar geometry.Formulation and Solution Strategies915.6Saint-Venant’s PrincipleConsider the set of three identical rectangular strips under compressive loadings as shown in Figure 5-7.As indicated,the only difference between each problem is the loading.Because the total resultant load applied to each problem is identical (statically equivalent loadings),it is expected that the resulting stress,strain,and displacement ﬁelds near the bottom of each strip would be approximately the same.This behavior can be generalized by considering an elastic solid with an arbitrary loading T (n )over a boundary portion S *,as shown in Figure 5-8.Based on experience from other ﬁeld problems in engineering science,it seems logical that the particular boundary loading would produce detailed and characteristic effects only in the vicinity of S *.In other words,we expect that at points far away from S *the stresses generally depend more on the resultant F R of the tractions rather than on the exact distribution.Thus,the characteristic signature of the generated stress,strain,and displacement ﬁelds from a given boundary loading tend to disappear as we move away from the boundary loading points.These concepts form the principle of Saint-Venant ,which can be stated as follows:Saint-Venant’s Principle:The stress,strain,and displacement ﬁelds caused by two different statically equivalent force distributions on parts of the body far away from the loading points are approximately the same.=+ ij ij i ij ij i {s (1)ij + s (2) ij , e (1) ij , + e (2) ij , u (1) i + u (2) i }FIGURE 5-6Two-dimensional superposition example.2P 2P3P 3P 3P FIGURE 5-7Statically equivalent loadings.92FOUNDATIONS AND ELEMENTARY APPLICATIONSThis statement of the principle includes qualitative terms such as far away and approxi-mately the same ,and thus does not provide quantitative estimates of the differences between the two elastic ﬁelds in question.Quantitative results have been developed by von Mises (1945),Sternberg (1954),and Toupin (1965),while Horgan (1989)has presented a recent review of related work.Some of this work is summarized in Boresi and Chong (2000).If we restrict our solution to points away from the boundary loading,Saint-Venant’s principle allows us to change the given boundary conditions to a simpler statically equivalent statement and not affect the resulting solution.Such a simpliﬁcation of the boundary conditions greatly increases our chances of ﬁnding an analytical solution to the problem.This concept therefore proves to be very useful,and we formally outline this solution scheme in the next section.5.7General Solution StrategiesHaving completed our formulation and related solution principles,we now wish to examine some general solution strategies commonly used to solve elasticity problems.At this stage we categorize particular methods and outline only typical techniques that are commonly used.As we move further along in the text,many of these methods are developed in detail and are applied in speciﬁc problem solution.We ﬁrst distinguish three general methods of solution called direct,inverse ,and semi-inverse .5.7.1Direct MethodThis method seeks to determine the solution by direct integration of the ﬁeld equations (5.1.5)or equivalently the stress and/or displacement formulations given in Figure 5-5.Boundary conditions are to be satisﬁed exactly.This method normally encounters signiﬁcant mathemat-ical difﬁculties,thus limiting its application to problems with simple geometry.EXAMPLE 5-1:Direct Integration Example:Stretching of Prismatic Bar Under Its Own WeightAs an example of a simple direct integration problem,consider the case of a uniform prismatic bar stretched by its own weight,as shown in Figure 5-9.The body forces forContinuedFIGURE 5-8Saint-Venant concept.5.7.2Inverse MethodFor this technique,particular displacements or stresses are selected that satisfy the basicﬁeld equations.A search is then conducted to identify a speciﬁc problem that would be solved by this solutionﬁeld.This amounts to determine appropriate problem geometry,boundary conditions,and body forces that would enable the solution to satisfy all conditions on the ing this scheme it is sometimes difﬁcult to construct solutions to a speciﬁc problem of practical interest.5.7.3Semi-Inverse MethodIn this scheme part of the displacement and/or stressﬁeld is speciﬁed,and the other remaining portion is determined by the fundamentalﬁeld equations(normally using direct integration) and the boundary conditions.It is often the case that constructing appropriate displacement and/or stress solutionﬁelds can be guided by approximate strength of materials theory.The usefulness of this approach is greatly enhanced by employing Saint-Venant’s principle, whereby a complicated boundary condition can be replaced by a simpler statically equivalent distribution.EXAMPLE5-3:Semi-Inverse Example:Torsion of Prismatic BarsA simple semi-inverse example may be borrowed from the torsion problem that isdiscussed in detail in Chapter9.Skipping for now the developmental details,we propose the following displacementﬁeld:ContinuedThere are numerous mathematical techniques used to solve the elasticity ﬁeld equations.Many techniques involve the development of exact analytical solutions ,while others involve the construction of approximate solution schemes .A third procedure involves the establishment of numerical solution methods .We now brieﬂy provide an overview of each of these techniques.5.7.4Analytical Solution ProceduresA variety of analytical solution methods are used to solve the elasticity ﬁeld equations.The following sections outline some of the more common methods.Power Series MethodFor many two-dimensional elasticity problems,the stress formulation leads to the use of a stress function f (x ,y ).It is shown that the entire set of ﬁeld equations reduces to a single partial differential equation (biharmonic equation)in terms of this stress function.A general mathematical scheme to solve this equation is to look for solutions in terms of a power series in the independent variables,that is,f (x ,y )¼P C mn x m y n (see Neou 1957).Use of the boundary conditions determines the coefﬁcients and number of terms to be used in the series.This method is employed to develop two-dimensional solutions in Section 8.1.Fourier MethodA general scheme to solve a large variety of elasticity problems employs the Fourier method.This procedure is normally applied to the governing partial differential equations by using。

自动化专业英语词汇大全

自动化专业英语词汇大全accelerationtransducer加速度传感器basecoordinatesystem基座坐标系acceptancetesting验收测试Bayesclassifier贝叶斯分类器accessibility可及性bearingalignment方位对准accumulatederror累积误差bellowspressuregauge波纹管压力表AC-DC-ACfrequencyconverter交-直-交变频器benefit-costanalysis收益本钱分析AC(alternatingcurrent)electricdrive交流电子传bilinearsystem双线性系统动biocybernetics生物控制论activeattitudestabilization主动姿态稳定biologicalfeedbacksystem生物反响系统actuator驱动器，执行机构blackboxtestingapproach黑箱测试法adaline线性适应元blindsearch盲目搜索adaptationlayer适应层blockdiagonalization块对角化adaptivetelemetersystem适应遥测系统Boltzmanmachine玻耳兹曼机adjointoperator伴随算子bottom-updevelopment自下而上开发admissibleerror容许误差boundaryvalueanalysis边界值分析aggregationmatrix集结矩阵brainstormingmethod头脑风暴法AHP(analytichierarchyprocess)层次分析法breadth-firstsearch广度优先搜索amplifyingelement放大环节butterflyvalve蝶阀analog-digitalconversion模数转换CAE(computeraidedengineering)计算机辅助工annunciator信号器程antennapointingcontrol天线指向控制CAM(computeraidedmanufacturing)计算机辅助anti-integralwindup抗积分饱卷制造aperiodicdecomposition非周期分解Camflexvalve偏心旋转阀aposterioriestimate后验估计canonicalstatevariable标准化状态变量approximatereasoning近似推理capacitivedisplacementtransducer电容式位移传aprioriestimate先验估计感器articulatedrobot关节型机器人capsulepressuregauge膜盒压力表assignmentproblem配置问题，分配问题CARD计算机辅助研究开发associativememorymodel联想记忆模型Cartesianrobot直角坐标型机器人associatron联想机cascadecompensation串联补偿asymptoticstability渐进稳定性catastrophetheory突变论attainedposedrift实际位姿漂移centrality集中性attitudeacquisition姿态捕获chainedaggregation链式集结AOCS(attritudeandorbitcontrolsystem)姿态轨chaos混沌道控制系统characteristiclocus特征轨迹attitudeangularvelocity姿态角速度chemicalpropulsion化学推进attitudedisturbance姿态扰动calrity清晰性attitudemaneuver姿态机动classicalinformationpattern经典信息模式attractor吸引子classifier分类器augmentability可扩大性clinicalcontrolsystem临床控制系统augmentedsystem增广系统closedlooppole闭环极点automaticmanualstation自动-手动操作器closedlooptransferfunction闭环传递函数automaton自动机clusteranalysis聚类分析autonomoussystem自治系统coarse-finecontrol粗-精控制backlashcharacteristics间隙特性cobwebmodel蛛网模型coefficientmatrix系数矩阵costatevariable共态变量cognitivescience认知科学cost-effectivenessanalysis费用效益分析cognitron认知机couplingoforbitandattitude轨道和姿态耦合coherentsystem单调关联系统criticaldamping临界阻尼combinationdecision组合决策criticalstability临界稳定性combinatorialexplosion组合爆炸cross-overfrequency穿越频率，交越频率combinedpressureandvacuumgauge压力真空currentsourceinverter电流[源]型逆变器表cut-offfrequency截止频率commandpose指令位姿cybernetics控制论companionmatrix相伴矩阵cyclicremotecontrol循环遥控compartmentalmodel房室模型cylindricalrobot圆柱坐标型机器人compatibility相容性，兼容性dampedoscillation阻尼振荡compensatingnetwork补偿网络damper阻尼器compensation补偿，矫正dampingratio阻尼比compliance柔顺，顺应dataacquisition数据采集compositecontrol组合控制dataencryption数据加密computablegeneralequilibriummodel可计算一datapreprocessing数据预处理般均衡模型dataprocessor数据处理器conditionallyinstability条件不稳定性DCgenerator-motorsetdrive直流发电机-电动机configuration组态组传动connectionism连接机制Dcontroller微分控制器connectivity连接性decentrality分散性conservativesystem守恒系统decentralizedstochasticcontrol分散随机控制consistency一致性decisionspace决策空间constraintcondition约束条件decisionsupportsystem决策支持系统consumptionfunction消费函数decomposition-aggregationapproach分解集结法context-freegrammar上下文无关语法decouplingparameter解耦参数continuousdiscreteeventhybridsystemdeductive-inductivehybridmodelingmethod演simulation连续离散事件混合系统仿真绎与归纳混合建模法continuousduty连续工作制delayedtelemetry延时遥测controlaccuracy控制精度derivationtree导出树controlcabinet控制柜derivativefeedback微分反响controllabilityindex可控指数describingfunction描述函数controllablecanonicalform可控标准型desiredvalue希望值[control]plant控制对象,被控对象despinner消旋体controllinginstrument控制仪表destination目的站controlmomentgyro控制力矩陀螺detector检出器controlpanel控制屏，控制盘deterministicautomaton确定性自动机controlsynchro控制[式]自整角机deviation偏差controlsystemsynthesis控制系统综合deviationalarm偏差报警器controltimehorizon控制时程DFD数据流图cooperativegame合作对策diagnosticmodel诊断模型coordinabilitycondition可协调条件diagonallydominantmatrix对角主导矩阵coordinationstrategy协调策略diaphragmpressuregauge膜片压力表coordinator协调器differenceequationmodel差分方程模型cornerfrequency转折频率differentialdynamicalsystem微分动力学系统----differentialgame微分对策economicindicator经济指标differentialpressurelevelmeter差压液位计eddycurrentthicknessmeter电涡流厚度计differentialpressuretransmitter差压变送器effectiveness有效性differentialtransformerdisplacementtransducereffectivenesstheory效益理论差动变压器式位移传感器elasticityofdemand需求弹性differentiationelement微分环节electricactuator电动执行机构digitalfiler数字滤波器electricconductancelevelmeter电导液位计digitalsignalprocessing数字信号处理electricdrivecontrolgear电动传动控制设备digitization数字化electrichydraulicconverter电-液转换器digitizer数字化仪electricpneumaticconverter电-气转换器dimensiontransducer尺度传感器electrohydraulicservovale电液伺服阀directcoordination直接协调electromagneticflowtransducer电磁流量传感器disaggregation解裂electronicbatchingscale电子配料秤discoordination失协调electronicbeltconveyorscale电子皮带秤discreteeventdynamicsystem离散事件动态系统electronichopperscale电子料斗秤discretesystemsimulationlanguage离散系统仿elevation仰角真语言emergencystop异常停顿discriminantfunction判别函数empiricaldistribution经历分布displacementvibrationamplitudetransducer位endogenousvariable内生变量移振幅传感器equilibriumgrowth均衡增长dissipativestructure耗散构造equilibriumpoint平衡点distributedparametercontrolsystem分布参数控equivalencepartitioning等价类划分制系统ergonomics工效学distrubance扰动error误差disturbancecompensation扰动补偿error-correctionparsing纠错剖析diversity多样性estimate估计量divisibility可分性estimationtheory估计理论domainknowledge领域知识evaluationtechnique评价技术dominantpole主导极点eventchain事件链dose-responsemodel剂量反响模型evolutionarysystem进化系统dualmodulationtelemeteringsystem双重调制遥exogenousvariable外生变量测系统expectedcharacteristics希望特性dualprinciple对偶原理externaldisturbance外扰dualspinstabilization双自旋稳定factbase事实dutyratio负载比failurediagnosis故障诊断dynamicbraking能耗制动fastmode快变模态dynamiccharacteristics动态特性feasibilitystudy可行性研究dynamicdeviation动态偏差feasiblecoordination可行协调dynamicerrorcoefficient动态误差系数feasibleregion可行域dynamicexactness动它吻合性featuredetection特征检测dynamicinput-outputmodel动态投入产出模型featureextraction特征抽取econometricmodel计量经济模型feedbackcompensation反响补偿economiccybernetics经济控制论feedforwardpath前馈通路economiceffectiveness经济效益fieldbus现场总线economicevaluation经济评价finiteautomaton有限自动机economicindex经济指数FIP(factoryinformationprotocol)工厂信息协议firstorderpredicatelogic一阶谓词逻辑harmoniousstrategy和谐策略fixedsequencemanipulator固定顺序机械手heuristicinference启发式推理fixedsetpointcontrol定值控制hiddenoscillation隐蔽振荡FMS(flexiblemanufacturingsystem)柔性制造系hierarchicalchart层次构造图统hierarchicalplanning递阶规划flowsensor/transducer流量传感器hierarchicalcontrol递阶控制flowtransmitter流量变送器homeostasis内稳态fluctuation涨落homomorphicmodel同态系统forcedoscillation强迫振荡horizontaldecomposition横向分解formallanguagetheory形式语言理论hormonalcontrol内分泌控制formalneuron形式神经元hydraulicstepmotor液压步进马达forwardpath正向通路hypercycletheory超循环理论forwardreasoning正向推理Icontroller积分控制器fractal分形体，分维体identifiability可辨识性frequencyconverter变频器IDSS(intelligentdecisionsupportsystem)智能frequencydomainmodelreductionmethod频域决策支持系统模型降阶法imagerecognition图像识别frequencyresponse频域响应impulse冲量fullorderobserver全阶观测器impulsefunction冲击函数，脉冲函数functionaldecomposition功能分解inching点动FES(functionalelectricalstimulation)功能电刺激incompatibilityprinciple不相容原理functionalsimularity功能相似incrementalmotioncontrol增量运动控制fuzzylogic模糊逻辑indexofmerit品质因数gametree对策树inductiveforcetransducer电感式位移传感器gatevalve闸阀inductivemodelingmethod归纳建模法generalequilibriumtheory一般均衡理论industrialautomation工业自动化generalizedleastsquaresestimation广义最小二inertialattitudesensor惯性姿态敏感器乘估计inertialcoordinatesystem惯性坐标系generationfunction生成函数inertialwheel惯性轮geomagnetictorque地磁力矩inferenceengine推理机geometricsimilarity几何相似infinitedimensionalsystem无穷维系统gimbaledwheel框架轮informationacquisition信息采集globalasymptoticstability全局渐进稳定性infraredgasanalyzer红外线气体分析器globaloptimum全局最优inherentnonlinearity固有非线性globevalve球形阀inherentregulation固有调节goalcoordinationmethod目标协调法initialdeviation初始偏差grammaticalinference文法推断initiator发起站graphicsearch图搜索injectionattitude入轨姿势gravitygradienttorque重力梯度力矩input-outputmodel投入产出模型grouptechnology成组技术instability不稳定性guidancesystem制导系统instructionlevellanguage指令级语言gyrodriftrate陀螺漂移率integralofabsolutevalueoferrorcriterion绝对gyrostat陀螺体误差积分准那么Halldisplacementtransducer霍尔式位移传感器integralofsquarederrorcriterion平方误差积分准hardware-in-the-loopsimulation半实物仿真那么harmoniousdeviation和谐偏差integralperformancecriterion积分性能准那么integrationinstrument积算仪器localasymptoticstability局部渐近稳定性integrity整体性localoptimum局部最优intelligentterminal智能终端logmagnitude-phasediagram对数幅相图interactedsystem互联系统，关联系统longtermmemory长期记忆interactivepredictionapproach互联预估法，关联lumpedparametermodel集总参数模型预估法Lyapunovtheoremofasymptoticstability李雅普interconnection互联诺夫渐近稳定性定理intermittentduty断续工作制macro-economicsystem宏观经济系统internaldisturbance内扰magneticdumping磁卸载ISM(interpretivestructuremodeling)解释构造建magnetoelasticweighingcell磁致弹性称重传感器模法magnitude-frequencycharacteristic幅频特性invariantembeddingprinciple不变嵌入原理magnitudemargin幅值裕度inventorytheory库伦论magnitudescalefactor幅值比例尺inverseNyquistdiagram逆奈奎斯特图manipulator机械手inverter逆变器man-machinecoordination人机协调investmentdecision投资决策manualstation手动操作器isomorphicmodel同构模型MAP(manufacturingautomationprotocol)制造iterativecoordination迭代协调自动化协议jetpropulsion喷气推进marginaleffectiveness边际效益job-lotcontrol分批控制Mason'sgainformula梅森增益公式joint关节masterstation主站Kalman-Bucyfiler卡尔曼-布西滤波器matchingcriterion匹配准那么knowledgeaccomodation知识顺应maximumlikelihoodestimation最大似然估计knowledgeacquisition知识获取maximumovershoot最大超调量knowledgeassimilation知识同化maximumprinciple极大值原理KBMS(knowledgebasemanagementsystem)知mean-squareerrorcriterion均方误差准那么识库管理系统mechanismmodel机理模型knowledgerepresentation知识表达meta-knowledge元知识ladderdiagram梯形图metallurgicalautomation冶金自动化lag-leadcompensation滞后超前补偿minimalrealization最小实现Lagrangeduality拉格朗日对偶性minimumphasesystem最小相位系统Laplacetransform拉普拉斯变换minimumvarianceestimation最小方差估计largescalesystem大系统minorloop副回路lateralinhibitionnetwork侧抑制网络missile-targetrelativemovementsimulator弹体leastcostinput最小本钱投入-目标相对运动仿真器leastsquarescriterion最小二乘准那么modalaggregation模态集结levelswitch物位开关modaltransformation模态变换librationdamping天平动阻尼MB(modelbase)模型库limitcycle极限环modelconfidence模型置信度linearizationtechnique线性化方法modelfidelity模型逼真度linearmotionelectricdrive直线运动电气传动modelreferenceadaptivecontrolsystem模型参linearmotionvalve直行程阀考适应控制系统linearprogramming线性规划modelverification模型验证LQR(linearquadraticregulatorproblem)线性二modularization模块化次调节器问题MEC(mosteconomiccontrol)最经济控制loadcell称重传感器motionspace可动空间MTBF(meantimebetweenfailures)平均故障间隔orderparameter序参数时间orientationcontrol定向控制MTTF(meantimetofailures)平均无故障时间originator始发站multi-attributiveutilityfunction多属性效用函数oscillatingperiod振荡周期multicriteria多重判据outputpredictionmethod输出预估法multilevelhierarchicalstructure多级递阶构造ovalwheelflowmeter椭圆齿轮流量计multiloopcontrol多回路控制overalldesign总体设计multi-objectivedecision多目标决策overdamping过阻尼multistatelogic多态逻辑overlappingdecomposition交叠分解multistratumhierarchicalcontrol多段递阶控制Padeapproximation帕德近似multivariablecontrolsystem多变量控制系统Paretooptimality帕雷托最优性myoelectriccontrol肌电控制passiveattitudestabilization被动姿态稳定Nashoptimality纳什最优性pathrepeatability路径可重复性naturallanguagegeneration自然语言生成patternprimitive模式基元nearest-neighbor最近邻PR(patternrecognition)模式识别necessitymeasure必然性侧度Pcontrol比例控制器negativefeedback负反响peaktime峰值时间neuralassembly神经集合penaltyfunctionmethod罚函数法neuralnetworkcomputer神经网络计算机perceptron感知器Nicholschart尼科尔斯图periodicduty周期工作制noeticscience思维科学perturbationtheory摄动理论noncoherentsystem非单调关联系统pessimisticvalue悲观值noncooperativegame非合作博弈phaselocus相轨迹nonequilibriumstate非平衡态phasetrajectory相轨迹nonlinearelement非线性环节phaselead相位超前nonmonotoniclogic非单调逻辑photoelectrictachometrictransducer光电式转速nonparametrictraining非参数训练传感器nonreversibleelectricdrive不可逆电气传动phrase-structuregrammar短句构造文法nonsingularperturbation非奇异摄动physicalsymbolsystem物理符号系统non-stationaryrandomprocess非平稳随机过程piezoelectricforcetransducer压电式力传感器nuclearradiationlevelmeter核辐射物位计playbackrobot示教再现式机器人nutationsensor章动敏感器PLC(programmablelogiccontroller)可编程序逻Nyquiststabilitycriterion奈奎斯特稳定判据辑控制器objectivefunction目标函数plugbraking反接制动observabilityindex可观测指数plugvalve旋塞阀observablecanonicalform可观测标准型pneumaticactuator气动执行机构on-lineassistance在线帮助point-to-pointcontrol点位控制on-offcontrol通断控制polarrobot极坐标型机器人openlooppole开环极点poleassignment极点配置operationalresearchmodel运筹学模型pole-zerocancellation零极点相消opticfibertachometer光纤式转速表polynomialinput多项式输入optimaltrajectory最优轨迹portfoliotheory投资搭配理论optimizationtechnique最优化技术poseovershoot位姿过调量orbitalrendezvous轨道交会positionmeasuringinstrument位置测量仪orbitgyrocompass轨道陀螺罗盘posentiometricdisplacementtransducer电位器orbitperturbation轨道摄动式位移传感器positivefeedback正反响realizability可实现性,能实现性powersystemautomation电力系统自动化realtimetelemetry实时遥测predicatelogic谓词逻辑receptivefield感受野pressuregaugewithelectriccontact电接点压力表rectangularrobot直角坐标型机器人pressuretransmitter压力变送器rectifier整流器pricecoordination价格协调recursiveestimation递推估计primalcoordination主协调reducedorderobserver降阶观测器primaryfrequencyzone主频区redundantinformation冗余信息PCA(principalcomponentanalysis)主成分分析法reentrycontrol再入控制principleofturnpike大道原理regenerativebraking回馈制动，再生制动priority优先级regionalplanningmodel区域规划模型process-orientedsimulation面向过程的仿真regulatingdevice调节装载productionbudget生产预算regulation调节productionrule产生式规那么relationalalgebra关系代数profitforecast利润预测relaycharacteristic继电器特性remotemanipulator遥控操作器PERT(programevaluationandreviewtechnique)方案评审技术remoteregulating遥调programsetstation程序设定操作器remotesetpointadjuster远程设定点调整器proportionalcontrol比例控制rendezvousanddocking交会和对接proportionalplusderivativecontroller比例微分控reproducibility再现性制器resistancethermometersensor热电阻protocolengineering协议工程resolutionprinciple归结原理prototype原型resourceallocation资源分配pseudorandomsequence伪随机序列responsecurve响应曲线pseudo-rate-incrementcontrol伪速率增量控制returndifferencematrix回差矩阵pulseduration脉冲持续时间returnratiomatrix回比矩阵pulsefrequencymodulationcontrolsystem脉冲reverberation回响调频控制系统reversibleelectricdrive可逆电气传动pulsewidthmodulationcontrolsystem脉冲调宽revoluterobot关节型机器人控制系统revolutionspeedtransducer转速传感器PWMinverter脉宽调制逆变器rewritingrule重写规那么pushdownautomaton下推自动机rigidspacecraftdynamics刚性航天动力学QC(qualitycontrol)质量管理riskdecision风险分析quadraticperformanceindex二次型性能指标robotics机器人学qualitativephysicalmodel定性物理模型robotprogramminglanguage机器人编程语言quantizednoise量化噪声robustcontrol鲁棒控制quasilinearcharacteristics准线性特性robustness鲁棒性queuingtheory排队论rollgapmeasuringinstrument辊缝测量仪radiofrequencysensor射频敏感器rootlocus根轨迹rampfunction斜坡函数rootsflowmeter腰轮流量计randomdisturbance随机扰动rotameter浮子流量计，转子流量计randomprocess随机过程rotaryeccentricplugvalve偏心旋转阀rateintegratinggyro速率积分陀螺rotarymotionvalve角行程阀ratiostation比值操作器rotatingtransformer旋转变压器reachability可达性Routhapproximationmethod劳思近似判据reactionwheelcontrol反作用轮控制routingproblem路径问题sampled-datacontrolsystem采样控制系统socioeconomicsystem社会经济系统samplingcontrolsystem采样控制系统softwarepsychology软件心理学saturationcharacteristics饱和特性solararraypointingcontrol太阳帆板指向控制scalarLyapunovfunction标量李雅普诺夫函数solenoidvalve电磁阀SCARA(selectivecomplianceassemblyrobotsource源点arm)平面关节型机器人specificimpulse比冲scenarioanalysismethod情景分析法speedcontrolsystem调速系统sceneanalysis物景分析spinaxis自旋轴s-domains域spinner自旋体self-operatedcontroller自力式控制器stabilitycriterion稳定性判据self-organizingsystem自组织系统stabilitylimit稳定极限self-reproducingsystem自繁殖系统stabilization镇定，稳定self-tuningcontrol自校正控制Stackelbergdecisiontheory施塔克尔贝格决策理论semanticnetwork语义网络stateequationmodel状态方程模型semi-physicalsimulation半实物仿真statespacedescription状态空间描述sensingelement敏感元件staticcharacteristicscurve静态特性曲线sensitivityanalysis灵敏度分析stationaccuracy定点精度sensorycontrol感觉控制stationaryrandomprocess平稳随机过程sequentialdecomposition顺序分解statisticalanalysis统计分析sequentialleastsquaresestimation序贯最小二乘statisticpatternrecognition统计模式识别估计steadystatedeviation稳态偏差servocontrol伺服控制，随动控制steadystateerrorcoefficient稳态误差系数servomotor伺服马达step-by-stepcontrol步进控制settlingtime过渡时间stepfunction阶跃函数sextant六分仪stepwiserefinement逐步精化shorttermplanning短期方案stochasticfiniteautomaton随机有限自动机shorttimehorizoncoordination短时程协调straingaugeloadcell应变式称重传感器signaldetectionandestimation信号检测和估计strategicfunction策略函数signalreconstruction信号重构stronglycoupledsystem强耦合系统similarity相似性subjectiveprobability主观频率simulatedinterrupt仿真中断suboptimality次优性simulationblockdiagram仿真框图supervisedtraining监视学习simulationexperiment仿真实验supervisorycomputercontrolsystem计算机监控simulationvelocity仿真速度系统simulator仿真器sustainedoscillation自持振荡singleaxletable单轴转台swirlmeter旋进流量计singledegreeoffreedomgyro单自由度陀螺switchingpoint切换点singlelevelprocess单级过程symbolicprocessing符号处理singlevaluenonlinearity单值非线性synapticplasticity突触可塑性singularattractor奇异吸引子synergetics协同学singularperturbation奇异摄动syntacticanalysis句法分析sink汇点systemassessment系统评价slavedsystem受役系统systematology系统学slower-than-real-timesimulation欠实时仿真systemhomomorphism系统同态slowsubsystem慢变子系统systemisomorphism系统同构socio-cybernetics社会控制论systemengineering系统工程----tachometer转速表turbineflowmeter涡轮流量计targetflowtransmitter靶式流量变送器Turingmachine图灵机taskcycle作业周期two-timescalesystem双时标系统teachingprogramming示教编程ultrasoniclevelmeter超声物位计telemechanics远动学unadjustablespeedelectricdrive非调速电气传动telemeteringsystemoffrequencydivisiontypeunbiasedestimation无偏估计频分遥测系统underdamping欠阻尼telemetry遥测uniformlyasymptoticstability一致渐近稳定性teleologicalsystem目的系统uninterruptedduty不连续工作制，长期工作制teleology目的论unitcircle单位圆temperaturetransducer温度传感器unittesting单元测试templatebase模版库unsupervisedlearing非监视学习tensiometer X力计upperlevelproblem上级问题texture纹理urbanplanning城市规划theoremproving定理证明utilityfunction效用函数therapymodel治疗模型valueengineering价值工程thermocouple热电偶variablegain可变增益，可变放大系数thermometer温度计variablestructurecontrolsystem变构造控制thicknessmeter厚度计vectorLyapunovfunction向量李雅普诺夫函数three-axisattitudestabilization三轴姿态稳定velocityerrorcoefficient速度误差系数threestatecontroller三位控制器velocitytransducer速度传感器thrustvectorcontrolsystem推力矢量控制系统verticaldecomposition纵向分解thruster推力器vibratingwireforcetransducer振弦式力传感器timeconstant时间常数vibrometer振动计time-invariantsystem定常系统，非时变系统viscousdamping粘性阻尼timeschedulecontroller时序控制器voltagesourceinverter电压源型逆变器time-sharingcontrol分时控制vortexprecessionflowmeter旋进流量计time-varyingparameter时变参数vortexsheddingflowmeter涡街流量计top-downtesting自上而下测试WB(waybase)方法库topologicalstructure拓扑构造weighingcell称重传感器TQC(totalqualitycontrol)全面质量管理weightingfactor权因子trackingerror跟踪误差weightingmethod加权法trade-offanalysis权衡分析Whittaker-Shannonsamplingtheorem惠特克-香transferfunctionmatrix传递函数矩阵农采样定理transformationgrammar转换文法Wienerfiltering维纳滤波transientdeviation瞬态偏差workstationforcomputeraideddesign计算机辅助设计工作站transientprocess过渡过程transitiondiagram转移图w-planew平面transmissiblepressuregauge电远传压力表zero-basedbudget零基预算transmitter变送器zero-inputresponse零输入响应trendanalysis趋势分析zero-stateresponse零状态响应triplemodulationtelemeteringsystem三重调制zerosumgamemodel零和对策模型遥测系统z-transformz变换。

On the stochastic pendulum with Ornstein-Uhlenbeck noise

a r X i v :c o n d -m a t /0407198v 1 [c o n d -m a t .s t a t -m e c h ] 8 J u l 2004On the stochastic pendulum with Ornstein-Uhlenbeck noiseKirone MallickService de Physique Th´e orique,Centre d’´Etudes de Saclay,91191Gif-sur-Yvette Cedex,France ∗Philippe MarcqInstitut de Recherche sur les Ph´e nom`e nes Hors ´Equilibre,Universit´e de Provence,49rue Joliot-Curie,BP 146,13384Marseille Cedex 13,France †(Dated:March 16,2004)We study a frictionless pendulum subject to multiplicative random noise.Because of destructive interference between the angular displacement of the system and the noise term,the energy ﬂuctua-tions are reduced when the noise has a non-zero correlation time.We derive the long time behavior of the pendulum in the case of Ornstein-Uhlenbeck noise by a recursive adiabatic elimination pro-cedure.An analytical expression for the asymptotic probability distribution function of the energy is obtained and the results agree with numerical stly,we compare our method to other approximation schemes.PACS numbers:05.10.Gg,05.40.-a,05.45.-aI.INTRODUCTIONThe behavior of a nonlinear dynamical system is strongly modiﬁed when randomness is taken into account:noise can shift bifurcation thresholds,create new phases (noise-induced transitions),or even generate spatial patterns [1,2,3,4,5].The interplay of noise with nonlinearity gives rise to a variety of phenomena that constantly motivate new research of theoretical and practical signiﬁcance.Stochastic resonance [6]and biomolecular Brownian motors [7,8]are celebrated examples of nonlinear random systems of current interest.In particular,stochastic ratchets have generated a renewed interest in the study of simple mechanical systems subject to random interactions,the common ancestor of such models being Langevin’s description of Brownian motion.Many unexpected phenomena appear when one generalizes Langevin’s equations to include,e.g.,inertial terms,nonlinearities,external (multiplicative)noise or noise with ﬁnite correlation time;each of these new features opens a ﬁeld of investigations that calls for speciﬁc techniques or approximation schemes [9].Nonlinear oscillators with parametric noise are often used as paradigms for the study of these various eﬀects and related mathematical methods [10].The advantage of such models is that they have an appealing physical interpretation and appear as building blocks in many diﬀerent ﬁelds;they can be simulated on a computer or constructed as real electronic or mechanical systems [11,12].Moreover,the mathematical apparatus needed to analyze them remains relatively elementary (as compared to the perturbative ﬁeld-theoretical methods required for spatio-temporal systems [13])and can be expected to yield exact and rigorous results.In the present work,we study the motion of a frictionless pendulum with parametric noise,which can be physically interpreted as a randomly vibrating suspension axis.We show that the long time behavior of a stochastic pendulum driven by a colored noise with ﬁnite correlation time is drastically diﬀerent from that of a pendulum subject to white noise.Whereas the average energy of the white-noise pendulum is a linear function of time,that of the colored-noise pendulum grows only as the the square-root of time.Our analysis is based on a generalization of the averaging technique that we have used previouly for nonlinear oscillators subject to white noise:in [14,15,16]an eﬀective dynamics for the action variable of the system is derived after integrating out the fast angular variable,and is then exactly solved.However,this averaging technique as such ceases to apply to systems with colored noise when the time scale of the fast variable becomes smaller than the correlation time of the noise.Correlations between the fast variable and the noise modify the long time scaling behavior of the system and therefore must be taken into account.We shall develop here a method that systematically retains these correlation terms before the fast variable is averaged out.This will allow us to derive analytical expressions for the asymptotic probability distribution function (P.D.F.)of the energy of the stochastic pendulum,and to deduce the long time behavior of the system.Our analytical results are veriﬁed by numerical simulations.Finally,we shall compare our method and results with some known approximation schemes used for multivariate systems with colored noise.We shall show in particular thatsmallcorrelation time expansions cannot explain the anomalous diﬀusion exponent when truncated at anyﬁnite order.The partial summation of Fokker-Planck type terms,used to derive a‘best eﬀective Fokker-Planck equation’for colored noise[17,18]leads to results that agree with ours.We emphasize that the noise considered in this work has aﬁnite correlation time and its auto-correlation function does not have long time tails.This article is organized as follows.In section II,we analyse the case where the parametricﬂuctuations of the pendulum are modeled by Gaussian white noise,and explain heuristically why colored noise leads to anomalous scaling of the energy.In section III,we study the case of Ornstein-Uhlenbeck noise and explain how a recursive adiabatic elimination of the fast variable can be performed.This allows us to derive analytical results for the P.D.F. of the energy,which we validate with direct numerical simulations.In section IV,we compare our results with eﬀective Fokker-Planck approaches.Concluding remarks are presented in section V.II.THE PENDULUM WITH PARAMETRIC NOISEThe dynamics of a non-dissipative classical pendulum with parametric noise can be described by the following system of stochastic diﬀerential equations˙Ω=−(ω2+ξ(t))sinθ,(1)˙θ=Ω,(2) whereθrepresents the angular displacement andΩthe angular velocity.The energy E of the system isE=Ω2∂t =−∂∂Ω ω2sinθP t +D∂Ω2.(5)From Eq.(2),we observe that the angular variableθvaries rapidly as compared toΩ.Thus,following[14],we assume that,in the long time limit,the angleθis uniformly distributed over[0,2π].This allows us to average Eq.(5)over the angular variable and to derive an eﬀective Fokker-Planck equation for the marginal distribution˜P t(Ω):∂˜P t4∂2˜P twhere we have replaced sin2θby its mean value1/2.From this eﬀective Fokker-Planck equation,we readily deduce thatΩis a Gaussian variable with P.D.F.˜P t (Ω)=1πD texp −Ω222exp −2E4t,(9)and also the skewness andﬂatness factorsS(E)= E3 2)Γ(1Γ(5√E2 2=Γ(92)2)2=35whereαis an unknown exponent to be determined.Weﬁnd from Eq.(2)thatθ∼tα+1,and Eq.(1)can then be written as˙Ω≃ξ(t)sin tα+1.(13) (We could have retained the deterministic term−ω2sinθbut it would not aﬀect this scaling analysis.)We now take ξ(t)to be a discrete dichotomous noise with correlation timeτand with values±1.The previous equation,then, becomesΩ(t)∼t/τk=1ǫk kτ(k−1)τsin xα+1d x,withǫk=±1.(14)We estimate the last integral by an integration by parts:(α+1) t+τt xαsin xα+1xα t+τt+O(t−α−1),(15) and obtainΩ2 ∼t/τk=1 kτ(k−1)τsin xα+1d x 2∼t/τ k=112τe−|t−t′|/τ,(17)whereτis the correlation time of the noise.This noiseξcan be generated from white noise via the Ornstein-Uhlenbeck equation˙ξ=−1τη(t),(18) whereη(t)is a white noise of auto-correlation function Dδ(t−t′).In the stationary limit,t,t′≫τ,the solution of Eq.(18)satisﬁes Eq.(17).The pendulum with Ornstein-Uhlenbeck noise is thus written as a three-dimensional stochastic dynamical system coupled to a white noiseη(t):˙Ω=−ω2sinθ−ξsinθ,(19)˙θ=Ω,(20)˙ξ=−1τη(t).(21) The Fokker-Planck equation for the three-dimensional P.D.F.P t(θ,Ω,ξ)is given by∂P t∂θ(ΩP t)+∂τ∂2τ2∂2P tA.Zeroth-order averagingWe show here that the averaging procedureused in section II A forwhite noiseleads toerroneous results for colored noise.Averaging the Fokker-Planck equation (22)over the fast angular variable θ,we ﬁnd that the marginal P.D.F.˜Pt (Ω,ξ)obeys the Ornstein-Uhlenbeck diﬀusion equation ∂˜Pt τ∂2τ2∂2˜Pt τξ+12to yield Eq.(6).In the next subsections,we develop an averaging scheme that allows us to eliminate adiabatically the fast variable while retaining the correlation terms.The idea is to deﬁne recursively a new set of dynamical variables that embodies the correlations order by order.This will enable us to derive sound asymptotic results for the pendulum with colored noise.In this scheme,Eq.(24)appears as a zeroth order approximation,and its correct interpretation is not that Ωis conserved but that its variations are slower than that of normal diﬀusion.B.First-order averagingMultiplying both sides of Eq.(19)by Ω,and using Eq.(20),we obtainΩ˙Ω=−Ω ω2sin θ+ξsin θ=−ω2˙θsin θ−ξ(˙θsin θ).(25)Introducing the energy E of the system deﬁned in Eq.(3),this equation becomesdE dtΩ2dt=ddt(E −ξcos θ)=cos θτη.(28)This leads us to deﬁne a new dynamical variable E 1E 1=E −ξcos θ=Ω2τξ−cos θ2 E 1+(ω2+ξ)cos θ ,(31)˙ξ=−1τη(t ).(32)FIG.2:Stochastic pendulum for ω=1.0and (D ,τ)=(1,0.1),(1,1),(1,10),(10over 104realizations.We plot the ratio E 2 /(D t/τ2)vs.time t .The dotted line in the ﬁgure corresponds to the asymptotic value 0.25predicted by second-order averaging (Eq.(55)).Note that convergence is slower for smaller values of the correlation time τ.The advantage of this system as compared to the previous one (19,20,21)is that the white noise η(t )now appears in the equation for the dynamical variable E 1and this white noise contribution will survive the averaging process.The Fokker-Planck equation for the P.D.F.P t (E 1,θ,ξ)associated with Eqs.(30,31and 32)reads:∂P t ∂E 1cos θ∂θ(Ω(E 1,θ,ξ)P t )+1∂ξ(ξP t )+D ∂E 21−2cos θ∂2P t ∂ξ2 .(33)Averaging this equation with respect to θleads to the following evolution equation for the marginal distribution˜Pt (E 1,ξ):∂˜Pt τ∂4τ2∂2˜Pt 2τ2∂2˜Pt √τξ+1√D twith E ≥0.(37)From this expression we calculate the ﬁrst two moments of the energyE =1π D t 2≃0.564D t2,(38) E 2 =1τ2.(39)These expressions provide scaling relations between the averages,the time t and the dimensional parameters of the problem,D and τ.We have veriﬁed numerically that these scalings are correct (see Fig.2).However,the prefactorsthat appear in Eqs.(38)and(39)are pure numbers and do not agree with the results of our numerical simulations. We conclude that Eq.(38)is exact at leading order as it gives the correct asymptotic scaling for the energy,E∝t1/2 but fails to provide the prefactors.The reason is that some correlations betweenθandξhave still been neglected in the averaging procedure.Carrying out the calculations to the next higher order will enable us to derive the correct expressions for the prefactors.C.Second-order averagingMore precise results can indeed be derived by applying recursively the procedure described above.In the Langevin equation(30)for E1,we perform one more‘integration by parts’and obtain˙E 1=ξcosθΩ−cosθdt ξsinθτΩ˙ξ+˙Ωτξ−cosθτΩ=Ω2τΩ.(41)Using Eqs.(19)and(21),Eq.(40)becomes˙E 2=−ξsinθτΩ2(ω2+ξ)− cosθ+sinθτ.(42)In this equation we must express the variableΩin terms of E2,θandξ.Inverting the relation(41),we deduce thatΩ=(2E2)1(2E2)1τ(2E2)+O 122 ,(43) where we have retained terms up to the order1/E2.From Eq.(42),we deduce the Langevin equation for E2˙E 2=J E(E2,θ,ξ)+D E(E2,θ,ξ)η(t)E3τ2(2E2)1τ(2E2)(ω2+ξ),(45)D E(E2,θ,ξ)=− cosθ+sinθ2 .(46)This equation,combined with Eqs.(20and21),deﬁnes a three-dimensional stochastic system for the variables (E2,θ,ξ).The Fokker-Planck equation for the P.D.F.P t(E2,θ,ξ)is∂P t∂E2(J E P t)−∂τ∂2τ2 ∂∂E2(D E P t)+∂2∂E2D E∂P t∂ξ2 .(47)We now integrate out the fast angular variableθfrom Eq.(47),retaining only the leading term in the average of the expression∂∂E2(D E P t),(recalling that∂/∂E2scales as E−12,the contribution of the subdominant terms is ofthe order of E−5/22and is negligible in the long time limit).We thus obtain the following evolution equation for themarginal distribution˜P t(E2,ξ):∂˜P t∂E2 ω2ξ+ξ2τ∂2τ21∂E22+∂2˜P tAlthough the cross-derivative terms between E2andξvanish,the two variables are coupled through the drift term. From Eq.(48)we derive an eﬀective,two-dimensional,stochastic system in E2andξ:˙E 2=−ω2ξ+ξ2√τξ+18τ2E2+12τη2(t)−18τ2E2+12τη2(t).(52)The variables E2andξare decoupled at large times:the eﬀective problem is thus one-dimensional.In the long time limit,the variable E2is identical to the energy E up toﬁnite terms.We thus obtain the asymptotic P.D.F.of the pendulum’s energy by explicitely solving the Fokker-Planck associated with Eq.(52)˜P t (E)=2√Γ 12exp−τ2E22π4)2 D tτ21/2,(54)E2 =1τ2.(55) Besides the skewness and theﬂatness factors areS(E)=Γ(74)4)3/2=6√Γ(1 4)Γ(1Γ(5FIG.3:τ=1.(a)the average E and the ratio E /(D t/τ2)1/2(inset),(b)the skewness andﬂatness factors of E vs.time t.The asymptotic behavior of all observables presented agrees with Eqs.(54–57)(dotted lines in theﬁgures),irrespective of the value ofω.the predictions of Eqs.(54-57).In particular,we notice that the asymptotic P.D.F.˜P t(E),and therefore the moments of the energy,do not depend on the value of the mean frequencyω(see Fig.3).Our averaging technique thus provides sound asymptotic results for the energy of the stochastic pendulum:this technique not only yields the correct scalings but also leads to analytical formulae for the large time behavior of the energy.This averaging method could be carried over to the third order to calculate the subdominant corrections to the P.D.F.of the energy.However,we shall not pursue this course any further:the calculations become very unwieldy and the agreement between the analytical results and the numerical computations is already very satisfactory at the second order.We emphasize that the white noise and the colored noise cases fall in two distinct universality classes because the long time scaling exponents are diﬀerent.The colored noise scaling will always be observed after a suﬃciently long time provided that the correlation timeτis non-zero.However,the eﬀect of this correlation time appears only when the period T of the pendulum is less thanτ.This period,which is proportional toΩ−1,decreases with time.At short times,the period T is much greater thanτand white noise scalings are observed.At large times,T≪τand colored noise scalings are satisﬁed.The crossover time t c is reached when T∼Ω−1∼τ;using Eq.(9),which is valid for t<t c,we obtaint c∼(Dτ2)−1.(58) Hence,when the correlation timeτbecomes vanishingly small,the crossover time diverges to inﬁnity and the colored noise regime is not reached(simulation times needed to observe the colored noise scalings become increasingly long). In Fig.4,we plot the behavior of the mean energy vs time for D=1.0andτ=0.01,0.1and1.Forτ=1,the colored noise scaling regime is obtained from the very beginning and the curve for E has a slope1/2in log-log scale.For τ=0.1,at short times E ∝t whereas at long times E ∝t1/2.The crossover is observed around t c∼100=τ−2. Forτ=0.01,the curve for E has a slope1and the colored noise regime is not reached in this simulation(t c∼104). The averaging method provides analytical results for t≫t c and t≪t c.The intermediate time regime is described by a crossover function of the scaled variable t/t c[15],which cannot be analyzed by our technique.In the next section,we compare our method with two well-known approximations based on eﬀective colored noise Fokker-Planck equations.One of these approaches will enable us to derive approximate formulae for t c and for the crossover function.PARISON WITH OTHER APPROXIMATION SCHEMESThe main diﬃculty for the study of a Langevin equation with colored noise stems from its non-Markovian character [1]:there exists no closed Fokker-Planck equation that describes the evolution of the P.D.F.of the dynamical variables. The process can be embedded into a Markov system if the colored noise itself is treated as a variable.However,this mathematical trick increases the dimension of the problem by one and the noise has to be integrated out in the end: exact calculations can be carried out with this method only in the case of linear problems.FIG.4:Stochastic pendulum forω=0.0, D=1.0andτ=0.01,0.1and energy vs. time.The crossover between white noise behavior( E ∼t)and colored noise behavior( E ∼t1/2)occurs later for smaller correlation timeτ.A colored noise master equation can be rigorously derived(e.g.,with the help of functional methods)but it involves correlations between the dynamical variables and the noise[23,24].The equation of motion for these correlations involves higher order correlations,and so on.Since this hierarchy must be stopped at some stage,this question is a genuine closure problem.Many diﬀerent approaches have been devoted to derive eﬀective Fokker-Planck equations, such as short correlation time expansions[17,18,25,26],uniﬁed colored noise approximation[27],projection methods [28]and self-consistent decoupling Ans¨a tze[29](for a general review see[30]).In section III,to derive the long time behavior of the stochastic pendulum with colored noise,we did not use any closure approximation but started from the exact Fokker-Planck equation for the system,treating the noise as an auxiliary variable.Thus,we did not make any hypothesis onτand our analytical formulae are valid for any value of the correlation time(for t larger than the crossover time given in Eq.(58)).In this section,we compare our results with those that can be derived from some well-known approximation schemes.A.Small correlation time expansionAn approximate evolution equation for the P.D.F.of a Langevin equation with colored noise can be derived in the case of short correlation times(i.e.,in the white noise limit)by expanding the colored noise master equation around the Markovian point[26,30].This procedure leads to a Fokker-Planck type equation,with eﬀective drift and diﬀusion coeﬃcients.Applying to our system(19,20,21)the smallτexpansion derived in[26]for arbitrary stochastic equations with colored noise,we obtain,atﬁrst order inτ,the eﬀective Fokker-Planck equation for P t(θ,Ω)∂P t∂θ(ΩP t)−3Dτ∂Ω+D(1−τΩ)∂Ω2+Dτ∂θ∂Ω sin2θP t .(59)To simplify the discussion,we have taken the mean frequencyωequal to zero.After integrating out the rapid variations ofθan averaged Fokker-Planck equation is obtained which is similar to Eq.(6),but with an eﬀective diﬀusion given by D eﬀ=D(1−τΩ)(because D eﬀbecomes negative for large values ofΩ,Eq.(59)is valid only over the restricted region of positive diﬀusion).From dimensional analysis,we observe that such an eﬀective Fokker-Planck equation leads to a normal diﬀusive behavior,Ω∼t1/2and E∼t,and therefore cannot account for the results we have obtained.B.Best Eﬀective Fokker-Planck equationWe now turn to another smallτapproximation,which intends to improve theﬁrst-order eﬀective Fokker-Planck equation(59)by summing contributions of the type Dτn(where n is an integer larger than1).The resulting equation has been christened‘Best Fokker-Planck Equation’(B.F.P.E.)by its proponents[17,18,31].Although this approach is not free from drawbacks and is known to lead in some cases to unphysical results[32],we show that for the system studied in this work,the B.F.P.E.leads to results that agree with ours.An approximate evolution equation for the P.D.F.P t(Ω,θ)is given by the second-order cumulant expansion[1]of the(stochastic)Liouville equation associated with Eqs.(19and20):∂P t∂θ(ΩP t),(61)L1(t)P t=−∂∂t =−∂2∂21+(τΩ)2P t +Dτ∂Ωsinθ∂(1+(τΩ)2)2P t .(63)Integrating out the fast variableθ,we obtain an averaged B.F.P.E for˜P t(Ω),the probability distribution of the slow variable,∂˜P t4∂1+(τΩ)2∂˜P t4 Ω4 +14t.(65)When t is small,the quadratic term dominates over the quartic term,and we recover Ω2 =2 E ≃D4 Ω4 ≃1τ2t.(66)This result is identical to our Eq.(55),which was validated by numerical results(see Fig.2).The identity(65)can also be used to derive an approximate scaling function for the mean energy.Let us deﬁne theﬂatnessφof˜P t(Ω)asΩ4 =φ Ω2 2.(67) Rigorously speakingφis a function of time,but it remains a number of order1.For simplicity,let us assume thatφis constant.Substituting Eq.(67)in Eq.(65),and solving for Ω2 ,we obtainE =1Dτ2φt+1−1V.CONCLUSIONA nonlinear pendulum subject to parametric noise undergoes a noise-induced diﬀusion in phase space.The charac-teristics of this motion depend on the nature of the randomness:when the noise is white the energy of the pendulum grows linearly with time,whereas it varies only as the square root of time when the noise is colored.This change of behavior is due to destructive interference between the displacement of the pendulum and the noise term:the eﬀect of the colored noise is partially averaged out by fast angular variations as soon as the period of the pendulum becomes smaller than the correlation time τof the noise.We have carried out an analytical study of this model by deﬁning recursively new coordinates in the phase space and averaging out the fast angular variable.At zeroth order,the only information obtained is that the motion is subdiﬀusive;at ﬁrst order,this procedure provides the correct scalings;at second order,a quantitative agreement with numerical simulations is reached.We emphasize that our method is diﬀerent from the usual approximations that involve eﬀective Fokker-Planck equations in which the colored noise does not appear as an auxiliary variable.Our averaging procedure integrates out the fast dynamical variable and leads to an eﬀective stochastic dynamics for the slow variables with colored noise.Whereas usual eﬀective Fokker-Planck equations are valid only for small noise and for short correlation times,we do not make any hypothesis on the amplitude of the noise or on the correlation time τ.However,the asymptotic subdiﬀusive regime is reached earlier for larger values of τ.Our results agree with those derived after averaging the ‘Best Fokker-Planck Equation’(B.F.P.E.)approximation (which is obtained from a summation of a truncated cumulant expansion of the Liouville equation).This approximation is also useful to draw a qualitative physical picture of the system and allows to calculate approximate crossover functions between short time (white noise)and long time (colored noise)regimes.The recursive averaging scheme that we have used here for the stochastic pendulum can be extended to other systems coupled with colored noise.In particular,we believe that thanks to this method a precise mathematical analysis of the long time behavior of a nonlinear oscillator subject to multiplicative or additive colored noise can be carried out [33].AcknowledgmentsWe thank S.Mallick for his helpful comments on the manuscript and F.Moulay for useful discussions.APPENDIX A:DERIV ATION OF EQ.(63)In this appendix,we derive the B.F.P.E.(Eq.(63))following the procedure of [17,18].We evaluate the right hand side of Eq.(60)by applying the following operator formulaexp(A )B exp(−A )=B +[A,B ]+13![A,[A,[A,B ]]]+...,(A1)with A =L 0and B =L 1(t −x ).We thus have to calculate some commutators of the two operators L 0and L 1(t −x )deﬁned in Eqs.(61)and (62).By induction,we derive the following expression for the n -th commutatorT n =[L 0,[...,[L 0,[L 0,L 1(t −x )]]...]=ξ(t −x ) ∂∂θH (n )2(Ω,θ) ,(A2)where the functions H (n )1and H (n )2satisfy the recursion relationsH (n )1=−Ω∂H (n −1)1∂θ.(A4)The ﬁrst few terms can be calculated explicitely and we obtainH (0)1=−sin θand H (0)2=0,(A5)H (1)1=Ωcos θand H (1)2=−sin θ,(A6)H (2)1=Ω2sin θandH (2)2=2Ωcos θ.(A7)The general solution for the recursion(A4)is readily found:H(n)1=(−1)n−1Ωn sin(θ+nπ2).(A9)From Eqs.(A1)and(A2),we deduce the following identityL1(t)exp(L0x)L1(t−x)exp(−L0x) = ∞n=0x n∂Ωsinθ ∞n=0x n∂ΩH(n)1(Ω,θ)+∂2τ∂∂ΩH1(Ω,θ,t)+∂n!H(n)1=∞n=0 t/τ0d x x n e−x2),(A12)H2(Ω,θ,t)=∞n=0 t0d xx n e−x/τn!(−1)n−1τn+1nΩn−1cos(θ+nπ1+(τΩ)2,(A14) H2(Ω,θ,∞)=−τ2(1−(τΩ)2)sinθ−2τΩcosθ[18]Lindenberg K and West B J1984Physica A12825[19]Bourret R C,Frisch U and Pouquet A1973Physica65303[20]Van Kampen N G1976Phys.Rep.24171[21]Lindenberg K,Seshadri V and West B J1981Physica A105445[22]Lindenberg K,Seshadri V and West B J1980Phys.Rev.A222171[23]H¨a nggi P1985in Stochastic Processes Applied to Physics edited by Pesquera L and Rodriguez M A(Singapore:WorldScientiﬁc)[24]H¨a nggi P1989in Noise in Dynamical Systems Vol.1edited by Moss F and Mc Clintock P V E(Cambridge:CambridgeUniversity Press)[25]San Miguel M and Sancho J M1980Phys.Lett.A7697[26]Ramirez-Piscina L and Sancho J M1988Phys.Rev.A374469[27]H’walisz L,Jung P,H¨a nggi P,Talkner P and Schimansky-Geier L1989Z.Phys.B77471[28]Fox R F1986Phys.Rev.A33467[29]Fronzoni L,Grigolini P,H¨a nggi P,Moss F,Mannella R and Mc Clintock P V E1986Phys.Rev.A333320[30]H¨a nggi P and Jung P1995Adv.Chem.Phys.89239[31]Peacock-Lopez E,de la Rubia F J,Lindenberg K and West B J1989Phys.Lett.A13696[32]H¨a nggi P,Marchesoni F and Grigolini P1984Z.Phys.B56333[33]Mallick K and Marcq P in preparation。

the standard model

Discrete linear objects in dimension n :the standard modelEric Andres *IRCOM-SIC,SP2MI,BP 30179,F-86962Futuroscope Cedex,FranceReceived 15May 2002;received in revised form 4September 2002;accepted 17October 2002AbstractA new analytical description model,called the standard model,for the discretization of Eu-clidean linear objects (point,m -ﬂat,m -simplex)in dimension n is proposed.The objects are deﬁned analytically by inequalities.This allows a global deﬁnition independent of the number of discrete points.A method is provided to compute the analytical description for a given lin-ear object.A discrete standard model has many properties in common with the supercover model from which it derives.However,contrary to supercover objects,a standard object does not have bubbles.A standard object is ðn À1Þ-connected,tunnel-free and bubble-free.The standard model is geometrically consistent.The standard model is well suited for modelling applications.Ó2003Elsevier Science (USA).All rights reserved.Keywords:Discrete geometry;Digitization;Dimension n ;Simplex;m -ﬂat1.IntroductionWhen working in discrete geometry,aside from considering an object simply as a set of discrete points,the problem of deﬁning discrete geometrical objects arises.A discrete 2D line segment can be deﬁned as 8-connected,4-connected or even discon-nected as a dotted line.There is not a unique way of deﬁning a discrete object or of digitizing a Euclidean object.This problem has been around for 40years and many diﬀerent discrete object deﬁnitions have been proposed.One can say thatauthors Graphical Models 65(2003)92–111*Fax:+33-5-49-49-6570.E-mail address:andres@sic.univ-poitiers.fr.1524-0703/03/$-see front matter Ó2003Elsevier Science (USA).All rights reserved.doi:10.1016/S1524-0703(03)00004-3E.Andres/Graphical Models65(2003)92–11193 have followed three main approaches to deﬁne discrete geometrical objects:an algo-rithmic approach,a topological approach,and a more recent analytical approach followed in this paper.In the algorithmic approach[1,10,13,16,21–24,34]a discrete object is the result of a generation algorithm.Historically,theﬁrst approach that has been used,it has shown a number of limitations.It is often diﬃcult to control the properties of the so deﬁned discrete objects.For instance,the discrete objects might not be geometrically consistent:the edge of a3D triangle is typically not nec-essary a3D line segment or the3D triangle is not a piece of3D plane[21,22].It is also diﬃcult to propose generation algorithms for discrete objects in dimension high-er than three.Except for n-dimensional lines[34],to the best authors knowledge,no discrete object,in dimensions higher than three,has been algorithmically deﬁned.In the topological approach,a discrete object is typically deﬁned as a class of objects verifying local properties,often topological in nature[18–20,25,28].While it is,by deﬁnition,easier to obtain the desired properties,it is diﬃcult to be sure with such an approach,that the class of objects deﬁned by a given set of properties is not larger than what is initially expected.A third,more recent approach,deﬁnes a discrete ob-ject by a global analytical deﬁnition[2–5,7,9,17,18,25,30,32].This approach has many advantages such as providing a compact deﬁnition(independent on the num-ber of points forming the discrete object),a global control of the discrete object.It has also an advantage that is not immediately visible when one is not familiar with this approach.It allows a good control of the local topological properties of the ob-ject.The many links with mathematical morphology are also an interesting property of some analytically deﬁned models such as the supercover model[7,19,26,29,31,33]. One of the main advantages is that it is relatively easy to deﬁne discrete objects in an arbitrary dimension[3,4,7,30,32].The standard model introduced in the following pages is analytically deﬁned.A new analytical description model for all linear objects in dimension n(discrete points,m-ﬂats,and geometrical simplices)is presented in this paper.The analytical model is called the standard model.The names derives from the name given by Francßon[18]toðnÀ1Þ-connected analytical discrete3D planes(see also[4]for gen-eral details on discrete analytical hyperplanes).To the best authors knowledge,it is theﬁrst time that a discrete model is proposed that deﬁnes a large class of discrete objects in arbitrary dimensions.The standard model is called a discrete analytical model because the discrete objects(points,m-ﬂats,simplices)are deﬁned analytically by inequalities.The analytical deﬁnition is independent of the number of discrete points of the object.For instance,a3D standard triangle is deﬁned by17or less in-equalities independently of its size.The model we propose has many interesting properties.The model is geometri-cally consistent:for instance,the vertices of a3D standard polygon are3D standard points,the edges of a3D standard polygon are3D standard line segments and the 3D standard polygon is a piece of a3D standard plane.It has been shown that the standard model is in fact a0-discretization of Brimkov et al.[12]and therefore isðnÀ1Þ-connected and tunnel-free.In3D,ðnÀ1Þ-connectivity in our notations corresponds to the classical6-connectivity.Contrary to the supercover model,from which it derives,the standard objects are bubble-free.One of the problems of the su-percover model is that it is not topologically consistent.A supercover m -ﬂat is always ðn À1Þ-connected but sometimes it has simple points (located on so-called bubbles on the object).This makes the model diﬃcult to use in practice [14,15].For instance,a supercover of a Euclidean n D point can be composed of any 2i discrete points,06i 6n .A standard m -ﬂat is almost identical to the supercover m -ﬂat,it remains ðn À1Þ-connected and tunnel-free,except for the simple points in the bubbles that are removed.The standard digitization of a n D Euclidean point is always composed only of one discrete point.Finally,the standard model has a very important property in the framework of discrete modelling:S t ðF [G Þ¼S t ðF Þ[S t ðG Þ.This means that,for instance,the deﬁnition of the standard 3D polygon is suﬃcient to deﬁne the standard model of an arbitrary Euclidean polygonal 3D object.The deﬁnition of the standard model is derived from the supercover model [2,5–7,14,15,31].A standard object is obtained by a simple rewriting process of the in-equalities deﬁning analytically a supercover object [7].A supercover linear object is deﬁned by a set of inequalities ‘‘P n i ¼1a i X i 6a 0.’’The simple points in the bubbles are points that verify ‘‘P n i ¼1a i X i ¼a 0.’’In order to remove the simple points,and thus bubbles,some of the inequalities need simply to be rewritten into ‘‘P n i ¼1a i X i <a 0.’’The selection of inequalities that are modiﬁed is based on an ori-entation convention.Depending on the orientation of the half-space,the corre-sponding inequality is modiﬁed or not.In Section 2,we introduce our notations and the principal properties of the super-cover model on which the standard model is based.In Section 3the standard model is introduced and deﬁned.We start,in Section 3.1,by explaining why such a ‘‘heavy’’mathematical machinery is necessary to deﬁne ðn À1Þ-connected discrete objects.We show in particular why a classical,misleading,approach does not work.In Sec-tion 3.2,we explain the basic ideas behind the standard model.In Section 3.3,we introduce the orientation convention that forms the basis of the deﬁnition of the standard model.The standard model is deﬁned for all linear primitives in dimension n in Section 3.4.The properties of the standard primitives,especially the tunnel-free-ness and the ðn À1Þ-connectivity,are presented in Section 3.5.In Section 4,we ex-amine the diﬀerent classes of standard linear objects to see how the deﬁnition is translated in practice and how the diﬀerent inequalities deﬁning the objects are estab-lished.Conclusion and several perspectives are presented in Section 5.2.Preliminaries2.1.Basic notations in discrete geometryMost of the following notations correspond to those given by Cohen and Kauf-man in [14,15]and those given by Andres in [7].We provide only a short recall of these notions.Let Z n be the subset of the n D Euclidean space R n that consists of all the integer coordinate points.A discrete (resp.Euclidean )point is an element of Z n (resp.R n ).A discrete (resp.Euclidean )object is a set of discrete (resp.Euclidean )points.A discrete94 E.Andres /Graphical Models 65(2003)92–111inequality is an inequality with coeﬃcients in R from which we retain only the integer coordinate solutions.A discrete analytical object is a discrete object deﬁned by a ﬁ-nite set of discrete inequalities .An m -ﬂat is a Euclidean aﬃne subspace of dimension m .Let us consider a set P of m þ1linearly independent Euclidean points P 0;...;P m .We denote A m ðP Þthe m -ﬂat induced by P (i.e.,the m -ﬂat containing P ).We denote S m ðP Þthe geometrical simplex of dimension m in R n induced by P (i.e.,the convex hull of P ).For S ¼S m ðP Þa geometrical simplex,we denote S ¼A m ðP Þthe corre-sponding m -ﬂat.For an n -simplex S ¼S n ðP Þ,we denote E ðS ;P i Þthe half-space of boundary A n À1ðP n P i Þthat contains P i (see Fig.1).We denote p i the i th coordinate of a point or vector p .Two discrete points p and q are k -neighbours ,with 06k 6n ,if j p i Àq i j 61for 16i 6n ,and k 6n ÀP ni ¼1j p i Àq i j .The voxel V ðp Þ&R n of a discrete n D point p is deﬁnedby V ðp Þ¼½p 1À12;p 1þ12 ÂÁÁÁÂ½p n À12;p n þ12 .For a discrete object F ,V ðF Þ¼S p 2F V ðp Þ.We denote r n the set of all the permutations of f 1;...;n g .Let us de-note J nm the set of all the strictly growing sequences of m integers all between 1and n :J n m ¼f j 2Z m j 16j 1<j 2<ÁÁÁ<j m 6n g .This deﬁnes a set of multi-indices .Let us consider an object F in the n -dimensional Euclidean space R n ,with n >1.The orthogonal projection is deﬁned by:p i ðF Þ¼fðq 1;...;q i À1;q i þ1;...;q n Þj q 2R n gfor 16i 6n ;p j ðF Þ¼ðp j 1 p j 2 ÁÁÁ p j m ÞðF Þfor j 2J n m :The orthogonal extrusion is deﬁned bye j ðF Þ¼p À1j ðp j ðF ÞÞfor j 2J n m :Example.Let us consider the set of points P ¼f P 0ð0;0;0Þ;P 1ð9;1;1Þ;P 2ð3;8;4Þg .The corresponding simplex T ¼S 2ðP Þis a 3D triangle.The orthogonal projection p 2ðT Þ¼S 2ðp 2ðP ÞÞ¼S 2ðfð0;0Þ;ð9;1Þ;ð3;4ÞgÞis a 2D triangle.The orthogonal ex-trusion e 2ðT Þ¼fð0;t ;0Þ;ð9;t ;1Þ;ð3;t ;4Þj t 2R g is a 3D Euclidean object deﬁned by threehalf-spaces.Fig.1.Triangle T ¼S 2ðf P 0;P 1;P 2gÞ,edge S 1ðf P 0;P 1gÞ,straight line A 1ðf P 0;P 1gÞ,and half-space E ðf P 0;P 1;P 2g ;P 2Þ.E.Andres /Graphical Models 65(2003)92–11195We deﬁne an axis arrangement application r j,for j2J nm,by: r j:R n!R nx!ðx rjð1Þ;x rjð2Þ;...;x rjðnÞÞ;where the permutation r j2r n is deﬁned byr j¼for16i6m;r jðj iÞ¼i else for m<i6n;r jðk rÞ¼iso that k r<k rþ1and k r¼j s for all16r6nÀm and for all16s6m.The axis arrangement application has been speciﬁcally designed so that it veriﬁes the twofollowing properties:p jðFÞ¼pð1;2;...;mÞðrÀ1j ðFÞÞand e jðFÞ¼r jðeð1;2;...;mÞðrÀ1jðFÞÞÞfor allF in R n and j2J nm.Example.Let us consider the5D point Pð1;2;3;4;5Þand j¼ð2;4Þ2J52.Thecorresponding axis arrangement application is deﬁned by rð2;4Þ:x!ðx3;x1;x4;x2;x5Þand rÀ1ð2;4Þ:x!ðx2;x4;x1;x3;x5Þ.The orthogonal projection veriﬁes pð2;4ÞðPÞ¼pð1;2ÞðrÀ1ð2;4ÞðPÞÞ¼pð1;2Þð2;4;1;3;5Þ¼ð1;3;5Þ.The orthogonal extrusion veriﬁes eð2;4Þð1;3;5Þ¼rð2;4Þðeð1;2ÞðrÀ1ð2;4ÞðPÞÞÞ¼rð2;4Þðeð1;2Þð2;4;1;3;5ÞÞ¼rð2;4ÞðpÀ1ð1;2Þð1;3;5ÞÞand therefore eð2;4Þð1;3;5Þ¼rð2;4Þðfðt;u;1;3;5Þjðt;uÞ2R2gÞ¼fð1;t;3;u;5Þjðt;uÞ2R2g.2.2.Geometric properties of the supercoverA discrete object G is a cover of a Euclidean object F if F&VðGÞand 8p2G;VðpÞ\F¼£.The supercover SðFÞof a Euclidean object F is deﬁned by SðFÞ¼f p2Z n j VðpÞ\F¼£g(see Fig.2a).SðFÞis by deﬁnition a cover of F.It is easy to see that if G is a cover of F,then G&SðFÞ.The supercover of FcanFig.2.Supercover deﬁnitions.96 E.Andres/Graphical Models65(2003)92–111be deﬁned in diﬀerent ways:S ðF Þ¼ðF ÈB 1ð12ÞÞ\Z n ¼f p 2Z n j d 1ðp ;F Þ612g (see Fig.2b)where B 1ðr Þif the ball centered on the origin,of radius r for the distance d 1.This links the supercover to mathematical morphology [7,26,29,31].The supercover has many properties.Let us consider two Euclidean objects F andG ,and a multi-index j 2J n m ,then:S ðF Þ¼S a 2F S ða Þ,S ðF [G Þ¼S ðF Þ[S ðG Þ,ifF &G ,then S ðF Þ&S ðG Þ.These properties are well known [14,15].The following properties are more recent and are useful in the framework of this paper:S ðF ÂG Þ¼S ðF ÞÂS ðG Þ,r j ðS ðF ÞÞ¼S ðr j ðF ÞÞ,p j ðS ðF ÞÞ¼S ðp j ðF ÞÞ,and e j ðS ðF ÞÞ¼S ðe j ðF ÞÞ¼r j ðZ m ÂS ðp j ðF ÞÞÞ[7].Deﬁnition 1(Bubble ).A k -bubble,with 16k 6n ,is the supercover of a Euclidean point that has exactly k half-integer coordinates.A half-integer is a real l þ12,with l an integer.A k -bubble is formed of 2k discrete points.A 2-bubble can be seen in Fig.2a (marked by the black circle).The two white dots are what we call here ‘‘simple’’points.This corresponds to an extension of the notion of simple points that ﬁts a supercover simplex.A point P belonging to the supercover simplex S is said to be a simple point if it is a simple point for S with the classical deﬁnition given in Section 2.1.Deﬁnition 2(Bubble-free ).The cover of an m -ﬂat is said to be bubble-free if it has no k -bubbles for k >m .The cover of a simplex S is said to be bubble-free if S is bubble-free.There are two types of bubbles in the supercover of an m -ﬂat F .The k -bub-bles,for k 6m ,are discrete points that are part of all the covers of F .If we remove any of these points,the discrete object is not a cover anymore.In the k -bubbles,for k >m ,there are discrete points that are ‘‘simple’’points.The aim of this paper is to propose discrete analytical objects that are bubble-free and ðn À1Þ-connected by removing some of the simple points.In Fig.2a,by removing one of the two simple points,we obtain a bubble-free,1-connected discrete 2D line segment.Lemma 1.A discrete point p belongs to a k -bubble,k >m ,of the supercover of an m -flat F if and only if there exists a point a 2F with k half-integer coordinates such that p 2S ða Þ.The proof of this lemma is obvious.3.Standard modelThe aim of this paper is to propose a new cover class,called the standard cover.The standard cover is so far only deﬁned for linear objects in all dimensions.The dis-crete analytical model has been designed to conserve most of the properties of theE.Andres /Graphical Models 65(2003)92–11197supercover,to be bubble-free andðnÀ1Þ-connected.The supercover model has al-most all the properties we are looking for:tunnel-freeness,ðnÀ1Þ-connectivity,sta-bility for union,etc.The only property that is missing is the bubble-freeness.Some supercover objects have simple points.The model is therefore not topologically con-sistent and this is a problem for several applications such as,for instance,polygonal-ization.For this reason several attempts have been made to modify the supercover discretization by modifying the deﬁnition of a pixel[14,15,27].We show in the fol-lowing section that such attempts cannot work.In our approach,presented in Sec-tion3.2,we explain how,by studying the analytical description of linear objects,it is possible to remove selectively the simple points in the supercover model while pre-serving the modeling properties.In the section that follow the standard model and its properties are introduced.3.1.What does not work with the classical approachSeveral unsuccessful attempts have been made to deﬁne discrete objects that have supercover type modeling properties with bubble-freeness andðnÀ1Þ-connectivity properties[14,15,27].All these ideas basically modify,in various ways,the deﬁnition of a voxel in order to avoid bubbles.We give here a simple such example and show why it does not work that way(see[14]for some other examples).In Fig.3,the pixel deﬁnition has been changed.A pixel is now formed of the SW vertex(black disk),the two corresponding edges(bold edges)and itsÕinterior.The three other vertices and two other edges do not belong to the pixel.This deﬁnition derives thatSp2Z n VðpÞ¼R n with VðpÞ\VðqÞ¼£for p¼q:The discretization of a discreteline is necessarily bubble-free.However,as we see in Fig.3,the discretised line x1Àx2¼0is not1-connected.In fact,it has been shown as early as in1970[27],that no change in the deﬁnition of the pixel or voxel can lead to a correct solution.This means that a simple pixel deﬁnition modiﬁcation avoids bubbles but creates primi-tives that are not topologically consistent.This makes such a model useless for ap-plications such as polygonalization.Tunnel-freeness property is also lost with such an approach.3.2.Standard model approach:a modiﬁcation of the supercover deﬁnitionThe discrete analytical description of the supercover of a linear convex is deﬁned as intersection of half-spaces deﬁned by discrete inequalities P n i ¼1a i x i 6a 0[2,5–7].A linear concave object is simply considered as union of convexes.The orientation of each half-space is checked with an orientation convention and depending on it,its inequality ‘‘P n i ¼1a i x i 6a 0’’remains unchanged or is replaced by ‘‘P n i ¼1a i x i <a 0.’’Let us give a simple example,the 2D straight line D :3x 1À7x 2¼0shown in Fig.4,to illustrate why and how this works.The general case in dimension n works ex-actly in the same way.The supercover of the Euclidean line D is described by the two inequalities S ðD Þ¼fðx 1;x 2Þ2Z 2j À563x 1À7x 265g .A bubble occurs only when the straight line D contains half-integer coordinate points.We have then (and only then)discrete points verifying on one side 3x 1À7x 2¼À5and on the other side 3x 1À7x 2¼5.All these points are simple points.Removing the points on one side only leads to a discrete straight line that is 1-connected,separating,1-minimal and bubble-free.This can be done simply by replacing a ‘‘6’’by a ‘‘<’’for one of the two inequalities in the supercover analytical description.In the case of Fig.4,we have S t ðD Þ¼fðx 1;x 2Þ2Z 2j À563x 1À7x 2<5g .The change is based on an orien-tation convention.Opposing half-spaces such as ‘‘3x 1À7x 265’’and ‘‘À3x 1þ7x 265’’have a diﬀerent orientation in this convention and thus only one of them will have its Õ‘‘6’’changed into ‘‘<.’’This ensures that only one simple point for the 2D line will be removed.3.3.Orientation conventionThe standard model,contrary to the supercover,is not unique [7,9].For instance,in example of Fig.4,one of two possible simple points can be removed.Each selec-tion leads to another standard model deﬁnition.It depends on the orientation con-vention selection.One orientation convention per dimension R m ,m >0,isrequired.Fig.4.Standard and supercover straight line.The black points belong to both line.The white point be-longs only to the supercover.E.Andres /Graphical Models 65(2003)92–11199This selection must then remain unchanged for all the primitives handled.The selec-tion of an orientation convention per dimension has to be coherent with the operator p .The property S t ðp j ðF ÞÞ¼p j ðS t ðF ÞÞfor the operator p should be veriﬁed.If this is not the case,the modelling properties would not be veriﬁed (such as S t ðF [G Þ¼S t ðF Þ[S t ðG Þ,etc.).In general,with arbitrary orientation conventions there is no reason for this property to be veriﬁed.We propose a set of orientation conventions,denoted O n and called the basic orientation conventions.The basic ori-entation conventions verify the above mentioned property.Deﬁnition 3(Standard orientation ).Let us consider a discrete analytical half-space E :P n i ¼1C i X i 6B and the basic orientation convention O n .We say that E has a standard orientation if:•C 1>0;•or if C 1¼0and C 2>0;•...•or if C 1¼ÁÁÁ¼C n À1¼0and C n >0:If E has not a standard orientation,then we say that E has a supercover orientation.We consider from now on,without loss of generality,only the basic orientation conventions for all n >0.All the standard primitives are deﬁned with these basic ori-entation conventions.The basic orientation conventions are coherent with respect to the operators p .After p j ,for j 2J nm ,the orientation convention O n in R n becomesO n Àm in R n Àm .3.4.Standard model deﬁnitionAll the elements required to deﬁne the standard discretization model of linear ob-jects in R n are available.Deﬁnition 4(Standard model ).Let F be a linear Euclidean object in R n whose su-percover is described analytically by a ﬁnite set of inequalities F k :P n i ¼1C i ;k X i 6B k .The standard model S t ðF Þof F ,for the basic orientation convention O n ,is the discrete object described analytically by a ﬁnite set of discrete inequalities F 0k ob-tained by substituting each inequality F k by F 0k deﬁned as follows:•If F k has a standard orientation,then F 0k :P n i ¼1C i ;k X i <B k ;•else F 0k :P n i ¼1C i ;k X i 6B k .This deﬁnition is algorithmically easy to set up.Once a discrete analytical descrip-tion of an object is available,the transition from the supercover model to the stan-dard model and vice versa is trivial.3.5.Geometric properties of the standard modelIn this section,some properties of the standard model are presented.These prop-erties are very important for the derivation of our model description.Let us consider100 E.Andres /Graphical Models 65(2003)92–111a Euclidean linear object F of topological dimension m in R n .We have by deﬁnition S t ðF Þ&S ðF Þeven more precisely,if p 2S ðF Þn S t ðF Þ,then d 1ðp ;F Þ¼12.A stan-dard object is a supercover object from which some discrete points have been re-moved.These points are all at a distance 1from the Euclidean primitive.We have S t ðF Þ¼S ðF Þif no point,with at least m þ1half-integer coordinates,belongs to the boundary of F .The diﬀerences between the supercover of F and the standard model of F are located in the k -bubbles of F ,for k >m .Fig.4illustrates this in di-mension 2.One of the immediate consequences of this last property,is that the stan-dard model remains a cover:F &V ðS t ðF ÞÞ.That is why the standard model is also sometimes called standard cover [33,31].The standard model retains most of the set properties of the supercover.It is easy to deduce from Deﬁnition 4,that if we consider two Euclidean linear objects F and G in R n ,then:S t ðF [G Þ¼S t ðF Þ[S t ðG Þ;S t ðF \G Þ&S t ðF Þ\S t ðG Þ;F &G )S t ðF Þ&S t ðG Þ;S t ðF ÂG Þ¼S t ðF ÞÂS t ðG Þ;S t ðp j ðF ÞÞ¼p j ðS t ðF ÞÞ;S t ðe j ðF ÞÞ¼e j ðS t ðF ÞÞ:The ﬁrst property ensures that we will be able to construct complex discrete ob-jects out of basic elements such as simplices.These last properties are characteristic of correct orientation conventions.The properties are only veriﬁed if the orientation conventions are deﬁned for all dimensions lower or equal to n and if they are coher-ent with respect to the operator p .This is the case for the basic orientation conven-tions O k ,for k 6n .It is important to notice that,in general,S t ðF Þ¼S a 2F S t ða Þ.This property of the supercover is not conserved.We have S t ðF [G Þ¼S t ðF Þ[S t ðG Þfor a union of a ﬁnite number of objects.This comes simply from the fact that the standard model is not deﬁned for an analytical description that has an inﬁnite number of discrete in-equalities.One simple example for that is given by the 2D line D :x 1Àx 2¼0:The standard model of the line is S t ðD Þ:À16x 1Àx 2<1while S a 2F S t ða Þ:À1<x 1Àx 2<1.One of the main properties of the standard model concerns the connectivity and the tunnel-freeness.Theorem 2(Connectivity and tunnel-freeness).Let F be a Euclidean linear object of topological dimension m in R n .Its standard model S t ðF Þis ðn À1Þ-connected and tunnel-free.The standard model is a particular case of k -discretizations as introduced by Brim-kov et al.in [12].It is shown that the standard model is in fact a 0-discretization (The-orem 3in [12])and that 0-discretizations are ðn À1Þ-connected and tunnel-free E.Andres /Graphical Models 65(2003)92–111101(Proposition 3in [11]and Theorem 4in [12]).Another property proved in [12,31,33]is that the standard model minimizes the Hausdorﬀdistance with the Euclidean object.4.Des cription ofs tandard primitivesWe will examine now the discrete analytical description of the diﬀerent classes of standard linear primitives (half-space,point,m -ﬂat and m -simplex)and how they can be computed.Our purpose here is to propose a discretization scheme that can be used in practical applications.As stated in Deﬁnition 4,every analytical description of a standard linear primitive is based on the analytical description of a standard half-space.That is the one we present ﬁrst.We deduce from it the discrete analytical for-mulas describing a standard point,m -ﬂat and m -simplex in the sections that follow.4.1.Standard half-spaceThe standard half-space is given by:Proposition 3(Standard half-space).Let us consider a Euclidean half-space E :P n i ¼1C i X i 6B .The standard model S t ðE Þof E ,according to an orientation convention,isanalytically described by:•If E has a standard orientation,then S t ðE Þ¼p 2Z n X n i ¼1C i p i (<B þP n i ¼1j C i j 2);•elseS t ðE Þ¼p 2Z n X n i ¼1C i p i (6B þP n i ¼1j C i j 2):The proposition is an immediate extension to dimension n of results on the super-cover [2,4,5,7]and of Deﬁnition 4.4.2.Standard pointThe analytical description of a standard point can easily be deduced from the one of the standard half-space.It is however interesting to notice that the standard dis-cretization of a Euclidean point is always composed of one and only one discrete point contrary to what happens with a supercover discretization of a Euclidean point that can be formed of 2k points,06k 6n (in case of a k -bubble).Proposition 4(Standard point).Let us consider a Euclidean point a 2R n and the basic orientation convention O n .The standard model S t ða Þof a is the discrete point102 E.Andres /Graphical Models 65(2003)92–111。

自动化专业英语常用词汇

自动化专业英语常用词汇acceleration transducer 加速度传感器accumulatederror 累积误差AC-DC-AC frequency converter 交 -直 -交变频器AC (alternating current)electric drive 交流电子传动active attitudestabilization 主动姿态稳定adjointoperator 伴随算子admissibleerror 容许误差amplifyingelement 放大环节analog-digital conversion 模数转换operationalamplifiers 运算放大器aperiodic decomposition 非周期分解approximate reasoning 近似推理a prioriestimate 先验估计articulatedrobot 关节型机器人asymptoticstability 渐进稳定性attained posedrift 实际位姿漂移attitudeacquisition 姿态捕获AOCS ( attitude and orbit control system) 姿态轨道控制系统attitude angular velocity 姿态角速度attitude disturbance 姿态扰动automatic manual station 自动 -手动操作器automaton 自动机base coordinate system 基座坐标系bellows pressure gauge 波纹管压力表gauge 测量仪器black box testingapproach 黑箱测试法bottom-up development 自下而上开发boundary value analysis 边界值分析brainstorming method 头脑风暴法CAE (computer aided engineering) 计算机辅助工程CAM (computer aided manufacturing) 计算机辅助制造capacitive displacement transducer 电容式位移传感器capacity 电容displacement 位移capsule pressure gauge 膜盒压力表rectangular coordinatesystem 直角坐标系cascade compensation 串联补偿using series or parallel capacitors 用串联或者并联的电容chaos 混沌calrity 清晰性classical informationpattern 经典信息模式classifier 分类器clinical control system 临床控制系统closed loop pole 闭环极点open loop 开环closed loop transfer function 闭环传递函数combined pressure and vacuum gauge 压力真空表command pose 指令位姿companion matrix 相伴矩阵compatibility 相容性，兼容性compensating network 补偿网络Energy is conserved in all of its forms 能量是守恒的compensation 补偿，矫正conditionally instability 条件不稳定性configuration 组态connectivity 连接性conservative system 守恒系统consistency 一致性constraint condition 约束条件control accuracy 控制精度Gyroscope 陀螺仪control panel 控制屏，控制盘control system synthesis 控制系统综合corner frequency 转折频率coupling of orbit and attitude 轨道和姿态耦合critical damping 临界阻尼Damper 阻尼器临界 criticalcritical stability 临界稳定性cross-overfrequency 穿越频率，交越频率cut-off frequency 截止频率cybernetics 控制论cyclic remotecontrol 循环遥控cycle 循环 cyclic cylindrical robot 圆柱坐标型机器人damped oscillation 阻尼振荡oscillation 振荡；振动；摆动damper 阻尼器damping ratio 阻尼比ratio 比data acquisition 数据采集data preprocessing 数据预处理data processor 数据处理器D controller 微分控制器微分控制： Differentialcontrol积分控制： integralcontrol 比例控制： proportional controldescribing function 描述函数desired value 希望值真值： truthvalues 参考值： reference valuedestination 目的站detector 检出器deviation 偏差deviation alarm 偏差报警器differential dynamicalsystem 微differential pressure level meter差压液位计 meter=gauge仪表differen tial差别的微分的differential pressure transmitter 差压变送器differential transformer displacement transducerdifferentiation element 微分环节差动变压器式位移传感器digital filer 数字滤波器 fil ter滤波器digital signal processing数字信号处理dimension transducer 尺度传感器discrete system simulation language 离散系统仿真语言discrete 离散的不连续的displacement vibration amplitude transducer 位移振幅传感器幅度： amplitude distrubance 扰动disturbance compensation 扰动补偿diversit y 多样性 divisibi lity 可分性domain knowledge 领域知识dominant pole 主导极点零点 zero 调制： modulation ； modulate 解调： demodulation countermodulatio n duty ratio 负载比 dynamic characteristics 动态特性 dynamic deviation 动态偏差dynamic error coefficient 动态误差系数 dynamic input-output model 动态投入产出模型Index 指数eddy current thickness meter 电涡流厚度计 meter 翻译成计 gauge 翻译成表 electric conductance level meter 电导液位计 electromagnetic flow transducer 电磁流量传感器electronic batching scale 电子配料秤 scale 秤electronic belt conveyor scale 电子皮带秤 electronic hopper scale 电子料斗秤elevation 仰角depression 俯角equilibrium point 平衡点error 误差estimate 估计量estimation theory 估计理论expected characteristics 希望特性failure diagnosis 故障诊断feasibility study 可行性研究feasible 可行的feasible region 可行域feature detection 特征检测feature extraction 特征抽取feedback compensation 反馈补偿Feed forward path 前馈通路前馈： feed forward 反馈 feedbackFMS ( flexible manufacturing system) 柔性制造系统柔性： flexible 刚性： rigiditybending deflection 弯曲挠度deflect 偏向偏离flow sensor/transducer流量传感器flow transmitter 流量变送器forward path 正向通路frequency converter 变频器frequency domain model reduction me thod 频域模型降阶法频域frequency response 频域响应functional decomposition 功能分解FES (functional electrical stimulation ) 功能电刺激stimulate 刺激functional simularity 功能相似fuzzy logic 模糊逻辑generalized least squares estimation 广义最小二乘估计geometric similarity 几何相似global optimum 全局最优goal coordinationmethod 目标协调法graphic search 图搜索guidance system 制导系统gyro drift rate 陀螺漂移率gyrostat 陀螺体Hall displacement transducer 霍尔式位移传感器horizontaldecomposition 横向分解hydraulic step motor 液压步进马达Icontroller 积分控制器integral 积分identifiability 可辨识性imagerecognition 图像识别impulse 冲量impulsefunction 冲击函数，脉冲函数index of merit 品质因数index 指数inductive force transducer 电感式位移传感器感应的inductive电感：inductanceindustrial automation 工业自动化inertial attitude sensor 惯性姿态敏感器inertial coordinate system 惯性坐标系information acquisition 信息采集infrared gas analyzer 红外线气体分析器infrared 红外线红外线的ultraviolet ray 紫外线的visible light可见光inherent nonlinearity 固有非线性inherent regulation 固有调节initial deviation 初始偏差input-output model 投入产出模型instability 不稳定性integrity 整体性intelligent terminal 智能终端internal disturbance 内扰invariant embedding principle 不变嵌入原理inverse Nyquist diagram 逆奈奎斯特图investment decision 投资决策joint 关节knowledge acquisition 知识获取knowledge assimilation 知识同化knowledge representation 知识表达lag-lead compensation滞后超前补偿Laplacetransform 拉普拉斯变换large scale system 大系统least squares criterion 最小二乘准则criterion 准则linearizationtechnique 线性化方法linear motion electricdrive 直线运动电气传动linear motionvalve 直行程阀linearprogramming 线性规划load cell 称重传感器local optimum 局部最优local 局部log magnitude-phase diagram对数幅相图magnitude大小的程度amplitude 振幅long term memory 长期记忆Lyapunov theorem of asymptotic stability 李雅普诺夫渐近稳定性定理magnetoelastic weighing cell 磁致弹性称重传感器magnitude-frequency characteristic 幅频特性magnitude margin幅值裕度margin边缘magnitude scalefactor幅值比例尺manipulator机械手man-machine coordination人机协调MAP (manufacturing automation protocol) 制造自动化协议protocol 协议marginal effectiveness 边际效益Mason‘‘ s gain formula 梅森增益公式matchingcriterion匹配准则maximum likelihood estimation 最大似然估计maximum overshoot 最大超调量maximum principle 极大值原理mean-square error criterion 均方误差准则minimal realization 最小实现minimum phase system 最小相位系统minimum variance estimation 最小方差估计model reference adaptive control system 模型参考适应控制系统 model verification 模型验证modularization 模块化mean 平均MTBF (mean time between failures) 平均故障间隔时间 MTTF (mean time to failures) 平均无故障时间multiloop control 多回路控制multi-objective decision 多目标决策Nash optimality 纳什最优性nearest-neighbor 最近邻necessity measure 必然性侧度negative feedback 负反馈neural assembly 神经集合neural network computer 神经网络计算机Nichols chart 尼科尔斯图Nyquist stability criterion 奈奎斯特稳定判据objective function 目标函数on-line assistance 在线帮助on-off control 通断控制optic fiber tachometer 光纤式转速表optimal trajectory 最优轨迹optimization technique 最优化技术order parameter 序参数orientation control 定向控制oscillating period 振荡周期周期：period cycleoutput prediction method 输出预估法oval wheel flowmeter 椭圆齿轮流量计Over damping 过阻尼underdamping 欠阻尼PR (pattern recognition) 模式识别P control 比例控制器peak time 峰值时间penalty function method 罚函数法perceptron 感知器phase lead 相位超前phase lag相位滞后Photoelectri c光电tachometric transducer光电式转速传感器piezoelectric force transducer压电式力传感器PLC (programmable logic controller) 可编程序逻辑控制器plug braking 反接制动pole assignment 极点配置pole-zero cancellation零极点相消polynomial input 多项式输入portfolio theory 投资搭配理论pose overshoot位姿过调量position measuring instrument 位置测量仪posentiometric displacement transducer 电位器式位移传感器positive feedback 正反馈power system automation 电力系统自动化pressure transmitter 压力变送器primary frequency zone 主频区priority 优先级process-oriented simulation 面向过程的仿真proportional control 比例控制proportional plus derivative controller 比例微分控制器pulse duration 脉冲持续时间pulse frequency modulation control system 脉冲调频控制系统： frequency modulation 频率调制调频pulse width modulation control system 脉冲调宽控制系统PWM inverter 脉宽调制逆变器QC (qualitycontrol) 质量管理quantized noise 量化噪声ramp function 斜坡函数randomdisturbance 随机扰动random process 随机过程rate integratinggyro 速率积分陀螺real time telemetry 实时遥测receptive field 感受野rectangular robot 直角坐标型机器人redundantinformation 冗余信息regional planningmodel 区域规划模型regulatingdevice 调节装载regulation 调节relationalalgebra 关系代数remoteregulating 遥调reproducibility 再现性resistance thermometer sensor 热电阻电阻温度计传感器response curve 响应曲线return difference matrix 回差矩阵 return ratio matrix回比矩阵revolute robot 关节型机器人revolution speed transducer 转速传感器 rewriting rule重写规则rigid spacecraft dynamics 刚性航天动力学 dynamics 动力学robotics 机器人学robot programming language 机器人编程语言 robust control 鲁棒控制 robustness 鲁棒性 root locus 根轨迹 roots flowmeter腰轮流量计rotameter 浮子流量计，转子流量计sampled-data control system 采样控制系统sampling control system 采样控制系统saturation characteristics 饱和特性 scalar Lyapunov function 标量李雅普诺夫函数s-domain s 域self-operated controller 自力式控制器 self-organizing system 自组织系统self-reproducing system 自繁殖系统self-tuning control 自校正控制sensing element 敏感元件 sensitivity analysis 灵敏度分析sensory control 感觉控制 sequential decomposition顺序分解sequential least squares estimation 序贯最小二乘估计 servo control 伺服控制，随动控制servomotor settling time伺服马达过渡时间 sextan t六分仪short term planning短期计划short time horizon coordinationsignal detection and estimation短时程协调信号检测和估计signal reconstruction 信号重构similarity 相似性simulated interrupt 仿真中断simulation block diagram 仿真框图simulation experiment 仿真实验simulation velocity 仿真速度simulator 仿真器single axle table 单轴转台single degree of freedom gyro 单自由度陀螺翻译顺序呵呵spin axis 自旋轴spinner 自旋体stability criterion 稳定性判据stabilitylimit 稳定极限stabilization 镇定，稳定state equation model 状态方程模型state space description 状态空间描述static characteristicscurve 静态特性曲线station accuracy 定点精度stationary randomprocess 平稳随机过程statistical analysis 统计分析statistic patternrecognition 统计模式识别steady state deviation稳态偏差顺序翻译即可steady state error coefficient稳态误差系数step-by-step control步进控制step function 阶跃函数strain gauge load cell 应变式称重传感器subjective probability 主观频率supervisory computer control system 计算机监控系统sustained oscillation 自持振荡swirlmeter 旋进流量计switching point 切换点systematology 系统学system homomorphism 系统同态system isomorphism 系统同构system engineering 系统工程tachometer 转速表target flow transmitter 靶式流量变送器task cycle 作业周期temperature transducer 温度传感器tensiometer 张力计texture 纹理theorem proving 定理证明therapy model 治疗模型thermocouple 热电偶thermometer 温度计thickness meter 厚度计three-axis attitude stabilization 三轴姿态稳定three state controller 三位控制器thrust vector control system 推力矢量控制系统推力器thrustertime constant 时间常数time-invariant system 定常系统，非时变系统invariant不变的时序控制器time schedulecontrollertime-sharing control 分时控制time-varying parameter 时变参数top-down testing 自上而下测试全面质量管理TQC (total qualitycontrol)tracking error 跟踪误差trade-off analysis 权衡分析transfer function matrix传递函数矩阵transformation grammar 转换文法transient deviation 瞬态偏差短暂的瞬间的transient process过渡过程transition diagram 转移图transmissible pressure gauge 电远传压力表transmitter 变送器trend analysis 趋势分析triple modulation telemetering system 三重调制遥测系统turbine flowmeter 涡轮流量计Turing machine 图灵机two-time scale system 双时标系统ultrasonic levelmeter 超声物位计unadjustable speed electric drive 非调速电气传动unbiased estimation 无偏估计underdamping 欠阻尼uniformly asymptotic stability 一致渐近稳定性uninterrupted duty 不间断工作制，长期工作制unit circle 单位圆unit testing 单元测试unsupervised learing 非监督学习upper level problem 上级问题urban planning 城市规划value engineering 价值工程variable gain 可变增益，可变放大系数variable structure control system 变结构控制function 函数vector Lyapunov function 向量李雅普诺夫函数velocity error coefficient 速度误差系数velocity transducer 速度传感器 vertical decomposition纵向分解vibrating wire force transducer 振弦式力传感器vibrometer 振动计vibrationVibrate 振动viscousdamping 粘性阻尼voltage source inverter 电压源型逆变器vortex precessionflowmeter 旋进流量计vortex sheddingflowmeter 涡街流量计WB (way base) 方法库weighing cell 称重传感器weightingfactor 权因子weightingmethod 加权法Whittaker-Shannon sampling theorem 惠特克 -香农采样定理Wiener filtering维纳滤波w- plane w 平面zero-based budget 零基预算zero-input response零输入响应zero-state response零状态响应z-transform z 变换《信号与系统》专业术语中英文对照表第1章绪论信号（ signal）系统（ system）电压（ voltage）电流（ current）信息（ information）电路（ circuit ）网络（ network）确定性信号（ determinate signal）随机信号（ random signal）一维信号（ one–dimensional signal）多维信号（ multi –dimensional signal）连续时间信号（ continuous time signal）离散时间信号（ discrete time signal）取样信号（ sampling signal）数字信号（ digital signal）周期信号（ periodic signal）非周期信号（ nonperiodic（aperiodic） signal）能量（ energy）功率（ power）能量信号（ energy signal）功率信号（ power signal）平均功率（ average power）平均能量（ average energy）指数信号（ exponential signal）时间常数（ time constant）正弦信号（ sine signal）余弦信号（ cosine signal）振幅（ amplitude）角频率（ angular frequency）初相位（ initial phase）周期（ period）频率（ frequency）欧拉公式（ Euler ’s formula）复指数信号（ complex exponential signal）复频率（ complex frequency）实部（ real part）虚部（ imaginary part）抽样函数Sa(t)（sampling（Sa） function）偶函数（ even function）奇异函数（ singularity function ）奇异信号（ singularity signal）单位斜变信号（ unit ramp signal）斜率（ slope）单位阶跃信号（ unit step signal）符号函数（ signum function）单位冲激信号（ unit impulse signal）广义函数（ generalized function）取样特性（ sampling property）冲激偶信号（ impulse doublet signal）奇函数（ odd function）偶分量（even component）偶数even 奇数odd 奇分量（odd component）正交函数（ orthogonal function）正交函数集（ set of orthogonal function）数学模型（ mathematics model）电压源（ voltage source）基尔霍夫电压定律（ Kirchhoff ’s voltage law（KVL ））电流源（ current source）连续时间系统（ continuous time system）离散时间系统（ discrete time system）微分方程（ differential function）差分方程（ difference function）线性系统（ linear system）非线性系统（ nonlinear system）时变系统（ time–varying system）时不变系统（ time–invariant system）集总参数系统（ lumped–parameter system）分布参数系统（ distributed–parameter system）偏微分方程（ partial differential function ）因果系统（ causal system）非因果系统（ noncausal system）因果信号（ causal signal）叠加性（ superposition property）均匀性（ homogeneity）积分（ integral）输入–输出描述法（ input–output analysis）状态变量描述法（ state variable analysis）单输入单输出系统（ single–input and single–output system）状态方程（ state equation）输出方程（ output equation）多输入多输出系统（ multi –input and multi–output system）时域分析法（ time domain method）变换域分析法（ transform domain method）卷积（ convolution）傅里叶变换（ Fourier transform）拉普拉斯变换（ Laplace transform）第 2 章连续时间系统的时域分析齐次解（ homogeneous solution）特解（ particular solution）特征方程（ characteristic function）特征根（ characteristic root）固有（自由）解（ natural solution）强迫解（ forced solution）起始条件（ original condition）初始条件（ initial condition）自由响应（ natural response）强迫响应（ forced response）零输入响应（ zero-input response）零状态响应（ zero-state response）冲激响应（ impulse response）阶跃响应（ step response）卷积积分（ convolution integral）交换律（ exchange law）分配律（ distribute law）结合律（ combine law）第3 章傅里叶变换频谱（ frequency spectrum）频域（ frequency domain）三角形式的傅里叶级数（trigonomitric Fourier series）指数形式的傅里叶级数（exponential Fourier series）傅里叶系数（ Fourier coefficient）直流分量（ direct component）基波分量（ fundamental component）component分量n 次谐波分量（ nth harmonic component）复振幅（ complex amplitude）频谱图（ spectrum plot（diagram））幅度谱（ amplitude spectrum）相位谱（ phase spectrum）包络（ envelop）离散性（ discrete property）谐波性（ harmonic property）收敛性（ convergence property）奇谐函数（ odd harmonic function）吉伯斯现象（ Gibbs phenomenon）周期矩形脉冲信号（ periodic rectangular pulse signal）直角的周期锯齿脉冲信号（ periodic sawtooth pulse signal）周期三角脉冲信号（ periodic triangular pulse signal）三角的周期半波余弦信号（ periodic half–cosine signal）周期全波余弦信号（ periodic full –cosine signal）傅里叶逆变换（inverse Fourier transform）inverse 相反的频谱密度函数（ spectrum density function）单边指数信号（ single–sided exponential signal）双边指数信号（ two–sided exponential signal）对称矩形脉冲信号（ symmetry rectangular pulse signal）线性（ linearity ）对称性（ symmetry）对偶性（ duality）位移特性（ shifting）时移特性（ time–shifting）频移特性（ frequency–shifting ）调制定理（ modulation theorem）调制（ modulation）解调（ demodulation）变频（ frequency conversion）尺度变换特性（ scaling）微分与积分特性（ differentiation and integration）时域微分特性（ differentiation in the time domain）时域积分特性（ integration in the time domain）频域微分特性（ differentiation in the frequency domain）频域积分特性（ integration in the frequency domain）卷积定理（ convolution theorem）时域卷积定理（ convolution theorem in the time domain）频域卷积定理（ convolution theorem in the frequency domain）取样信号（ sampling signal）矩形脉冲取样（ rectangular pulse sampling）自然取样（ nature sampling）冲激取样（ impulse sampling）理想取样（ ideal sampling）取样定理（ sampling theorem）调制信号（ modulation signal）载波信号（ carrier signal）已调制信号（ modulated signal）模拟调制（ analog modulation）数字调制（ digital modulation）连续波调制（ continuous wave modulation）脉冲调制（ pulse modulation）幅度调制（ amplitude modulation）频率调制（ frequency modulation）相位调制（ phase modulation）角度调制（ angle modulation）频分多路复用（ frequency–division multiplex （FDM ））时分多路复用（ time–division multiplex （TDM ））相干（同步）解调（ synchronous detection）本地载波（ local carrier）载波系统函数（ system function）网络函数（ network function）频响特性（ frequency response）幅频特性（ amplitude frequency response）幅频响应相频特性（ phase frequency response）无失真传输（ distortionless transmission）理想低通滤波器（ideal low–pass filter）截止频率（ cutoff frequency）正弦积分（ sine integral）上升时间（ rise time）窗函数（ window function ）理想带通滤波器（ ideal band–pass filter）太直译了第 4 章拉普拉斯变换代数方程（ algebraic equation）双边拉普拉斯变换（ two-sided Laplace transform）双边拉普拉斯逆变换（ inverse two-sided Laplace transform）单边拉普拉斯变换（ single-sided Laplace transform）拉普拉斯逆变换（ inverse Laplace transform）收敛域（ region of convergence（ ROC））延时特性（ time delay）s 域平移特性（ shifting in the s-domain）s域微分特性（ differentiation in the s-domain）s 域积分特性（ integration in the s-domain）初值定理（ initial-value theorem）终值定理（ expiration-value）复频域卷积定理（ convolution theorem in the complex frequency domain）部分分式展开法（ partial fraction expansion）留数法（ residue method）第 5 章策动点函数（ driving function ）转移函数（ transfer function）极点（ pole）零点（ zero）零极点图（ zero-pole plot）暂态响应（ transient response）稳态响应（ stable response）稳定系统（ stable system）一阶系统（ first order system）高通滤波网络（ high-pass filter）低通滤波网络（ low-pass filter）二阶系统（ second order system）最小相位系统（ minimum-phase system）高通（ high-pass）带通（ band-pass）带阻（ band-stop）有源（ active）无源（ passive）模拟（ analog）数字（ digital）通带（ pass-band）阻带（ stop-band）佩利－维纳准则（ Paley-Winner criterion）最佳逼近（ optimum approximation）过渡带（ transition-band）通带公差带（ tolerance band）巴特沃兹滤波器（ Butterworth filter ）切比雪夫滤波器（ Chebyshew filter）方框图（ block diagram）信号流图（ signal flow graph）节点（ node）支路（ branch）输入节点（ source node）输出节点（ sink node）混合节点（ mix node）通路（ path）开通路（ open path）闭通路（ close path）环路（ loop）自环路（ self-loop）环路增益（ loop gain）不接触环路（ disconnect loop）前向通路（ forward path）前向通路增益（ forward path gain）梅森公式（ Mason formula）劳斯准则（ Routh criterion）第 6 章数字系统（ digital system）数字信号处理（ digital signal processing）差分方程（ difference equation）单位样值响应（ unit sample response）卷积和（ convolution sum）Z 变换（ Z transform）序列（ sequence）样值（ sample）单位样值信号（ unit sample signal）单位阶跃序列（ unit step sequence）矩形序列(rectangular sequence)单边实指数序列（ single sided real exponential sequence）单边正弦序列（ single sided exponential sequence）斜边序列（ ramp sequence）复指数序列（ complex exponential sequence）线性时不变离散系统（ linear time-invariant discrete-time system）常系数线性差分方程（ linear constant-coefficient difference equation）后向差分方程（ backward difference equation）前向差分方程（ forward difference equation）海诺塔（ Tower of Hanoi）菲波纳西（ Fibonacci）冲激函数串（ impulse train）第7 章数字滤波器（ digital filter ）单边 Z 变换（ single-sided Z transform）双边 Z 变换 (two-sided (bilateral) Z transform)幂级数（ power series）收敛（ convergence）有界序列（ limitary-amplitude sequence）正项级数（ positive series）有限长序列（ limitary-duration sequence）右边序列（ right-sided sequence）左边序列（ left-sided sequence）双边序列（ two-sided sequence）Z逆变换（ inverse Z transform）围线积分法（ contour integral method）幂级数展开法（ power series expansion）z域微分（ differentiation in the z-domain）序列指数加权（ multiplication by an exponential sequence）z域卷积定理（ z-domain convolution theorem）帕斯瓦尔定理（ Parseval theorem）传输函数（ transfer function）序列的傅里叶变换（ discrete-time Fourier transform：DTFT）序列的傅里叶逆变换（ inverse discrete-time Fourier transform：IDTFT ）幅度响应（ magnitude response）相位响应（ phase response）量化（ quantization）编码（ coding）模数变换（ A/D 变换： analog-to-digital conversion）数模变换（ D/A 变换： digital-to- analog conversion）第8 章端口分析法（ port analysis）状态变量（ state variable）无记忆系统（ memoryless system）有记忆系统（ memory system）矢量矩阵（ vector-matrix ）常量矩阵（ constant matrix ）输入矢量（input vector）输出矢量（ output vector）直接法（ direct method）间接法（ indirect method）状态转移矩阵（ state transition matrix）系统函数矩阵（ system function matrix）冲激响应矩阵（ impulse response matrix）光学专业词汇大全Accelaration 加速度Myopia-near-sighted 近视Sensitivity to Light 感光灵敏度boost 推进lag behind 落后于Hyperopic-far-sighted 远视visual sensation 视觉ar Pattern 条状图形approximate 近似adjacent 邻近的normal 法线Color Difference 色差V Signal Processing 电视信号处理back and forth 前后vibrant 震动quantum leap 量子越迁derive from 起源自inhibit 抑制 ,约束stride 大幅前进obstruction 障碍物substance 物质实质主旨residue 杂质criteria 标准parameter 参数parallax 视差凸面镜convex mirror凹面镜concave mirror分光镜 spectroscope入射角angle of incidence 出射角emergent angle平面镜plane mirror放大率角度放大率 angularmagnification放大率：magnification折射refraction反射reflect干涉 interfere衍射diffraction干涉条纹interference fringe衍射图像diffraction fringe 衍射条纹偏振polarize polarization透射 transmission透射光transmission light光强度 ] light intensity电磁波electromagnetic wave振动杨氏干涉夫琅和费衍射焦距brewster Angle 布鲁斯特角quarter Waveplates 四分之一波片ripple 波纹capacitor 电容器vertical 垂直的horizontal 水平的airy disk 艾里斑exit pupil 出[ 射光 ]瞳Entrance pupil 入瞳optical path difference 光称差radius of curvature 曲率半径spherical mirror 球面镜reflected beam 反射束YI= or your information 供参考phase difference 相差interferometer 干涉仪ye lens 物镜 /目镜spherical 球的field information 场信息standard Lens 标准透镜refracting Surface 折射面principal plane 主平面vertex 顶点 ,最高点fuzzy 失真 ,模糊light source 光源wavelength 波长angle 角度spectrum 光谱diffraction grating 衍射光栅sphere 半球的DE= ens data editor Surface radius of curvature 表面曲率半径surface thickness 表面厚度semi-diameter 半径focal length 焦距field of view 视场stop 光阑refractive 折射reflective 反射机械专业英语词汇（大全）金属切削metal cutting机床machine tool tool 机床金属工艺学technology of metals刀具 cutter摩擦 friction传动 drive/transmission轴shaft弹性 elasticity频率特性frequency characteristic误差 error响应 response定位 allocation动力学dynamic运动学kinematic静力学static分析力学analyse mechanics 力学拉伸 pulling压缩 hitting compress剪切 shear扭转 twist弯曲应力bending stress强度 intensity几何形状geometricalUltrasonic 超声波精度 precision交流电路AC circuit机械加工余量machining allowance变形力deforming force变形 deformation应力 stress硬度 rigidity热处理heat treatment电路 circuit半导体元件semiconductor element反馈 feedback发生器generator直流电源DC electrical source门电路 gate circuit逻辑代数logic algebra磨削grinding螺钉 screw铣削 mill铣刀 milling cutter功率 power装配 assembling流体动力学fluid dynamics流体力学fluid mechanics加工 machining稳定性 stability介质 medium强度 intensity载荷 load应力 stress可靠性 reliability精加工 finish machining粗加工 rough machining腐蚀 rust氧化 oxidation磨损 wear耐用度durability随机信号random signal离散信号discrete signal超声传感器ultrasonic sensor摄像头CCD cameraLead rail 导轨合成纤维synthetic fibre电化学腐蚀electrochemical corrosion车架 automotive chassis悬架 suspension转向器redirector变速器speed changer车间 workshop工程技术人员engineer数学模型mathematical model标准件standard component零件图part drawing装配图assembly drawing刚度 rigidity内力 internal force位移 displacement截面 section疲劳极限fatigue limit断裂 fracture 破裂塑性变形plastic distortionelastic deformation 弹性变形脆性材料brittleness material刚度准则rigidity criterion齿轮gearGrain 磨粒转折频率corner frequency =break frequencyConvolution Convolution integral Convolution property Convolution sum 卷积卷积积分卷积性质卷积和Correlation function Critically damped systems Crosss-correlation functions Cutoff frequencies 相关函数临界阻尼系统互相关函数截至频率transistor diode semiconduct or nn晶体管二极管n半导体resistor n 电阻器capacitor n 电容器alternating adj 交互的amplifier n 扩音器，放大器integrated circuit 集成电路linear time invariant systems 线性时不变系统voltage n 电压，伏特数Condenser= capacitor n 电容器dielectric electromagnetic adj 电磁的adj 非传导性的deflection n 偏斜；偏转；偏差linear device 线性器件the insulation resistance 绝缘电阻anode n 阳极，正极cathode n 阴极breakdown n 故障；崩溃terminal n 终点站；终端，接线端emitter n 发射器collect v 收集，集聚，集中insulator n 绝缘体，绝热器oscilloscope n 示波镜；示波器gain n 增益，放大倍数forward biased 正向偏置reverse biased 反向偏置P-N junction PN 结MOS（ metal-oxide semiconductor ）金属氧化物半导体enhancement and exhausted 增强型和耗尽型integrated circuits 集成电路analog n 模拟digital adj 数字的，数位的horizontal adj, 水平的，地平线的vertical adj 垂直的，顶点的amplitude n 振幅，广阔，丰富multimeter n 万用表frequency n 频率，周率the cathode-ray tube 阴极射线管dual-trace oscilloscope 双踪示波器signal generating device 信号发生器peak-to-peak output voltage 输出电压峰峰值sine wave 正弦波triangle wave 三角波square wave 方波amplifier 放大器，扩音器oscillator n 振荡器feedback n 反馈，回应phase n 相，阶段，状态filter n 滤波器，过滤器rectifier n 整流器；纠正者band-stop filter 带阻滤波器band-pass filter 带通滤波器decimal adj 十进制的，小数的hexadecimal adj/n 十六进制的binary adj 二进制的；二元的octaladj八进制的n绝缘体；电解质domain n 域；领域code n 代码，密码，编码 v 编码 the Fourier transform 傅里叶变换 Fast Fourier Transform 快速傅里叶变换 microcontro ller n 微处理器；微控制器 assembly language instrucions n 汇编语言指令 chip n 芯片，碎片modular adj 模块化的；模数的 sensor n 传感器plugvt 堵，塞，插上 n 塞子，插头，插销coaxial adj 同轴的，共轴的fiber n 光纤 relay contact 继电接触器 ArtificialIntelligence 人工智能 Perceptive Systems 感知系统 neural network 神经网络 fuzzy logic 模糊逻辑intelligent agent 智能代理 electromagn etic adj 电磁的coaxial adj 同轴的，共轴的 microwav e n 微波charge v 充电，使充电 insulato r n 绝缘体，绝缘物 nonconducti ve adj 非导体的，绝缘的 simulati on n 仿真；模拟 prototyp e n 原型 array n 排队，编队 vector n 向量，矢量inverse adj 倒转的，反转的 n 反面；相反 v倒转 high-performance 高精确性，高性能 two-dimensional 二维的；缺乏深度的 three-dimensional 三维的；立体的；真实的。

Divergent Fields, Charge, and Capacitance in FDTD Simulations

Divergent Fields,Charge,and Capacitance in FDTDSimulationsChristopher L.Wagner and John B.SchneiderAugust2,1998AbstractFinite-difference time-domain(FDTD)grids are often described as being divergence-free in a source-free region of space.However,in the presence of a source,the continuity equationstates that charges may be deposited in the grid,while Gauss’s law dictates that theﬁelds mustdiverge from any deposited charge.The FDTD method will accurately predict the(diverging)ﬁelds associated with charges deposited by a source embedded in the grid.However,thebehavior of the charge differs from that of charge in the physical world unless the FDTDimplementation is explicitly modiﬁed to include charge dynamics.Indeed,the way in whichcharge behaves in an FDTD grid naturally leads to the deﬁnition of grid capacitance.This gridcapacitance,though small,is an intrinsic property of the grid and is independent of the wayin which energy is introduced.To account for this grid capacitance one should use a slightlymodiﬁed form of the lumped-element capacitor model currently used.1IntroductionIt is well established that theﬁnite-difference time-domain(FDTD)method can model accu-rately a wide range of wave propagation and scattering problems.There are,however,signiﬁ-cant differences between the behavior of a system governed byﬁnite-difference formulations ofMaxwell’s equations and one governed by the complete formulation of continuum physics.In the “discretized world”ﬁnite-difference calculus needs to be used to compute quantities and the con-tinuous forms are not directly applicable to data extracted from the grid[1].Another distinction between the discretized FDTD world and the physical world is a byproduct of the way in which the FDTD method is implemented.A typical FDTD simulation does not explicitly include charge dynamics and thus has properties not expected from the physics.For example,in an FDTD simu-lation positive and negative charges can be deposited in free space and,though theﬁelds associated with these charges are correct,the charges do not move and are,in a sense,inﬁnitely massive.In free-space regions in the absence of a source,the FDTD method can be shown to be divergence-free[2],although the derivation of this property of the FDTD grid assumes inﬁnite precision.In practice,FDTD simulations useﬁnite precision and thus theﬁelds are not completely divergence-free.However,the amount thatﬁelds diverge,representing a failure of charge conser-vation,is near the numeric noise-ﬂoor of the simulation and is of little practical concern.On the other hand,substantialﬁeld divergence can be(and,indeed,should be)produced by certain sources embedded in the computational domain.Current sources with a dc component can deposit persistent charges while current sources with no dc component can produce tempo-rary charging.The geometry of the source,as well as the temporal form of the source function, ultimately dictates the amount of charging.Open-endedﬁlamentary radiators can deposit charge because the current diverges at the ends.The charge is not represented by a separately stored quantity,but only by the divergence of aﬁeld.The relationship between the electricﬁeld and charge density is given by Gauss’s law:(1)When theﬁeld diverges from a point,(1)states that the charge density is nonzero.The Yee space lattice staggers theﬁeld components in space to allow the spatial derivatives in Maxwell’s equa-tions to be computed with second-order accurate central differences.This arrangement of the ﬁeld vector components also allows the divergence to be computed with central differences,thus preserving the second-order accuracy of charge density computations.The charge density(ﬁeld divergence)is therefore deﬁned between theﬁeld vectors.The geometry of Yee-grid cells with implicit charge locations is shown in Fig.1.In the following section the deposition of charge in an FDTD grid and the associatedﬁelds are examined.Both“harmonic”and transient sources are considered.Because the FDTD grid can store charge,it is natural to deﬁne a grid capacitance.This capacitance is considered in Section3.2Sources and ChargingIn this section the relationship between the currents,ﬁelds,and charge is examined usingﬁnite-difference calculus.Examples are given that show how charge may exist in the FDTD grid even though there is no explicit storage location for charge.Indeed,the charge“exists”only insofar as divergingﬁelds exist.These divergingﬁelds,which are required to satisfy the continuity equation, may persist indeﬁnitely and hence remain in the computational domain even after all the radiated ﬁelds are gone.The equations relevant to the discussion here are Maxwell’s curl equations and the continuity equation:(3)one end of the current source,i.e.,the amount of charge enclosed within a surface surrounding the end of theﬁlament,is given by the volume and temporal integration of(4):enc(5) where is the total current entering the volume and enc is the enclosed charge.For an inte-grable source function,enc can be determined analytically using continuum calculus.Alterna-tively,enc can be determined in an FDTD simulation by the discrete-calculus volume integration of Gauss’s law[1].When the volume containing the charge is a single-cell cube with an edge length,integration of(1)yieldssix faces face enc(6)where face is the totalﬁeld on a face.To illustrate the deposition of charge and to conﬁrm the correspondence between the“ex-pected”charge given by the temporal integral of the current and the“measured”charge obtained from theﬂux integral of the electricﬁeld,consider a single element current source driven by a Gaussian pulse.The charge as a function of time at the two ends of a single-element(Hertzian) current source is shown in Fig.2.Results were obtained using an cubic cell domain with a one meter cell spacing.(Note that the results shown in Fig.2are independent of this di-mension provided the current density is scaled to the grid size,i.e.,the amount of charging is the same if the same is used.)The current source was a single elementﬁlament at grid coordinate (40,40,40)and a Higdon third-order absorbing boundary condition(ABC)was used to terminate the domain.Note that the measured FDTD results correspond precisely to those predicted by the temporal integral of the current.Thus,although the FDTD method is not considered a dc analysis technique,it does,nevertheless,properly predict theﬁelds associated with the rearrangement of ﬁxed(dc)charge.2.1Harmonic SourcesFDTD simulations sometimes employ“harmonic”sources.However,since a source must beturned on at some time,it is not truly harmonic.Consequently the time integral of the current source function(i.e.,the charge)can have a dc value.A speciﬁc example is a sinusoidal current, turned on at,which will deposit charge into the domain:(7)where unit frequency and amplitude are used for simplicity.The charge oscillates between0and 2,and has an average value of one.This average charge which,for example,might be deposited at the ends of a Hertzian radiator,will produce nonzero dcﬁelds throughout unshielded portions of the domain.Cosine currents,turned on at,do not analytically deposit nonzero average charge.How-ever the large turn-on discontinuity of the cosine source function at contains signiﬁcant high frequency components which will suffer large numerical dispersion.In the analytic case we have:(8)The deposited charge oscillates between and1,with an average value of zero.To demonstrate harmonic source charging,Fig.3shows the charge at the ends of a Hertzian radiator driven with a sinusoidal current that has a strength of A and a period of64time steps (about8.1MHz).Figure4shows the electricﬁeld at the radiator(between the charges).The domain for this example is the same as that used for Fig.2.The nonzero mean in the charge and the electricﬁeld is a direct consequence of turning on the source at.These offsets are distinct from true harmonic solutions which would have a zero mean.2.2Transient Pulse SourcesBecause the FDTD method is a time-domain technique,it is possible to compute the model response at several frequencies in a single run by using a transient source.As was shown in Fig.2, if a Hertzian radiator consisting of a single element of electric current is driven by a pulse that has a nonzero dc component,then charge will be deposited into the domain.Here the dc behavior oftheﬁelds in the vicinity of multi-element radiators,driven either by electric or magnetic currents, is considered.Consider the domain depicted in Fig.5of size.This domain contains a-directed 20-element long electric current source centered at and a-directed magnetic current source20elements long centered at.Magnetic charges and currents,while non-physical,can be modeled in FDTD simulations as simply as electric charges and currents.The current on both sources is given by a Gaussian.Early in the simulation both of these sources have associated radiatedﬁelds,i.e.,time-varying electric and magneticﬁelds that are coupled through Maxwell’s equations.Figures6and7show the magnitude of and,respectively,measured over a plane that includes the source.A logarithmic gray scale is used to depict theﬁelds where the darkness of a pixel is indicative of theﬁeld strength—the darker the pixel,the stronger theﬁeld. Theseﬁgures show the radiatedﬁelds at a time when they are still close to their respective sources. Eventually,the radiatedﬁelds are absorbed by the ABC,and only the staticﬁelds due to the charges remain.As evidence of this,the electricﬁeld magnitude is shown in Fig.8and the magneticﬁeld magnitude is shown in Fig.9at a time when the radiatedﬁelds have exited the computational space.(The data in Figs.6and8have been normalized to the same value so that theseﬁgures can be compared qualitatively.The data in Figs.7and9have been similarly normalized.)Theﬁelds in Figs.8and9are static and purely divergent from the implied electric or magnetic charge at the ends of each source.Figure8clearly shows that the electric current source has deposited electric charge in the computational domain and hence perturbs the electricﬁeld.The perturbation is,of course, strongest near the deposited charge.On the other hand,the magnetic source does not deposit electric charge and hence does not affect the dc electricﬁelds.Nevertheless,the magnetic source does,as shown in Fig.9,deposit magnetic charge and does perturb the dc magneticﬁeld.In these simulations the Gaussian current pulse had a half width()of eight time steps(s)and a peak of1A.The peak magnetic current strength was377V/m and the domain boundaries were terminated with a third-order Higdon ABC applied to the tangentialﬁeld components.3Grid CapacitanceIn the physical world,a positive and negative charge exert on each other an electrical force of attraction.In free space these charges will move under the inﬂuence of this electrical force.In an FDTD model without charge dynamics,positive and negative charges in free space will not move. The charges are subject to coulomb forces,but no motion occurs.Consequently,since adjacent cells can store charge,it is natural to deﬁne a capacitance between cells of the grid.The capacitance of adjacent cells in free space in the Yee grid can be derived in the following manner.The standard deﬁnition of capacitance is(9)where is the stored charge and is the voltage between the charges.The charge stored in the grid can be found either from the temporal integral of the current that deposited the charge or from theﬁnite-differenceﬂux integral of(Gauss’s law).By expressing the charge in terms of theﬂux integral,an electricﬁeld appears in the numerator of(9)that can be canceled with the electricﬁeld that subsequently appears in the denominator.When computing theﬂux integral, the totalﬁeld at any point on theﬂux surface can be decomposed into two parts,face enc .The contribution to the integral by theﬁelds from“distant”sources(i.e.,sources external distantto the surrounding surface)is zero.Theﬁeld from the enclosed charge,enc,makes a nonzero contribution to theﬂux integral provided the total enclosed charge is nonzero.Consider a cubic-cell Gaussian surface that encloses a charge.The relationship between the charge and theﬁeld on the faces of the cell surrounding the charge is,from(6),enc distant enc(10)six faceswhere is the length of one side of the cubic cell.The expression on the right-hand side is a consequence of symmetry which dictates that enc is the same over all six faces of a single-cell cube and of distant not contributing to the integral.The difference in potential between two adjacent cells containing charges of equal magnitude and opposite sign ispos Qenc(11)neg QThe factor of two is a consequence of the opposite charges doubling the total electricﬁeld over the face common to both ing(10)and(11)in(9),the capacitance between adjacent nodes in the FDTD grid is found to begrid(12) Thus,for example,the grid capacitance between adjacent nodes of a one meter cubic unit cell will be about pF.To illustrate the effect of this grid capacitance and to verify(12),we discharge deposited charge through a conductance introduced into the grid.The rate of discharge is easily measured and can be used to obtain the associated time constant.From this time constant and the known resistance, one can obtain the capacitance.Charge deposited into the grid will discharge through conductance with a time constantgrid loadwhere load is the resistance associated with the conductivity at the grid location.Discharging of deposited charge is shown in Fig.10.The grid spacing is m with a load resistance of k [2]so the expected time constant,using the capacitance of(12),is ns.The measured time constant that characterizes the decay shown in Fig.10is precisely this amount and hence veriﬁes the capacitance predicted by(12).We have only considered the capacity to store electric charge but a similar(dual)effect exists for storage of magnetic charge.While it may be of theoretical or pedagogical interest to pro-duce magnetic charge in a simulation,magnetic charge is not physical.Simulations which deposit magnetic charge(static or transient)are not modeling real-world electromagnetic problems.Nev-ertheless,magnetic sources that do not deposit magnetic charge,such as loops,may correspond to physical problems if the source is equivalent to an electric source.When implementing lumped-element capacitors in an FDTD grid,the update equation at the location of the capacitor is modiﬁed to include the capacitive effects[3].Previous formulations of lumped-elements have neglected the inherent grid capacitance.As described in[3],a lumped capacitor in a cubic-cell grid can be realized by changing the coefﬁcients of the curl of the magneticﬁeld.The new coefﬁcient islump gridA capacitor with a parallel load resistor should discharge with a time constant of.In the FDTD grid the total capacitance at a node is the sum of the lumped element and the inherent grid capacitance(15).To illustrate this point Fig.11shows the potential,measured between charge locations,as a function of time for two simulations.(The potential is indicative of the amount of charge present.)In theﬁrst simulation,deposited charge is discharged through a conductance and no lumped-element capactior is added,i.e.,the only capacitance present is the inherent grid capacitance.In the second,a lumped-element capacitor with a capacitance equal to the inherent grid capacitance is introduced in accordance with the method described in[3].Since the lumped element and the inherent grid capacitance are equal,the addition of the lumped element doubles the time constant.Viewed another way,if one were to ignore the inherent grid capacitance,the time constant would be incorrect by a factor of two.4ConclusionsSources in the FDTD computational domain can deposit charge which produce dc offsets in theﬁelds.The charge is evident by the divergence of theﬁeld.Theﬁeld associated with de-posited(static)charge is accurately predicted by the FDTD method in accordance with Gauss’s law.However,because charge cannot move within the FDTD grid,there is an inherent grid ca-pacitance which is a function of the grid spacing.One can account for this grid capacitance when introducing lumped elements to obtain the desired capacitance. AcknowledgmentsThis work was supported by the Ofﬁce of Naval Research,Code321OA.References[1]W.C.Chew,“Electromagnetic theory on a lattice,”J.Appl.Phys.,vol.75,pp.4843–4850,May1994.[2]A.Taﬂove,Computational Electrodynamics:The Finite-Difference Time-Domain Method.Boston,MA:Artech House,1995.[3]M.Piket-May,A.Taﬂove,and J.Baron,“FD-TD modeling of digital signal propagation in3-D circuits with passive and active loads,”IEEE Trans.Microwave Theory Tech.,vol.42, pp.1514–1523,Aug.1994.H centered unit cell EzEyE centered unit cell Magnetic ChargeElectric ChargeFigure 1:Yee grid showing location of implicit magnetic or electric charge.Time [ns]C h a r g e [n C ]Figure 2:Charge deposited at the ends of a dipole radiator vs.time.The charge is “measured”from the FDTD grid using the ﬁnite-difference form of Gauss’s law.The expected charge is computed from the time integral of the current waveform.For reference,the current source waveform is shown and is normalized to the peak expected charge.The actual peak current strength is A.Time [ns]C h a r g e [n C ]Figure 3:Normalized source current and charge at ends of radiator vs.time.The source is a sinewave turned on atand the frequency is about 8.1MHz.The charge at both ends of the radiator is computed from the divergence of .The charge has a nonzero average about which the instantaneous charge oscillates.50100150Time [ns]-1500-1000-500E l e c t r i cF i e l d [ V /m ]Electric FieldFigure 4:Electric ﬁeld at the radiator (between the charges)vs.time.The same parameters are used as pertained to Fig.3.(160,80,80)(0,0,0)Figure 5:Domain geometry for electric/magnetic charging example.Figure6:Magnitude of over a plane that includes the two sources.This snapshot was taken after 100time steps.The electric current source is to the left.Figure7:Magnitude of over a plane that includes the two sources.This snapshot was also taken after100time steps and is the dual of Fig.6.Figure8:Magnitude of recorded after300time steps.The radiatedﬁeld has left the domain and only the staticﬁelds(and charge)remain.Figure 9:Magnitude ofrecorded after 300time steps.2004006008001000Time [ns]-20-101020C h a r g e [n C ]Positive End Negative EndNormalized SourceFigure 10:Charge vs.time when a conductivity is present.The same domain and source are used as in Fig.2.The conductivity is S/m which is equivalent to 5k .The time constant which characterizes the discharge is ns.200400600800Time [ns]101102P o t e n t i a l [V ]Grid LumpedFigure 11:Potential vs.time when a conductivity is present.In one simulation,labeled “Grid,”no additional capacitance is introduced so that only the inherent grid capacitance is present.In the other,labeled “Lumped,”a lumped-element capacitor is introduced that has a capacitance equal to that of the inherent grid capacitance.Since inherent grid capacitance acts in paralled with the lumped-element capacitance,the time constant doubles.The plot shows the potential after the sources have been turned off.The strengths of the source functions were chosen so that curves were approximately equal when the sources were turned off (this has no effect on the time constants,but was done to facilitate interpretation of the plot).。

冲压模具英文翻译原文

j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y209(2009)3532–3541j o u r n a l h o m e p a g e:w w w.e l s e v i e r.c o m/l o c a t e/j m a t p r o t ecContact pressure evolution at the die radius in sheet metal stampingMichael P.Pereira a,∗,John L.Duncan b,Wenyi Yan c,Bernard F.Rolfe da Centre for Material and Fibre Innovation,Deakin University,Pigdons Road,Geelong,VIC3217,Australiab Professor Emeritus,The University of Auckland,284Glenmore Road,RD3,Albany0793,New Zealandc Department of Mechanical and Aerospace Engineering,Monash University,Clayton,VIC3800,Australiad School of Engineering and IT,Deakin University,Geelong,VIC3217,Australiaa r t i c l e i n f oArticle history:Received27March2008 Received in revised form 18July2008Accepted17August2008Keywords:Contact pressureSheet metal stamping Tool wearBending-under-tension a b s t r a c tThe contact conditions at the die radius are of primary importance to the wear response for many sheet metal forming processes.In particular,a detailed understanding of the con-tact pressure at the wearing interface is essential for the application of representative wear tests,the use of wear resistant materials and coatings,the development of suitable wear models,and for the ultimate goal of predicting tool life.However,there is a lack of infor-mation concerning the time-dependant nature of the contact pressure response in sheet metal stamping.This work provides a qualitative description of the evolution and distribu-tion of contact pressure at the die radius for a typical channel forming process.Through an analysis of the deformation conditions,contact phenomena and underlying mechanics, it was identiﬁed that three distinct phases exist.Signiﬁcantly,the initial and intermediate stages resulted in severe and localised contact conditions,with contact pressures signif-icantly greater than the blank material yield strength.Theﬁnal phase corresponds to a larger contact area,with steady and smaller contact pressures.The proposed contact pres-sure behaviour was compared to other results available in the literature and also discussed with respect to tool wear.©2008Elsevier B.V.All rights reserved.1.IntroductionIn recent years,there has been an increase in wear-related problems associated with the die radius of automotive sheet metal forming tools(Sandberg et al.,2004).These problems have mainly been a consequence of the implementation of higher strength steels to meet crash requirements,and the reduced use of lubricants owing to environmental concerns. As a result,forming tools,and the die radii in particular, are required to withstand higher forming forces and more severe tribological stresses.This can result in high costs due∗Corresponding author.Tel.:+61352273353;fax:+61352271103.E-mail address:michael.pereira@.au(M.P.Pereira).to unscheduled stoppages and maintenance,and lead to poor part quality in terms of surfaceﬁnish,geometric accuracy and possible part failure.If the side-wall of a part is examined after forming,a demarcation known as the‘die impact line’is easily visible (Karima,1994).This line separates the burnished material that has travelled over the die radius and the free surface that has not contacted the tooling,clearly indicating that severe sur-face effects exist at the die radius.It is therefore important to understand the contact phenomena at this location of the tooling.0924-0136/$–see front matter©2008Elsevier B.V.All rights reserved. doi:10.1016/j.jmatprotec.2008.08.010j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 209(2009)3532–354135331.1.Bending-under-tension testThe bending-under-tension test –in which a strip is bent over a cylindrical tool surface and pulled against a speci-ﬁed back tension –has been used in the laboratory for many years to simulate conditions at the die radius (Ranta-Eskola et al.,1982).The literature contains numerous experimental investigations that examine surface degradation over the die radius after repeated or continuous bending-under-tension operations.For example,in independent studies with differ-ing test conditions and materials,Mortensen et al.(1994),Hortig and Schmoeckel (2001)and Attaf et al.(2002),each visu-ally observed wear in two localised regions on the die radius.More detailed examination of the worn die radius surface,through measurement of surface roughness (Christiansen and De Chiffre,1997),determination of wear depth (Eriksen,1997)and scanning electron microscope imaging (Boher et al.,2005),has also conﬁrmed the existence of similar localised wear regions.In addition to the experimental analyses,Mortensen et al.(1994),Hortig and Schmoeckel (2001)and Attaf et al.(2002),each conducted ﬁnite element analyses of the bending-under-tension process.In all cases,the ﬁnite ele-ment models predicted the existence of distinct contact pressure peaks on the die radius surface,correlating well with the regions of localised ing in situ sensors Hanaki and Kato (1984)and more recently Coubrough et al.(2002)experimentally demonstrated that similar contact pressure peaks exist at locations on the die radius near the entry and exit of the strip during the bending-under-tension test.It is evident that despite covering a wide range of die materials (both coated and un-coated),lubrication,surface roughness,bend ratio and work-piece materials,each of thestudies discussed in the preceding paragraphs were found to exhibit similar characteristic two-peak contact pressure distributions and localised regions of wear over the die radius.These results,and the documented power law rela-tion between wear and normal load for sliding contacts (Rhee,1970),indicate that contact pressure is of primary signiﬁcance to the wear response.1.2.Sheet metal stampingThe contact conditions occurring during sheet metal stamping operations have not been studied as extensively as those of the bending-under-tension process.Through ﬁnite element anal-yses of axisymmetric cup-drawing processes,Mortensen et al.(1994)and Jensen et al.(1998)identiﬁed that time-dependant contact conditions occur at the die radius,as opposed to the ‘stationary’conditions of the bending-under-tension test (Hortig and Schmoeckel,2001).In recent numerical studies on a plane strain channel forming process,Pereira et al.(2007,2008)also reported time-dependant plex contact conditions over the die radius were found to occur,with regions of highly localised and severe contact pressure.Selected results of the ﬁnite element analysis by Pereira et al.(2008)are given in Fig.1,where the dynamic nature of the con-tact pressure distribution can be seen.Additionally,the Mises stress contours show the corresponding deformation of the blank and provide an indication of where yielding occurs.Although each of the above investigations report time-dependant contact conditions for sheet metal stamping processes,the authors in each case provide little explanation into the reasons for the identiﬁed contact behaviour.Further analysis of this phenomenon has not been found in the liter-ature.Fig.1–Mises stress contours and normalised contact pressure distributions predicted by ﬁnite element analysis at the three distinct stages during a channel forming process (see Section 4.1for more details).The regions in white in the Mises contours indicate values of stress below the blank material initial yield strength.3534j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y209(2009)3532–3541 1.3.MotivationIn order to understand tool wear in sheet metal stamp-ing,or to use representative tests(bending-under-tension,slider-on-sheet,etc.)to characterise the wear response of toolmaterials and coatings,knowledge of the local contact condi-tions that occur during the stamping operation is essential.Asdiscussed,the contact pressure is of particular signiﬁcance.However,a description of the evolution and distribution ofcontact stresses experienced by sheet metal forming tool-ing,including an explanation for this behaviour,has not beenfound in the literature.In this work,a qualitative description of the contact pres-sure evolution at the die radius and the associated stressdistributions in the blank during a channel forming processis given.The description is based on experimental observa-tions and the results ofﬁnite element analyses.Through ananalysis of the deformation conditions,contact phenomenaand underlying mechanics,it will be shown that three dis-tinct phases exist.Due to the unique deformation and contactconditions that are found to occur,the initial and intermedi-ate stages exhibit localised regions of severe contact pressure,with peak contact stresses that are signiﬁcantly greater thanthe blank material yield strength.Theﬁnal stage,which canbe considered as steady state with regards to the conditions atthe die radius,corresponds to a larger contact area with stableand smaller contact pressures.It is noted that the magnitude of the contact stress peakswill depend on variables such as back tension on the sheet,thedie radius to sheet thickness ratio,and the clearance betweenthe punch and die.These effects are not investigated in thiswork.The objective of this work is to provide an understandingof an important aspect of sheet metal forming,rather thana quantitative analysis of a speciﬁc case.This should assistin understanding die wear,which is an increasing problemwith the implementation of higher strength sheet in stampedautomotive components.2.The sheet metal stamping processThe stamping or draw die process is shown schematically inFig.2.Sheet metal is clamped between the die and blank-holder and stretched over the punch.The sheet slides overthe die radius surface with high velocity in the presence ofcontact pressure and friction,as it undergoes complex bend-ing,thinning and straightening deformation(Fig.2c).In themost rudimentary analysis of sheet metal forming,bending isneglected and the deformation is studied under the action ofprincipal tensions(Marciniak et al.,2002).The tension is theforce per unit width transmitted in the sheet and is a prod-uct of stress and thickness.For two-dimensional plane straindeformation around the die radius,the well-known analysisindicates that the contact pressure p isp=TR=1R/t(1)where 1is the longitudinal principal stress,T is the longitu-dinal tension,R is the die radius,t is the sheet thickness,and Fig.2–(a)The beginning of a typical sheet metal stamping process.(b)The motion and forces exerted by the tools cause the blank to be formed into a channel shape during the stamping process.(c)Forces acting on the sheet at the die radius region.R/t the bend ratio.Due to the effect of friction,the longitudinal tension in the sheet varies along the die radius.If the tension at one point,j,on the die radius is known,then the tension at some other point,k,further along the radius can be found according to:T k=T j exp( Âjk)(2)whereÂjk is the angle turned through between the two points, and is the coefﬁcient of friction between the tool and sheet surfaces.j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y209(2009)3532–35413535Eq.(1)provides a useful relationship that shows the contactpressure is inversely proportional to the bend ratio.Given thatthe tension is usually close to the yield tension and that thebend ratio in typical tooling is often less than10,Eq.(1)indi-cates that the contact stress is an appreciable fraction of theyield stress.This implies that the assumption of plane stressin the strip may not be valid.Additionally,a numerical studyof a bending-under-tension process with a bend ratio of3.3revealed that the restraint forces attributed to bending(andunbending)were almost50%of the total restraint forces onthe sheet(Groche and Nitzsche,2006).Although Eqs.(1)and(2)can be modiﬁed to include the work done in bending andstraightening,these simple models are unlikely to adequatelydescribe the contact pressure distribution.Furthermore,such an analysis assumes that the sheetslides continuously over the die radius under steady-state-type conditions analogous to a bending-under-tensionprocess.However,as discussed in Section1,several studies inthe literature have shown that the contact conditions are notsteady during typical sheet metal stamping.For these reasons,it is evident that a more detailed analysis,including examina-tion of the stress states and yielding in the sheet,is required inorder to understand the complex and time-dependant contactconditions at the die radius.3.Contact pressure at the die radiusIn this work,a qualitative description of the developmentof peak contact pressures at the die radius for the channelforming process shown in Fig.2is given.For simplicity,thedeformation of the sheet is considered as a two-dimensional,plane strain process.A linear-elastic,perfectly plastic sheetmaterial model,obeying a Tresca yield criterion is used.Thematerial curve is shown in Fig.3,where theﬂow stress is S,with zero Bauschinger effect on reverse loading.It is assumedthat if there is a draw-bead,it is at some distance from the dieradius so that the sheet entering the die radius is undeformedbut has some tension applied.In this study,the deformation and contact conditions at thedie radius for a typical sheet metal forming process are dividedinto three distinct phases(Fig.4).A material element on theblank,Point A,is initially located at the beginning of the dieradius,as shown in Fig.4a.At this instant,contact islimitedFig.3–Simpliﬁed plane strain material response with reverseloading.Fig.4–Three distinct phases of deformation and contact, which occur during the channel forming process:(a)initial deformation,(b)intermediate conditions,and(c)steady-state conditions at die radius.to a line across the die radius.During the next stage,Point A has travelled around the die radius,but has not yet reached the exit or tangent point(Fig.4b).At this instant,the material in the side-wall(between the die radius and punch radius) remains straight and has not previously contacted the tools.A state of approximately steady conditions at the die radius is reached in Fig.4c,where Point A is now in the side-wall region.3.1.Initial deformationAt the start of the forming stroke,contact between the blank and die occurs near the start of the die radius at an angle of Â=˛,as shown in Fig.5a.The Mohr circle of stress at the con-tacting inner surface and the stress distribution through the thickness of the sheet are given schematically in this diagram. The regions of plastic deformation in the sheet are indicated by shading.The sheet is bent by the transverse force F shown,so that a compressive bending stress 1exists on the upper surface.Due to the initial lack of conformance of the blank to the radius, contact occurs almost along a line,resulting in a contact pres-sure P˛that can be very high.As a result,the normal stress 3, which is equal to−P˛,is greatest at the surface and diminishes to zero at the outer,free surface.At this location,approx-3536j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 209(2009)3532–3541Fig.5–(a)Schematic of the blank to die radius interface during the initial deformation stage—the stress distribution through the thickness and the Mohr’s circle at the surface of the contact zone are shown.Corresponding distributions around the die radius of (b)contact pressure and (c)bending moment in the sheet.imately plane stress conditions exist and the sheet yields under tension at the plane strain yield stress S .The transverse stress 2at the inner surface will have an intermediate value,since the process is plane strain.In the plastic case,this is the mean of the other principal stresses.In the elastic case,this is only approximately so.The bending stress and contact pressure at the inner sur-face generate a high compressive hydrostatic stress,such that yielding can be suppressed (the diameter of the Mohr circle is <S ).This phenomenon is supported by the ﬁnite element simulation results of the case study shown in Fig.1a.The bending moment m is greatest at the contact line,as shown in Fig.5c;yet plastic bending only takes place either side of thisregion,where the inhibiting compressive hydrostatic stress is lower.The result is that a very high-pressure peak occurs at the contact line,greater in magnitude than the sheet yield stress (Fig.5b).This initial line contact,causing a localised peak contact pressure,is a momentary event.3.2.Intermediate conditionsAs the punch draws the sheet to slide into the die cavity,Point A moves away from the start of the radius,as shown in Fig.6a.Due to the plastic bending of the sheet that occurs near the beginning of the die radius,in the vicinity of Â=0◦,the mate-rial entering the die radius has greater conformance with thej o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y209(2009)3532–35413537Fig.6–(a)Schematic of the blank to die radius interface during the intermediate conditions—the stress distribution through the thickness and the Mohr’s circle at the surface of the contact zones are shown.Corresponding distributions around the die radius of(b)contact pressure and(c)bending moment in the sheet.die radius surface.This causes a reduction in contact pressure, due to the change from line contact in Fig.5to a broader con-tact area in Fig.6.Consequently,the compressive hydrostatic stress is reduced and plastic deformation at the blank surface occurs(the diameter of the Mohr circle is S).The bending moment on the sheet is greatest near the Point A,as shown in Fig.6c,such that the strip may be over-bent at this point,causing a loss of contact between the sheet and the die radius.A similar effect can exist over the nose of the punch in vee-die bending(Marciniak et al.,2002).As such,a second contact point with the die occurs further along the radius,at Â=ˇ.Point A,which began at the start of the radius,has not yet reached the tangent point atˇ.Hence,the material currently atˇis largely undeformed,despite the fact that the angle of wrap of the blank over the die radius is relatively large.With similar contact conditions to the initial deformation stage,line contact occurs atˇ.As seen previously,these conditions result in high contact pressure,large compressive hydrostatic stress, and can suppress plastic deformation at the blank surface as supported by the case study in Fig.1b.Fig.6b shows the contact pressure distribution for the inter-mediate stage.The magnitude of the contact pressure at the start of the radius is less than the yield stress,where con-tact is distributed over a wider area.Conversely,a sharp peak exists at the tangent point atˇ,where the sheet is still being bent and the contact area is small.In many punch and die3538j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y209(2009)3532–3541conﬁgurations,the punch displacement needed to draw the material from the beginning of the die radius(Point A in this case)around to the tangent point is signiﬁcant.Therefore,the intermediate phase may be long and the maximum contact angle,ˇmax,quite large.3.3.Steady-state conditions at the die radiusSteady-state conditions at the die radius are reached when Point A,which began at the start of the die radius,has moved around and become part of the side-wall,as shown in Fig.7a. New material is plastically bent as it enters the die radius from the blank-holder region.Here,the contact pressure and stress distributions are similar to those of the intermediate stage, due to the bending and conformance of the blank to the die radius.Beyond this region,the sheet remains in contact with the die without further plastic deformation,and the resulting contact pressure is small.Further along the radius,under the action of an increasing opposite moment,the sheet is partially straightened,whereFig.7–(a)Schematic of the blank to die radius interface during the steady-state deformation stage—the stress distribution through the thickness and the Mohr’s circle at the surface of the contact zones are shown.The stress distribution through the thickness at two locations in the side-wall region is also shown.Corresponding distributions around the die radius of (b)contact pressure and(c)bending moment in the sheet.j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y209(2009)3532–35413539it loses contact with the die radius.A second,smaller con-tact pressure peak occurs at the locationÂ= .This peak can be explained,at least in part,by examining the sim-pliﬁed analysis presented in Section2.According to Eq.(1), the contact pressure is proportional to the tension in the sheet—which itself increases with increasing angleÂalong the radius,according to Eq.(2).Therefore,the contact pressure increases with angle along the radius,causing a peak pressure near the sheet exit point,indicated by P in Fig.7b.Here,the sheet unloads elastically and the stress distribution is shown (the diameter of the Mohr circle is<S).Beyond the contact pressure peak,the bending moment on the sheet becomes reversed,as shown in Fig.7c,and straightening begins at the tangent point.The straightening process continues beyond the contact point;the extent of which depends on the tooling conditions and the tension gen-erated by the blank-holder.‘Side-wall curl’is a well-known phenomenon in channel forming and is greatest with smaller blank-holder tension.As a result of the curl in the side-wall,the angle of contact is less than in the intermediate stage,where the entire side-wall was approximately straight. This indicates that there is a region on the die radius that only makes contact with the blank during the intermediate stage—i.e.an intermediate-only contact region.It is worth emphasizing that,despite the approximately steady contact conditions that occur at the die radius during this stage,the forming process itself does not reach a true steady state.This is because the blank continues to experi-ence signiﬁcant deformation and displacement as it is drawn over the die radius by the action of the moving punch.As a result,there will be a continual reduction in theﬂange length and a subsequent changing of contact conditions in the blank-holder region.4.DiscussionIn Section3,a qualitative description of the deformation and contact pressure response at the die radius of a sheet metal stamping process was given.This section will discuss the identiﬁed response,with particular reference to results from other analyses in the literature,comparison to the bending-under-tension process,and wear at the die radius.4.1.Correlation withﬁnite element model predictionsIn recent studies,Pereira et al.(2007,2008)usedﬁnite element analysis to examine the contact pressure at the die radius for a channel forming process.A2mm thick high strength steel blank was formed over an R5mm die radius(R/t=2.5), with a punch stroke of50mm.The contact pressure response predicted by Pereira et al.(2008)was re-plotted at three dis-tinct instances in Fig.1.In thisﬁgure,the contact pressure is normalised by the constant Y,which can be considered as theﬂow stress of the blank material if a perfectly plas-tic approximation of the material stress–strain response was adopted(see Marciniak et al.(2002)for an explanation of the approximation method and calculation of Y).As such,the use of the normalised contact pressure allows better comparison between the analysis employing a blank material with con-siderable strain hardening(Fig.1)to that which assumes the blank material has zero strain hardening(Figs.5–7).The normalised contact pressure distributions in Fig.1 clearly demonstrate the existence of the three phases iden-tiﬁed in Section3.Notably,theﬁrst two stages in Section3 correspond to the single transient phase reported in the pre-vious numerical study(Pereira et al.,2008).The discrepancy is caused by the fact that the initial contact stage,which is a momentary event,is easily overlooked without a detailed analysis of the deformation and contact conditions occurring at the die radius.The results by Pereira et al.(2007,2008)verify that the ini-tial and intermediate phases of the process result in the most severe and localised contact loads.Fig.1shows that at the regions of line contact,identiﬁed in Sections3.1and3.2,the peak contact pressures are well in excess of Y.In fact,the maximum contact pressure for the entire process was found to occur during the intermediate stage,with a magnitude of approximately3times the material’s initial yield strength (Pereira et al.,2008).Examination of the Mises stress plots in Fig.1at the regions of line contact also conﬁrm the hypothesis of suppressed plasticity due the localised zones of large con-tact pressure,and hence large compressive hydrostatic stress.The results in Fig.1c conﬁrm that the contact pressure is signiﬁcantly reduced during the steady phase,with the mag-nitude of pressure less than Y due to the increased contact area.Theﬁnite element results also show that the maximum angles of contact between the blank and die radius during the intermediate and steady phases are approximately80◦and 45◦,respectively(Pereira et al.,2008).This conﬁrms the exis-tence of an intermediate-only contact region,corresponding to the region of45◦<Â≤80◦for the case examined.parison to the bending-under-tension testThe identiﬁed steady-state behaviour at the die radius during the stamping process shows numerous similarities to a typical bending-under-tension test.For example,the stress distribu-tions through the thickness of the sheet shown in Fig.7a, compare well to those proposed by Swift(1948),in his analysis of a plastic bending-under-tension process for a rigid,per-fectly plastic strip.Additionally,the angle of contact and shape of contact pressure distributions presented in Figs.7b and1c, show good correlation with the results recorded by Hanaki and Kato(1984)for experimental bending-under-tension tests.The separateﬁnite element studies of bending-under-tension processes by Hortig and Schmoeckel(2001)and by Boher et al.(2005)also show similarly shaped two-peak contact pressure distributions.The distributions are char-acterised by large and relatively localised pressure peaks at the beginning of the contact zone,with smaller and more distributed secondary peaks at the end of the con-tact zone.Additionally,these investigations each show that the angle of contact is signiﬁcantly less than the geomet-ric angle of wrap,conﬁrming the existence of the unbending of the blank and curl that occurs in the side-wall region. These attributes of the bending-under-tension test have direct similarities to the contact pressure response predicted by Pereira et al.(2008)and described previously in Section 3.3,despite the obvious differences in materials,processes,3540j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y209(2009)3532–3541bend ratios and back tensions considered.Although there are numerous similarities,direct quantitative comparison between the bending-under-tension test and the steady-state phase of the channel forming process cannot be made,due to the differences in the application of the back and forward tensions.4.3.Contradictions withﬁnite element model predictionsAs stated in Section1,there are a limited number of other investigations in the literature that examine the time-dependant contact pressure response of sheet metal stamping processes.Finite element analyses by Mortensen et al.(1994) and Jensen et al.(1998)predicted that time-dependant contact conditions do occur.However,these results do not show the same trends as presented in this study and shown by Pereira et al.(2007,2008)in previousﬁnite element investigations. This section will brieﬂy discuss the possible reasons for such discrepancies.Firstly,considering theﬁnite element analysis of a cup-drawing process by Mortensen et al.(1994),the predicted contact pressure over the die radius was presented at only three distinct intervals during the process.By comparison, Pereira et al.(2008)recorded the contact pressure at approx-imately140intervals throughout theﬁnite element results history,in order to completely characterise the complex pressure evolution.Therefore,it is likely that the transient effects,which are reported in this study,were not captured by Mortensen et al.(1994)due to the limited number of instances at which the contact pressure was recorded.Theﬁnite element investigation by Jensen et al.(1998) examined the contact conditions at approximately100inter-vals during a cup-drawing process,but also did not observe a severe and localised transient response,as seen in this study. (Signiﬁcantly varied and localised contact conditions were observed at the end of the process,but these were identi-ﬁed to be due to the blank-rim effect,and are not relevant to this study.)Close examination of the results by Jensen et al. (1998)show that some localised contact conditions do occur at the beginning of the process—however,these appear rela-tively mild and were not discussed in the text.This reduced severity of the transient response,compared to that predicted by Pereira et al.(2008),can be partly explained by the fact that the actual contact pressure at the die radius was not shown by Jensen et al.(1998).Instead,Z xt,which was deﬁned to be a function of contact pressure and sliding velocity,was used to characterise the contact conditions.This could have effec-tively reduced the appearance of the initial localised contact conditions,due to the slower sliding velocity shown to exist during the initial stage.Additionally,Jensen et al.(1998)used 20ﬁnite elements to describe the die radius surface,compared to240elements used by Pereira et al.(2008).The reduced num-ber of elements at the die radius surface can have the effect of averaging the extremely localised contact loads over a larger area,thus reducing the magnitude of the observed contact pressure peaks.Finally,the different processes examined(cup drawing vs.channel forming)may also result in a different transient response.4.4.Relevance to tool wearWear is related to contact pressure through a power law rela-tionship(Rhee,1970).Therefore,the regions of severe contact pressure during the initial and intermediate stages may be particularly relevant to tool wear at the die radius.Theﬁnite element investigations by Pereira et al.(2007,2008)showed that the maximum contact pressure for the entire process occurs in the intermediate-only contact region,at approximately Â=59◦,indicating that the intermediate stage is likely to be of primary signiﬁcance to the wear response.This result was val-idated by laboratory-based channel forming wear tests,for the particular case examined(Pereira et al.,2008).However,for each stamping operation,it can be seen that the relative sliding distance between the blank and die radius associated with the initial and intermediate stages is small—i.e.no greater than the arc length of the die radius surface.In comparison,the steady contact pressure phase cor-responds to a much larger sliding distance—i.e.the sliding distance will be approximately in the same order of magnitude as the punch travel.Therefore,despite the smaller contact pressures,it is possible that the steady phase may also inﬂu-ence the tool life;depending on the process conditions used (e.g.materials,surface conditions,sliding speed,lubrication) and the resulting wear mechanisms that occur.The existence of an intermediate-only contact zone(i.e.the region <Â≤ˇmax),is convenient for future wear analyses.Due to the lack of sliding contact in this region during the steady-state phase,any surface degradation of the die radius at angles ofÂ> must be attributed to the intermediate stage of the sheet metal stamping process.Therefore,it is recommended that future wear analysis examine this region to assess the importance of the intermediate contact conditions on the overall tool wear response of the sheet metal stamping pro-cess.The existence of the initial and intermediate stages high-light that the bending-under-tension test,due to its inherently steady nature,is unable to capture the complete contact con-ditions that exists during a typical sheet metal stamping process.Therefore,the applicability of the bending-under-tension test for sheet metal stamping wear simulation may be questionable.5.SummaryIn this work,a qualitative description of the development of peak contact pressures at the die radius for a sheet metal stamping process was given.It was shown that three distinct phases exist:(i)At the start of the process,the blank is bent by the actionof the punch and a high contact pressure peak exists at the start of the die radius.(ii)During the intermediate stage,the region of the sheet that was deformed at the start of the die radius has not reached the side-wall.Therefore,the side-wall remains straight and the arc of contact is a maximum.The largest pressure,which is signiﬁcantly greater than the sheet materialﬂow stress,exists towards the end of the die。

Meshfree and particle methods and their applications

Meshfree and particle methods and their applicationsShaofan LiDepartment of Civil&Environmental Engineering,University of California,Berkeley CA94720;li@Wing Kam LiuDepartment of Mechanical Engineering,Northwestern University,2145Sheridan Rd,Evanston IL60208;w-liu@Recent developments of meshfree and particle methods and their applications in applied mechan-ics are surveyed.Three major methodologies have been reviewed.First,smoothed particlehydrodynamics͑SPH͒is discussed as a representative of a non-local kernel,strong form collo-cation approach.Second,mesh-free Galerkin methods,which have been an active researcharea in recent years,are reviewed.Third,some applications of molecular dynamics͑MD͒inapplied mechanics are discussed.The emphases of this survey are placed on simulations ofﬁnite deformations,fracture,strain localization of solids;incompressible as well as compress-ibleﬂows;and applications of multiscale methods and nano-scale mechanics.This review ar-ticle includes397references.͓DOI:10.1115/1.1431547͔1INTRODUCTIONSince the invention of theﬁnite element method͑FEM͒in the1950s,FEM has become the most popular and widely used method in engineering computations.A salient feature of the FEM is that it divides a continuum into discrete ele-ments.This subdivision is called discretization.In FEM,the individual elements are connected together by a topological map,which is usually called a mesh.Theﬁnite element in-terpolation functions are then built upon the mesh,which ensures the compatibility of the interpolation.However,this procedure is not always advantageous,because the numerical compatibility condition is not the same as the physical com-patibility condition of a continuum.For instance,in a La-grangian type of computations,one may experience mesh distortion,which can either end the computation altogether or result in drastic deterioration of accuracy.In addition, FEM often requires a veryﬁne mesh in problems with high gradients or a distinct local character,which can be compu-tationally expensive.For this reason,adaptive FEM has be-come a necessity.Today,adaptive remeshing procedures for simulations of impact/penetration problems,explosion/fragmentation prob-lems,ﬂow pass obstacles,andﬂuid-structure interaction problems etc have become formidable tasks to undertake. The difﬁculties involved are not only remeshing,but also mapping the state variables from the old mesh to the new mesh.This process often introduces numerical errors,and frequent remeshing is thus not desirable.Therefore,the so called Arbitrary Lagrangian Eulerian͑ALE͒formulations have been developed͑see,eg͓1–4͔͒.For a complete descrip-tion on this subject,readers may consult Chapter7of the book by Belytschko,Liu,and Moran͓5͔.The objective of the ALE formulation is to make the mesh independent of the material so that the mesh distortion can be minimized.Un-fortunately,in computer simulations of very large deforma-tion and/or high-speed mechanical and structural systems, even with the ALE formulation,a distorted mesh introduces severe errors in numerical computations.Furthermore,the convective transport effects in ALE often lead to spurious oscillation that needs to be stabilized by artiﬁcial diffusion or a Petrov-Galerkin stabilization.In other cases,a mesh may carry inherent bias in numerical simulations,and its presence becomes a nuisance in computations.A well known example is the simulation of the strain localization problem,which is notorious for its mesh alignment sensitivity͓6,7͔.Therefore, it would be computationally efﬁcacious to discretize a con-tinuum by only a set of nodal points,or particles,without mesh constraints.This is the leitmotif of contemporary mesh-free Galerkin methods.The advantages of the meshfree particle methods may be summarized as follows:1͒They can easily handle very large deformations,since the connectivity among nodes is generated as part of the computation and can change with time;2͒The methodology can be linked more easily with a CAD database thanﬁnite elements,since it is not necessary to generate an element mesh;3͒The method can easily handle damage of the components, such as fracture,which should prove very useful in mod-elings of material failure;Transmitted by Associate Editor JN ReddyASME Reprint No AMR319$26.00Appl Mech Rev vol55,no1,January2002©2002American Society of Mechanical Engineers14͒Accuracy can be controlled more easily,since in areas where more reﬁnement is needed,nodes can be added quite easily͑h-adaptivity͒;5͒The continuum meshfree methods can be used to model large deformations of thin shell structures,such as nano-tubes;6͒The method can incorporate an enrichment ofﬁne scale solutions of features,such as discontinuities as a function of current stress states,into the coarse scale;and7͒Meshfree discretization can provide accurate representa-tion of geometric object.In general,particle methods can be classiﬁed based on two different criteria:physical principles,or computational formulations.According to the physical modeling,they may be categorized into two classes:those based on deterministic models,and those based on probabilistic models.On the other hand,according to computational modelings,they may be categorized into two different types as well:those serving as approximations of the strong forms of partial differential equations͑PDEs͒,and those serving as approximations of the weak forms of PDEs.In this survey,the classiﬁcation based on computational strategies is adopted.To approximate the strong form of a PDE using a particle method,the partial differential equation is usually discretized by a speciﬁc collocation technique.Examples are smoothed particle hydrodynamics͑SPH͓͒8–12͔,the vortex method ͓13–18͔,the generalizedﬁnite difference method͓19,20͔, and many others.It is worth mentioning that some particle methods,such as SPH and vortex methods,were initially developed as probabilistic methods͓10,14͔,and it turns out that both SPH and the vortex method are most frequently used as deterministic methods today.Nevertheless,the ma-jority of particle methods in this category are based on probabilistic principles,or used as probabilistic simulation tools.There are three major methods in this category:1͒molecular dynamics͑both quantum molecular dynamics ͓21–26͔and classical molecular dynamics͓27–32͔͒;2͒di-rect simulation Monte Carlo͑DSMC͒,or Monte Carlo method based molecular dynamics,such as quantum Monte Carlo methods͓33–41͔͒͑It is noted that not all the Monte Carlo methods are meshfree methods,for instance,a proba-bilisticﬁnite element method is a mesh-based method͓42–44͔͒;and3͒the lattice gas automaton͑LGA͒,or lattice gas cellular automaton͓45–49͔and its later derivative,the Lat-tice Boltzmann Equation method͑LBE͓͒50–54͔.It may be pointed out that the Lattice Boltzmann Equation method is not a meshfree method,and it requires a grid;this example shows that particle methods are not always meshfree.The second class of particle methods is used with various Galerkin weak formulations,which are called meshfree Galerkin methods.Examples in this class are Diffuse Ele-ment Method͑DEM͓͒55–58͔,Element Free Galerkin Method͑EFGM͓͒59–63͔,Reproducing Kernel Particle Method͑RKPM͓͒64–72͔,h-p Cloud Method͓73–76͔,Par-tition of Unity Method͓77–79͔,Meshless Local Petrov-Galerkin Method͑MLPG͓͒80–83͔,Free Mesh Method͓84–88͔,and others.There are exceptions to this classiﬁcation,because someparticle methods can be used in both strong form collocationas well as weak form discretization.The particle-in-cell ͑PIC͒method is such an exception.The strong form colloca-tion PIC is often called theﬁnite-volume particle-in-cellmethod͓89–91͔,and the weak form PIC is often called thematerial point method͓92͔,or simply particle-in-cell method ͓93–95͔.RKPM also has two versions as well:a collocation version͓96͔and a Galerkin weak form version͓66͔.In areas such as astrophysics,solid state physics,biophys-ics,biochemistry and biomedical research,one may encoun-ter situations where the object under consideration is not acontinuum,but a set of particles.There is no need for dis-cretization to begin with.A particle method is the naturalchoice in numerical simulations.Relevant examples are thesimulation of formation of a star system,the nano-scalemovement of millions of atoms in a non-equilibrium state,folding and unfolding of DNA,and dynamic interactions ofvarious molecules,etc.In fact,the current trend is not onlyto use particle methods as discretization tools to solve con-tinuum problems͑such as SPH,vortex method͓14,15,97͔and meshfree Galerkin methods͒,but also to use particlemethods as a physical model͑statistical model,or atomisticmodel͒to simulate continuum behavior of physics.The latestexamples are using the Lattice Boltzmann method to solveﬂuid mechanics problems,and using molecular dynamics tosolve fracture mechanics problems in solid mechanics͓98–103͔.This survey is organized as follows:Theﬁrst part is acritical review of smoothed particle hydrodynamics͑SPH͒.The emphasis is placed on the recent development of correc-tive SPH.The second part is a summary of meshfree Galer-kin methods,which includes DEM,EFGM,RKPM,hp-Cloud method,partition of unity method,MLPGM,andmeshfree nodal integration methods.The third part reviewsrecent applications of molecular dynamics in fracture me-chanics as well as nanomechanics.The last part is a surveyon some other meshfree/particle methods,such as vortexmethods,the Lattice Boltzmann method,the natural elementmethod,the particle-in-cell method,etc.The survey is con-cluded with the discussions of some emerging meshfree/particle methods.2SMOOTHED PARTICLE HYDRODYNAMICS2.1OverviewSmoothed Particle Hydrodynamics is one of the earliest par-ticle methods in computational mechanics.Early contribu-tions have been reviewed in several articles͓8,12,104͔.In1977,Lucy͓10͔and Gingold and Monaghan͓9͔simulta-neously formulated the so-called Smoothed Particle Hydro-dynamics,which is known today as SPH.Both of them wereinterested in the astrophysical problems,such as the forma-tion and evolution of proto-stars or galaxies.The collectivemovement of those particles is similar to the movement of aliquid,or gasﬂow,and it may be modeled by the governingequations of classical Newtonian hydrodynamics.Today,SPH is being used in simulations of supernovas͓105͔,col-2Li and Liu:Meshfree and particle methods and applications Appl Mech Rev vol55,no1,January2002lapse as well as formation of galaxies͓106–109͔,coales-cence of black holes with neutron stars͓110,111͔,single and multiple detonations in white dwarfs͓112͔,and even in ‘‘Modeling the Universe’’͓113͔.Because of the distinct ad-vantages of the particle method,soon after its debut,the SPH method was widely adopted as one of the efﬁcient computa-tional techniques to solve applied mechanics problems. Therefore,the term hydrodynamics really should be inter-preted as mechanics in general,if the methodology is applied to other branches of mechanics rather than classical hydro-dynamics.To make distinction with the classical hydrody-namics,some authors,eg Kum et al͓114,115͔,called it Smoothed Particle Applied Mechanics.This idea of the method is somewhat contrary to the con-cepts of the conventional discretization methods,which dis-cretize a continuum system into a discrete algebraic system. In astrophysical applications,the real physical system is dis-crete;in order to avoid singularity,a local continuousﬁeld is generated by introducing a localized kernel function,which can serve as a smoothing interpolationﬁeld.If one wishes to interpret the physical meaning of the kernel function as the probability of a particle’s position,one is dealing with a probabilistic method.Otherwise,it is only a smoothing tech-nique.Thus,the essence of the method is to choose a smooth kernel,W(x,h)͑h is the smoothing length͒,and to use it to localize the strong form of a partial differential equation through a convoluted integration.Deﬁne SPH averaging/ localization operator asA k͑x͒ϭ͗A͑x͒͘ϭ͵R n W͑xϪxЈ,h͒A͑xЈ͒d⍀xЈϷ͚Iϭ1N W͑xϪx I,h͒A͑x I͒⌬V I(1) One may derive a SPH discrete equation of motion from its continuous counterpart͓12,116͔,ͳ␳d v dtʹI ϭϪٌ͗•␴͘I⇒␳Id v IdtϷϪ͚Jϭ1N͑␴Iϩ␴J͒•ٌW͑x IϪx J,h͒⌬V J(2)where␴is Cauchy stress,␳is density,v is velocity,and⌬V J is the volume element carried by the particle J.Usually a positive function,such as the Gaussian func-tion,is chosen as the kernel functionW͑x,h͒ϭ1͑␲h2͒n/2expͫϪx2h2ͬ,1рnр3(3)where the parameter h is the smoothing length.In general, the kernel function has to satisfy the following conditions:i)W͑x,h͒у0(4) ii)͵R n W͑u,h͒d⍀uϭ1(5) iii)W͑u,h͒→␦͑u͒,h→0(6) i v)W͑u,h͒෈C p͑R n͒,pу1(7)The third property ensures the convergence,and the last property comes from the requirement that the smoothing ker-nel must be differentiable at least once.This is because the derivative of the kernel function should be continuous to prevent a largeﬂuctuation in the force felt by the particle. The latter feature gives rise to the name smoothed particle hydrodynamics.In computations,compact supported kernel functions such as spline functions are usually employed͓117͔.In this case, the smoothing length becomes the radius of the compact sup-port.Two examples of smooth kernel functions are depicted in Fig.1.The advantage of using an analytical kernel is that one can evaluate a kernel function at any spatial point without knowing the local particle distribution.This is no longer true for the latest corrective smoothed particle hydrodynamics methods͓66,118͔,because the corrective kernel function de-pends on the local particle distribution.The kernel representation is not only an instrument that can smoothly discretize a partial differential equation,but it also furnishes an interpolant scheme on a set of moving par-ticles.By utilizing this property,SPH can serve as a La-grangian type method to solve problems in continuum me-chanics.Libersky and his co-workers apply the method to solid mechanics͓117,119,120͔,and they successfully simu-late3D thick-wall bomb explosion/fragmentation problem, tungsten/plate impact/penetration problem,etc.The impact and penetration simulation has also been conducted by Johnson and his co-workers͓121–123͔,and an SPH option is implemented in EPIC code for modeling inelastic,dam-age,large deformation problems.Attaway et al͓124͔devel-oped a coupling technique to combine SPH with theﬁnite element method,and an SPH option is also included in PR-ONTO2D͑Taylor and Flanagan͓125͔͒.SPH technology has been employed to solve problems of both compressibleﬂow͓126͔and incompressibleﬂow Fig.1Examples of kernel functionsAppl Mech Rev vol55,no1,January2002Li and Liu:Meshfree and particle methods and applications3͓116,127–129͔,multiple phaseﬂow and surface tension ͓114,115,129,130,131,132,133͔,heat conduction͓134͔, electro-magnetic͑Maxwell equations͓͒90,104,135͔,plasma/ﬂuid motion͓135͔,general relativistic hydrodynamics͓136–138͔,heat conduction͓134,139͔,and nonlinear dynamics ͓140͔.2.2Corrective SPH and other improvementsin SPH formulationsVarious improvements of SPH have been developed through the years͓104,141–149͔.Most of these improvement are aimed at the following shortcomings,or pathologies,in nu-merical computations:•tensile instability͓150–154͔;•lack of interpolation consistency,or completeness ͓66,155,156͔;•zero-energy mode͓157͔;•difﬁculty in enforcing essential boundary condition ͓120,128,131͔.2.2.1Tensile instabilitySo-called tensile instability is the situation where particles are under a certain tensile͑hydrostatic͒stress state,and the motion of the particles become unstable.To identify the cul-prit,a von Neumann stability analysis was carried out by Swegle et al͓150͔,and by Balsara͓158͔.Swegle and his co-workers have identiﬁed and explained the source of the tensile instability.Recently,by using von Neumann and Cou-rant stability criterion,Belytschko et al͓151͔revisited the problem in the general framework of meshfree particle meth-ods.In their analysis,ﬁnite deformation effects are also con-sidered.Several remedies have been proposed to avoid such ten-sile instability.Morris proposed using special kernel func-tions.While successful in some cases,they do not always yield satisfactory results͓152͔.Randles and Libersky͓120͔proposed adding dissipative terms,which is related to con-servative smoothing.Notably,Dyka et al͓153,154͔proposed a so-called stress point method.The essential idea of this approach is to add additional points other than SPH particles when evaluating,or sampling,stress and other state vari-ables.Whereas the kinematic variables such as displacement,velocity,and acceleration are still sampled at particle points.In fact,the stress point plays a similar role as the‘‘Gaussquadrature point’’does in the numerical integration of theGalerkin weak form.This analogy wasﬁrst pointed out byLiu et al͓66͔.This problem was revisited again recently byChen et al͓159͔as well as Monaghan͓148͔.The formerproposes a special corrective smoothed-particle method ͑CSPM͒to address the tensile instability problem by enforc-ing the higher order consistency,and the latter proposes toadd an artiﬁcial force to stabilize the computation.Randlesand Libersky͓160͔combined normalization with the usualstress point approach to achieve better stability as well aslinear consistency.Apparently,the SPH tensile instability isrelated to the lack of consistency of the SPH interpolant.A2D stress point deployment is shown in Fig.2.2.2.2Zero-energy modeThe zero energy mode has been discovered in bothﬁnitedifference andﬁnite element computations.A comprehensivediscussion of the subject can be found in the book by Be-lytschko et al͓5͔.The reason that SPH suffers similar zeroenergy mode deﬁciency is due to the fact that the derivativesof kinematic variables are evaluated at particle points by ana-lytical differentiation rather than by differentiation of inter-polants.In many cases,the kernel function reaches a maxi-mum at its nodal position,and its spatial derivatives becomezero.To avoid a zero-energy mode,or spurious stress oscil-lation,an efﬁcient remedy is to adopt the stress point ap-proach͓157͔.2.2.3Corrective SPHAs an interpolation among moving particles,SPH is not apartition of unity,which means that SPH interpolants cannotrepresent rigid body motion correctly.This problem wasﬁrstnoticed by Liu et al͓64–66͔.They then set forth a key no-tion,a correction function,which has become the centraltheme of the so-called corrective SPH.The idea of a correc-tive SPH is to construct a corrective kernel,a product of thecorrection function with the original kernel.By doing so,theconsistency,or completeness,of the SPH interpolant can beenforced.This new interpolant is named the reproducing ker-nel particle method͓64–66͔.SPH kernel functions satisfy zero-th order moment condi-tion͑5͒.Most kernel functions satisfy higher order momentcondition as well͓104͔,for instance͵R xW͑x,h͒dxϭ0.(8)These conditions only hold in the continuous form.In gen-eral they are not valid after discretization,ie͚Iϭ1NPW͑xϪx I,h͒⌬x I 1(9)͚Iϭ1NP͑xϪx I͒W͑xϪx I,h͒⌬x I 0(10)where NP is the total number of the particles.Note that con-dition͑9͒is the condition of partition of unity.Sincethe Fig.2A2D Stress point distribution4Li and Liu:Meshfree and particle methods and applications Appl Mech Rev vol55,no1,January2002kernel function can not satisfy the discrete moment condi-tions,a modiﬁed kernel function is introduced to enforce the discrete consistency conditionsW˜h ͑x Ϫx I ;x ͒ϭC h ͑x Ϫx I ;x ͒W ͑x Ϫx I ,h ͒(11)where C h (x ;x Ϫx I )is the correction function,which can be expressed asC h ͑x ;x Ϫx I ͒ϭb 0͑x ,h ͒ϩb 1͑x ,h ͒x Ϫx Ihϩb 2͑x ,h ͒ϫͩx Ϫx I hͪ2ϩ¯¯(12)where b 0(x ),b 1(x ),¯.,b n (x )are unknown functions.Wecan determine them to correct the original kernel function.Suppose f (x )is a sufﬁciently smooth function.By Taylor expansion,f I ϭf ͑x I ͒ϭf ͑x ͒ϩf Ј͑x ͒ͩx I Ϫxhͪh ϩf Љ͑x ͒2!ͩx I Ϫx hͪ2h 2ϩ¯¯(13)the modiﬁed kernel approximation can be written as,f h ͑x ͒ϭ͚I ϭ1NPW˜h ͑x Ϫx I ;x ͒f I ⌬x I ϭ͚ͩI ϭ1NP W˜h ͑x Ϫx I ,x ͒⌬x I ͪf ͑x ͒h 0Ϫ͚ͩI ϭ1NP ͩx Ϫx IhͪW ˜h ͑x Ϫx I ,x ͒⌬x I ͪf Ј͑x ͒h ϩ¯¯ϩ͚ͩI ϭ1NP ͑Ϫ1͒nͩx Ϫx I hͪnW˜h ϫ͑x Ϫx I ,x ͒⌬x Iͪf n ͑x ͒n !h nϩO ͑h n ϩ1͒.(14)To obtain an n -th order reproducing condition,the moments of the modiﬁed kernel function must satisfy the following conditions:M 0͑x ͒ϭ͚I ϭ1NPW ˜h ͑x Ϫx I ,x ͒⌬x I ϭ1;M 1͑x ͒ϭ͚I ϭ1NPͩx Ϫx IhͪW ˜h ͑x Ϫx I ,x ͒⌬x I ϭ0;ӇM n ͑x ͒ϭ͚I ϭ1NPͩx Ϫx IhͪnW˜h ͑x Ϫx I ,x ͒⌬x I ϭ0;·(15)Substituting the modiﬁed kernel expressions,͑11͒and ͑12͒into Eq.͑15͒,we can determine the n ϩ1coefﬁcients,b i (x ),by solving the following moment equations :ͩm 0͑x ͒m 1͑x ͒¯m n ͑x ͒m 1͑x ͒m 2͑x ͒¯m n ϩ1͑x ͒ӇӇӇӇm n ͑x ͒m n ϩ1͑x ͒¯m 2n ͑x ͒ͪͩb 0͑x ,h ͒b 1͑x ,h ͒Ӈb n ͑x ,h ͒ͪϭͩ10Ӈ0ͪ(16)It is worth mentioning that after introducing the correction function,the modiﬁed kernel function may not be a positive function anymore,K ͑x Ϫx I ͒у”0.(17)Within the compact support,K (x Ϫx I )may become negative.This is the reason why Duarte and Oden refer to it as the signed partition of unity ͓73,74,76͔.There are other approaches to restoring completeness of the SPH approximation.Their emphases are not only consis-tency,but also on cost ing RKPM,or a moving-least-squares interpolant ͓155,156͔to construct modiﬁed kernels,one has to know all the neighboring par-ticles that are adjacent to a spatial point where the kernel function is in evaluation.This will require an additional CPU to search,update the connectivity array,and calculate the modiﬁed kernel function pointwise.It should be noted that the calculation of the modiﬁed kernel function requires pointwise matrix inversions at each time step,since particles are moving and the connectivity map is changing as well.Thus,using a moving least square interpolant as the kernel function may not be cost-effective,and it destroys the sim-plicity of SPH formulation.Several compromises have been proposed throughout the years,which are listed as follows:1͒Monaghan’s symmetrization on derivative approximation ͓104,145͔;2͒Johnson-Beissel correction ͓123͔;3͒Randles-Libersky correction ͓120͔;4͒Krongauz-Belytschko correction ͓61͔;5͒Chen-Beraun correction ͓139,140,161͔;6͒Bonet-Kulasegaram integration correction ͓118͔;7͒Aluru’s collocation RKPM ͓96͔.Since the linear reproducing condition in the interpolation is equivalent to the constant reproducing condition in the de-rivative of the interpolant,some of the algorithms directly correct derivatives instead of the interpolant.The Chen-Beraun correction corrects even higher order derivatives,but it may require more computational effort in multi-dimensions.Completeness,or consistency,closely relates to conver-gence.There are two types of error estimates:interpolation error and the error between exact solution and the numerical solution.The former usually dictates the latter.In conven-tional SPH formulations,there is no requirement for the completeness of interpolation.The particle distribution is as-sumed to be randomly distributed and the summations areAppl Mech Rev vol 55,no 1,January 2002Li and Liu:Meshfree and particle methods and applications5Monte Carlo estimates of integral interpolants.The error of random interpolation wasﬁrst estimated by Niedereiter͓162͔as beingϰNϪ1log N nϪ1where N is total particle number and n is the dimension of space.This result was further improved by Wozniakowski͓163͔as beingϰNϪ1log N nϪ1/2.Accord-ing to reference͓104͔,‘‘this remarkable result was produced by a challenge with a payoff of sixty-four dollars!’’Twenty-one years after its invention,in1998Di Lisio et al͓164͔gave a convergence proof of smoothed particle hydrodynam-ics method for regularized Eulerﬂow equations.Besides consistency conditions,the conservation proper-ties of a SPH formulation also strongly inﬂuence its perfor-mance.This has been a critical theme throughout SPH re-search,see͓12,104,120,145,155,165͔.It is well known that classical SPH enjoys Galilean invariance,and if certain de-rivative approximations,or Golden rules as Monaghan puts it,are chosen,the corresponding SPH formulations can pre-serve some discrete conservation laws.This issue was re-cently revisited by Bonet et al͓166͔,and they set forth a discrete variational SPH formulation,which can automati-cally satisfy the balance of linear momentum and balance of angular momentum conservation laws.Here is the basic idea. Assume the discrete potential energy in a SPH system is ⌸͑x͒ϭ͚I V I0U͑J I͒(18) where V I0is the initial volume element,and U(J I)is the internal energy density,which is assumed to be the function of determinant of the Jacobian—ratio between the initial and current volume element,JϭV IV I0ϭ␳I0␳I(19)where␳I0and␳I are pointwise density in initial conﬁguration and in current conﬁguration.For adiabatic processes,the pressure can be obtained from ץU I/ץJϭp I.Thus,the stationary condition of potential en-ergy gives␦⌸ϭD⌸͓␦v͔ϭ͚I V I0DU I͓␦v͔ϭ͚I͚ͭJ m I m Jͩp I␳I2ϩp J␳J2ٌͪW I͑x J͒ͮ␦v I(20)where m I is the mass associated with particle I.On the other hand,D⌸͓␦v͔ϭ͚Iץ⌸ץx I␦v Iϭ͚I T I•␦v I(21)where T is the internal force͑summation of stress͒.Then through the variational principle,one can identify,T Iϭ͚I m I m Jͩp I␳I2ϩp J␳J2ٌͪW I͑x J͒(22)and establish the discrete SPH equation of motion͑balance of linear momentum͒,m Id v IdtϭϪ͚I m I m Jͩp I␳I2ϩp J␳J2ٌͪW I͑x J͒.(23)2.2.4Boundary conditionsSPH,and in fact particle methods in general,have difﬁculties in enforcing essential boundary condition.For SPH,some effort has been devoted to address the issue.Takeda’s image particle method͓131͔is designed to satisfy the no-slip boundary condition;it is further generalized by Morris et al ͓128͔to satisfy boundary conditions along a curved bound-ary.Based on the same philosophy,Randles and Libersky ͓120͔proposed a so-called ghost particle approach,which is outlined as follows:Suppose particle i is a boundary particle. All the other particles within its support,N(i),can be di-vided into three subsets:1͒I(i):all the interior points that are the neighbors of i;2͒B(i):all the boundary points that are the neighbors of i; 3͒G(i):all the exterior points that are the neighbors of i,ie, all the ghost particles.Therefore N(i)ϭI(i)ഫB(i)ഫG(i).Figure3illustrates such an arrangement.In the ghost particle approach,the boundary correction formula for general scalarﬁeld f is given as followsf iϭf bcϩ͚j෈I…i…͑f jϪf bc͒⌬V j W i jͩ1Ϫ͚j෈B…i…⌬V j W i jͪ(24)where f bc is the prescribed boundary value at xϭx i.One of the advantages of the above formula is that the sampling formula only depends on interior particles.2.3Other related issues and applicationsBesides resolving the above fundamental issues,there have been some other progresses in improving the performance of SPH,which have focused on applications as well as algorith-mic efﬁciency.How to choose an interpolation kernel to en-sure successful simulations is discussed in͓167͔;how to modify the kernel functions without correction isdiscussed Fig.3The Ghost particle approach for boundary treatment6Li and Liu:Meshfree and particle methods and applications Appl Mech Rev vol55,no1,January2002。

An equivalent method for blasting vibration simulation

An equivalent method for blasting vibration simulationWenbo Lu,Jianhua Yang ⇑,Ming Chen,Chuangbing ZhouState Key Laboratory of Water Resources and Hydropower Engineering Science,Wuhan University,Wuhan 430072,ChinaKey Laboratory of Rock Mechanics in Hydraulic Structural Engineering,Wuhan University,Ministry of Education,Wuhan 430072,Chinaa r t i c l e i n f o Article history:Received 8December 2010Received in revised form 18May 2011Accepted 19May 2011Available online 23June 2011Keywords:Explosion mechanics Blasting vibrationEquivalent simulationDynamic ﬁnite element method Blasting loada b s t r a c tDue to the complicated blasting load,the diversiﬁed medium models and various constitu-tive relations of the rock mass,and a huge job for simulating blasting of multiple holes,it is very difﬁcult and costly to simulate the blasting vibration accurately in numerical compu-tation.This paper presents an equivalent simulation method so as to transform this com-plex dynamic problem into an approximate initial-boundary problem.The equivalent elastic boundary applied by the blasting load was developed for multiple holes according to the spatial distribution of rock damage around each blasthole.The equivalent mechanics process of the complex blasting load was performed through analysis of the expansion of the borehole volume,the growth of cracks,the movement of stemming and the outburst of detonation gases.In combination with the blasting excavation of the tailrace tunnel in the Pubugou Hydropower Station,particle vibration velocities in the surrounding rock at dif-ferent distances from the explosion source were simulated by applying this equivalent method based on the dynamic ﬁnite element method.The comparison with ﬁeld monitor-ing data indicates that this equivalent simulation method is applicable to predicting the far-ﬁeld dynamic response of the ground subjected to blasting load,and the selection of rock mass properties near the equivalent elastic boundary has a signiﬁcant impact on sim-ulation results.Ó2011Elsevier B.V.All rights reserved.1.IntroductionBecause of better adaptability to different geological conditions,drill and blast is still a common excavation technique for rock foundations and underground caverns in the ﬁelds of hydropower,transportation and mining.The prediction and con-trol of blasting vibration has long been considered to be an important issue in the blasting design and construction,however,thumb-rule formulas derived by different authors according to their own data obtained from different sites are still the most commonly used method to predict the propagation laws of blasting vibration [1–3].Recently,some new control theories,such as the neural network technology,have also been used in this problem [4].However,these methods depend on a large number of test data and cannot take into account the inﬂuence of vibration frequency and duration [5],so they have some limitations in practice.In recent years,with the rapid development of explosion theory and computer technology,numerical simulation has be-come a promising approach to studying blast and wave propagation.Researchers worldwide tried to investigate the stress wave propagation in the rock mass with different numerical methods and gained a series of achievements.Ma et al.[6],Wu1569-190X/$-see front matter Ó2011Elsevier B.V.All rights reserved.⇑Corresponding author at:State Key Laboratory of Water Resources and Hydropower Engineering Science,Wuhan University,Wuhan 430072,China.Tel.:+8602768772221,fax:+8602768772310.E-mail address:whuyjh@ (J.Yang).W.Lu et al./Simulation Modelling Practice and Theory19(2011)2050–20622051 et al.[7],Toraño et al.[8]and Saharan and Mitri[9],using theﬁnite element method(FEM),predicted blast waves from bench blasting or underground explosion,evaluated and quantiﬁed some factors affecting the blasting vibration to simulate complex waves accurately enough in real blasts.Chen and Zhao[10]applied the discrete element method(DEM)for the study of blast wave propagation in jointed rock masses with the universal distinct element code(UDEC).However,propa-gation of explosive detonation,possible phase transition of rock mass induced by the shock wave,breakage of rock mass, piling of fragments,andﬁnally excitation of seismic waves,all of these processes occur instantaneously and are accompanied by complicated physical and mechanical action procedure.It is still very difﬁcult to simulate blasting vibration precisely dur-ing rock fragmentation by blast.First of all,after the initiation of explosive at the blasthole bottom,the detonation wave rapidly transits to the oriﬁce along the axis and the whole blasthole isﬁlled with detonation gases under a very high pressure and temperature.Under blasting load the blasthole volume expands dynamically in radial direction,fractures occur and further grow driven by explosive gases,the stemming moves outward,and gases escape rapidly from the oriﬁce and gaps,leading to an attenua-tion of the blasting load.This complex mechanical process is hard to describe by using current numerical simulation meth-ods;an attempt to model the pressure variation applied to the blasthole wall for a typical bench blast was made by Mortazavi and Katsabanis[11]employing discontinuous deformation analysis(DDA),but their study is limited to the expansion of blast chamber in connection with pressure decay.Thus the semi-empirical formula as a direct input of dy-namic pressure[12–15]and the Jones–Wilkens–Lee(JWL)equation of state for detonation products[6,7,10]are normally used to simulate the operating load due to their simple forms.Whereas,empirical formulas including the decay function and the triangular load function require assumptions of some parameters[9],and fail to fully consider blasting parameters and the quasi-static load by detonation gases;the JWL equation can only accurately describe the relationship between the load attenuation and the expansion of the blast chamber under an effective stemming without taking the outburst of gases into account.Some authors,e.g.Ding and Zheng[16],demonstrated that the impulse,the form of the time function,is more important in some cases,but the methods mentioned above cannot get close to the real physical characteristics of the blasting load.Secondly,in the rock mass blasting,the shock wave and detonation gases pressurize on the blasthole wall immediately, and induce the crushed zone,the fractured zone and the elastic seismic zone of surrounding rock mass depending on the distance away from the blasthole.The pressure on the blasthole wall is much higher than the rock compressive strength, and thus,the rock mass in the crushed zone behaves similarly to aﬂuid.In the fractured zone,the stress wave and the per-meation of gases at high pressure encourage the development and growth of radial cracks,while tangential cracks come about as the stress releases in this zone.In terms of constitutive relation,the rock mass in this zone can be considered an elasto-plastic body.In the elastic seismic zone the stress wave has attenuated to a seismic wave,where no direct damage happens to the rock mass.It is clear that different medium models are required in simulation because during the whole blast process the rock mass near the blasthole is broken into discontinuum andﬁnally into fragments from continuum,yet in the far area can still be regarded as a continuum.Furthermore,different constitutive equations need to be adopted as the rock mass around charge are respectively hydro-plastic,elasto-plastic and elastic medium in accordance with its stress state at different distance.Moreover,in the elasto-plastic constitutive relation it is very difﬁcult to judge the loading and unloading process.Considerable efforts have been directed towards developing a continuum–discontinuum model of the rock mass. Chen and Zhao[10],using a coupled method of UDEC(discontinuum-based approach)and AUTODYN(continuum-based ap-proach),modeled aﬁeld explosion test and investigated the response of the jointed rock mass.Elasto-plastic model such as Mohr–Coulomb criterion and Drucker–Prager criterion,both isotropic and anisotropic damage models were used in the past to simulate the behavior of rock mass subjected to blasting load[7].Medium models and constitutive equations near the charge correlate well with the propagation characteristics of the blasting seismic wave,but unfortunately,few of the previ-ous studies have taken a global view of different constitutive equations in the blasting vibration simulation.Finally,in practice it is a common occurrence to detonate an array of blastholes in the same delay simultaneously,while the dimension of blasthole is much smaller than that of engineering rock mass.This presents a considerable challenge when developing the model as to how to mesh and how to deal with the contact of explosive and rock mass.Previous simulation studies of blasting vibration were always limited to a single blasthole or chamber,and cannot be directly applied to actual engineering applications.As indicated above,due to the complicated blasting load,the diversiﬁed medium models and various constitutive rela-tions of the rock mass,and the difﬁculty of simulating the blasting of multiple holes,it is uneconomical,even impossible to accurately simulate the whole rock fragmentation by blast from the explosive detonation to the propagation of seismic waves.Seeking a simple and practical equivalent method is necessary for blasting vibration simulation in engineering appli-cation.In this regard,Toraño et al.[8]suggested that the pressure pulse lasting for a few milliseconds can be applied against the bench face in bench blasting;Ding and Zheng[16]and Gibson et al.[17]proposed an equivalent source model for the blasting vibrations of fragmentation blasting.Inspired by these works,this study focuses on an equivalent boundary for mul-tiple hole blasting to which the blasting load is applied as an initial and boundary condition.In addition,we will also discuss in theory the equivalent process of the complicated blasting load.As an actual case study of applying this equivalent sim-ulation method,in combination with the No.1tailrace tunnel blasting excavation in the Pubugou Hydropower Station,par-ticle vibration velocities of the surrounding rock at different distances from the explosion source are simulated by employing the dynamicﬁnite element software ANSYS/LS-DYNA.Results are compared withﬁeld monitoring data.2.Equivalent boundary of blasting load applicationIt is well known that the far-ﬁeld vibration induced by blasting excavation results from the propagation of elastic seismic waves,to which the elastic model is appropriate.For the purpose of adopting a uniﬁed constitutive relation based on con-tinuum mechanics to simulate the blasting vibration,the crushed zone and the fractured zone are treated as parts of the blasting vibration source,and the blasting load is applied to the boundary of equivalent vibration source which is called the equivalent elastic boundary in the following text.2.1.Determination of equivalent elastic boundaryConsidering rock mass surrounding the charge to be linearly elastic and incompressible,and assuming the explosive en-ergy to be fully converted into kinetic energy of the rock mass,the behavior of surrounding rock mass for the cylindrical charge in a half space is described by Neyman and Hustrulid [18].In this case,the radial velocity V is given by Eq.(1)accord-ing to the hydrodynamic theory.In the actual rock fragmentation process,the boundary of fractured zone can be obtained by establishing the relationship between critical velocity V cr and destruction.V ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃq 0q q r v sr Lr ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃð r Þ2þL 22r ð1Þwithv s¼ln þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1þ2p ÀL þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1þL 2p ;L ¼L 12r 0;and r ¼r2r 0:where q 0and q are respectively the explosive density and the heat of explosion,q r is the rock mass density,L 1is the chargelength,r 0is the blasthole radius,and r is the distance from the blasthole center.Studies by Xa y y r ae d [19]show that when blasting a single hole in a semi-inﬁnite rock medium and a known blasting load,the crushed zone radius r 1and fractured zone radius r 2take the form:r 1¼q r c 2pr c!1Pr Ã1r 0ð2Þr 2¼l Pð1Àl Þr t1a r 0ð3Þwhere l is the Poisson ratio of the rock mass,c p is the P-wave velocity of the rock mass,r c and r t are respectively the uncon-ﬁned compressive strength and tensile strength of the rock,r ⁄is compressive strength of the rock under multiaxial stress,P is the radial blasting load,and a is the attenuation exponent of the stress wave.Research and experiments indicate that the radius of crushed zone is about 3–5times the blasthole radius,and the frac-tured zone is around 10–15times the blasthole radius [18–20].If the interaction between holes is ignored,the blast of each hole in the initial cut section of underground blasting is similar to an explosion in a half space.So for multiple cut hole blasts,the equivalent elastic boundary is the envelope of the formed fractured zones of these blastholes.Blasting of stopping holes,buffer holes and contour holes is accomplished through cracks interconnecting between adjacent holes in the same delay under the condition of an existing free surface,which is provided by the blast in the previous delay.So for these blastholes,the equivalent elastic boundary is a newly-formed free surface.Equivalent elastic boundaries shown diagrammatically in Fig.1can be plotted.2.2.Blasting load on equivalent elastic boundaryDeﬁning the blasting load as P (x ,t )applied to any section of blasthole wall at any time,the attenuation law with distance for the shock wave and stress wave around a single blasthole has the following mathematical expression [21]:P a ðx ;t Þ¼P ðx ;t Þr 0ra ð4Þwhere P a (x ,t )is the attenuated blasting load with distance;a is the attenuation exponent,for the shock wave a =2+l /(1Àl ),and for the stress wave a =2Àl /(1Àl ).Thus,when blasting multiple holes in an initial cut for underground blasting,the blasting load on the equivalent elastic boundary is given by P e (x ,t ):P e ðx ;t Þ¼kP ðx ;t Þr 0r 1 2þl 1Àl r 1r 22Àl 1Àl ð5Þ2052W.Lu et al./Simulation Modelling Practice and Theory 19(2011)2050–2062For multiple stopping holes,buffer holes and contour holes,the blasting load is exerted on the area which consists of the connecting line of borehole centers and the borehole axis in the same delay.It meets the Saint–Venant’s principle,so calcu-lated far-ﬁeld dynamic response is consistent with the condition that the blasting load is directly on blasthole walls.The load P e (x ,t )on the equivalent elastic boundary is shown in Eq.(6):P e ðx ;t Þ¼2r 0r 3P ðx ;t Þð6Þwhere r 3is the distance between two adjacent blastholes.2.3.Selection of rock mass parametersCorresponding dynamic parameters of rock mass should be adopted in dynamic calculations.Note that although the pres-sure of the stress wave has attenuated too much small to cause crushing or plastic deformation of rock mass when it is prop-agated to the elastic boundary,an excavation damage zone (EDZ)would be induced in the surrounding rock mass when an underground opening is excavated with the method of drill and blast.The Underground Research Laboratory (URL)of Atomic Energy of Canada subdivided the EDZ into an inner and outer zone.Particularly the inner damage zone,close to the excava-tion perimeter,results more from blasting than from stress redistribution,and the outer damage zone,in the outer portion,is attributed to the effects of stress redistribution alone.Martino and Chandler [22]at the URL measured ultrasonic wave veloc-ities in the damage zone and found that the micro-cracks mainly extended to 0.5m close to the tunnel wall,but did persist to a depth of at least 1.0m into the rock mass.Therefore,rock mass around the excavation perimeter including the formed tun-nel wall and the forming equivalent elastic boundary,would deteriorate and be different from virgin rock mass.Rock mass properties near the tunnel wall and the equivalent elastic boundary will be reduced in numerical simulation.3.Equivalent process of blasting loadThe lasting load variation on a blasthole wall is a complex mechanical process.With the propagation of the detonation wave,the gas pressure in a hole rises to a maximum.It drives the expansion of borehole volume,the growth of cracks,the movement of stemming,which in turn cause the expansion of high-pressure gases and initially reduce the blasting load.After the ejection out of stemming and the interconnecting of cracks,gases escape out quickly,producing a bunch of rare-faction waves.The propagation of rarefaction waves from the blasthole oriﬁce to the bottom results in a gradual diminution of the gas pressure.According to the wave theory,when the previous waves arrive at the solid end of bottom,rarefaction wave reﬂections would emerge and spread to the oriﬁce,leading to further decline of the pressure.3.1.Peak blasting loadAccording to the Chapman–Jouguet model for the detonation wave in a condensed explosive,the parameters at the det-onation front are guided by the widely known equation [21]:P D ¼1c þ1q 0D 2c D ¼cD8>><>>:ð7ÞW.Lu et al./Simulation Modelling Practice and Theory 19(2011)2050–20622053where P D and c D are the detonation pressure and the acoustic velocity respectively,D is the detonation velocity and c is the ratio of the speciﬁc heats for the detonation gases.The initial explosion pressure P0which denotes the gas pressure applied to the blasthole wall just after detonation is approximately half of this detonation pressure for coupled charges.P0¼qD2cð8ÞFor decoupled charges,if the decoupling is smaller,the initial explosion pressure P0involved is:P0¼qD22ðcþ1Þab2cð9Þwhere a is the charge diameter and b is the blasthole diameter.If the decoupling is larger,the explosion pressure decreases to below P k from above it,where P k is the critical pressure of explosion gases.The value of c is3.0when P P P k,while c is m=4/3when P<P k.So Eq.(9)can be rearranged to yield the following expression for the explosion pressure:P0¼qD22ðcþ1Þ"#m cPcÀmckab2mð10Þ3.2.Blasting load variation with time on the blasthole wallThe variation with time of the pressure applied to the blasthole wall can be reduced to three procedures in light of the previous analysis,the rise of load,the initial expansion of gases,and the outburst of gases.When the detonation wave spreads to the involved blasthole cross-section from the bottom,blasting load at this section rises to a maximum.In this case,the rising time t r is:t r¼LÀxDð11Þwhere L is the blasthole length,and x is the location of any cross-section in the coordinate axis with origin being at the oriﬁce of blasthole as plotted in Fig.2.Lu et al.[23]studied the initial expansion of gases in detail.The increment of gas volume D V(t)at a time of t is available under a given initial-boundary condition of stress.D VðtÞ¼2p rðtÞuðtÞLdtþ2L Z L axðfÞd fþ1p rðtÞ2yðtÞð12Þwhere r(t)is the hole radius at the time of t,u(t)is the expansion velocity of the hole wall,x(f)is the crack width,L a is the crack length,and y(t)is the displacement of stemming.2054W.Lu et al./Simulation Modelling Practice and Theory19(2011)2050–2062The pressure P1(t)in the blasthole after the initial expansion can be obtained through the widely used Ideal Gas Law equation:P1ðtÞ¼V0V0þVðtÞcP0ð13Þwhere V0is the initial blasthole volume.Assuming that the expansion occurs adiabatically in this moment,the isentropic equation can give the wave velocity after the initial expansion:c1¼P1P DcÀ12cc Dð14ÞBecause of the slow movement of gases before outburst,the value of gas velocity is supposed to be v1=0.The outburst process can be generalized into the following model:a pipeline whose face(x=0)is a membrane and end(x=L)is a solid wall isﬁlled with gases at high temperature and under a high pressure,with P=P1(gas pressure after initial expansion), c=c1and v1=0.At the instant removal of the membrane,gases escape out at the speed of acoustic velocity on the free face, generating a bunch of rarefaction waves.A combination of the incident waves propagating to the end and reﬂected waves to the face forms three wave zones:undisturbed wave zone(1),simple wave zone(2)and multiple wave zone(3),as can be seen in Fig.2.This problem can be solved by the wave equation of one-dimensional unsteady isentropicﬂow(Eq.(15))[24].@q @t þv@q@xþq@v@x¼0@v @t þv@v@xþc2q@q@x¼08>><>>:ð15Þwhere q is the gas density,v is the gas velocity,and c is the wave velocity.The undisturbed wave zone(1):c¼c1ð16ÞThe simple wave zone(2):c¼cÀ1cxþ2c c1ð17ÞThe multiple wave zone(3):x¼v tÀLðnÀ2Þ!@nÀ2@h nÀ2ﬃﬃﬃhpÀv2Àh0nÀ2ﬃﬃﬃhpﬃﬃﬃhpÀv9>>>=>>>;8>>><>>>:t¼L4c@nÀ1@hﬃﬃﬃhpÀv2Àh0nÀ1ﬃﬃﬃhp9>>>=>>;8>>><>>>:8>>>>>>>>>>>>><>>>>>>>>>>>>>:ð18Þwith n¼cþ12ðcÀ1Þand h¼2c cÀ12.One would determine the wave velocity c at any time t and any section x using Eqs.(16)–(18).Substitution of the value of c into isentropic Eq.(19)gives the blasting load in the outburst process of detonation gases.The whole blasting load variation with time P(x,t)applied to blasthole wall is obtained from Eqs.(7)–(19).P2ðx;tÞ¼cc12c cÀ1P1ð19Þ4.Practical use of the equivalent method4.1.Project background andﬁeld testThe Pubugou Hydropower Station is located in the western part of Sichuan Province in China and in the midstream of Tatu River,a branch of Yangtze River.Its total installed capacity is3300MW.This project includes two parallel tailrace tunnelsW.Lu et al./Simulation Modelling Practice and Theory19(2011)2050–20622055wave velocity value of 4500m/s.The maximum principal in situ stress is horizontal and perpendicular to the longitudinal axis of the tailrace tunnel and the intermediate one approaches the vertical direction,and both are 20MPa.We monitored blasting vibration by using TOPBOX self-recording instrument (developed and manufactured by Top Dig-ital Equipment Company Limited of Sichuan)during the tailrace tunnel excavation in this project.The tailrace tunnel is exca-vated in three layers,each of which has a height of about 8.0m.In the upper layer,using No.2rock emulsion explosive,an 8.0m wide middle pilot tunnel with 8.0m in height is excavated ahead of the two sides,with a designed cycle footage of 3.0m and a drilling depth of 3.5m.Fig.3shows the general situation of the blasting area and the arrangement of measure-ment points for the blast of upper middle pilot tunnel in No.1tailrace tunnel.Vibration sensors are arranged on the ﬂoor of the tunnel being excavated.For simplicity,here only vibration induced by the blast of holes in the initial cut is calculated during the upper middle pilot tunnel excavation.The detailed blasting design is shown in Fig.4and the blasting parameters of the holes in the initial cut section are listed in Table 1.4.2.Numerical model and calculation conditionsIn this study,the model is developed by using commercial dynamic FEM software ANSYS/LS-DYNA.It is an underground block measuring 120m long by 100m wide by 100m high,where a 21.2m wide tunnel is excavated in its middle area,as shown in Fig.5.To reduce the inﬂuence of possible reﬂected waves under the dynamic load,the model limits are set to trans-n n e lTunnel face 6.015.53 5.525.0No.1 tailrace tunnel No.2 tailrace tunnelBlasting area2056W.Lu et al./Simulation Modelling Practice and Theory 19(2011)2050–2062calculations.Fig.5.Dynamicﬁnite element model used for themitting boundaries.The following parameters are considered in the calculations:the crushed zone radius r1=3r0and the fractured zone radius r2=10r0,while the equivalent elastic boundary for blasting multiple initial cut holes is available from Section2.1,as the rectangle in the model center shows.In FEM simulation,the block is transformed into a mesh of hexahedron-shapedﬁnite elements.The SOLID164element included in the software is adopted for the current study,which is an8-node brick element.Mesh size in the computational model has a great inﬂuence on numerical modelling accuracy.Kuhlemeyer and Lysmer[25]suggested the mesh size should be shorter than1/8–1/10of the wave length.This requires a maximum element size of3.0m given the P-wave velocity of 4395.6m/s and the main frequency of100$150Hz byﬁeld monitoring.The element sizes vary from0.2m on the tunnel wall to2.5m in the border,where the element size is limited to a maximum of0.5m near pressuring point,thus obtaining theinexistence of wave distortions.The model mesh has527080elements and548820nodes.The time step size correspondsTable2Rock properties in different zones of the model.Special zones Virgin rock mass Tunnel wall perimeter Equivalent elastic boundary perimeterInner EDZ Outer EDZ Inner EDZ Outer EDZ Dynamic elastic modulus E r(MPa)47,20037,80042,50023,60033,000Density q r(kg/m3)270027002700270027002058W.Lu et al./Simulation Modelling Practice and Theory19(2011)2050–2062roughly to the transient time of an acoustic wave through an element using the shortest characteristic distance.According to the shortest characteristic distance of0.2m and P-wave velocity of4395.6m/s,the critical time step is calculated as0.04ms.W.Lu et al./Simulation Modelling Practice and Theory19(2011)2050–20622059 Therefore the selected time step of0.02ms is enough in the present simulation.For these geometrical parameters and thetime step of this model,about100min is required to solve the dynamic response with a duration of80ms.Because of the unsteadyﬂow of detonation gases along blasthole during outburst process,the gas pressures acted on thehole wall at different depth are nonuniform.At different depths of the equivalent elastic boundary of three-dimensionalnumerical model,different pressure variations with time should be exerted.Thus at intervals of0.75m along borehole axis,ﬁve pressure variation curves for different sections are computed and plotted in Fig.6.For the case of decoupled charge,substituting these values listed in Table1into Eqs.(7)and(9)gives parameters of thedetonation front P D=3240MPa,c D=2700m/s and the explosion pressure applied to the blasthole wall P0=ingEq.(11),the rising time of pressure for different cross-sections ranges from0to0.7ms.Detonation gases at different sectionsare assumed to expand initially in equal amounts after the detonation propagation,so the pressure curves overlap at thisprocess.Solving Eqs.(12)and(13)gives the gas pressure P1(t)and P1=358MPa at the termination of1.54ms of quasi-staticexpansion.The wave velocity c1is determined by substituting the known P1,P D and C D into Eq.(14),in this casec1=1295.6m/s.Substitution the values of P1,c1and v1=0into Eqs.(16)–(18)yields the rarefaction wave velocity c at any time t and any section x through computer program,further yields pressure P2(t)from Eq.(19).Before the rarefactionwaves arrive at the calculated section,the blasting loads remain a constant for a period of time.Since the rarefaction wavevelocity c is a dependent variable on the section location x,theseﬁve pressure curves vary differently during the outburst ofdetonation gases.After obtaining the pressure variations P(x,t)applied to the blasthole wall,the blasting loads on the equiv-alent elastic boundary are obtained by substitution of k=10,P(x,t),r1=3r0,r2=10r0and l=0.23into Eq.(5).The peak blast-ing loads P e0are56MPa and the variations are consistent with the pressures on the blasthole wall,as plotted in Fig.6.。

Numerical modelling of crack propagation automatic remeshing and comparison of different criteria

Numerical modelling of crack propagation:automaticremeshing and comparison of diﬀerent criteriaP.O.Bouchard *,F.Bay,Y.ChastelCentre de Mise en Forme des Mat eria ux,Ecole des Mines de Pa ris,UMR 7635,B.P.207,06904Sophia -Antipolis Cedex,Fra nce Received 13December 2002;received in revised form 18April 2003;accepted 23May 2003AbstractModelling of a crack propagating through a ﬁnite element mesh under mixed mode conditions is of prime importance in fracture mechanics.Three diﬀerent crack growth criteria and the respective crack paths prediction for several test cases are compared.The maximal circumferential stress criterion,the strain energy density fracture criterion and the criterion of the maximal strain energy release rate are implemented using advanced ﬁnite element techniques.A fully automatic remesher enables to deal with multiple boundaries and multiple materials.The propagation of the crack is calculated with both remeshing and nodal relaxation.Several examples are presented to check for the robustness of the numerical techniques,and to study speciﬁc features of each criterion.Ó2003Elsevier B.V.All rights reserved.Keywords:Finite element method ;Crack propagation criteria;Automatic remeshing;Strain energy release rate;G h method;Strain energy density1.IntroductionSince the ﬁrst studies in the early 1920s by Inglis [1],Griﬃth [2]and Irwin [3],the research in linear elastic fracture mechanics has led to the development of a vast number of theories and applications.The prop-agation of a crack in a part leads to an important displacement discontinuity.The more accurate way of modelling such a discontinuity in a ﬁnite element mesh is to modify the part topology (due to the crack propagation)and to perform an automatic remeshing.However,several methods have been proposed in the literature to model this discontinuity without any remeshing.Belytschko has suggested a meshless method (Element free Galerkin method [4])where the discretisation is achieved by a model which consists of nodes and a description of the surfaces of the model.More recently,two interesting ﬁnite element techniques have been presented to deal with such discontinuities without any remeshing stage.The Strong Discontinuity Approach (SDA)[5–7]in which displacement jumps due to the presence of the crack are *Corresponding author.Tel.:+33-4-93-957432;fax:+33-4-92-389752.E-mail addresses:pierre-olivier.bouchard@ensmp.fr,bouchard@cemef.cma.fr (P.O.Bouchard).0045-7825/$-see front matter Ó2003Elsevier B.V.All rights reserved.doi:10.1016/S0045-7825(03)00391-8Comput.Methods Appl.Mech.Engrg.192(2003)3887–3908/locate/cma3888P.O.Bouchard et al./Comput.Methods Appl.Mech.Engrg.192(2003)3887–3908embedded locally in each crackedﬁnite element without aﬀecting neighbouring elements.The amplitude of displacement jumps across the crack are deﬁned using additional degrees of freedom related to the plastic multiplier,leading to a formulation similar to standard non-associative plasticity models.The Ex-tended Finite Element Method(XFEM)[8–10]in which the displacement-based approximation is enriched near a crack by incorporating both discontinuousﬁelds and the near tip asymptoticﬁelds through a Partition of Unity method[11].However,these recent techniques still have to be improved in order to deal with complex conﬁgurations such as multiple cracks,large deformation crack propagation,etc.When remeshing is possible,a real mesh discontinuity representing the crack seems to be more accurate.This technique is used,in particular,in the2D and3D fracture numerical code FRANC[12,13].However, whatever the technique used,the accuracy of the crack path directly depends on the accuracy of kinking criteria.In a recent paper,Bouchard et al.[14]have introduced an interesting remeshing technique to model crack propagation accurately using the discrete crack approach.This technique is used here for three diﬀerent criteria.First the numerical tools used to propagate a crack through a mesh are described.The theoretical aspects and the numerical diﬃculties when dealing with the three criteria are then studied.In the third part several examples are presented to compare each criterion and to show the robustness of the numerical schemes.Finally a discussion about advantages and disadvantages of the three criteria is presented and examples of propagation in multimaterial structures are investigated.2.Numerical tools for crack propagationThe technique presented in[14]has been implemented in aﬁnite element code based on a2D model for viscoplastic or elastic–viscoplastic behaviour,FORGE2[15].It has been developed for plane strain or axisymmetric cases in large strain.The mesher and automatic remesher of FORGE2is based on P2/P0 triangles––quadratic velocity and constant pressure for each element––and allows to deal with multiple edges and multiple materials.When a crack propagates through a mesh,the accuracy at the crack tip is of prime importance.In FORGE2,a nodal relaxation is combined with a remeshing technique.This enables to avoid the problem of distortion at the crack tip and to continue with a new,undistorted and well suited mesh.The transfer of data from the old to the new mesh at each remeshing stage is performed so as to preserve theﬁelds of the various state variables.This transfer is a closeness-based interpolation technique which respects the plastic criterion.Fracture may be decomposed into two steps:the crack initiation and the crack propagation.The stage of crack initiation is crucial but quite problematic.General fracture mechanics codes avoid this problem because their aim is to study the evolution of a pre-existing crack.Damage-based numerical models are more adapted to this problem because they study the evolution of damage continuously and a crack is initiated for a critical damage value.However these codes are often unable to model the crack propagation without a local collapse criterion.In fact,it is very diﬃcult to determine the location of a new crack.Micro-failure and inclusions always induce local stress concentrations which are at the origin of failure and cracks.As all these defects cannot be taken into account numerically(unless if we use statistical approaches),either the material is assumed to be perfect or homogeneised,or the initiation location is imposed by positioning a pre-crack.Moreover,the initiation of a crack in a mesh induces a severe topological change which is rarely supported by numerical codes.Many initiation criteria have been proposed in the literature.They often depend on the materials studied. Some of them are based on critical values for mechanical state variables,stress or strain.An other pos-sibility is to use a damage law and a critical damage parameter to localize the crack initiation.In this paper,the selected crack initiation criterion is based on a critical stress value,which is a char-acteristic of the material.So,at each time step,the maximum principal stress value is computed and its position determined on the boundary.Once this value reaches the critical one,a crack is initiated in two steps:•an internal outline is added to the geometrical deﬁnition of the domain in order to model the crack ini-tiation;•the automatic remeshing procedure is carried out on the new domain,and the nodes of the new outlineare split to make the opening of the crack possible (see Fig.1).Moreover,many numerical tools have been developed to improve the accuracy at the crack tip.A concentric mesh around the crack tip may be coupled with singular elements [16]to model the stress ﬁeld singularity (Fig.2a).These elements are studied in detail in [14].A ring of elements (Fig.2b)can also be introduced in order to compute the strain energy release rate [17,18].Finally mesh reﬁnement around the crack tip enables to keep a good precision in the vicinity of the crack.As the crack tip moves along,the areas which need to be reﬁned will change;a new mesh is created and reﬁned only in the areas where it is needed in order to optimise calculation time (Fig.2c).More details about the mesh structure at the crack tip may be found in [14].3.Crack growth criteriaWhen a crack is initiated,one needs to check,at each time step,if the crack is going to propagate (crack propagation criteria),and in which direction (crack kinkingcriteria).Fig.1.Crackinitiation.Fig.2.(a)Concentric mesh with singular elements,(b)ring of elements and (c)evolutionary mesh reﬁnement at the tip of the crack,and unreﬁnement elsewhere.P.O.Bouchard et al./Comput.Methods Appl.Mech.Engrg.192(2003)3887–39083889Stress intensity factors––strength singularity at the crack tip––are often used for crack propagation.In mode I loading,they are compared with a critical value K Ic of the material.For example,in the maximum circumferential stress criterion proposed by Erdogan [19],the crack propagates when ﬃﬃﬃﬃﬃﬃﬃ2p r p r hh ¼K Ic ,where r hh is the circumferential stress,and r represents the distance to the crack tip.Other criteria,based on energetic parameters––strain energy release rate [2],strain energy density [20]––are also available.For elastic–plastic materials,the crack tip is blunted by plasticity.Then the crack tip opening dis-placement (CTOD)introduced by A.A.Wells in 1961,may be used as a material crack propagation para-meter.Then,determination of the crack direction may be obtained by kinking criteria:•Some criteria are based on the local ﬁelds at the crack tip following a local approach ,e.g.the maximumcircumferential stress criterion introduced by Erdogan and Sih [19],or the maximum strain criterion [21].•Other criteria are based on the energy distribution throughout the cracked part,following a global ap-proach ,e.g.the maximal strain energy release rate criterion [22].•Some authors have also proposed to determine the crack extension using a micro-void continuum dam-age model [23,24].These theories are based on the assumption that the void initialisation and the void growth control the crack growth direction [25].When selecting one of these criteria,one has to consider two main points:•Does the criterion comply with the physical characteristics of the considered material?•Can the criterion be implemented so as to be calculated accurately?More often,the accuracy of the direction computation is directly linked to the accuracy of the numerical computation of local or energetic parameters.In the following,we brieﬂy present the maximum circumferential stress criterion,the minimum strain energy density fracture criterion and the maximum strain energy release rate criterion.For each of them,the inﬂuence of numerical parameters on the results is pointed out.3.1.Maximum circumferential stress criterion (MCSC)This criterion,introduced by Erdogan [19]for elastic materials,states that the crack propagates in the direction for which the circumferential stress is maximum.It is a local approach since the direction of crack growth is directly determined by the local stress ﬁeld along a small circle of radius r centered at the crack tip.The kinking angle of the propagating crack is computed by solving the following system:K I sin ðh ÞþK II ð3cos ðh ÞÀ1Þ¼0with K II sin ðh =2Þ<0;h 2 Àp ;p ½;K I >0;8<:ð1Þwhere K I and K II are respectively the stress intensity factors corresponding to mode I and mode II loading,and h is the kinking angle.This criterion also proves the existence of a limit angle corresponding to pure shear:h 0¼Æ70:54°.This criterion is widely used in the literature,but numerical details concerning its implementation are rarely mentioned.When stress intensity factors are not computed by the ﬁnite element software,the computation of the kinking angle has to be based on the circumferential stress r hh at each integration point at the crack tip (Fig.3).The crack propagation is then performed toward the integration point that3890P.O.Bouchard et al./Comput.Methods Appl.Mech.Engrg.192(2003)3887–3908maximises r hh .However a direct application of this method would be mesh-dependent since the crack direction would then directly depend on the number of elements at the crack tip (Fig.4a)––the crack di-rection being provided by the integration point location with respect to the crack tip.We propose here an alternative method based on the maximal principal stress evaluated at each inte-gration point nearest to the crack tip (Fig.3):•The integration points nearest to the crack tip are identiﬁed.•For each of them,eigenvalues and eigenvectors of the stress tensor are computed.•These eigenvalues provide the principal stresses and enable to ﬁnd the direction of propagation for eachintegration point.•The ﬁnal direction of the crack propagation is obtained by a weighted average of each direction withrespect to the distance between the integration point and the crack tip.This leads to a mesh indepen-dence at the crack tip since the kinking direction is not directly associated to a speciﬁc integration point at the crack tip.This way of computing the crack propagation direction may be questionable since the direction is merely determined using the maximal principal stress.The direction of the maximal principal stress at the crack tip corresponds to the direction of the maximal tensile stress.The approximation performed here is based on the fact that a crack propagates perpendicularly to the maximum tensile stress.Moreover,the advantage of such a technique is to improve the mesh-independence since the direction of crack propagation is not di-rectly associated to the position of an integration point (Fig.4b).These two diﬀerent implementations of the same criterion can thus lead to diﬀerent crack paths,even for simple applications.Fig.4shows the crack path in a part containing two symmetric holes during a tensiletest.Fig.3.Direction of propagation with the maximum circumferential stresscriterion.Fig.4.Two diﬀerent implementations of the MCSC leading to two diﬀerent crack paths.P.O.Bouchard et al./Comput.Methods Appl.Mech.Engrg.192(2003)3887–39083891Details about diﬀerent ways of implementing the maximum circumferential stress criterion can be found in [26].The maximum circumferential stress criterion is one of the most used criteria,because it is easy to im-plement.However it may be questionable because the stress ﬁeld in the immediate vicinity of the crack tip is only approximated.Moreover the existence of a non-elastic zone at the crack tip undoubtedly modiﬁes the stress distribution.That is why energy-based criteria,based on global values of the structure,may be more adequate.Their implementation is generally more complex,but they lead to more accurate results since the crack propagation direction is computed further away from the crack tip.3.2.Minimum strain energy density criterion (MSEDC)Sih [20]considers that high values of strain energy,W e ,tend to prevent crack growth.Then the crack grows in the direction that minimises this energy.Let w be the strain energy density:w ¼d W e ÀÁ,where W e is the sum of the volumetric part of the strain energy W v and the distortion energy W d .This quantity is proportional to the square of stress,and since the stress has a 1=ﬃﬃr p singularity at the crack tip (where r represents the distance to the crack tip),the strain energy density w has a 1=r singularity.Therefore the so-called strain energy density factor,S ¼rw ,remains bounded.Parameter S can be computed using two diﬀerent techniques:•An analytical formulation .The strain energy density is inversely proportional to the distance r to thecrack tip [17].Then S represents the intensity of the local energy ﬁeld:S ¼rw ¼r 1þm 2E r 211 þr 222þr 233Àm 1þmðr 11þr 22þr 33Þ2þ2r 212!:•A numerical formulation .In the simulation,the strain power P e is computed at each time step and foreach integration point.A time integration of P e gives the strain energy for each integration point of the mesh:W e ¼R t 0P e d t .In the numerical formulation,parameter S is computed by introducing a ring of elements around the crack tip (Fig.2b).The curve S ðh Þis plotted for each element of this ring as a function of the angle between the centre of the element and the crack axis (Fig.5).The kinking angle h 0is the angle corresponding to the local minimum of the curve S ðh Þ:o S o h h ¼h 0¼0;o 2S o h 2 h ¼h 0P 0:8>>><>>>:3892P.O.Bouchard et al./Comput.Methods Appl.Mech.Engrg.192(2003)3887–3908One should stress that it is important to compute the local minimum of the curve,and not the global minimum which can lead to results corresponding to angles which do not belong to ½À70:54°;70:54° ,and therefore are not relevant.Moreover,the computation accuracy is quantitatively linked to the number of elements in the ring around the crack tip.At last,this criterion may not exactly be considered as a global criterion since the computation of the strain energy density is based on local mechanical parameters.3.3.Maximum strain energy release rate criterion (MSERRC)The strain energy release rate G represents the energy required to increase the crack length by one unity.The criterion states that among all virtual and kinematically admissible crack length displacements,the real increase is the one which maximises the strain energy release rate.The kinking angle is then deter-mined byd G d h h ¼h 0¼0;d 2G d h 2 h ¼h 060:8>>><>>>:Numerous numerical techniques can be used to compute G .The most commonly used methods are brieﬂy described in the following.3.3.1.Real crack extensionBased on the deﬁnition of G ,this method consists in computing the total potential energy W p for the initial crack length a ,and for the increased crack length (a þd a ).As the strain energy release rate is the decrease of the total potential energy W p ,G can be approximated byG ¼Ào W p o A%ÀW p ða þd a ÞÀW p ða Þb d a ;where o A is the surface increment corresponding to the crack increase d a ,and b represents the thickness in the out-of-plane direction.This technique,based on the physical deﬁnition of G ,is easy to implement.However it is expensive in terms of computational time since it requires a signiﬁcant reﬁnement at the crack tip as well as two mechanical computations for each crack length increase.3.3.2.Path independent J integralAmong all the path independent integral methods [27,28],the J integral method introduced by Rice [29]is the most commonly used:J ¼Z Cw ðe Þn 1 Àr ij n j o u i o x d s ;where w is the elastic strain density,u is the displacement ﬁeld at point M of path C of outward normal n .r and e are respectively the stress and strain ﬁelds.Rice showed that J is path-independent in quasi-static isothermal conditions,if no load is applied on the crack edges and for self-similar crack growth.Moreover,J is equal to the strain energy release rate.In plasticity,this statement is only veriﬁed in experiments without unloading.In practice,it is shown that the accuracy of the computation is path-dependent.The most accurate results are obtained with paths far away from the crack tip [30].P.O.Bouchard et al./Comput.Methods Appl.Mech.Engrg.192(2003)3887–390838933.3.3.Virtual crack extensionHellen [18]and Parks [31]suggest a computation of G by using its deﬁnition and a single crack length computation.The crack is propagated by moving nodal points rather than by removing nodal tensions at the crack tip and by performing a second analysis.The strain energy release rate can then be computed using the rigidity matrix K ,the displacement vector u and the load vector f of the numerical system ½K f u g ¼f f g :G ¼Àd W p d a¼À12f u g t d K d a !f u g þf u g t d f d a &':3.3.4.Surface integralDe Lorenzi [32]introduces a method based on a continuum mechanics formulation of the virtual crack extension formulation.The strain energy release rate can be computed as a surface integral––in 2D:G ¼J ¼1d a Z Z Ar ij o u j o x 1 Àw d i 1 o D x 1o x i d A ;where A is the surface between paths C 0and C 1,and D x 1represents the virtual crack extension.Numerical applications of this method [28,33]show its great precision,and its independence to surface integration.3.3.5.G h methodIn 1983,Destuynder [17]has introduced a new method based on a virtual displacement ﬁeld h .The strain energy release rate is the decrease in the total potential energy during a growth of area d A of the crack.All derivatives with respect to the crack growth can be computed by using the Lagrangian method[17,34].Let X be a cracked solid (Fig.6)and F d an inﬁnitesimal geometrical perturbation d in the vicinity of the crack tip:F d :R 3!R 38M 2X ;F d ðM Þ¼M d ¼M þed ðM Þ;where the ﬁeld h gives the location of each point of the perturbated solid using its initial position before perturbation (Fig.7).If perturbation d is suﬃciently small,Destuynder showed that the stress ﬁeld r and the displacement ﬁeld u on the perturbed conﬁguration may be expressedas3894P.O.Bouchard et al./Comput.Methods Appl.Mech.Engrg.192(2003)3887–3908r d ¼r þdr 1;u d ¼u þd u 1;where r 1and u 1are the ﬁrst variations of the stress and displacement ﬁelds during the inﬁnitesimal per-turbation d on X .The total potential energy variation during a crack extension is then obtained when e aims towards 0:d W p d a ¼lim d !0W d pÀWp d .The virtual displacement ﬁeld h represents the virtual kinematics of the crack.This ﬁeld has the following properties:•h is parallel to the crack plan (obvious in 2D);•h is normal to the crack front;•the support of h is only needed in the vicinity of the crack;•k h k is constant in a deﬁned area around the crack tip.In practice,we deﬁne two paths C 1and C 2around the crack tip.These paths divide the part into three domains (Fig.8).Fig.8.Contours and domains used to compute G with the G hmethod.Fig.7.Cracked solid.P.O.Bouchard et al./Comput.Methods Appl.Mech.Engrg.192(2003)3887–39083895For example,the virtual displacement ﬁeld h ðh 1;h 2Þmay be expressed as h 1¼1ÀIM IJ cos ðh Þ;h 2¼1ÀIM IJsin ðh Þ;8>>><>>>:where O is the crack tip,M is an integration point,I and J are the intersections between OM and C 1and C 2respectively.Thus the values of h in the three domains are:•in C int ,the norm of h is constant and equal to 1;•in C ext ,h is 0;•in the ring C ring ,the norm of h varies continuously from 1to 0.Then,if there is neither thermal strain nor load applied directly to the crack,the total potential energy may be expressed asW p ¼12Z X Tr ðr r U Þd X ÀZ XfU d X ;where r and U are respectively the stress and displacement ﬁeld,and f the external loads.If an inﬁnitesimal perturbation d is performed,derivatives and integrals on the perturbed part may be expressed using a ﬁrst order ‘‘limited development’’of operations related to the non-perturbed part.This technique enables to express the strain energy release rate as G ¼Z X Tr ðr r U r h Þd X ÀZ Xw div ðh Þd X ;where w is the strain energy density.For an elastic material,we obtain G ¼Z XTr ðr r U r h Þd X À12Z X Tr ðr r U Þdiv ðh Þd X :Besides,as h varies only inside C ring ,the integration may be performed only over C ring .The strain energy release rate computation is then performed by integration over the seven integration points of each element of the ring:G ¼Z ringTr ðr r U r h Þ À12Tr ðr r U Þdiv ðh Þ!d C ;G ¼X elements X int p ðr 11u 1;1&þr 12u 1;2Þh 1;1þðr 11u 2;1þr 12u 2;2Þh 1;2þðr 12u 1;1þr 22u 1;2Þh 2;1þðr 12u 2;1þr 22u 2;2Þh 2;2À12r 11u 1;1ð½þr 12u 1;2þr 12u 2;1þr 22u 2;2Þðh 1;1þh 2;2Þ 'w int dArea int ;where w int is the Gauss weight of the integration point int and dArea int the associated area.All these ﬁelds are computed in a ﬁnite element analysis,so that it is possible to evaluate G accurately,using a single mesh and a single mechanical computation.Comparisons to other techniques [26]show that the G h method is very accurate and completely mesh independent.The G h method has been implemented in FORGE2.At each crack increment,G ðh Þis computed for h varying from )70°to +70°with degrees steps of 1°,5°and 10°.The G ðh Þcurve is increasing and then 3896P.O.Bouchard et al./Comput.Methods Appl.Mech.Engrg.192(2003)3887–3908decreasing(Fig.9),so that the determination of the angle h0corresponding to the maximum strain energy release rate is straightforward.The G h method can also be used for elastic–plastic materials.However,it is then restricted to stationary cracks.Indeed,the computation of G for elastic–plastic materials is based on an analogy between an elastic–plastic behaviour and a non-linear elastic behaviour.This analogy is only possible without any global unloading.A crack propagation itself represents a local unloading of the material,therefore G is only available for a unique crack extension with an elastic–plastic behaviour.4.ApplicationsThe maximum circumferential stress criterion(MCSC),the minimum strain energy density criterion (MSEDC),and the maximum strain energy release rate criterion(MSERRC)––using the G h method––have been implemented in FORGE2and tests have been performed successfully.In this section,we compare the diﬀerent crack trajectories for various applications.In each example,the material is purely elastic with a Young modulus E¼98,000MPa,and a Poisson ratio m¼0:3.4.1.Rectangular part with an oblique pre-crackA rectangular part with an oblique crack is submitted to a vertical tensile test(Fig.10).4.1.1.Maximum circumferential stress criterionThe maximum circumferential stress criterion is used to compute the direction of the crack propagation at each time step.As expected,the crack propagates in the cleavage mode(mode I),perpendicularly to the maximal principal stress which is vertical due to the applied load(Fig.11).Fig.12shows the shape of the equivalent stress(MPa)ﬁeld at the crack tip.It is similar to the theoretical shape in plane strain.4.1.2.Minimum strain energy density criterionIn this criterion,the accuracy is directly dependent on the number of elements in the ring around the crack tip.The strain energy density is computed for each element of this ring and the local minimum is then evaluated using the SðhÞcurve.Fig.11.Crack trajectory with theMCSC.parison between the theoretical maximum stress zone (a)and the numerical simulation(b).Fig.10.Rectangular part with an oblique pre-crack.However,there is a slight diﬀerence between values from external elements of the ring(ext in Fig.13a) and values from internal elements(int in Fig.13a).This diﬀerence makes the computation of the local minimum diﬃcult.In this case,it is recommended to separate values for external and internal elements(Fig. 13b).The computation of the local minimum may be improved by taking the local minimum and the values in the two neighbour elements.The local minimum is then computed as the minimum of the parabolaﬁtting these three points.As previously observed,the crack propagates horizontally(Fig.14b).Besides it is possible to visualise the strain energy during propagation(Fig.14a).It is concentrated around the crack tip,and in the direction for which this energy is minimum.Fig.13.(a)Ring elements and(b)calculated SðhÞcurve for the MSEDC with the numerical formulation.Fig.14.(a)Strain energyﬁeld and(b)crack propagation predicted by the MSEDC.。

自动化专业英语词汇表

自动化专业英语词汇表acceleration transducer 加速度传感器acceptance testing 验收测试accessibility 可及性accumulated error 累积误差AC-DC-AC frequency converter 交-直-交变频器AC (alternating current) electric drive 交流电子传动active attitude stabilization 主动姿态稳定actuator 驱动器，执行机构adaline 线性适应元adaptation layer 适应层adaptive telemeter system 适应遥测系统adjoint operator 伴随算子admissible error 容许误差aggregation matrix 集结矩阵AHP (analytic hierarchy process) 层次分析法amplifying element 放大环节analog-digital conversion 模数转换annunciator 信号器antenna pointing control 天线指向控制anti-integral windup 抗积分饱卷aperiodic decomposition 非周期分解a posteriori estimate 后验估计approximate reasoning 近似推理a priori estimate 先验估计articulated robot 关节型机器人assignment problem 配置问题，分配问题associative memory model 联想记忆模型associatron 联想机asymptotic stability 渐进稳定性attained pose drift 实际位姿漂移attitude acquisition 姿态捕获AOCS (attritude and orbit control system) 姿态轨道控制系统attitude angular velocity 姿态角速度attitude disturbance 姿态扰动attitude maneuver 姿态机动attractor 吸引子augment ability 可扩充性augmented system 增广系统automatic manual station 自动-手动操作器automaton 自动机autonomous system 自治系统backlash characteristics 间隙特性base coordinate system 基座坐标系Bayes classifier 贝叶斯分类器bearing alignment 方位对准bellows pressure gauge 波纹管压力表benefit-cost analysis 收益成本分析bilinear system 双线性系统biocybernetics 生物控制论biological feedback system 生物反馈系统black box testing approach 黑箱测试法blind search 盲目搜索block diagonalization 块对角化Boltzman machine 玻耳兹曼机bottom-up development 自下而上开发boundary value analysis 边界值分析brainstorming method 头脑风暴法breadth-first search 广度优先搜索butterfly valve 蝶阀CAE (computer aided engineering) 计算机辅助工程CAM (computer aided manufacturing) 计算机辅助制造Camflex valve 偏心旋转阀canonical state variable 规范化状态变量capacitive displacement transducer 电容式位移传感器capsule pressure gauge 膜盒压力表CARD 计算机辅助研究开发Cartesian robot 直角坐标型机器人cascade compensation 串联补偿catastrophe theory 突变论centrality 集中性chained aggregation 链式集结chaos 混沌characteristic locus 特征轨迹chemical propulsion 化学推进calrity 清晰性classical information pattern 经典信息模式classifier 分类器clinical control system 临床控制系统closed loop pole 闭环极点closed loop transfer function 闭环传递函数cluster analysis 聚类分析coarse-fine control 粗-精控制cobweb model 蛛网模型coefficient matrix 系数矩阵cognitive science 认知科学cognitron 认知机coherent system 单调关联系统combination decision 组合决策combinatorial explosion 组合爆炸combined pressure and vacuum gauge 压力真空表command pose 指令位姿companion matrix 相伴矩阵compartmental model 房室模型compatibility 相容性，兼容性compensating network 补偿网络compensation 补偿，矫正compliance 柔顺，顺应composite control 组合控制computable general equilibrium model 可计算一般均衡模型conditionally instability 条件不稳定性configuration 组态connectionism 连接机制connectivity 连接性conservative system 守恒系统consistency 一致性constraint condition 约束条件consumption function 消费函数context-free grammar 上下文无关语法continuous discrete event hybrid system simulation 连续离散事件混合系统仿真continuous duty 连续工作制control accuracy 控制精度control cabinet 控制柜controllability index 可控指数controllable canonical form 可控规范型[control] plant 控制对象,被控对象controlling instrument 控制仪表control moment gyro 控制力矩陀螺control panel 控制屏，控制盘control synchro 控制[式]自整角机control system synthesis 控制系统综合control time horizon 控制时程cooperative game 合作对策coordinability condition 可协调条件coordination strategy 协调策略coordinator 协调器corner frequency 转折频率costate variable 共态变量cost-effectiveness analysis 费用效益分析coupling of orbit and attitude 轨道和姿态耦合critical damping 临界阻尼critical stability 临界稳定性cross-over frequency 穿越频率，交越频率current source inverter 电流[源]型逆变器cut-off frequency 截止频率cybernetics 控制论cyclic remote control 循环遥控cylindrical robot 圆柱坐标型机器人damped oscillation 阻尼振荡damper 阻尼器damping ratio 阻尼比data acquisition 数据采集data encryption 数据加密data preprocessing 数据预处理data processor 数据处理器DC generator-motor set drive 直流发电机-电动机组传动D controller 微分控制器decentrality 分散性decentralized stochastic control 分散随机控制decision space 决策空间decision support system 决策支持系统decomposition-aggregation approach 分解集结法decoupling parameter 解耦参数deductive-inductive hybrid modeling method 演绎与归纳混合建模法delayed telemetry 延时遥测derivation tree 导出树derivative feedback 微分反馈describing function 描述函数desired value 希望值despinner 消旋体destination 目的站detector 检出器deterministic automaton 确定性自动机deviation 偏差deviation alarm 偏差报警器DFD 数据流图diagnostic model 诊断模型diagonally dominant matrix 对角主导矩阵diaphragm pressure gauge 膜片压力表difference equation model 差分方程模型differential dynamical system 微分动力学系统differential game 微分对策differential pressure level meter 差压液位计differential pressure transmitter 差压变送器differential transformer displacement transducer 差动变压器式位移传感器differentiation element 微分环节digital filer 数字滤波器digital signal processing 数字信号处理digitization 数字化digitizer 数字化仪dimension transducer 尺度传感器direct coordination 直接协调disaggregation 解裂discoordination 失协调discrete event dynamic system 离散事件动态系统discrete system simulation language 离散系统仿真语言discriminant function 判别函数displacement vibration amplitude transducer 位移振幅传感器dissipative structure 耗散结构distributed parameter control system 分布参数控制系统distrubance 扰动disturbance compensation 扰动补偿diversity 多样性divisibility 可分性domain knowledge 领域知识dominant pole 主导极点dose-response model 剂量反应模型dual modulation telemetering system 双重调制遥测系统dual principle 对偶原理dual spin stabilization 双自旋稳定duty ratio 负载比dynamic braking 能耗制动dynamic characteristics 动态特性dynamic deviation 动态偏差dynamic error coefficient 动态误差系数dynamic exactness 动它吻合性dynamic input-output model 动态投入产出模型econometric model 计量经济模型economic cybernetics 经济控制论economic effectiveness 经济效益economic evaluation 经济评价economic index 经济指数economic indicator 经济指标eddy current thickness meter 电涡流厚度计effectiveness 有效性effectiveness theory 效益理论elasticity of demand 需求弹性electric actuator 电动执行机构electric conductance levelmeter 电导液位计electric drive control gear 电动传动控制设备electric hydraulic converter 电-液转换器electric pneumatic converter 电-气转换器electrohydraulic servo vale 电液伺服阀electromagnetic flow transducer 电磁流量传感器electronic batching scale 电子配料秤electronic belt conveyor scale 电子皮带秤electronic hopper scale 电子料斗秤elevation 仰角emergency stop 异常停止empirical distribution 经验分布endogenous variable 内生变量equilibrium growth 均衡增长equilibrium point 平衡点equivalence partitioning 等价类划分ergonomics 工效学error 误差error-correction parsing 纠错剖析estimate 估计量estimation theory 估计理论evaluation technique 评价技术event chain 事件链evolutionary system 进化系统exogenous variable 外生变量expected characteristics 希望特性external disturbance 外扰fact base 事实failure diagnosis 故障诊断fast mode 快变模态feasibility study 可行性研究feasible coordination 可行协调feasible region 可行域feature detection 特征检测feature extraction 特征抽取feedback compensation 反馈补偿feedforward path 前馈通路field bus 现场总线finite automaton 有限自动机FIP (factory information protocol) 工厂信息协议first order predicate logic 一阶谓词逻辑fixed sequence manipulator 固定顺序机械手fixed set point control 定值控制FMS (flexible manufacturing system) 柔性制造系统flow sensor/transducer 流量传感器flow transmitter 流量变送器fluctuation 涨落forced oscillation 强迫振荡formal language theory 形式语言理论formal neuron 形式神经元forward path 正向通路forward reasoning 正向推理fractal 分形体，分维体frequency converter 变频器frequency domain model reduction method 频域模型降阶法frequency response 频域响应full order observer 全阶观测器functional decomposition 功能分解FES (functional electrical stimulation) 功能电刺激functional simularity 功能相似fuzzy logic 模糊逻辑game tree 对策树gate valve 闸阀general equilibrium theory 一般均衡理论generalized least squares estimation 广义最小二乘估计generation function 生成函数geomagnetic torque 地磁力矩geometric similarity 几何相似gimbaled wheel 框架轮global asymptotic stability 全局渐进稳定性global optimum 全局最优globe valve 球形阀goal coordination method 目标协调法grammatical inference 文法推断graphic search 图搜索gravity gradient torque 重力梯度力矩group technology 成组技术guidance system 制导系统gyro drift rate 陀螺漂移率gyrostat 陀螺体Hall displacement transducer 霍尔式位移传感器hardware-in-the-loop simulation 半实物仿真harmonious deviation 和谐偏差harmonious strategy 和谐策略heuristic inference 启发式推理hidden oscillation 隐蔽振荡hierarchical chart 层次结构图hierarchical planning 递阶规划hierarchical control 递阶控制homeostasis 内稳态homomorphic model 同态系统horizontal decomposition 横向分解hormonal control 内分泌控制hydraulic step motor 液压步进马达hypercycle theory 超循环理论I controller 积分控制器identifiability 可辨识性IDSS (intelligent decision support system) 智能决策支持系统image recognition 图像识别impulse 冲量impulse function 冲击函数，脉冲函数inching 点动incompatibility principle 不相容原理incremental motion control 增量运动控制index of merit 品质因数inductive force transducer 电感式位移传感器inductive modeling method 归纳建模法industrial automation 工业自动化inertial attitude sensor 惯性姿态敏感器inertial coordinate system 惯性坐标系inertial wheel 惯性轮inference engine 推理机infinite dimensional system 无穷维系统information acquisition 信息采集infrared gas analyzer 红外线气体分析器inherent nonlinearity 固有非线性inherent regulation 固有调节initial deviation 初始偏差initiator 发起站injection attitude 入轨姿势input-output model 投入产出模型instability 不稳定性instruction level language 指令级语言integral of absolute value of error criterion 绝对误差积分准则integral of squared error criterion 平方误差积分准则integral performance criterion 积分性能准则integration instrument 积算仪器integrity 整体性intelligent terminal 智能终端interacted system 互联系统，关联系统interactive prediction approach 互联预估法，关联预估法interconnection 互联intermittent duty 断续工作制internal disturbance 内扰ISM (interpretive structure modeling) 解释结构建模法invariant embedding principle 不变嵌入原理inventory theory 库伦论inverse Nyquist diagram 逆奈奎斯特图inverter 逆变器investment decision 投资决策isomorphic model 同构模型iterative coordination 迭代协调jet propulsion 喷气推进job-lot control 分批控制joint 关节Kalman-Bucy filer 卡尔曼-布西滤波器knowledge accomodation 知识顺应knowledge acquisition 知识获取knowledge assimilation 知识同化KBMS (knowledge base management system) 知识库管理系统knowledge representation 知识表达ladder diagram 梯形图lag-lead compensation 滞后超前补偿Lagrange duality 拉格朗日对偶性Laplace transform 拉普拉斯变换large scale system 大系统lateral inhibition network 侧抑制网络least cost input 最小成本投入least squares criterion 最小二乘准则level switch 物位开关libration damping 天平动阻尼limit cycle 极限环linearization technique 线性化方法linear motion electric drive 直线运动电气传动linear motion valve 直行程阀linear programming 线性规划LQR (linear quadratic regulator problem) 线性二次调节器问题load cell 称重传感器local asymptotic stability 局部渐近稳定性local optimum 局部最优log magnitude-phase diagram 对数幅相图long term memory 长期记忆lumped parameter model 集总参数模型Lyapunov theorem of asymptotic stability 李雅普诺夫渐近稳定性定理macro-economic system 宏观经济系统magnetic dumping 磁卸载magnetoelastic weighing cell 磁致弹性称重传感器magnitude-frequency characteristic 幅频特性magnitude margin 幅值裕度magnitude scale factor 幅值比例尺manipulator 机械手man-machine coordination 人机协调manual station 手动操作器MAP (manufacturing automation protocol) 制造自动化协议marginal effectiveness 边际效益Mason's gain formula 梅森增益公式master station 主站matching criterion 匹配准则maximum likelihood estimation 最大似然估计maximum overshoot 最大超调量maximum principle 极大值原理mean-square error criterion 均方误差准则mechanism model 机理模型meta-knowledge 元知识metallurgical automation 冶金自动化minimal realization 最小实现minimum phase system 最小相位系统minimum variance estimation 最小方差估计minor loop 副回路missile-target relative movement simulator 弹体-目标相对运动仿真器modal aggregation 模态集结modal transformation 模态变换MB (model base) 模型库model confidence 模型置信度model fidelity 模型逼真度model reference adaptive control system 模型参考适应控制系统model verification 模型验证modularization 模块化MEC (most economic control) 最经济控制motion space 可动空间MTBF (mean time between failures) 平均故障间隔时间MTTF (mean time to failures) 平均无故障时间multi-attributive utility function 多属性效用函数multicriteria 多重判据multilevel hierarchical structure 多级递阶结构multiloop control 多回路控制multi-objective decision 多目标决策multistate logic 多态逻辑multistratum hierarchical control 多段递阶控制multivariable control system 多变量控制系统myoelectric control 肌电控制Nash optimality 纳什最优性natural language generation 自然语言生成nearest-neighbor 最近邻necessity measure 必然性侧度negative feedback 负反馈neural assembly 神经集合neural network computer 神经网络计算机Nichols chart 尼科尔斯图noetic science 思维科学noncoherent system 非单调关联系统noncooperative game 非合作博弈nonequilibrium state 非平衡态nonlinear element 非线性环节nonmonotonic logic 非单调逻辑nonparametric training 非参数训练nonreversible electric drive 不可逆电气传动nonsingular perturbation 非奇异摄动non-stationary random process 非平稳随机过程nuclear radiation levelmeter 核辐射物位计nutation sensor 章动敏感器Nyquist stability criterion 奈奎斯特稳定判据objective function 目标函数observability index 可观测指数observable canonical form 可观测规范型on-line assistance 在线帮助on-off control 通断控制open loop pole 开环极点operational research model 运筹学模型optic fiber tachometer 光纤式转速表optimal trajectory 最优轨迹optimization technique 最优化技术orbital rendezvous 轨道交会orbit gyrocompass 轨道陀螺罗盘orbit perturbation 轨道摄动order parameter 序参数orientation control 定向控制originator 始发站oscillating period 振荡周期output prediction method 输出预估法oval wheel flowmeter 椭圆齿轮流量计overall design 总体设计overdamping 过阻尼overlapping decomposition 交叠分解Pade approximation 帕德近似Pareto optimality 帕雷托最优性passive attitude stabilization 被动姿态稳定path repeatability 路径可重复性pattern primitive 模式基元PR (pattern recognition) 模式识别P control 比例控制器peak time 峰值时间penalty function method 罚函数法perceptron 感知器periodic duty 周期工作制perturbation theory 摄动理论pessimistic value 悲观值phase locus 相轨迹phase trajectory 相轨迹phase lead 相位超前photoelectric tachometric transducer 光电式转速传感器phrase-structure grammar 短句结构文法physical symbol system 物理符号系统piezoelectric force transducer 压电式力传感器playback robot 示教再现式机器人PLC (programmable logic controller) 可编程序逻辑控制器plug braking 反接制动plug valve 旋塞阀pneumatic actuator 气动执行机构point-to-point control 点位控制polar robot 极坐标型机器人pole assignment 极点配置pole-zero cancellation 零极点相消polynomial input 多项式输入portfolio theory 投资搭配理论pose overshoot 位姿过调量position measuring instrument 位置测量仪posentiometric displacement transducer 电位器式位移传感器positive feedback 正反馈power system automation 电力系统自动化predicate logic 谓词逻辑pressure gauge with electric contact 电接点压力表pressure transmitter 压力变送器price coordination 价格协调primal coordination 主协调primary frequency zone 主频区PCA (principal component analysis) 主成分分析法principle of turnpike 大道原理priority 优先级process-oriented simulation 面向过程的仿真production budget 生产预算production rule 产生式规则profit forecast 利润预测PERT (program evaluation and review technique) 计划评审技术program set station 程序设定操作器proportional control 比例控制proportional plus derivative controller 比例微分控制器protocol engineering 协议工程prototype 原型pseudo random sequence 伪随机序列pseudo-rate-increment control 伪速率增量控制pulse duration 脉冲持续时间pulse frequency modulation control system 脉冲调频控制系统pulse width modulation control system 脉冲调宽控制系统PWM inverter 脉宽调制逆变器pushdown automaton 下推自动机QC (quality control) 质量管理quadratic performance index 二次型性能指标qualitative physical model 定性物理模型quantized noise 量化噪声quasilinear characteristics 准线性特性queuing theory 排队论radio frequency sensor 射频敏感器ramp function 斜坡函数random disturbance 随机扰动random process 随机过程rate integrating gyro 速率积分陀螺ratio station 比值操作器reachability 可达性reaction wheel control 反作用轮控制realizability 可实现性,能实现性real time telemetry 实时遥测receptive field 感受野rectangular robot 直角坐标型机器人rectifier 整流器recursive estimation 递推估计reduced order observer 降阶观测器redundant information 冗余信息reentry control 再入控制regenerative braking 回馈制动，再生制动regional planning model 区域规划模型regulating device 调节装载regulation 调节relational algebra 关系代数relay characteristic 继电器特性remote manipulator 遥控操作器remote regulating 遥调remote set point adjuster 远程设定点调整器rendezvous and docking 交会和对接reproducibility 再现性resistance thermometer sensor 热电阻resolution principle 归结原理resource allocation 资源分配response curve 响应曲线return difference matrix 回差矩阵return ratio matrix 回比矩阵reverberation 回响reversible electric drive 可逆电气传动revolute robot 关节型机器人revolution speed transducer 转速传感器rewriting rule 重写规则rigid spacecraft dynamics 刚性航天动力学risk decision 风险分析robotics 机器人学robot programming language 机器人编程语言robust control 鲁棒控制robustness 鲁棒性roll gap measuring instrument 辊缝测量仪root locus 根轨迹roots flowmeter 腰轮流量计rotameter 浮子流量计，转子流量计rotary eccentric plug valve 偏心旋转阀rotary motion valve 角行程阀rotating transformer 旋转变压器Routh approximation method 劳思近似判据routing problem 路径问题sampled-data control system 采样控制系统sampling control system 采样控制系统saturation characteristics 饱和特性scalar Lyapunov function 标量李雅普诺夫函数SCARA (selective compliance assembly robot arm) 平面关节型机器人scenario analysis method 情景分析法scene analysis 物景分析s-domain s域self-operated controller 自力式控制器self-organizing system 自组织系统self-reproducing system 自繁殖系统self-tuning control 自校正控制semantic network 语义网络semi-physical simulation 半实物仿真sensing element 敏感元件sensitivity analysis 灵敏度分析sensory control 感觉控制sequential decomposition 顺序分解sequential least squares estimation 序贯最小二乘估计servo control 伺服控制，随动控制servomotor 伺服马达settling time 过渡时间sextant 六分仪short term planning 短期计划short time horizon coordination 短时程协调signal detection and estimation 信号检测和估计signal reconstruction 信号重构similarity 相似性simulated interrupt 仿真中断simulation block diagram 仿真框图simulation experiment 仿真实验simulation velocity 仿真速度simulator 仿真器single axle table 单轴转台single degree of freedom gyro 单自由度陀螺single level process 单级过程single value nonlinearity 单值非线性singular attractor 奇异吸引子singular perturbation 奇异摄动sink 汇点slaved system 受役系统slower-than-real-time simulation 欠实时仿真slow subsystem 慢变子系统socio-cybernetics 社会控制论socioeconomic system 社会经济系统software psychology 软件心理学solar array pointing control 太阳帆板指向控制solenoid valve 电磁阀source 源点specific impulse 比冲speed control system 调速系统spin axis 自旋轴spinner 自旋体stability criterion 稳定性判据stability limit 稳定极限stabilization 镇定，稳定Stackelberg decision theory 施塔克尔贝格决策理论state equation model 状态方程模型state space description 状态空间描述static characteristics curve 静态特性曲线station accuracy 定点精度stationary random process 平稳随机过程statistical analysis 统计分析statistic pattern recognition 统计模式识别steady state deviation 稳态偏差steady state error coefficient 稳态误差系数step-by-step control 步进控制step function 阶跃函数stepwise refinement 逐步精化stochastic finite automaton 随机有限自动机strain gauge load cell 应变式称重传感器strategic function 策略函数strongly coupled system 强耦合系统subjective probability 主观频率suboptimality 次优性supervised training 监督学习supervisory computer control system 计算机监控系统sustained oscillation 自持振荡swirlmeter 旋进流量计switching point 切换点symbolic processing 符号处理synaptic plasticity 突触可塑性synergetics 协同学syntactic analysis 句法分析system assessment 系统评价systematology 系统学system homomorphism 系统同态system isomorphism 系统同构system engineering 系统工程tachometer 转速表target flow transmitter 靶式流量变送器task cycle 作业周期teaching programming 示教编程telemechanics 远动学telemetering system of frequency division type 频分遥测系统telemetry 遥测teleological system 目的系统teleology 目的论temperature transducer 温度传感器template base 模版库tensiometer 张力计texture 纹理theorem proving 定理证明therapy model 治疗模型thermocouple 热电偶thermometer 温度计thickness meter 厚度计three-axis attitude stabilization 三轴姿态稳定three state controller 三位控制器thrust vector control system 推力矢量控制系统thruster 推力器time constant 时间常数time-invariant system 定常系统，非时变系统time schedule controller 时序控制器time-sharing control 分时控制time-varying parameter 时变参数top-down testing 自上而下测试topological structure 拓扑结构TQC (total quality control) 全面质量管理tracking error 跟踪误差trade-off analysis 权衡分析transfer function matrix 传递函数矩阵transformation grammar 转换文法transient deviation 瞬态偏差transient process 过渡过程transition diagram 转移图transmissible pressure gauge 电远传压力表transmitter 变送器trend analysis 趋势分析triple modulation telemetering system 三重调制遥测系统turbine flowmeter 涡轮流量计Turing machine 图灵机two-time scale system 双时标系统ultrasonic levelmeter 超声物位计unadjustable speed electric drive 非调速电气传动unbiased estimation 无偏估计underdamping 欠阻尼uniformly asymptotic stability 一致渐近稳定性uninterrupted duty 不间断工作制，长期工作制unit circle 单位圆unit testing 单元测试unsupervised learing 非监督学习upper level problem 上级问题urban planning 城市规划utility function 效用函数value engineering 价值工程variable gain 可变增益，可变放大系数variable structure control system 变结构控制vector Lyapunov function 向量李雅普诺夫函数velocity error coefficient 速度误差系数velocity transducer 速度传感器vertical decomposition 纵向分解vibrating wire force transducer 振弦式力传感器vibrometer 振动计viscous damping 粘性阻尼voltage source inverter 电压源型逆变器vortex precession flowmeter 旋进流量计vortex shedding flowmeter 涡街流量计WB (way base) 方法库weighing cell 称重传感器weighting factor 权因子weighting method 加权法Whittaker-Shannon sampling theorem 惠特克-香农采样定理Wiener filtering 维纳滤波work station for computer aided design 计算机辅助设计工作站w-plane w平面zero-based budget 零基预算zero-input response 零输入响应zero-state response 零状态响应zero sum game model 零和对策模型z-transform z变换。

工业自动化专业英语词汇

A－Zacceleration transducer 加速度传感器acceptance testing 验收测试accessibility 可及性accumulated error 累积误差AC-DC-AC frequency converter 交-直-交变频器AC (alternating current) electric drive 交流电子传动active attitude stabilization 主动姿态稳定actuator 驱动器，执行机构adaline 线性适应元adaptation layer 适应层adaptive telemeter system 适应遥测系统adjoint operator 伴随算子admissible error 容许误差aggregation matrix 集结矩阵AHP (analytic hierarchy process) 层次分析法amplifying element 放大环节analog-digital conversion 模数转换annunciator 信号器antenna pointing control 天线指向控制anti-integral windup 抗积分饱卷aperiodic decomposition 非周期分解 a posteriori estimate 后验估计approximate reasoning 近似推理 a priori estimate 先验估计articulated robot 关节型机器人assignment problem 配置问题，分配问题associative memory model 联想记忆模型associatron 联想机asymptotic stability 渐进稳定性attained pose drift 实际位姿漂移attitude acquisition 姿态捕获AOCS (attritude and orbit control system) 姿态轨道控制系统attitude angular velocity 姿态角速度attitude disturbance 姿态扰动attitude maneuver 姿态机动attractor 吸引子augment ability 可扩充性augmented system 增广系统automatic manual station 自动-手动操作器automaton 自动机backlash characteristics 间隙特性base coordinate system 基座坐标系Bayes classifier 贝叶斯分类器bearing alignment 方位对准bellows pressure gauge 波纹管压力表benefit-cost analysis 收益成本分析bilinear system 双线性系统biocybernetics 生物控制论biological feedback system 生物反馈系统black box testing approach 黑箱测试法blind search 盲目搜索block diagonalization 块对角化Boltzman machine 玻耳兹曼机bottom-up development 自下而上开发boundary value analysis 边界值分析brainstorming method 头脑风暴法breadth-first search 广度优先搜索butterfly valve 蝶阀CAE (computer aided engineering) 计算机辅助工程CAM (computer aided manufacturing) 计算机辅助制造Camflex valve 偏心旋转阀canonical state variable 规范化状态变量capacitive displacement transducer 电容式位移传感器capsule pressure gauge 膜盒压力表CARD 计算机辅助研究开发Cartesian robot 直角坐标型机器人cascade compensation 串联补偿catastrophe theory 突变论centrality 集中性chained aggregation 链式集结chaos 混沌characteristic locus 特征轨迹chemical propulsion 化学推进calrity 清晰性classical information pattern 经典信息模式classifier 分类器clinical control system 临床控制系统closed loop pole 闭环极点closed loop transfer function 闭环传递函数cluster analysis 聚类分析coarse-fine control 粗-精控制cobweb model 蛛网模型coefficient matrix 系数矩阵cognitive science 认知科学cognitron 认知机coherent system 单调关联系统combination decision 组合决策combinatorial explosion 组合爆炸combined pressure and vacuum gauge 压力真空表command pose 指令位姿companion matrix 相伴矩阵compartmental model 房室模型compatibility 相容性，兼容性compensating network 补偿网络compensation 补偿，矫正compliance 柔顺，顺应composite control 组合控制computable general equilibrium model 可计算一般均衡模型conditionally instability 条件不稳定性configuration 组态connectionism 连接机制connectivity 连接性conservative system 守恒系统consistency 一致性constraint condition 约束条件consumption function 消费函数context-free grammar 上下文无关语法continuous discrete event hybrid system simulation 连续离散事件混合系统仿真continuous duty 连续工作制control accuracy 控制精度control cabinet 控制柜controllability index 可控指数controllable canonical form 可控规范型plant 控制对象,被控对象controlling instrument 控制仪表control moment gyro 控制力矩陀螺control panel 控制屏，控制盘control synchro 控制<式>自整角机control system synthesis 控制系统综合control time horizon 控制时程cooperative game 合作对策coordinability condition 可协调条件coordination strategy 协调策略coordinator 协调器corner frequency 转折频率costate variable 共态变量cost-effectiveness analysis 费用效益分析coupling of orbit and attitude 轨道和姿态耦合critical damping 临界阻尼critical stability 临界稳定性cross-over frequency 穿越频率，交越频率current source inverter 电流<源>型逆变器cut-off frequency 截止频率cybernetics 控制论cyclic remote control 循环遥控cylindrical robot 圆柱坐标型机器人damped oscillation 阻尼振荡damper 阻尼器damping ratio 阻尼比data acquisition 数据采集data encryption 数据加密data preprocessing 数据预处理data processor 数据处理器DC generator-motor set drive 直流发电机-电动机组传动 D controller 微分控制器decentrality 分散性decentralized stochastic control 分散随机控制decision space 决策空间decision support system 决策支持系统decomposition-aggregation approach 分解集结法decoupling parameter 解耦参数deductive-inductive hybrid modeling method 演绎与归纳混合建模法delayed telemetry 延时遥测derivation tree 导出树derivative feedback 微分反馈describing function 描述函数desired value 希望值despinner 消旋体destination 目的站detector 检出器deterministic automaton 确定性自动机deviation 偏差舱deviation alarm 偏差报警器DFD 数据流图diagnostic model 诊断模型diagonally dominant matrix 对角主导矩阵diaphragm pressure gauge 膜片压力表difference equation model 差分方程模型differential dynamical system 微分动力学系统differential game 微分对策differential pressure level meter 差压液位计differential pressure transmitter 差压变送器differential transformer displacement transducer 差动变压器式位移传感器differentiation element 微分环节digital filer 数字滤波器digital signal processing 数字信号处理digitization 数字化digitizer 数字化仪dimension transducer 尺度传感器direct coordination 直接协调disaggregation 解裂discoordination 失协调discrete event dynamic system 离散事件动态系统discrete system simulation language 离散系统仿真语言discriminant function 判别函数displacement vibration amplitude transducer 位移振幅传感器dissipative structure 耗散结构distributed parameter control system 分布参数控制系统distrubance 扰动disturbance compensation 扰动补偿diversity 多样性divisibility 可分性domain knowledge 领域知识dominant pole 主导极点dose-response model 剂量反应模型dual modulation telemetering system 双重调制遥测系统dual principle 对偶原理dual spin stabilization 双自旋稳定duty ratio 负载比dynamic braking 能耗制动dynamic characteristics 动态特性dynamic deviation 动态偏差dynamic error coefficient 动态误差系数dynamic exactness 动它吻合性dynamic input-output model 动态投入产出模型econometric model 计量经济模型economic cybernetics 经济控制论economic effectiveness 经济效益economic evaluation 经济评价economic index 经济指数economic indicator 经济指标eddy current thickness meter 电涡流厚度计effectiveness 有效性effectiveness theory 效益理论elasticity of demand 需求弹性electric actuator 电动执行机构electric conductance levelmeter 电导液位计electric drive control gear 电动传动控制设备electric hydraulic converter 电-液转换器electric pneumatic converter 电-气转换器electrohydraulic servo vale 电液伺服阀electromagnetic flow transducer 电磁流量传感器electronic batching scale 电子配料秤electronic belt conveyor scale 电子皮带秤electronic hopper scale 电子料斗秤elevation 仰角emergency stop 异常停止empirical distribution 经验分布endogenous variable 内生变量equilibrium growth 均衡增长equilibrium point 平衡点equivalence partitioning 等价类划分ergonomics 工效学error 误差error-correction parsing 纠错剖析estimate 估计量estimation theory 估计理论evaluation technique 评价技术event chain 事件链evolutionary system 进化系统exogenous variable 外生变量expected characteristics 希望特性external disturbance 外扰fact base 事实failure diagnosis 故障诊断fast mode 快变模态feasibility study 可行性研究feasible coordination 可行协调feasible region 可行域feature detection 特征检测feature extraction 特征抽取feedback compensation 反馈补偿feedforward path 前馈通路field bus 现场总线finite automaton 有限自动机FIP (factory information protocol) 工厂信息协议first order predicate logic 一阶谓词逻辑fixed sequence manipulator 固定顺序机械手fixed set point control 定值控制FMS (flexible manufacturing system) 柔性制造系统flow sensor/transducer 流量传感器flow transmitter 流量变送器fluctuation 涨落forced oscillation 强迫振荡formal language theory 形式语言理论formal neuron 形式神经元forward path 正向通路forward reasoning 正向推理fractal 分形体，分维体frequency converter 变频器frequency domain model reduction method 频域模型降阶法frequency response 频域响应full order observer 全阶观测器functional decomposition 功能分解FES (functional electrical stimulation) 功能电刺激functional simularity 功能相似fuzzy logic模糊逻辑game tree 对策树gate valve 闸阀general equilibrium theory 一般均衡理论generalized least squares estimation 广义最小二乘估计generation function 生成函数geomagnetic torque 地磁力矩geometric similarity 几何相似gimbaled wheel 框架轮global asymptotic stability 全局渐进稳定性global optimum 全局最优globe valve 球形阀徢goal coordination method 目标协调法grammatical inference 文法推断graphic search 图搜索gravity gradient torque 重力梯度力矩group technology 成组技术guidance system 制导系统gyro drift rate 陀螺漂移率gyrostat 陀螺体Hall displacement transducer 霍尔式位移传感器hardware-in-the-loop simulation 半实物仿真harmonious deviation 和谐偏差harmonious strategy 和谐策略heuristic inference 启发式推理hidden oscillation 隐蔽振荡hierarchical chart 层次结构图hierarchical planning 递阶规划hierarchical control 递阶控制homeostasis 内稳态homomorphic model 同态系统horizontal decomposition 横向分解hormonal control 内分泌控制hydraulic step motor 液压步进马达hypercycle theory 超循环理论I controller 积分控制器identifiability 可辨识性IDSS (intelligent decision support system) 智能决策支持系统image recognition 图像识别impulse 冲量impulse function 冲击函数，脉冲函数inching 点动incompatibility principle 不相容原理incremental motion control 增量运动控制index of merit 品质因数inductive force transducer 电感式位移传感器inductive modeling method 归纳建模法industrial automation 工业自动化inertial attitude sensor 惯性姿态敏感器inertial coordinate system 惯性坐标系inertial wheel 惯性轮inference engine 推理机infinite dimensional system 无穷维系统information acquisition 信息采集infrared gas analyzer 红外线气体分析器inherent nonlinearity 固有非线性inherent regulation 固有调节initial deviation 初始偏差initiator 发起站injection attitude 入轨姿势input-output model 投入产出模型instability 不稳定性instruction level language 指令级语言integral of absolute value of error criterion 绝对误差积分准则integral of squared error criterion 平方误差积分准则integral performance criterion 积分性能准则integration instrument 积算仪器integrity 整体性intelligent terminal 智能终端interacted system 互联系统，关联系统interactive prediction approach 互联预估法，关联预估法interconnection 互联intermittent duty 断续工作制internal disturbance 内扰ISM (interpretive structure modeling) 解释结构建模法invariant embedding principle 不变嵌入原理inventory theory 库伦论inverse Nyquist diagram 逆奈奎斯特图inverter 逆变器investment decision 投资决策isomorphic model 同构模型iterative coordination 迭代协调jet propulsion 喷气推进job-lot control 分批控制joint 关节Kalman-Bucy filer 卡尔曼-布西滤波器knowledge accomodation 知识顺应knowledge acquisition 知识获取knowledge assimilation 知识同化KBMS (knowledge base management system) 知识库管理系统瓢knowledge representation 知识表达ladder diagram 梯形图lag-lead compensation 滞后超前补偿Lagrange duality 拉格朗日对偶性Laplace transform 拉普拉斯变换large scale system 大系统lateral inhibition network 侧抑制网络least cost input 最小成本投入least squares criterion 最小二乘准则level switch 物位开关libration damping 天平动阻尼limit cycle 极限环linearization technique 线性化方法linear motion electric drive 直线运动电气传动linear motion valve 直行程阀linear programming 线性规划LQR (linear quadratic regulator problem) 线性二次调节器问题load cell 称重传感器local asymptotic stability 局部渐近稳定性local optimum 局部最优log magnitude-phase diagram 对数幅相图long term memory 长期记忆lumped parameter model 集总参数模型Lyapunov theorem of asymptotic stability 李雅普诺夫渐近稳定性定理macro-economic system 宏观经济系统magnetic dumping 磁卸载magnetoelastic weighing cell 磁致弹性称重传感器magnitude-frequency characteristic 幅频特性magnitude margin 幅值裕度magnitude scale factor 幅值比例尺manipulator 机械手man-machine coordination 人机协调manual station 手动操作器MAP (manufacturing automation protocol) 制造自动化协议marginal effectiveness 边际效益Mason''s gain formula 梅森增益公式master station 主站matching criterion 匹配准则maximum likelihood estimation 最大似然估计maximum overshoot 最大超调量maximum principle 极大值原理mean-square error criterion 均方误差准则mechanism model 机理模型meta-knowledge 元知识metallurgical automation 冶金自动化minimal realization 最小实现minimum phase system 最小相位系统minimum variance estimation 最小方差估计minor loop 副回路missile-target relative movement simulator 弹体-目标相对运动仿真器modal aggregation 模态集结modal transformation 模态变换MB (model base) 模型库model confidence 模型置信度model fidelity 模型逼真度model reference adaptive control system 模型参考适应控制系统model verification 模型验证modularization 模块化MEC (most economic control) 最经济控制motion space 可动空间MTBF (mean time between failures) 平均故障间隔时间MTTF (mean time to failures) 平均无故障时间multi-attributive utility function 多属性效用函数multicriteria 多重判据multilevel hierarchical structure 多级递阶结构multiloop control 多回路控制multi-objective decision 多目标决策multistate logic 多态逻辑multistratum hierarchical control 多段递阶控制multivariable control system 多变量控制系统myoelectric control 肌电控制Nash optimality 纳什最优性natural language generation自然语言生成nearest-neighbor 最近邻necessity measure 必然性侧度negative feedback 负反馈neural assembly 神经集合neural network computer 神经网络计算机Nichols chart 尼科尔斯图noetic science 思维科学noncoherent system 非单调关联系统noncooperative game 非合作博弈nonequilibrium state 非平衡态nonlinear element 非线性环节nonmonotonic logic 非单调逻辑nonparametric training 非参数训练nonreversible electric drive 不可逆电气传动nonsingular perturbation 非奇异摄动non-stationary random process 非平稳随机过程nuclear radiation levelmeter 核辐射物位计nutation sensor 章动敏感器Nyquist stability criterion 奈奎斯特稳定判据objective function 目标函数observability index 可观测指数observable canonical form 可观测规范型on-line assistance 在线帮助on-off control 通断控制open loop pole 开环极点operational research model 运筹学模型optic fiber tachometer 光纤式转速表optimal trajectory 最优轨迹optimization technique 最优化技术orbital rendezvous 轨道交会orbit gyrocompass 轨道陀螺罗盘orbit perturbation 轨道摄动order parameter 序参数orientation control 定向控制originator 始发站oscillating period 振荡周期output prediction method 输出预估法oval wheel flowmeter 椭圆齿轮流量计overall design 总体设计overdamping 过阻尼overlapping decomposition 交叠分解Pade approximation 帕德近似Pareto optimality 帕雷托最优性passive attitude stabilization 被动姿态稳定path repeatability 路径可重复性pattern primitive 模式基元PR (pattern recognition) 模式识别P control 比例控制器peak time 峰值时间penalty function method 罚函数法perceptron 感知器periodic duty 周期工作制perturbation theory 摄动理论pessimistic value 悲观值phase locus 相轨迹phase trajectory 相轨迹phase lead 相位超前photoelectric tachometric transducer 光电式转速传感器phrase-structure grammar 短句结构文法physical symbol system 物理符号系统piezoelectric force transducer 压电式力传感器playback robot 示教再现式机器人PLC (programmable logic controller) 可编程序逻辑控制器plug braking 反接制动plug valve 旋塞阀pneumatic actuator 气动执行机构point-to-point control 点位控制polar robot 极坐标型机器人pole assignment 极点配置pole-zero cancellation 零极点相消polynomial input 多项式输入portfolio theory 投资搭配理论pose overshoot 位姿过调量position measuring instrument 位置测量仪posentiometric displacement transducer 电位器式位移传感器positive feedback 正反馈power system automation 电力系统自动化predicate logic 谓词逻辑pressure gauge with electric contact 电接点压力表pressure transmitter 压力变送器price coordination 价格协调primal coordination 主协调primary frequency zone 主频区PCA (principal component analysis) 主成分分析法principle of turnpike 大道原理priority 优先级process-oriented simulation 面向过程的仿真production budget 生产预算production rule 产生式规则profit forecast 利润预测PERT (program evaluation and review technique) 计划评审技术program set station 程序设定操作器proportional control 比例控制proportional plus derivative controller 比例微分控制器protocol engineering 协议工程prototype 原型pseudo random sequence 伪随机序列pseudo-rate-increment control 伪速率增量控制pulse duration 脉冲持续时间pulse frequency modulation control system 脉冲调频控制系统pulse width modulation control system 脉冲调宽控制系统PWM inverter 脉宽调制逆变器pushdown automaton 下推自动机QC (quality control) 质量管理quadratic performance index 二次型性能指标qualitative physical model 定性物理模型quantized noise 量化噪声quasilinear characteristics 准线性特性queuing theory 排队论radio frequency sensor 射频敏感器ramp function 斜坡函数random disturbance 随机扰动random process 随机过程rate integrating gyro 速率积分陀螺ratio station 比值操作器reachability 可达性reaction wheel control 反作用轮控制realizability 可实现性,能实现性real time telemetry 实时遥测receptive field 感受野rectangular robot 直角坐标型机器人rectifier 整流器recursive estimation 递推估计reduced order observer 降阶观测器redundant information 冗余信息reentry control 再入控制regenerative braking 回馈制动，再生制动regional planning model 区域规划模型regulating device 调节装载regulation 调节relational algebra 关系代数relay characteristic 继电器特性remote manipulator 遥控操作器remote regulating 遥调remote set point adjuster 远程设定点调整器rendezvous and docking 交会和对接reproducibility 再现性resistance thermometer sensor 热电阻resolution principle 归结原理resource allocation 资源分配response curve 响应曲线return difference matrix 回差矩阵return ratio matrix 回比矩阵reverberation 回响reversible electric drive 可逆电气传动revolute robot 关节型机器人revolution speed transducer 转速传感器rewriting rule 重写规则rigid spacecraft dynamics 刚性航天动力学risk decision 风险分析robotics 机器人学robot programming language 机器人编程语言robust control 鲁棒控制robustness 鲁棒性roll gap measuring instrument 辊缝测量仪root locus 根轨迹roots flowmeter 腰轮流量计rotameter 浮子流量计，转子流量计rotary eccentric plug valve 偏心旋转阀rotary motion valve 角行程阀rotating transformer 旋转变压器Routh approximation method 劳思近似判据routing problem 路径问题sampled-data control system 采样控制系统sampling control system 采样控制系统saturation characteristics 饱和特性scalar Lyapunov function 标量李雅普诺夫函数SCARA (selective compliance assembly robot arm) 平面关节型机器人scenario analysis method 情景分析法scene analysis 物景分析s-domain s域self-operated controller 自力式控制器self-organizing system 自组织系统self-reproducing system 自繁殖系统self-tuning control 自校正控制semantic network 语义网络semi-physical simulation 半实物仿真sensing element 敏感元件sensitivity analysis 灵敏度分析sensory control 感觉控制sequential decomposition 顺序分解sequential least squares estimation 序贯最小二乘估计servo control 伺服控制，随动控制servomotor 伺服马达settling time 过渡时间sextant 六分仪short term planning 短期计划short time horizon coordination 短时程协调signal detection and estimation 信号检测和估计signal reconstruction 信号重构similarity 相似性simulated interrupt 仿真中断simulation block diagram 仿真框图simulation experiment 仿真实验simulation velocity 仿真速度simulator 仿真器single axle table 单轴转台single degree of freedom gyro 单自由度陀螺single level process 单级过程single value nonlinearity 单值非线性singular attractor 奇异吸引子singular perturbation 奇异摄动sink 汇点slaved system 受役系统slower-than-real-time simulation 欠实时仿真slow subsystem 慢变子系统socio-cybernetics 社会控制论socioeconomic system 社会经济系统software psychology 软件心理学solar array pointing control 太阳帆板指向控制solenoid valve 电磁阀source 源点specific impulse 比冲speed control system 调速系统spin axis 自旋轴spinner 自旋体stability criterion 稳定性判据stability limit 稳定极限stabilization 镇定，稳定Stackelberg decision theory 施塔克尔贝格决策理论state equation model 状态方程模型state space description 状态空间描述static characteristics curve 静态特性曲线station accuracy 定点精度stationary random process 平稳随机过程statistical analysis 统计分析statistic pattern recognition 统计模式识别steady state deviation 稳态偏差steady state error coefficient 稳态误差系数step-by-step control 步进控制step function 阶跃函数stepwise refinement 逐步精化stochastic finite automaton 随机有限自动机strain gauge load cell 应变式称重传感器strategic function 策略函数strongly coupled system 强耦合系统subjective probability 主观频率suboptimality 次优性supervised training 监督学习supervisory computer control system 计算机监控系统sustained oscillation 自持振荡swirlmeter 旋进流量计switching point 切换点symbolic processing 符号处理synaptic plasticity 突触可塑性synergetics 协同学syntactic analysis 句法分析system assessment 系统评价systematology 系统学system homomorphism 系统同态system isomorphism 系统同构system engineering 系统工程tachometer 转速表target flow transmitter 靶式流量变送器task cycle 作业周期teaching programming 示教编程telemechanics 远动学telemetering system of frequency division type 频分遥测系统telemetry 遥测teleological system 目的系统teleology 目的论temperature transducer 温度传感器template base 模版库tensiometer 张力计texture 纹理theorem proving 定理证明therapy model 治疗模型thermocouple 热电偶thermometer 温度计thickness meter 厚度计three-axis attitude stabilization 三轴姿态稳定three state controller 三位控制器thrust vector control system 推力矢量控制系统thruster 推力器time constant 时间常数time-invariant system 定常系统，非时变系统time schedule controller 时序控制器time-sharing control 分时控制time-varying parameter 时变参数top-down testing 自上而下测试topological structure 拓扑结构TQC (total quality control) 全面质量管理tracking error 跟踪误差trade-off analysis 权衡分析transfer function matrix 传递函数矩阵transformation grammar 转换文法transient deviation 瞬态偏差transient process 过渡过程transition diagram 转移图transmissible pressure gauge 电远传压力表transmitter 变送器trend analysis 趋势分析triple modulation telemetering system 三重调制遥测系统turbine flowmeter 涡轮流量计T uring machine 图灵机two-time scale system 双时标系统ultrasonic levelmeter 超声物位计unadjustable speed electric drive 非调速电气传动unbiased estimation 无偏估计underdamping 欠阻尼uniformly asymptotic stability 一致渐近稳定性uninterrupted duty 不间断工作制，长期工作制unit circle 单位圆unit testing 单元测试unsupervised learing 非监督学习upper level problem 上级问题urban planning 城市规划utility function 效用函数value engineering 价值工程variable gain 可变增益，可变放大系数variable structure control system 变结构控制vector Lyapunov function 向量李雅普诺夫函数velocity error coefficient 速度误差系数velocity transducer 速度传感器vertical decomposition 纵向分解vibrating wire force transducer 振弦式力传感器vibrometer 振动计viscous damping 粘性阻尼voltage source inverter 电压源型逆变器vortex precession flowmeter 旋进流量计vortex shedding flowmeter 涡街流量计WB (way base) 方法库weighing cell 称重传感器weighting factor 权因子weighting method 加权法Whittaker-Shannon sampling theorem 惠特克-香农采样定理Wiener filtering 维纳滤波work station for computer aided design 计算机辅助设计工作站w-plane w平面zero-based budget 零基预算zero-input response 零输入响应zero-state response 零状态响应zero sum game model 零和对策模型z-transform z变换。

ResearchonContrastExp_省略_ationDesignBase

Research on Contrast Experiment of Metering Device Simulation Design Based onDEMYu Jianqun11College of Biological and Agricultural EngineeringJilin UniversityChangchun,Jilin Province,ChinaYan Hui1,22Department of Information Engineering Jilin Business and Technology CollegeChangchun,Jilin Province,Chinayanhui7125@Fu Hong33College of Computer Science and TechnologyJilin UniversityChangchun,Jilin Province,ChinaAbstract—The paper discusses the use of simulation softwarefor metering device design and evaluation based on DEM.Thesoftware simulates the working process of metering device.Thesimulation result is analyzed by comparing to experimentalmeasurement in different rotation speed of the metering device.Through step by step instruction of the software demonstrated,the practicality and stability of this DEM simulation software is verified.Index Terms–DEM;Simulation;Metering device;Corn SeedI.I NTRODUCTIONTraditionally,the development of agriculture machineslike metering device involves design,test,verification,re-design and re-test,prolong the time-to-market,and manydesign problems are not identified in time during productdesign phase,most parameters are chosen based on experience.Overall,the whole development process is timing-consuming and labor intensive.Design goal can not be achieved most of the time.The simulation software developed by the authors facilitates the design and test process.The design efficiency is improved dramatically.The index reflecting performance of a metering deviceincludes single particle rate,re-meter rate,voidage and etc.the working procedure of a metering device as follows:seed feeding,seed discharge,seed retaining and seed dropping. Feeding and discharge mostly determine the performance of a metering device.So,comparison will be made between the DEM simulation results with experiment results under different rotation speed based on the index mentioned above.II.DEM S IMULATION S OFTWAREThe basic idea of discrete element method is to simplify the granule group into a certain shape and quality of the collection of particles,giving access to some kind of mechanical model and the model parameters required of contacts between the particles and particles or boundaries.This article uses Visual C++combining with Pro/Efurthermore development of Pro/E,achieving a Pro/E software and self-developed three-dimensional discreteelement method simulation software platform integration,and thus constructs a set of granule material mechanical parts common digital design method,which integrates design and performance analysis and evaluation as a whole.A CAD modeling of mechanical parts is complicated.But,the surface of mechanical parts contacts with particles is usually composed of some basic geometric shapes:flat,sphere, cylinder,cone and etc.It is common practice to build a mechanical parts discrete element module by reading the basic geometric shapes contacting with discrete particle from CAD model.The control flow of3-D discrete element modeling is as the following:∙Build or using the existing3-D CAD model.∙Load Pro/E application development dynamic linking library.∙Human interaction to read basic geometric shapes.∙Compute modeling parameters.∙Add boundary condition.∙Store the3-D geometric motion information into database.∙Continue the computation until there is no more basic geometric shapes left.The discrete element modeling computation can be started once the boundary model is built.In this phase,object oriented method is applied.Each particle is treated as an object.The simulation is event driven.In every iteration, each particle’s stress,acceleration,velocity and displacement are computed based on pre-defined parameters[size of particle,the length of time step and position],the velocity and stress state of particles from previous iteration.Repeat this process until we get all the results.Store the result into database.The control flow is as Figure1.2010 International Conference on Computer, Mechatronics, Control and Electronic Engineering (CMCE)978-1-4244-7956-6/10/$26.00 ©2010 IEEE CMCE 2010Figure1.Iteration Loop of DEM SimulationThe software tool saves all the information of every time-step.The particle and boundary model is saved into resultfile in binary format.The simulation model reads the computation result to simulate.Due to3-D discrete elementmodeling need huge amount of data,the software tool uses separate files to store particle model and boundary model.By doing this,not only file overflow is avoided,but also it is more convenient and more accurate to read/write data files.Choosing the length of time-step is critical for gettingcorrect simulation result.The common practice is to choose alonger time-step which can guarantee the accuracy and stability of the simulation result and can also minimize simulation time.The software tool chooses10-5s as time step during discrete element simulation.III.C OMPARISON RESULTS BETWEEN DEM SIMULATIONAND ACTUAL EXPERIMENTA.experiment of Combination Inner dimple cell meteringdevice for cornThe metering device has a simple structure,better seed-feeding properties and low damage based on the experiment and experiment result analysis.The main parts of themetering device such as shells,metering device wheel andretaining plate are manufactured with polished plexi-glass in order to capture the seed trajectory inside it.The shell of the metering device is adjusted according to the characteristics of plexi-glass,processing requirement and cost reduction needs.The adjusted metering device is shown as Figure2. The main structural parameters:the diameter of metering device wheel is107mm,the number of feeding holes and inner dimple cells is18respectively,the diameter of the feeding holes is20mm;the length,width and height of the Inner dimple cell are9.5mm,7.2mm and 4.7mm.The working process of the metering device is showed in Figure 3.The seeds enter the inner cavity of metering device through the entrance.The seeds enter the feeding hole under the combined action of wheel rotation,the seed gravity and centrifugal force.The seeds fall into inner dimple cell from feeding hole under the friction and gravity.The extra seeds fall back into inner wheel of the metering device when the feeding hole enters the selection area.While,the seeds remain in the inner dimple cell enter the retaining area under the rotation of the metering device wheel.Once the seeds reach the discharge area,they fall into ground under gravity and centrifugal force.Thus,the full process of the metering device:feeding,selection,retaining and discharge is completed.The inner dimple cell corn metering device is tested through PSJ test platform under four different speeds.The state motion of the seeds inside the metering device was captured by high-speed camera system.We analysis those images using blasting motion analysis software BLASTER’MAS.We also test some functionalities of the metering device and record the seed movement and velocity so that we have comparison basis for discrete element simulation results. Thus,we can prove that discrete element simulation can simulate the working process of meteringdevice.Figure2.Inner Dimple Cell Metering DeviceFigure 3.Working Sketch of IDCMDB.Comparison between combination inner dimple cell metering device simulations results and experimental resultsWe simulate and analysis the seed feeding trajectory when sowing corn seeds of JiFeng 218#and the trajectory of seed after casting under different rotation speed.Those simulation results are compared with combination inner dimple cell metering device experiment results.Figure 4shows the trajectory of seed-feeding and figure 5shows the trajectory of seed dropping.Figure 5shows that the trajectory of casting sees is a parabola and the faster the speed,the greater the parabola opens.This is the same as experimental results.We analyze the seed trajectory using BLASTER’S MAS software.It also shows trajectories in the plane coordinates for each point.The seeds trajectory of experimental result and simulation result is converted into a new coordinates system:the center of the wheel as the origin,X axis is pointing to right,Y axis is pointing upward.We compare the casting trajectory curve of both simulation results and experimental results.Figure 6shows the comparison of feeding seed trajectory between simulation results and experimental results.Figure 7shows the comparison of seed dropping trajectory between simulation results and experimental results.Figure 6and Figure 7show that the simulation curve is consistent with the experimental curve.With the casting wheel speed increases,the bias is increased,too.But,overall,the deviationissmall.Figure 4.Simulation Analysis Diagram of Seed Feeding Trajectory ofIDCMD in DifferentSpeed ofRotationFigure 5.Simulation Analysis Diagram of Seed Dropping Trajectory ofIDCMD inDifferent Speed ofRotationFigure parison of Simulation and Experiment of Seed FeedingTrajectory of IDCMD in Different Speed ofRotationFigure parison of Simulation and Experiment of Seed DroppingTrajectory of IDCMD in Different Speed of RotationThe initial angle and terminating angles of seed discharge are measured for 10times in the 3D simulation and then took the average.At the same rotation speed,the simulated initial angle and terminating angel are smaller than the experiment result.However,the errors are small.The maximum error is 6.61%.The higher the rotation speed,the smaller the error.As the rotation rate increases,the seed discharge angle and terminating angle increase both in simulation and experiment result.This demonstrates that the discrete element simulation method is reliable in analyzing the corn seed discharge initial angle and termination angle.IV.C ONCLUSIONThis article compares the independent developed simulation software based on the discrete element method with the experiment results.The simulation and experiment results are basically consistent.This proves that the discrete element simulation software is feasible and practical.This work also explores a new method in metering device design.A CKNOWLEDGMENTThe authors gratefully acknowledge the funding of this study by the Jilin Province Department of Education research project (2009321)and the Jilin province social science fund project (2009B225).R EFERENCES[1]Cundall P A,Strack O L.A discrete numerical model for granular assembles.Geotechnique,1979,29(1):47~65.[2]Tanaka H,Momozu M,Inooku K,et al.Simulation of soil deformation and resistance at bar penetration by the distinct element method[J].Journal of Terramechanics,2000,37:41~56.[3]Shmulevich I,Asaf Z,Rubinstein D.Interaction between soil and a wide cutting blade using the discrete element method[J].Soil &Tillage Research,2007,97:37~50.[4]Coetzee C J,Els D N J.Calibration of discrete element parameters and the modeling of silo discharge and bucket feeding[J].Computers and Electronics in agriculture,2009,65:198~212.[5]Yang Mingfang ．Study on Digital Design Method of Corn Seed-metering Device based on DEM[D]．JiLin university ,China 2009.[6]Coetzee C J,Basson A H,Vermeer P A.Discrete and continuum modelling of excavator bucket feeding[J].Journal of Terramechanics,2007,44:177~186.[7]Yan Yueming ，Luan Yuzhen ．Research on Experiment of Corn Particle Mechanical Character[J]．Journal of JiLin Agriculture College ，1996，18(4)：75~78.。

Symbolic Discrete-Event Simulation

Abstract
The event-scheduling view of the discrete-event simulation technique is widely used to model and simulate dynamic systems. This paper presents DMOD, a formalization of this technique. It o ers two main advantages. First, it retains the powerful intuitions behind this technique yet makes it easier to use them. In particular, it proposes a new method for logically specifying event preemption. Second, DMOD permits temporal reasoning about modeled systems. This is done by means of symbolic simulation, i.e. simulation in which input parameters can be constrained variables. Symbolic simulation yields histories parameterized by input variables. Simple manipulation of symbolic histories allows proof of a range of temporal properties such as safety, possibility, or sensitivity. Restrictions upon models are outlined which allow symbolic simulation to proceed e ciently. DMOD is illustrated by means of a hybrid system, a railroad crossing with continuous, non-linear state and discrete events. DMOD has been implemented in Prolog and CLP(R) and applied to analysis of a variety of industrial systems.