Analysis of Hybrid field problems by the method of lines with nonequidistant discretization

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故障预测技术综述

故障预测技术综述一、本文概述Overview of this article随着现代工业系统日益复杂化和智能化，故障预测技术在保障系统安全、提升运行效率以及降低维护成本等方面的重要性日益凸显。

本文旨在全面综述故障预测技术的研究现状与发展趋势，以期为相关领域的研究人员和实践者提供有益的参考和启示。

With the increasing complexity and intelligence of modern industrial systems, the importance of fault prediction technology in ensuring system safety, improving operational efficiency, and reducing maintenance costs is becoming increasingly prominent. This article aims to comprehensively review the research status and development trends of fault prediction technology, in order to provide useful reference and inspiration for researchers and practitioners in related fields.本文将首先介绍故障预测技术的基本概念、原理及其在工程实践中的应用背景，阐述故障预测技术的重要性和必要性。

随后，文章将重点回顾和总结近年来在故障预测技术领域的研究成果和进展，包括基于数据驱动的故障预测方法、基于物理模型的故障预测方法以及混合方法等方面的研究现状。

文章还将探讨各种方法的优缺点以及适用范围，并对未来研究方向进行展望。

This article will first introduce the basic concepts, principles, and application background of fault prediction technology in engineering practice, and explain the importance and necessity of fault prediction technology. Subsequently, the article will focus on reviewing and summarizing the research achievements and progress in the field of fault prediction technology in recent years, including the research status of data-driven fault prediction methods, physical model-based fault prediction methods, and hybrid methods. The article will also explore the advantages, disadvantages, and applicability of various methods, and provide prospects for future research directions.通过本文的综述，读者将能够全面了解故障预测技术的发展历程、现状和未来趋势，为深入研究和应用故障预测技术提供有力的理论支撑和实践指导。

混合型本安电路短路瞬态能量分析

混合型本安电路短路瞬态能量分析聂鸿霖1，2，许春雨1，2，宋建成1，2，田慕琴1，2，宋单阳1，2，杨永锴1，2，张晓海1，2（1. 太原理工大学矿用智能电器技术国家地方联合工程实验室，山西太原　030024；2. 太原理工大学煤矿电气设备与智能控制山西省重点实验室，山西太原　030024）摘要：目前针对本安电路本安特性的研究大多以IEC 火花实验装置为实验平台，仅对单一电容电路或电感电路的放电特性进行分析，存在适用性差、实验条件要求高等问题，缺少对混合型本安电路本安特性的研究。

针对该问题，在GB/T 3836.4—2010《爆炸性环境第4部分：由本质安全型“i ”保护的设备》的基础上，以截流型保护方式下的混合型电路为实验对象进行短路瞬态能量实验，通过分析短路瞬态能量释放过程，建立了短路瞬态能量数学模型，分析了等效数学模型中电容、电感、电源电压和保护时间对短路瞬态能量的影响。

Matlab 仿真结果表明：随着电容和电感的增大，短路瞬态能量会逐渐增大，最后趋于一个稳定值；增大电源电压会显著增加短路瞬态能量；缩短动作保护时间可有效降低瞬态能量，但只有当保护时间小于临界时间时其作用才明显。

基于短路瞬态能量数学模型开发了本安电源，进行了短路实验。

实验结果表明：短路电流和电压波形与理论分析基本吻合，短路瞬态能量为33.22 µJ ，符合本安要求，可为本安电源的设计提供参考。

关键词：混合型电路；本安电路；截流型短路保护；短路瞬态能量；保护动作时间；本安电源中图分类号：TD60 文献标志码：AShort circuit transient power analysis of hybrid intrinsically safe circuitNIE Honglin 1,2, XU Chunyu 1,2, SONG Jiancheng 1,2, TIAN Muqin 1,2,SONG Danyang 1,2, YANG Yongkai 1,2, ZHANG Xiaohai 1,2(1. National & Pronvincial Joint Engineering Laboratory of Mining Intelligent Electrical Apparatus Technology,Taiyuan University of Technology, Taiyuan 030024, China ； 2. Shanxi Key Laboratory of Mining ElectricalEquipment and Intelligent Control, Taiyuan University of Technology, Taiyuan 030024, China)Abstract : Currently, research on the intrinsically safe features of intrinsically safe circuits mostly relies on the IEC spark experimental device as the experimental platform. The research only analyzes the discharge features of a single capacitor or inductance circuit. There are problems such as poor applicability and high requirements for experimental conditions. There is a lack of research on the intrinsically safe features of hybrid intrinsically safe circuits. To solve this problem, based on GB/T 3836.4-2010 Explosive Atmospheres - Part 4: Equipment Protected by Intrinsic safety Type "i ", a short circuit transient energy experiment is carried out with the hybrid circuit under the cutoff type protection mode as the experimental object. By analyzing the release process of short circuit transient energy, a mathematical model of short circuit transient energy is established. The paper analyzes the effects of capacitance, inductance, power supply voltage, and protection time on short circuit transient energy in the equivalent mathematical model. The Matlab simulation results show that as the capacitance and inductance increase, the transient energy of the short circuit will gradually increase and eventually approach a stable value.Increasing the power supply voltage will significantly increase the short circuit transient energy. Shortening the action protection time can effectively reduce transient energy. But its effect is only significant when the protection收稿日期：2023-03-27；修回日期：2023-07-10；责任编辑：胡娴。

何谓混杂系统

What Is a Hybrid System?Jan LunzeInstitute of Automation and Computer ControlRuhr-University BochumUniversitätsstraße150D-44780Bochumphone:+492343224071http://www.ruhr-uni-bochum.de/atpAbstract.Hybrid systems have become a major research topic in Control Engineering and other disciplines.Many different models have been proposed for describing them.However, the question what a hybrid system is has remained a matter of debate.This paper argues that state jumps are the basic hybrid phenomenon that cannot be represented and analysed by methods elaborated either in continuous or in discrete systems theory.Hence,a system has to be considered as a hybrid system if both the continuous movement and the state jumps are important for the control task to be solved.1IntroductionDuring the last decade,hybrid dynamical systems have become a major research topic.The conference proceedings like(Alur et al.,1996,Antsaklis et al.,1999, Antsaklis et al.,1995,Antsaklis et al.,1997,Grossman et al.,1993,Lynch and Krogh,2000),the special journal issues(Antsaklis and Nerode,1998a,DEDS’98, 1998)and(Automatica35(3),1999)and this book describe the different research directions and results obtained.The hybrid nature of such systems has attracted the interest of mathematicians,control engineers and computer scientists.The methods applied and the results obtained are as diverse as the backgrounds of these researchers. No common deﬁnition of a hybrid system is available.A major argument in the hybrid system literature says that a given dynamical system should be considered a hybrid system if(and only if)it is impossible to deal with it either as a purely continuous–variable system or as a purely discrete–event system without ignoring important phenomena that result from the combination of continuous and discrete movements of this system.Here and in the rest of the paper, the terms continuous and discrete are used with respect to the range of the signals and not with respect to the time over which the signals are deﬁned.This argument does not clarify what a hybrid system is.Most of the theoretical papers start with a given hybrid system and do not consider whether and why hybrid systems theory has to be applied to the system under investigation.Likewise,ap-plication papers use hybrid models and analysis tools but do not elaborate the main reasons why the system had to be dealt with as a hybrid system.Most of the chap-ters of this book adopt the same position and investigate different kinds of hybrid systems.S. Engell, G. Frehse, E. Schnieder (Eds.): Modelling, Analysis and Design of Hybrid Systems, LNCIS 279, pp. 3−14, 2002.Springer-Verlag Berlin Heidelberg 20024J.LunzeThis situation is due to the fact that the theories of continuous and discrete systems have been elaborated completely separately until recently.Hybrid systems pose the problem of bridging the gap between both theories.This has been done until now not only by considering a combination of continuous and discrete subsystems but also by investigating different extensions of either continuous or discrete systems.Timed or hybrid Petri Nets and timed automata generalise the untimed models that are typically used in discrete systems theory whereas switched systems extend continu-ous systems by discrete phenomena,to mention some example systems investigated in this book.From a theoretical point of view the deﬁnition of theﬁeld of hybrid systems by a collection of different kinds of systems or models is not satisfactory.It is important to know why the new class of hybrid systems has to be deﬁned,because this clariﬁes the novelty of dynamical phenomena to be investigated and the necessity of developing new concepts,methods and tools.This chapter gives an answer to the question of what a hybrid system is.It argues that a new notion should only be deﬁned if the class of systems under investigation does not fall within the framework of the already existing theory.The characterisation of hybrid systems given here is in some contradiction with deﬁnitions used in literature.It should initiate a thorough discussion of this new notion.The problem of deﬁning a hybrid system is rather involved because it is not the characteristics of a given dynamical system alone that distinguishes continuous from discrete and hybrid systems.For example,a tank system is usually considered to be continuous if a level controller is to be designed but it is tackled as a discrete system if it is analysed as a part of a batch process,in which only the discrete state of a full or an empty tank are distinguished.The intentions of modelling,analysis or control have a considerable inﬂuence on the fact whether a system has to be considered as a hybrid system or not.Hybrid dynamical systems have existed for long,but before the appearance of the currently developing theory on hybrid systems,these systems have either been considered as purely continuous or as purely discrete.The reason why they became a hot topic in research is given by the fact that many modern technological processes cannot be analysed and controlled by investigating only the continuous or only the discrete movements.As the theories of continuous or discrete systems have made contradictory basic assumptions,which have to be satisﬁed by the systems in order to make their representation in the framework of the respective theory possible, the consideration of both continuous and discrete phenomena makes it necessary to develop a new theory.Roughly speaking,if the main assumptions of both theories are not satisﬁed,a system has to be dealt with as a hybrid system.Deﬁnition1.A hybrid system is a dynamical system that cannot be represented and analysed with sufﬁcient precision either by the methods of the continuous systems theory or by the methods of the discrete systems theory.What Is a Hybrid System?5 Remarks.Continuous systems theory assumes that the system under consideration can be described by some differential equation˙x=f(x(t),u(t),t),x(0)=x0(1) y(t)=g(x(t),u(t),t)(2) where x∈R n is the state vector,u∈R m the input vector and y∈R r the output vector.x0denotes the initial state.More generally,(1)can be replaced by a set of difference and algebraic equations,which then is called a differential–algebraic system(DAE system).The key assumption of continuous systems theory concerns the fact that the functions f and g satisfy a Lipschitz condition.With respect to the state x this smoothness assumption means for the function f that a constant L has to exist for which the inequalityf(x,u,t)−f(ˆx,u,t) ≤L· x−ˆx (3) holds for all x,ˆx,u and t,where · symbolises a vector norm.A similar condition should be satisﬁed with respect to u.Under this assumption,uniqueness and existence results can be derived for the solution of the differential equation(1).Furthermore, many analysis methods assume the property(3).Discrete systems theory considers systems whose signals have discrete range. That is,all signals are assumed to be binary–valued or have values from aﬁnite or inﬁnite discrete value set.Due to this assumption,the system jumps from one discrete state to another but the continuous movements of the system cannot be described. The abrupt state changes are called events.These basic assumptions of the continuous and discrete systems theories are,on the one hand,contradictory,but on the other hand complementary.A system can be either continuous or discrete,because it either moves Lipschitz–continuously from one real–valued state to another or it jumps among different discrete states.In many situations it is satisfactory to consider either the continuous movement or the discrete jumps of the system.2Hybrid PhenomenaHybrid phenomena are state transitions that cannot be represented or analysed by the methods developed in continuous or discrete systems theory.That is,these phe-nomena do not satisfy the Lipschitz condition(3)and cannot be represented as a sequence of discrete state values.Hence,neither a purely continuous nor a purely discrete representation is appropriate for the task at hand.Hypothesis1Consider a dynamical system subject to some continuous input u.The basic hybrid phenomenon is a combination of continuous state changes and abrupt state jumps.6J.LunzeRemarks.Figure 1shows a state trajectory x (t )of an autonomous ﬁrst–order system which includes a state jump at time t k .The state jump is an autonomous jump which occurs after the state has reached a threshold x s .This trajectory cannot occur as the solution of models that are developed either in continuous or in discrete systems theory.On the one hand,the Lipschitz condition (3)is not satisﬁed for x =x s (and u =0)and,hence,the system cannot be dealt with by the methods of continuous systems theory.On the other hand,the continuous movement of the system cannot be represented by models with discrete–valued state,because such a model could only describe the discrete state jump between x (t k −0)=x s =3and x (t k +0)=1but not the system behaviour for time t =t k .12x s =3xk Fig.1.Hybrid trajectoryIn the literature on hybrid systems four phenomena are called hybrid (Branicky,1995),(Branicky,1996):autonomous and controlled jumps and autonomous and controlled switches.The following remarks show how these phenomena relate to Hypothesis 1.The discussion concentrates on the state evolution x (t ),but extensions to systems with outputs y (t )are mentioned later.Autonomous state jumps.The ﬁrst situation concerns the fact that the state of a system may jump after it has reached a threshold x s or,more generally,the border ofa subset X of the continuous state space.Hence,the derivative ˙xexceeds all bounds but has the form of a Dirac impulse.A simple example is the ﬁrst–order system˙x =−x (t )+δ(x (t )−x s ),x (0)=x 0(4)for which the functionf (x )=−x +δ(x −x s )violates the Lipschitz condition (3)for x =x s .If both the continuous movement between the jumps and the state jumps have to be considered when solving a given modelling,analysis or control task,the system has to be considered as hybrid.The jump occurs at time t k for which the conditionx (t k )=x s (5)is satisﬁed.As the jump has no duration,the time instances t k −0before the k -th jump and t k +0after the jump are often distinguished.The point here is that the timeWhat Is a Hybrid System?7 t k(or t k−0)is implicitly described by the system equation(4).It depends upon the movement of the system and,hence,upon the initial state x0.If the system trajectory is to be determined,the time t k has also to be determined.For the system(4)an explicit representation can be obtainedt k=−ln x sx0if|x0|>|x s|and x0x s>0,(6)but for higher–order linear systems and for most nonlinear systems,such an explicit formula cannot be found.State jumps occur in particular,if the dynamics of a DAE system change when the state reaches a hypersurface in the state space.Then the state vector may even change its length.Details about such systems are described in(Verghese et al.,1981). Controlled state jumps.The state of a system may change discontinuously if the continuous input reaches a given bound u s.Then the vectorﬁeld f(x,u,t)violates the Lipschitz condition for u=u s.A simple example is theﬁrst–order system˙x=−x(t)+δ(u(t)−u s),x(0)=x0(7)whose state jumps if the input reaches the threshold u s.The occurrence time of the jump depends upon the input and,more generally,may also depend upon the state, as in the system˙x=−x(t)+δ(x(t)−u(t)−u s),x(0)=x0.(8)Like autonomous state jumps,controlled jumps occur if the combined vector(x u ) reaches a threshold(or a hyperplane).Contrary to autonomous jumps,controlled jumps can be forced or prevented to occur by appropriately choosing the input u.For example,for the control u(t)=x(t)the state of the system(8)never jumps.These controlled jumps have to be distinguished from jumps that occur due to input jumps or impulses.In systems theory,Dirac impulses are considered as a possible input to the system.Then a state jump occurs due to the inﬁnite magnitude of the input which implies an inﬁnite magnitude of the vectorﬁeld f.Such jumps do not represent a hybrid phenomenon.Therefore,in Hypothesis1the input u has been restricted to be continuous.Then,state jumps are a result of the dynamical properties of the system.In summary,state jumps that are brought about by continuous input signals represent a hybrid phenomenon.Autonomous switching.An abrupt change of the vectorﬁeld f if the state x reaches a given bound is called switching.Formally,the system can be represented by two or more different vectorﬁelds f q together with conditions that describe the validity of these vectorﬁelds,for example by˙x=f(x)(9)8J.Lunzewithf=f1(x)for h(x)≤0f2(x)for h(x)>0.(10)If the system is currently described by the vectorﬁeld f1and the state reaches the border h(x)=0of the region of validity of this vectorﬁeld,the vectorﬁeld switches to f2which is valid until the border described by h(x)=0is reached from the other side.The notion of switching has been introduced in order to denote the fact that the system is governed by two(or more)different differential equations,which can be analysed separately by well known results from continuous systems theory but whose common analysis poses new problems.If the vectorﬁelds are linear and the system is described by˙x=A1x for h(x)≤0˙x=A2x for h(x)>0,the separate analysis of the two models is very easy but the analysis of the system as a whole is difﬁcult.It is known,for example,that the system may be unstable even if the two matrices A1and A2are stable.This consideration shows that the notion of switching has been introduced mainly due to the modelling phenomenon that the vectorﬁeld f cannot be described by a unique analytical expression.This,however,does not imply that the system is hybrid in the sense deﬁned above.A Lipschitz condition may be satisﬁed even if the vector ﬁeld switches.If x s denotes a state on the border between the two regions of validity of the vectorﬁelds f1and f2,the conditionf(x s)−f(ˆx) ≤L· x s−ˆx (11) can be satisﬁed for some constant L for allˆx and x s satisfying the relation h(x s)=0. Hence,switching does not imply state jumps and,therefore,does not describe a hybrid phenomenon.If the vectorﬁeld f in(10)satisﬁes the condition(11),the system can be dealt with by continuous systems theory.If the Lipschitz condition is not satisﬁed,because for some state x s the relationlimε→0f1(x s−ε)=f2(x s)holds,the vectorﬁeld includes some“step”and represents a piecewise continuous function.Then the trajectory x(t)is not differentiable but piecewise differentiable. Even in this case,the trajectory x(t)does not include any state jumps.The system is continuous although its analysis is more involved due to the properties of the vector ﬁeld f.However,the system is not hybrid.This consequence is in contrast with the fact that switching systems are currently considered as an important subject of hybrid systems theory.At this point of the argument it should be stressed that Hypothesis1has been formulated with the aim to investigate the necessity of introducing the new notionWhat Is a Hybrid System?9of hybrid systems.It does not question the fact that switching systems pose a lot of unsolved theoretical problems,which even represent a focus of the current literature of systems theory.Controlled switching.The same arguments apply to systems with controlled swit-ching where the vectorﬁeld also changes abruptly in response to an input command u.Here,the notion of switching is used for systems with piecewise constant input, where for a given time interval the input isﬁxed to some value¯u and,hence,thevectorﬁeld isﬁxed tof(x,¯u)=¯f(x).This kind of switching is nothing else than a change in the vectorﬁeld due to a given input.As the autonomous system can be analysed more easily than a system with(arbitrary)input,the system with piecewise constant input is considered as an autonomous system with switching dynamics,which does not imply that the system exhibits hybrid phenomena.Switched linear systems occur for this reason.If the input to the system˙x=A x(t)+B u(t)is piecewise constant and can assume only the discrete values u i(i=1,...,q),the system is governed by the afﬁne differential equation˙x=A x(t)+B u i=A x(t)+b ias long as the input does not change.The system is considered as a switching system, although it is merely a linear system with piecewise constant input.Similarly to systems with autonomous switching,it is an interesting and useful way to analyse the vectorﬁeld that drives the system between the switchings se-parately and to combine the results to get an analysis result for the overall system. However,from the viewpoint adopted here it becomes clear that this way of analysis does not make the system hybrid.Extension to systems with outputs.For systems where the output y is generated by some function g as described by(2)the considerations above have to be extended. Then,the Lipschitz condition has to be imposed on g as well.Depending on whether both f and g satisfy Lipschitz conditions or not,the system may have different kinds of state or output jumps.As shown in(Verghese et al.,1981),systems may have a continuous output y although the state x jumps.On the other hand,quantised systems(Lunze,2000)are systems with continuous state evolution but jumping output.Note also that the jumps considered here concern the state or the output of the system for continuous input.Output jumps may occur even if the function f and the input u remainﬁnite.An example for this is the linear system˙x=A x+B uy=C x+D u10J.Lunzewith D=O.This system simply transfers a jump in the input directly to the output. Hence,jumps of the output do not necessarily indicate a hybrid system behaviour. It has to be analysed whether the jumps originate from the input or from the system dynamics.Only in the latter case,the system is hybrid.3Representation of Hybrid Systems by Differential EquationsTime is continuously changing.Therefore,the most natural description of any dy-namical system includes signals that are deﬁned over the real time axis R+.As the hybrid systems are considered here with a continuous time axis,the question ari-ses whether they can be described by differential equations.The answer is in the afﬁrmative.Hypothesis2A hybrid dynamical system can be described by the differential equa-tion˙x=f(x,u,t)(12) where the function f includes a Lipschitz–continuous part˜f(x,u,t)and Dirac im-pulses:f(x,u,t)=˜f(x,u,t)+n xi=1f iδ(g i(x(t),u(t),t))(13)Remarks.A state jump occurs at time t k if the state derivative˙x includes a Dirac impulsef iδ(g i(x(t),u(t),t))=δ(t−t k).This impulse occurs whenever the state x and the input u reach at time t k the hyper-plane described by g i(x,u,t)=0.Note that the time t k is implicitly deﬁned by the movement of the system,the input and the deﬁnition of the hyperplane.For example,the system˙x=−x(t)+δ(x(t)−x s),x(0)=x0(14) has a continuous trajectory until the state assumes the value x s at time t k.In this case the system jumps to the new statex(t k+0)=x(t k−0)+1.Hypothesis2and this example point to the fact that a hybrid system can be repre-sented by a differential equation and that the hybrid nature of a system does not imply that the system can only be described by a combination of differential equations andWhat Is a Hybrid System?11 automata,which is often argued in hybrid systems theory(cf.Sect.4).However, they also show that hybrid systems are nonlinear systems,whose vectorﬁelds do not satisfy the smoothness assumption made in nonlinear systems theory.The exi-stence of Dirac impulses in the representation of the vectorﬁeld is a consequence of Hypothesis1.It should be mentioned that the arbiter,which has been used in(Branicky,1995) (p.110)as an example system that cannot be represented by any differential equation, can be described by a differential equation of the form(13).The argument given in(Branicky,1995)is only true with respect to Lipschitz–continuous vectorﬁelds.If the restriction concerning the continuity of the vectorﬁeld is relaxed,a differential equation also exists for the arbiter.4Decomposition of Hybrid SystemsHybrid systems theory has tackled the question of which structure a hybrid system may be appropriately represented.Many authors use the decomposition of the hybrid system into a continuous and a discrete subsystem.This decomposition is reasonable, because this deﬁnition makes it possible to use,at least in part,the models and analysis methods that have been elaborated in the two corresponding theories(Fig.2).The continuous input u and the continuous output y are associated with the continuous–variable subsystem whereas the discrete–valued input v and output w concern the discrete–event subsystem.Both subsystems have to be connected through interfaces that transform continuous signals into discrete and vice versa.These interfaces are called quantiser and injector.Note that all signals represented in Fig.2by some arrow are deﬁned over the continuous time axis.Fig.2.Decomposed hybrid systemDifferent models have been proposed in literature in order to satisfy two aims:•The model should be capable of describing dynamical systems that exhibit hybrid phenomena.•It should be possible to analyse the model by rigorous methods.12J.LunzeThese aims are contradictory.Whereas theﬁrst aim necessitates that the model shouldbe as general as possible,the second aim can be satisﬁed only if the model is as speciﬁcas possible.Whereas in(13)the continuous and the discrete movement of the state are descri-bed in a common model,the decomposition depicted in Fig.2describes the systemby two different mathematical models.As this decomposition should take advantageof the methods of discrete systems theory,untimed discrete-event models are oftenused.This,however,brings about a new theoretical difﬁculty which is to be explainedbelow.The discrete subsystem,like the continuous subsystem,changes its state over thecontinuous time t and can be described by a differential equation.The state z jumpsamong different state values if the vectorﬁeld includes impulses like ˙z=δ(t−t k)˙z=δ(z−v).Theﬁrst equation describes a discrete system in which a state jump occurs after acertain time t k is elapsed.The second equation represents a system whose state jumpoccurs at the time instant at which the input v assumes the discrete value z.Such a description is called a timed model in discrete–event systems theory.Many models proposed in hybrid systems theory assume that the discrete model isuntimed.Typically,(untimed)automata or Petri Nets are used.The time at which astate transition occurs is given by the discrete input v which generally consists ofdiscrete control inputs to the system and of quantised continuous-variable signals(cf.Fig.2).The quantiser is used not only to determine the discrete value of the input v but also to determine the time t k at which the untimed model changes its state.In a more precise representation,the connection between the discrete-valued con-tinuous-time input v and output w of the discrete subsystem has to use two signals:One that describes the values v(t k)or w(t k)of the input and output signal and the second which describes the time instant t k at which the discrete subsystem changesits state(Fig.3).Two blocks have to be used as interfaces between discrete-valuedcontinuous-time and discrete-valued discrete-time signals.This distinction is oftenignored in the hybrid system literature.Fig.3.Quantisation with discrete signal and clock signalWhat Is a Hybrid System?13 The representation of a hybrid system as a composition of a continuous and a discrete subsystem has several advantages:•The representation shows the hybrid nature of the system explicitly.•The methods available for continuous and discrete systems can be applied to the separate subsystems(although the results obtained for the isolated subsystems are not valid for the overall hybrid system.)For example,automata theory can be applied to the discrete subsystems whereas results from continuous systems theory(controllability analysis,stability analysis etc.)can be applied to the continuous subsystem.However,the representation of a system in the form depicted in Fig.2does not imply that the system is hybrid.Switched linear systems may be represented in this form where the quantiser determines the region in which the state resides and the discrete subsystem switches to a new model“number”after the boundary of a region is reached.As the discussion above has shown,such a system can be represented as a nonlinear system with a piecewise continuous vectorﬁeld.5ConclusionsHybrid systems has emerged as a newﬁeld of research.To see the novelty of this ﬁeld,it has to be precisely deﬁned what a hybrid system is.Only with this deﬁnition it becomes clear which new questions have to be answered.This paper shows that state jumps within a continuous movement is the basic and probably the only hybrid phenomenon.Hence only those modelling,analysis and control tools that deal with this particular phenomenon are speciﬁc for hybrid systems.This arguments can be used to evaluate the growing number of papers entitled hybrid systems and to elaborate interesting research topics that bring forward the main ideas and intentions of hybrid systems.For example,analysis methods for switched linear systems are certainly an interesting research topic but from the view-point adopted in this chapter this class of systems does not exhibit the main problems encountered in hybrid systems.Likewise timed Petri Nets or timed automata are in-teresting extensions of classical discrete-event systems,but they should be combined in future research with continuous state or output variables to contribute to the main ideas of hybrid systems theory.This chapter concentrated on the question of what hybrid systems are.It did not concern the problem whether in a speciﬁc situation a hybrid system should be really tackled by the methods developed in hybrid systems theory.For example,methods for the discrete-event representation of quantised or hybrid systems surveyed in the chapter on“Discrete models for hybrid systems”concern the question of how the hybrid nature of the system can be neglected in order to simplify the analysis or control tasks.The resulting discrete-event model ignores the continuous movement of the system and,hence,cannot describe the hybrid system precisely.On the other hand,it may be interesting to analyse continuous systems by using methods developed14J.Lunzein hybrid systems.The example of switching systems,which can be represented by coupled discrete and continuous subsystems,has already been mentioned.Here,the application of a hybrid representation scheme should not be confused with the hybrid nature of the system under investigation.Finally,the question of how to decide from an input-output viewpoint whether a given system is hybrid,is still open.A continuity index has been proposed in(Lichten-berg and Kamau,2001)to distinguish continuous systems from hybrid systems.This method is,however,only theﬁrst step towards a thorough analysis of this important identiﬁcation problem.。

高温下无乳链球菌感染吉奥罗非鱼转录组差异表达分析

高温下无乳链球菌感染吉奥罗非鱼转录组差异表达分析摘要吉奥罗非鱼是由新吉富罗非鱼和奥利亚罗非鱼杂交而来的新品种，具有雄性率高、生长速度快、抗逆能力强等优点。

在高温季节链球菌病频发，已严重影响罗非鱼养殖业的可持续发展。

因此，了解吉奥罗非鱼在高温下链球菌的免疫应答机制是扩大养殖规模亟需解决的问题。

本研究使用链球菌在高温胁迫（32℃）下感染吉奥罗非鱼分析其鳃组织的免疫应答机制。

基因的差异表达不仅与病原体类型和宿主的环境有关，随着感染时间延续其差异表达基因种类和数量也有较大差异。

本研究通过腹腔注射链球菌对感染初始期（B6h）、濒死期（B9h）、拐点（B15h）和平台期（B18h）的转录组进行差异表达分析，另外进行了对应感染期的病理切片显微观察。

在病理切片显微观察分析中发现，除濒死期（B9h）的组织出现明显病变外其它感染期没有明显变化。

转录组共获得差异表达基因8759个，其中上调基因3846个，下调基因4313个，下调的基因数远大于上调基因数。

Go注释中免疫反应和先天免疫在感染初始期（B6h/P6h）、濒死期（B9h/P9h）、拐点（B15h/P15h）和平台期（B18h/P18h）均显著下调，说明宿主的免疫功能在四个感染期都受到了抑制。

细胞因子信号通路、细胞黏附和对抗原的加工和呈递在感染初始期显著上调富集，说明在感染初始期体液免疫发挥重要作用。

其中感染初始期（B6h）和濒死期（B9h）下调显著富集的有对细菌的免疫反应和通过主要相容性复合物对抗原的呈递。

在拐点（B15h）以及平台期（B18h）下调显著富集的有防御反应和对抗原的加工和呈递。

通过KEGG通路富集分析发现四个感染期均显著富集的有phagosome和cell adhesion molecules，在濒死期、拐点和平台期均显著富集的有Intestinal immune network for igA production、cytokine-cytokine receptor interaction。

基于杂交基本解的正交各向异性材料热传导问题有限元法

文章编号： 1009 − 444X （2020）04 − 0305 − 09基于杂交基本解的正交各向异性材料热传导问题有限元法仇文凯，王克用（上海工程技术大学机械与汽车工程学院，上海 201620）摘要：采用基于杂交基本解的有限元法（HFS-FEM ）对二维正交各向异性材料进行热传导分析. 单元域内和单元边界上的温度分布由两个温度场独立描述. 采用基本解的线性组合来近似单元内部温度场，采用标准一维线单元形函数来定义网线温度场. 利用修正变分泛函和散度定理导得相应的有限元列式，通过2个算例与ABAQUS 结果对比，验证了该方法具有有效性. 数值结果表明，该方法在单元形状极度扭曲情形下仍能保持良好的精度，这是区别于传统有限元法的显著特点.关键词：热传导；有限元法；基本解；坐标变换；正交各向异性材料中图分类号： O 343.1 文献标志码： AHybrid Fundamental-Solution-Based FEM for Heat ConductionProblems in Orthotropic MaterialsQIU Wenkai ，WANG Keyong（ School of Mechanical and Automotive Engineering, Shanghai University of Engineering Science, Shanghai 201620, China ）Abstract ：A heat conduction analysis of two-dimensional orthotropic materials was carried out by the hybrid fundamental-solution-based finite element method (HFS-FEM). Temperature distributions within the element domain and on the element boundary were independently described by two temperature fields. A linear combination of fundamental solutions was utilized to approximate the intra-element temperature field while standard one-dimensional shape functions were employed to define the frame temperature field. By virtue of the modified variational functional and divergence theorem, the resultant finite element formulation was derived. The effectiveness of the proposed method was verified by comparing two numerical examples with ABAQUS result. The numerical results demonstrate that the proposed method can still keep excellent accuracy even when the element shape degenerates to a situation of extreme distortion. This is one of marked features which differs from conventional finite element methods.Key words ：heat conduction ；finite element method （FEM ）；fundamental solution ；coordinate transformation ；orthotropic materials材料按照性质和内部结构，一般可分为各向同性材料和各向异性材料. 各向同性材料具有简单、优良的特性，在材料工程领域得到广泛的应用[1 − 3]. Wang 等[1]采用基于杂交基本解的有限元收稿日期： 2020 − 06 − 01基金项目：上海市自然科学基金资助项目（19ZR1421400）作者简介：仇文凯（1994−），男，在读硕士，研究方向为杂交有限元法. E-mail ：*****************通信作者：王克用（1975−），男，副教授，博士，研究方向为Trefftz 有限元法和多孔介质传热. E-mail ：*******************第 34 卷第 4 期上海工程技术大学学报Vol. 34 No. 42020 年 12 月JOURNAL OF SHANGHAI UNIVERSITY OF ENGINEERING SCIENCEDec. 2020方法（HFS-FEM）研究各向同性材料的热传导问题. Gao[2]提出一种求解各向同性材料热传导问题的无网格边界元方法. 目前，在汽车、造船、机械加工、航空航天、军工等工程领域中，许多各向同性材料还不能满足性能需求，因此，研究各向异性材料仍具有重要的理论和实际意义.在热传导问题[4 − 5]中，各向异性材料可分为一般各向异性材料和正交各向异性材料[6 − 9]. 各向异性材料的导热系数在各个方向上是不同的：一般各向异性材料导热系数张量中的所有元素都不为零，而正交各向异性材料导热系数张量中只有主对角线上的元素不为零. 根据导热系数的不同形式，正交各向异性材料可以进一步细分为常系数或变系数两种情况. 目前，关于用边界元法研究正交各向异性热传导问题的报道有很多. Perez 等[10]研究一般积分方程公式并用于求解均匀正交各向异性位势问题. Divo等[11]推导正交各向异性问题基本解的形式. Zhou等[12]针对二维正交各向异性位势问题，建立一个新的势导数边界积分方程，称为自然边界积分方程（NBIE）. 通过边界元法以及其他数值方法分析此类问题已经开展了许多工作，而利用杂交基本解有限元法分析正交各向异性热传导问题的报道却非常少.杂交基本解有限元法是基于杂交Trefftz法的一种高效数值方法. Trefftz方法是由Trefftz于1926年提出的，利用满足控制方程的叠加函数来求解边值问题. 随后，Jirousek等[13]于1977年提出杂交Trefftz有限元法，将边界概念推广到单元间边界，并在单元内部构造满足非齐次Lagrange方程的坐标函数. 目前，杂交Trefftz有限元法已成功地应用于许多工程问题，如位势问题[14 − 15]、平面弹性问题[16]、夹杂分析[17 − 18]、接触问题[19]、轴对称问题[20 − 21]等. Wang等[22]采用杂交Trefftz有限元法（HT-FEM），以T-完备函数作为内部插值函数，研究轴对称位势问题. Wang等[23]基于完备解系提出分析正交各向异性位势问题的杂交Trefftz 有限元模型. 王克用等[24]利用杂交完备解有限元法分析功能梯度材料位势问题. 刘博等[25]利用含有特解的Poisson方程分析杂交Trefftz有限元法的轴对称问题. 杂交基本解有限元法的原始思想由Kompiš等[26]提出，其利用基本解近似位移场和应力场，并利用网线函数来实现相邻单元之间的连接. 高可乐等[27]采用杂交基本解有限元法分析考虑体力项的轴对称弹性问题，与杂交Trefftz 完备解有限元法相比，该方法可避免T-完备函数项选取困难，直接利用基本解来构造满足控制微分方程的单元内部插值函数. 此外，与边界元法相比，该方法消除了积分奇异性的缺点，在网格畸变方面表现出良好性能[28].本文基于文献[10 − 12, 23]的研究工作，利用杂交基本解有限元法分析正交各向异性材料的热传导问题.1 问题描述及基本方程u Qk=[k11k12k21k22]设为温度；为内部热源；k为各向异性材料的导热系数张量，，二维各向异性k12=k21=0k11 k22 0Q=0当，，时，式（1）可表示为式（2）即为二维正交各向异性热传导问题的控制方程. 考虑Dirichlet和Neumann两类边界条件，为¯u¯q n x n yΓ=Γu∪Γq 式中：和分别为给定的温度和热流；和分别为边界上任意点外法线向量的分量；为求解区域的整个边界.2 假定温度场与杂交Trefftz有限元法类似，杂交基本解有限元法采用两套假定的温度场来建立有限元模型，包括非协调单元内部温度场和辅助协调网线场. 精确满足控制方程的单元内部温度场，可保证单元内各点的计算精度，而相邻单元之间则由独立定义在单元边界上的辅助协调网线场连接，与杂交Trefftz有限元法不同的是，单元内部温度场由已知的基本解而不是T-完备函数构造.· 306 ·上海工程技术大学学报第 34 卷2.1 非协调单元内部温度场对于正交各向异性热传导问题，非协调的单元内部温度场可以表示为n s c e j N e (x ,y j )Ωe Γe 式中：为每个单元的源点个数；为待定参数；为二维正交各向异性热传导问题的基本解；为边界包围的单元域. 问题的基本解[10 − 12]应完全满足方程其中r =√(x P −x Q )2k 11+(y P −y Q )2k 22x P y P x Q y Q 式中：、和分别为场点坐标；和分别为源点坐标. x 向和y 向的采用以下关系确定源点的布局，为x c x b λ式中：为单元形心；为单元边界上的点；为无量纲参数. 特殊单元的源点分布如图1所示.中心点yxu e = N e c e (单元域内场)~~u e = N e d e (辅助网线场)源点节点图 1 两个假定温度场及其源点Fig. 1 Two assumed temperature fields with source points2.2 辅助协调网线温度场为保证相邻2个单元之间的连续性，在单元边界上建立一个辅助协调的网线温度场，为Ne (x )d e 式中：为形函数向量；为由单元的节点自由度组成的向量. 两节点单元边上的形函数如图2所示.12−1(1 + ξ)2−1(1 − ξ)2ξ = −1ξ = 0ξ = +1N 2~N 1~图 2 两节点单元边上的形函数Fig. 2 Shape functions on each two-node side of an element沿单元每两节点边上的温度分布为N 1 N 2ξ∈[−1,1]其中，和为在自然坐标系中定义的一维形函数，可表示为相应地，热流可表示为其中3 杂交基本解有限元列式3.1 修正的变分泛函∏m =∑e∏me∏me热传导问题总的杂交变分泛函可以通过得到，其中每个单元上的泛函可表示为第 4 期仇文凯等：基于杂交基本解的正交各向异性材料热传导问题有限元法· 307 ·K ε=G T εH −1εG εP ε式中：为单元刚度矩阵；为等效的节点矢量.3.2 刚体运动的恢复为保证矩阵满秩，在计算单元内部场变量时，需要恢复舍弃的刚体运动项. 为获得单元内任意点的真实温度，根据现有研究[1, 18 − 19]提出的方法，可以很容易地恢复单元内温度场中舍弃的刚体运动项. 因此，温度的最终表达式为c 0u e ˜u ei 式中：为刚体运动参数，该参数可由单元所有节点处的和的最小二乘匹配确定，可写成c 0进一步地，刚体运动参数可表示为m 式中：为单元节点数.4 数值算例ε为定量理解计算精度，对任意变量f 引入相对误差()，可得f HFS −FEM f reference 式中：和分别为杂交基本解有限元解和参考解.为方便表达，算例中所有参数都采用无量纲（没有单位的物理量）的形式表示.4.1 方形区域内的热传导k 11=1k 22=2u =0u =10q =10在第1个算例中，考虑边长为0.1的正方形区域，其中材料的导热系数为和. 对正方形区域左右边界分别施加温度和；上部边界施加热流，下部边界假设为绝热；将整个模型划分为16个四节点四边形单元进行求解计算，如图3所示.yq = −10u = 0u = 10q = 0x4 × 4网格图 3 正方形区域、边界条件和有限元网格Fig. 3 Square domain, boundary conditions andfinite element mesh· 308 ·上海工程技术大学学报第 34 卷γ=e /l γ=0为验证杂交基本解有限元法对网格畸变的不敏感性，定义5种网格变形方案，畸变参数()分别为0.1、0.3、0.5、0.7和0.9，与规则网格（，不变形）的计算结果对比如图4所示. 温度u 和热流q x ε(u )ε(q x )分量的相对误差如图5所示. 从图4中可以看出，即使对于γ = 0.9的极度扭曲网格，的最大值仍低于0.6%，且低于3%，这在工程实践中是可以接受的. 不同畸变程度下相同点的温度结果见表1.(a) γ = 0.1(b) γ = 0.3(c) γ = 0.5(d) γ = 0.7(e) γ = 0.9el图 4 网格畸变方案Fig. 4 Mesh distortion schemes1.0γ(a) 温度 u 的相对误差0.10.20.30.40.50.60.70.80.90.80.60.40.20Point 1 (0.050, 0.025)Point 2 (0.025, 0.025)Point 3 (0.035, 0.075)Point 4 (0.075, 0.075)Point 5 (0.015, 0.045)γ(b) 热流分量 q x 的相对误差0.10.20.30.40.50.60.70.80.94.03.53.02.52.01.51.00.50Point 1 (0.050, 0.025)Point 2 (0.025, 0.025)Point 3 (0.035, 0.075)Point 4 (0.075, 0.075)Point 5 (0.015, 0.045)q x 图 5 温度u 和热流分量的相对误差u q xFig. 5 Relative errors of temperature and heat flux component综上表明，该方法具有对网格畸变不敏感的优点. 将利用有限元软件ABAQUS 在划分841个单元网格时的计算结果作为参考解，杂交基本解有限元法在16个单元网格下的计算结果与之对比，两者能够较好地吻合，如图6所示.4.2 带圆孔的三角陀螺区域内的热传导在此算例中，研究包含圆孔的三角陀螺域的热传导，如图7所示. 模型中，圆孔半径0.1，小弧半径0.1，大弧半径0.4. 外边界上给定温度为u = 20，内边界上给定温度为u = 0. 考虑两种网格划分，分第 4 期仇文凯等：基于杂交基本解的正交各向异性材料热传导问题有限元法· 309 ·k 11=1k 22=3别包含147和1 960个四节点四边形单元. 材料的导热系数为和. 三角陀螺域的温度云图如图8所示. 在粗网格下（含147个单元），杂交基本解有限元计算结果与ABAQUS 计算结果相差不大. 而与1 960个单元下的ABAQUS 计算结果相比，该方法可以在不牺牲精度的前提下，用粗网格（147个单元）计算出几乎相同的结果，这表明了该方法的有效性.q x q y 热流分量和云图分别如图9和图10所示.结果表明，用147个单元的杂交基本解有限元计算结果与用1 960个单元的ABAQUS 解更接近.对比表明，在相同条件下杂交基本解有限元法表现出更好的性能.5 结　语本文利用基于杂交基本解有限元法研究正交各向异性介质中的热传导问题. 该方法采用基本解的线性组合来近似单元域内的温度场，并引入定义在单元边界上的网线场来保证单元间的连续性. 借鉴文献[10 − 12]的工作，构建正交各向异性热传导问题的基本解，通过修正变分泛函，将两个假定的温度场关联起来，并利用高斯散度定理和驻值定理，从而导得单元刚度方程. 该方法在处理一些工程问题和物理问题时，由于其高效灵活的特点受到广泛关注和应用.数值算例表明，该方法具有计算精度高，对网格畸变不敏感且收敛速度快的优势. 虽然该方法解决了稳态正交各向异性热传导问题，但是仍然可以方便地推广至瞬态情形.表 1 不同网格畸变下选定5个点的温度结果Table 1 Results of temperatures at selected five points under different mesh distortions坐标γ=0γ=0.1γ=0.3γ=0.5γ=0.7γ=0.9(0.05,0.025) 5.077 8 5.077 1 5.078 8 5.080 9 5.080 1 5.059 6(0.025,0.025) 2.538 6 2.547 8 2.550 8 2.552 5 2.552 3 2.548 7(0.035,0.075) 3.636 1 3.639 3 3.633 2 3.643 9 3.647 1 3.626 8(0.075,0.075)7.627 57.620 57.628 07.621 67.627 57.635 6(0.015,0.045)1.541 51.540 81.540 11.543 01.547 21.550 9NT1110.0009.1678.3337.5006.6675.8335.0004.1673.3332.5001.6670.8330.000(a) ABAQUS 841网格(b) HFS-FEM 16网格10.0009.1678.3337.5006.6675.8335.0004.1673.3332.5001.6670.8330.000图 6 方形区域温度云图Fig. 6 Cloud maps of temperature in square domainu = 20147网格1 960网格u = 0R 1R 2R 3xy图 7 三角陀螺域，边界条件及有限元网格Fig. 7 Trigonometric gyroscopic domain, boundary conditionsand finite element mesh· 310 ·上海工程技术大学学报第 34 卷(a) ABAQUS 147网格(b) HFS-FEM 147网格(c) ABAQUS 1 960网格20.000NT11NT1118.33316.66715.00013.33311.66710.0008.3336.6675.0003.3331.6670.00020.00018.33316.66715.00013.33311.66710.0008.3336.6675.0003.3331.6670.00020.00018.33316.66715.00013.33311.66710.0008.3336.6675.0003.3331.6670.000图 8 三角陀螺域温度云图Fig. 8 Cloud maps of temperature in the trigonometric gyroscopic domain(a) ABAQUS 147网格(b) HFS-FEM 147网格(c) ABAQUS 1 960网格315.036260.176205.316150.45595.59540.735−14.125−68.986−123.846−178.706−233.566−288.426−343.287387.894323.319258.744194.168129.59365.0180.443−64.133−128.708−193.283−257.859−322. 434−387.009387.894323.319258.744194.168129.59365.0180.443−64.133−128.708−193.283−257.859−322.434−387.009HFL, HFL1(Avg: 75%)HFL, HFL1(Avg: 75%)图 9 三角陀螺域热流分量q x 云图q x Fig. 9 Cloud maps of heat flux component in the trigonometric gyroscopic domain第 4 期仇文凯等：基于杂交基本解的正交各向异性材料热传导问题有限元法· 311 ·参考文献：WANG H, QIN Q H. 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CMA-BJ_2.0_版逐时快速更新追赶循环同化预报系统研发及应用Ⅰ：_资料同化及系统构建

doi:10.11676/qxxb2023.20220172气象学报CMA-BJ 2.0版逐时快速更新追赶循环同化预报系统研发及应用Ⅰ：资料同化及系统构建*陈敏仲跻芹卢冰童文雪冯琎张舒婷黄向宇范水勇CHEN Min ZHONG Jiqin LU Bing TONG Wenxue FENG Jin ZHANG ShutingHUANG Xiangyu FAN Shuiyong1. 北京城市气象研究院，北京，1000892. 中国气象局城市气象重点开放实验室，北京，1000891. Institute of Urban Meteorology，CMA，Beijing 100089，China2. Key Laboratory of Urban Meteorology，China Meteorological Administration，Beijing 100089，China2022-10-27收稿，2023-06-25改回.陈敏，仲跻芹，卢冰，童文雪，冯琎，张舒婷，黄向宇，范水勇. 2023. CMA-BJ 2.0版逐时快速更新追赶循环同化预报系统研发及应用Ⅰ：资料同化及系统构建. 气象学报，81（6）：911-925Chen Min， Zhong Jiqin， Lu Bing， Tong Wenxue， Feng Jin， Zhang Shuting， Huang Xiangyu， Fan Shuiyong. 2023. On the CMA-BJ v2.0 hourly rapid update catch-up cycling assimilation and forecast system. Part Ⅰ: Data assimilation and system attributes. Acta Meteorologica Sinica， 81（6）:911-925Abstract In this paper, the key technical features of CMA-BJ v2.0, the Hourly Rapid Catch-up Cycling Assimilation and Forecast System, are introduced in detail. This system uses IAU (Incremental Analysis Update) as the initialization scheme, which effectively suppresses the initial noise accumulation problem. By fully considering the actual truncated time of all kinds of observations' arrival, the two coupling parts, e.g., the cycle analysis and forecast update implementing data with different cut-off times, run in turn within each hourly cycling to meet the high demands of nowcasting and short-term forecast services. By applying the dynamic forecast hybrid scheme to the assimilated background field, the dynamic constraint of large-scale fields from global models on the development of micro- and meso-scale thermal and dynamic fields in the regional model is realized, and the deformation of large-scale prediction fields caused by the accumulation of rapid update cycle prediction errors is effectively suppressed. In terms of data assimilation, the national-wide mosaic radar reflectivity is only assimilated at the stage of forecast update to avoid continuous accumulation of water vapor. The radar assimilation background field error variance and length scale strategy optimization effectively promote the application of radar assimilation effect. In addition, real-time assimilation of national wind profile radar observations is realized in the CMA-BJ v2.0 system.Key words Rapid updated cycling， Initialization， Radar data assimilation， Wind profile data assimilation， Incremental analysis update摘要介绍了CMA-BJ 2.0版区域逐时快速更新循环同化分析及短时预报业务系统在逐时更新循环和资料同化方面的关键技术特点。

已发表的学术论文

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SCI. , V ol. 2, No. 3, pp. 477-495. 2004.(12)Jiansheng Yang, Qiang Kong, Tie zhou, Ming Jiang, "Cone Beam Cover Method: An Approach toPerforming Backprojection in Katsevich’s Exact Algorithm for Spiral Cone Beam CT", Journal of X-ray Science and Technology，V ol.9 , 1-16, 2004. (EI)(13)蔚喜军，周铁，“流体力学方程的间断有限元方法”，计算物理，V ol. 22, No. 2 (2005), pp108-116. (EI)(14)DanFeng Lu, HongKai Zhao, Ming Jiang, ShuLin Zhou, Tie Zhou, "A Surface ReconstructionMethod for Highly Noisy Point Clouds", N. Paragios et al. (Eds.): VLSM 2005, LNCS 3752, pp.283–294. Springer-Verlag Berlin Heidelberg 2005. (SCI)(15)郭晓虎，杨建生，孔强，周铁，姜明，“锥束覆盖方法的并行实现及性能分析”，中国体视学与图像分析，V ol.10, No.3, 165-169, 2005.(16)Jiansheng Yang, Xiaohu Guo, Qiang Kong, Tie Zhou, Ming Jiang, "Parallel Implementation of theKatsevich's FBP Algorithm", International Journal of Biomedical Imaging, Special Issue on"Development of Computed Tomography Algorithms", 2006, Article ID 17463.(17)Ge Wang, Ming Jiang, Jie Tian,Wenxiang Cong, Yi Li, Weimin Han, Durai Kumar, Xin Qian, HaiouShen, Tie Zhou, Jiantao Cheng, Y ujie Lv, Hui Li, Jie Luo, "Recent development in bioluminescence tomography", 2006 3RD IEEE INTERNA TIONAL SYMPOSIUM ON BIOMEDICAL IMAGING: FROM MACRO TO NANO, VOLS 1-3, IEEE International Symposium on Biomedical Imaging, pages: 678-681, 2006. (SCI)(18)Ming Jiang, Tie Zhou, Jiantao Cheng, Wengxiang Cong, Durairaj Kumar, Ge Wang, "ImageReconstruction for Bioluminescence Tomography", RSNA 2005.(19)Ming Jiang, Tie Zhou, Jiantao Cheng, Wenxiang Cong, Ge Wang, "Development of bioluminescencetomography", art. no. 63180E, Developments in X-Ray Tomography V, PROCEEDINGS OF THE SOCIETY OF PHOTO-OPTICAL INSTRUMENTATION ENGINEERS (SPIE), vol. 6318, pages: E3180-E3180, 2006. (SCI)(20)Jinxiao Pan, Tie Zhou, Yan Han, Ming Jiang, “Variable Weighted Ordered Subset ImageReconstruction Algorithm”, International Journal of Biomedical Imaging, V olume 2006 (2006), doi:10.1155/IJBI/2006/10398 Article ID 10398, 7 pages(21)Seung Wook Lee, Jinxiao Pan, Chunhua Liu, Tie Zhou, Cheul-Muu Sim, Ming Jiang, A PreliminaryStudy of Iterative Reconstruction Algorithms for Neutron Tomography, The 8th World Conference on Neutron Radiography, National Institute of Standards and Technology, Gaithersburg, MD, 16 - 19 October, 2006. (SCI)(22)G. Wang, X. Qian, W. 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径向基函数-稀疏多项式混沌展开混合代理模型可靠性分析方法

Journal of Mechanical Strength2023,45(5):1108-1116DOI :10.16579/j.issn.1001.9669.2023.05.014∗20211225收到初稿,20220217收到修改稿㊂国家自然科学基金项目(51905146,12102122),河北省博士后择优资助基金(B2021005004)资助㊂∗∗赵子达,男,1997年生,河北保定人,汉族,河北工业大学硕士研究生,主要研究方向为工业机器人可靠性分析㊂∗∗∗欧阳衡(通信作者),男,1992年生,湖南衡阳人,汉族,河北工业大学机械工程学院讲师,主要研究方向为可靠性分析与设计理论㊂径向基函数-稀疏多项式混沌展开混合代理模型可靠性分析方法∗RELIABILITY ANALYSIS ON HYBRID SURROGATE MODEL OF RADIAL BASIS FUNCTION AND SPARSE POLYNOMIALCHAOS EXPANSION赵子达∗∗1㊀张德权1㊀欧阳衡∗∗∗1㊀武泽平2(1.河北工业大学机械工程学院省部共建电工装备可靠性与智能化国家重点实验室,天津300401)(2.国防科技大学空天科学学院,长沙410073)ZHAO ZiDa 1㊀ZHANG DeQuan 1㊀OUYANG Heng 1㊀WU ZePing 2(1.Key Laboratory of Reliability and Intelligence of Electrical Equipment ,School of Mechanical Engineering ,Hebei University of Technology ,Tianjin 300401,China )(2.College of Aerospace Science and Engineering ,National University of Defense Technology ,Changsha 410073,China )摘要㊀为解决现有代理模型在可靠性分析中存在的普适性差㊁分析精度低的问题,提出一种融合径向基函数(Radial Basis Function,RBF)和稀疏多项式混沌(Sparse Polynomial Chaotic Expansion,SPCE)展开的混合代理模型,实现功能函数的快速准确预示,从而提高可靠性分析的工程适用性和计算精度㊂采用正交匹配追踪技术求解多项式混沌(Polynomial Chaotic Expansion,PCE)展开中的重要项,获得SPCE 展开模型,并将其增广到RBF 代理模型中,形成RBF-SPCE 展开混合代理模型,提高代理模型的预测精度,进而结合蒙特卡洛模拟(Monte Carlo Simulation,MCS)方法开展复杂结构的可靠性分析㊂三个数值算例对比了所提方法与传统RBF 方法及增广RBF 方法的具体差异㊂结果表明,所提方法对结构可靠性分析具有更高的精度和效率㊂最后,汽车侧面碰撞的工程算例说明所提方法对复杂问题具有良好的工程适用性㊂关键词㊀径向基函数㊀稀疏多项式混沌展开㊀混合代理模型㊀可靠性分析㊀计算效率中图分类号㊀TB114.3Abstract ㊀To resolve the poor universality and low accuracy of the existing surrogate models for reliability analysis,a hybridsurrogate model based on radial basis function (RBF )and sparse polynomial chaotic expansion (SPCE )was proposed.Itrealized rapid and accurate prediction of performance functions to improve the engineering applicability and the accuracy of structural reliability analysis.Importantly,the orthogonal matching pursuit technology was applied to obtain the important terms inPCE,and an SPCE model could be established directly to form the RBF-SPCE model for improving the prediction accuracy of surrogate model.Subsequently,the reliability analysis of complex structures is carried out based on Monte Carlo simulation(MCS).In this work,three simulation cases were implemented to compare the performance of the proposed method with the traditional RBF model and augmented RBF model.The results illustrated that the proposed method has higher accuracy andefficiency for structural reliability analysis.Finally,a vehicle side impact engineering example illustrated that the proposedmethod has good engineering applicability for complex problems.Key words ㊀Radial basis function ;Sparse polynomial chaotic expansion ;Hybrid surrogate model ;Reliabilityanalysis ;Computational efficiencyCorresponding author :OUYANG Heng ,E-mail :ouyangheng @ ,Tel :+86-22-60202050The project supported by the National Natural Science Foundation of China (No.51905146,12102122),and the HebeiProvincial Department of Human Resources and Social Security of China (No.B2021005004).Manuscript received 20211225,in revised form 20220217.㊀第45卷第5期赵子达等:径向基函数-稀疏多项式混沌展开混合代理模型可靠性分析方法1109㊀㊀0㊀引言㊀㊀在航空航天设备㊁特种装备㊁土木结构等复杂工程设计问题中,广泛存在着因材料差异㊁几何特性和边界条件所引入的各类不确定性因素,导致产品实际性能与设计状态之间存在偏差,进而可能影响结构可靠性与安全性[1-4]㊂结构可靠性通过定量分析这些不确定性的影响,合理评估失效概率,以保证结构安全㊂大型工程结构中由于涉及的各种不确定性变量与实际模型之间的响应关系复杂㊂因此,在对复杂工程结构进行可靠性分析时,通常需要采用有限元分析方法来得到实际模型的对应响应,计算成本较高㊂为降低有限元分析所导致的昂贵成本,各类代理模型迅速发展起来,并在实际工程中得到了广泛应用[5]㊂经典的代理模型包括响应面模型(Response Surface Method,RSM)㊁Kriging模型㊁径向基函数(Radial Basis Function,RBF)㊁多项式混沌展开(Polynomial Chaos Expansion,PCE)等[6-7],其中,RBF 能够以较高的拟合精度拟合高度非线性的极限状态函数,从而有效地处理高维非线性问题㊂张天龙等[8]引入了基于主动学习的RBF代理模型方法,提高了模型计算的效率与稳定性,解决了强度折减法在边坡系统可靠性分析当中计算成本过高的问题㊂刘鑫等[9]基于RBF建立了乘员约束系统的代理模型,并应用于近似可靠性优化设计问题中,高效准确地为汽车乘员约束系统匹配最佳的设计参数㊂然而,传统的RBF方法无法较好地拟合线性系统[10],将多项式增广到RBF 模型可以有效解决该问题,该方法能够充分利用所有样本点,广泛应用于结构可靠性分析和设计领域㊂WEI Y等[11]针对含隐式和非线性极限状态函数的可靠性分析问题,提出了一种基于协同主动学习策略的增广RBF代理模型,用于含隐式和非线性功能函数的可靠性分析㊂胡常福等[12]针对结构极限承载力的可靠性分析中计算成本过高的问题,将不同次数增广基多项式引入到RBF响应面模型,有效减少了计算成本㊂WU Z P等[13]为了解决工程中的全局敏感性分析问题,提出一种正交增广RBF方法来估计Sobol指数,提高了RBF模型拟合复杂极限状态函数的适应性㊂近年来,PCE方法作为一种流行的代理模型,采用不同正交多项式近似替代原复杂模型,广泛应用于不确定量化分析及敏感性分析等领域㊂黄悦琛等[14]为了对无人机飞行性能展开不确定性分析,基于广义PCE方法建立了代理模型,较蒙特卡洛方法大幅提升了计算效率㊂刘安民等[15]针对气动力参数对翼伞飞行性能的不确定性量化评估问题,采用PCE方法建立代理模型,在不降低精度的前提下提高了计算效率㊂李阳天等[16]针对求解PCE系数中出现的过拟合问题,提出一种改进的PCE方法,有效解决了PCE系数过拟合的问题㊂ZHANG X F等[17]提出一种有效的结构不确定性分析的PCE方法,克服了一般多项式混沌展开过程中存在的维数问题㊂赵威等[18]为解决传统PCE方法的多重共线性问题,提出一种稀疏偏最小二乘回归-多项式混沌展开代理模型方法,实现了较高精度的结构可靠度分析㊂TORII A J等[19]提出了一种解决基于风险和可靠性的设计优化问题的方法,该方法结合PCE与梯度算法进行失效概率评估和灵敏度分析,从而减轻此类问题所需的计算负担㊂PAN Q等[20]为了克服最小二乘法求解容易产生过拟合的缺点,利用贝叶斯分析给出的预测均值和方差,提出了一个学习函数选择信息量最大的样本构建PCE,以提升计算效率㊂然而,目前代理模型研究领域中,仍缺乏将RBF 和PCE两者优点相结合所发展的可靠性分析方法㊂PCE方法通过正交多项式的线性组合逼近真实模型,具有正交特性,能够表征模型响应的全局行为,并与输入随机变量的联合分布相联系㊂同样,RBF方法采用基函数的线性组合来近似模型响应㊂因此,融合RBF 与PCE的代理模型方法能够兼具两种方法的优点,提高模型预测能力㊂本文提出一种径向基函数和稀疏多项式混沌(RBF and Sparse Polynomial Chaotic Expansion,RBF-SPCE)相融合的混合代理模型的可靠性分析方法,将SPCE中重要的基函数项增广到RBF 代理模型,结合两种模型各自优势,在减少样本点的前提下提高计算效率㊂1㊀多项式混沌展开及径向基函数代理模型1.1㊀多项式混沌展开代理模型㊀㊀PCE是用一组与输入变量分布类型相应的正交多项式之和来近似隐式函数的高精度代理模型㊂该模型在概率论框架下具有表达任意有限方差随机响应的能力,且对于光滑的输入输出关系能够迅速收敛㊂PCE模型利用不同正交多项式来分别对应不同变量分布,对应关系如表1所示㊂因此,PCE模型可表示为[21]597-617y=f(x)=ðm j=1λj p j(x)(1)式中,j为PCE项数;λj为第j个待求解的PCE系数;m为PCE的总项数;pj(x)为第j维标准随机变量所对应的一维正交多项式基函数的乘积,可采用式(2)表征㊂pj(x)=ᵑd k=1ϕj k(x k)(2)式中,k为随机变量维数;ϕj k(x k)为第k维随机变量x k㊀1110㊀机㊀㊀械㊀㊀强㊀㊀度2023年㊀所对应的一维正交多项式基函数;d 为随机变量维数㊂PCE 的总项数m 由PCE 的最高阶数p 和随机变量维数d 共同决定,具体函数关系可表示为m =(d +p )!d !p !(3)表1㊀不同分布类型对应的典型正交多项式Tab.1㊀Typical orthogonal polynomials correspondingto different distribution types分布类型Distribution type概率密度函数Probability density function正交多项式Orthogonal polynomial高斯分布Gaussian distribution1/2πexp(-x 2/2)埃尔米特Hermite 均匀分布Uniform distribution1/2勒让德Legendre 伽玛分布Gamma distributionx a e-x拉盖尔Laguerre 贝塔分布Beta distribution(1-x )a (1+x )bB(a )B(b )雅可比Jacobi依据变量的不同分布类型选取对应的正交基函数构建PCE 模型,其中PCE 模型系数的求解是该方法中关键步骤,决定了代理模型的精度,从而影响到可靠性分析的精度㊂一种方法是基于Galerkin 投影法,利用正交多项式基函数的正交性,将PCE 模型依次投影到混合多项式上,获得PCE 模型的系数;另一种方法是基于线性回归法,利用最小二次回归估算PCE 模型系数㊂回归法相比于Galerkin 投影法,PCE 模型收敛速度更快,算法步骤如下:步骤1:实验设计㊂通过拉丁超立方抽样(LatinHypercube Sampling,LHS)得到一组输入变量的样本X =(x 1,x 2, ,x n )T (n 为样本点个数),并计算所得样本点处原模型的响应值F =(f (x 1),f (x 2), ,f (x n ))T ㊂步骤2:求取PCE 模型的展开系数㊂将样本X =(x 1,x 2, ,x n )T 和对应的响应值F =(f (x 1),f (x 2), ,f (x n ))T分别代入式(1)PCE 模型左端和右端得p 1(x 1)p 2(x 1) p m (x 1)p 1(x 2)p 2(x 2) p m (x 2)︙︙︙p 1(x n )p 2(x n )p m (x n )éëêêêêêêùûúúúúúúλ1λ2︙λm éëêêêêêêùûúúúúúú=f (x 1)f (x 2)︙f (x n )éëêêêêêêùûúúúúúú(4)式(4)可简写为Pλ=F(5)式中,P =p 1(x 1)p 2(x 1) p m (x 1)p 1(x 2)p 2(x 2) p m (x 2)︙︙︙p 1(x n )p 2(x n )p m (x n )éëêêêêêêùûúúúúúú,λ=λ1λ2︙λm éëêêêêêêùûúúúúúú,F =f (x 1)f (x 2)︙f (x n )éëêêêêêêùûúúúúúú(6)㊀㊀PCE 模型的展开系数向量λ的计算式为λ=(PP T )-1P T F(7)㊀㊀步骤3:采取蒙特卡洛模拟(Monte CarloSimulation,MCS)方法,利用步骤2获得的PCE 模型进行可靠性分析㊂在求解PCE 模型系数过程中,PCE 模型总项数随着随机变量维数和最高阶数的增加而急剧增加,从而导致维数灾难 ,大幅增加了计算成本㊂为平衡计算效率和计算精度,应采取更加高效的方法求解PCE 展开系数㊂根据稀疏效应准则[22],对PCE 模型结果影响较大的展开式主要是单一输入变量的多项式基函数和变量之间的低阶交叉项,其余混合多项式对应的PCE 模型系数趋近于0,对PCE 模型精度影响较小,从而减少混合基函数的选取数量,提高建立PCE 模型的效率㊂为了高效地求解PCE 模型展开系数,采取正交匹配追踪算法[23]进行重要基函数的选取㊂1.2㊀RBF 代理模型㊀㊀RBF 代理模型可表示为[24]f R (x )=ðn i =1w iφ( x-x i )(8)式中,φ为径向基函数; x -x i 为两点间的欧氏距离;w i 为待求解未知权重系数㊂将样本点代入RBF 模型后,可得AW =F(9)其中,A =φ( x 1-x 1 )φ( x 1-x 2 ) φ( x 1-x n )φ( x 2-x 1 )φ( x 2-x 2 )φ( x 2-x n )︙︙︙φ( x n -x 1 )φ( x n -x 2 )φ( x n -x n )éëêêêêêùûúúúúú,W =λ1λ2︙λn éëêêêêêêùûúúúúúú,F =f (x 1)f (x 2)︙f (x n )éëêêêêêêùûúúúúúú(10)㊀㊀RBF 代理模型的待求解权重系数向量W 为W =(AA T )-1A T F(11)㊀㊀由于使用了高度非线性的径向基函数,式(8)中的RBF 模型可以精确拟合非线性响应,但仍无法较好拟合线性响应㊂为了解决该问题,将多项式增广到RBF 代理模型中,表示如下:f (x )=ðni =1w iφ( x-x i )+ðqj =1b jc j(x )(12)㊀第45卷第5期赵子达等:径向基函数-稀疏多项式混沌展开混合代理模型可靠性分析方法1111㊀㊀式中,b j 为由于多项式项而在插值中引入的q 个未知系数;c j (x )为增广多项式中的第j 项,对于二维问题,增广多项式中的单项依次为[1,x ,y ,x 2,xy ,y 2, ]T ㊂式(12)中引入的额外q 个未知系数的数量取决于多项式阶数和问题维数d ,其函数关系为q =1常数q =d +1线性多项式q =(d +1)(d +2)/2二次多项式q =(d +1)(d +2)(d +3)/6三次多项式ìîíïïïïï(13)㊀㊀RBF 模型的增广可采用线性或二次多项式等函数,本文只研究增广线性多项式(Linear Polynomial,LP)函数的RBF 模型㊂在本文余下部分,皆用RBF-LP 表示增广线性多项式函数的RBF 模型㊂式(12)中存在n 个方程,n +q 个未知系数,未知系数数量大于可用方程数量㊂因此,由下列q 个正交约束条件可得到附加的q 个方程,从而求解附加未知系数:ðni =1w i c j(x i)=0(14)式中,j =1,2, ,q ㊂结合式(12)㊁式(14),可得到(n +p )个方程,其矩阵形式表示为A C C T()W B()=F()(15)式中,C =c 1(x 1)c 2(x 1) c q (x 1)c 1(x 2)c 2(x 2) c q (x 2)︙︙︙c 1(x n )c 2(x n )c q (x n )éëêêêêêêùûúúúúúú,B =b 1b 2︙b q éëêêêêêêùûúúúúúú(16)㊀㊀求解式(15)的方程组得到W ㊁B ,表示为W B()=A C CT()-1F()(17)㊀㊀由式(8)㊁式(12)可以看出,RBF 模型在抽样点上预测的函数值与对应的真实函数值相等㊂因此,方差分析法无法检验RBF 模型的准确性,应使用非设计点对RBF 模型进行评估㊂非设计点的均方根误差R RMSE 为R RMSE =ðti =1[f (x i)-f ᶄ(x i )]2/t(18)式中,t 为非设计点的个数;f (x i )为第i 个非设计点的真实函数值;f ᶄ(x i )为RBF 模型在第i 个非设计点处的预测函数值㊂式(8)㊁式(12)分别为RBF 和RBF-LP 代理模型的表达式,在构建模型并求出未知系数后,这两个代理模型都具有显式的函数表达式,任意输入都可以通过构建好的RBF 模型求出对应的响应,可采用MCS 方法求解失效概率[25]:P ^f ʈ1N ðNi =1I [f (x i )ɤ0](19)式中,P ^f 为预测失效概率;N 为MCS 的样本数;I [㊃]为指示函数,表示为I [㊃]=1f (x i )ɤ0f (x i )>0{(20)2㊀RBF-SPCE 代理模型㊀㊀现有径向基函数代理模型研究中主要采取多项式增广㊂PCE 模型不仅能够拟合线性响应,还具有良好的全局拟合能力,与径向基函数代理模型相结合的方法已在全局敏感性分析和可靠性优化领域得到应用[21]597-617[26]㊂为了减少样本点数量并提高代理模型预测精度,从而保证可靠性分析的求解效率,本文采用SPCE 模型代替RBF-LP 模型中的线性多项式项,RBF-SPCE 模型的表达式为f (x )=ðn i =1w iφ( x-x i )+ðmj =1λj p j(x )(21)式中,φ( x -x i )㊁p j (x )分别为径向基函数和正交PCE 多项式㊂在本方法中,利用RBF 模型逼近强非线性多项式,利用PCE 模型逼近弱非线性多项式㊂在RBF 项中,最优形状参数c 的确定是一个全局优化问题,采用粒子群优化[27](Particle Swarm Optimization,PSO)算法求解㊂该方法通过模拟鸟类群集或鱼群的行为从而在问题空间中寻找最优解决方案㊂具有惯性权值的PSO 算法能保证较好的全局收敛性,因此本文采用该算法,主要步骤如下:v ij (t +1)=wv ij (t )+c 1r 1(t )[p ij (t )-x ij (t )]+c 2r 2(t )[p g j (t )-x ij (t )](22)式中,x ij (t +1)=x ij (t )+v ij (t +1)(23)式中,c 1㊁c 2均为学习因子;r 1㊁r 2均为[0,1]范围内的均匀随机数;i =1,2, ,N ,其中N 为粒子个数;j =1,2, ,D ,其中D 为搜索空间维数;v ij 为粒子速度;x ij为当前迭代中的粒子;p ij 为存储第i 个粒子当前搜索到的最优位置,即个体极值;p g j 为存储整个粒子群当前搜索到的最优位置,即全局极值;w 为惯性权重,表示当前迭代速度与上一次迭代速度的关系㊂采用较多的是动态惯性权重,表达式为w =w max -(w max -w min )tT max(24)式中,T max 为最大进化代数;w max ㊁w min 分别为最大和最小惯性权重,在本文中取w max =0.9,w min =0.4;t 为当㊀1112㊀机㊀㊀械㊀㊀强㊀㊀度2023年㊀前迭代次数㊂式(21)存有n 个方程,n +m 个未知系数,未知量大于可用方程㊂因此,由下列m 个正交约束条件可得出附加的m 个方程,从而求解附加系数为ðni =1w i p j(x i)=0(25)式中,j =1,2, ,m ㊂结合式(21)和式(25),得到(n +m )个方程,其矩阵形式表示为A P P T()W λ()=F0()(26)式中,P =p 1(x 1)p 2(x 1) p m (x 1)p 1(x 2)p 2(x 2) p m (x 2)︙︙︙p 1(x n )p 2(x n )p m (x n )éëêêêêêêùûúúúúúún ˑm,λ=λ1λ2︙λm éëêêêêêêùûúúúúúú,F =f (x 1)f (x 2)︙f (x n )éëêêêêêêùûúúúúúú(27)㊀㊀RBF-SPCE 代理模型的待求解系数向量W 和展开系数向量λ可求得W λ()=A P P T()-1F()(28)㊀㊀在求解出RBF-SPCE 代理模型系数后,采用MCS 方法求解失效概率,具体算法过程如下:步骤1:生成具有n 个样本点的初始样本集,初始迭代次数设置为k =1,采用LHS 方法生成RBF-SPCE 代理模型的初始样本集㊂步骤2:对步骤1中生成的初始样本集求得原计算模型对应的响应值㊂对于实际问题,需采用有限元法计算模型响应㊂步骤3:更新样本集从而包含所有样本点,n =n +m ㊂在第一次迭代中,k =1,m =0,不添加额外的样本点㊂步骤4:采用正交匹配追踪算法筛选出RBF-SPCE 模型中重要的基函数㊂步骤5:使用样本点集n ,采用式(17)构建RBF-SPCE 代理模型㊂步骤6:采用MCS 方法计算第k 次迭代时RBF-SPCE 代理模型的失效概率P f ㊂步骤7:判断是否满足收敛准则㊂如果满足收敛准则,则迭代停止;否则,继续执行步骤8㊂本文中采用的收敛准则是失效概率P f 在两个连续迭代步的相对误差小于容许值,容许值设置为1%㊂步骤8:生成带有m 个样本点的额外样本集,更新迭代次数k ,进行下一轮迭代㊂步骤9:对步骤8生成的额外样本集m 求取原模型对应的响应值,返回步骤3㊂基于RBF-SPCE 代理模型的可靠性分析方法具体流程如图1所示㊂图1㊀RBF-SPCE 代理模型运算流程Fig.1㊀Flowchart of RBF-SPCE surrogate model3㊀数值算例㊀㊀算例1:该算例为二维问题,功能函数为g X ()=exp(0.2x 1+6.2)-exp(0.47x 2+5.0)(29)式中,x 1㊁x 2均为不确定变量,具体分布如表2所示㊂表3中给出了不同可靠性方法在求解算例1问题的结果㊂由表3可以看出,直接采用MCS 方法得到的失效概率为0.9372%,相比于RBF 模型和RBF-LP 模型的22次功能函数调用次数,RBF-SPCE 模型在调用20次㊀第45卷第5期赵子达等:径向基函数-稀疏多项式混沌展开混合代理模型可靠性分析方法1113㊀㊀功能函数后就达到稳定,且相对误差0.619%也小于RBF 和RBF-LP 模型㊂图2给出了在初始样本点相同的情况下,随着不断迭代增加样本点,三种方法计算得到失效概率的变化情况㊂表2㊀算例1不确定变量分布Tab.2㊀Distributions of uncertain variables for example 1变量Variable 均值Mean 标准差Standard deviation分布类型Distribution x 11正态分布Gaussian distributionx 201正态分布Gaussian distribution表3㊀不同可靠性方法求解算例1问题的对比Tab.3㊀Comparison of different reliability methods in example 1方法Method 失效概率Failure probability /%相对误差Relative error /%功能函数调用次数Number of performance functionRBF0.89804.18322RBF-LP 0.92201.62222RBF-SPCE 0.94300.61920MCS0.9372图2㊀算例1迭代过程中失效概率的变化情况Fig.2㊀Failure probability in the iteration process of example 1算例2:该算例包含交叉项二次多项式和线性多项式的二维问题,其不确定变量分布如表4所示,功能函数表达式为[28]g X ()=(x 1-x 2)210-x 1+x 22+52(30)㊀㊀表5中给出了不同可靠性方法求解算例2问题的结果,图3给出了在初始样本点相同的情况下,随着不断迭代增加样本点,三种方法计算得到失效概率的变化情况㊂由此可以看出,由于该算例中包含线性多项式,RBF 模型计算得到的失效概率相比于MCS 方法误差较大㊂RBF-LP 模型计算得到的结果相对准确,但两种模型的功能函数调用次数相当㊂相比于RBF 模型和RBF-LP 模型,RBF-SPCE 模型仅调用8次功能函数就达到稳定的计算结果,所得失效概率的相对误差也最小,仅为0.523%㊂表4㊀算例2不确定变量分布Tab.4㊀Distributions of uncertain variables for example 2变量Variable 均值Mean 标准差Standard deviation分布类型Distribution x 11正态分布Gaussian distributionx 201正态分布Gaussian distribution表5㊀不同可靠性方法求解算例2问题的对比Tab.5㊀Comparison of different reliability methods in example 2方法Method 失效概率Failure probability /%相对误差Relative error /%功能函数调用次数Number of performance functionRBF0.39805.42026RBF-LP0.42400.76025RBF-SPCE 0.42300.5238MCS0.4208图3㊀算例2迭代过程中失效概率的变化情况Fig.3㊀Failure probability in the iteration process of example 2算例3:该算例是一个包含线性多项式的七维可靠性分析问题,其不确定变量的分布如表6所示,功能函数为g X ()=x 2x 3x 4-x 23x 24x 5x 6x 7-x 1(31)㊀㊀表7中给出了不同可靠性方法在求解算例3问题的结果,图4给出了在初始样本点相同的情况下,随着不断迭代增加样本点,三种方法得到的失效概率的变化㊂由此可以看出,RBF㊁RBF-LP㊁RBF-SPCE 三种方法都在调用40次功能函数以后达到稳定,其中,RBF-SPCE 方法得到的失效概率与MCS 方法的相对误差最小,仅为1.128%㊂㊀1114㊀机㊀㊀械㊀㊀强㊀㊀度2023年㊀表6㊀算例3不确定变量分布Tab.6㊀Distributions of uncertain variables for example 3变量Variable 均值Mean 标准差Standard deviation分布类型Distribution x 10.010.003正态分布Gaussian distributionx 20.30.015正态分布Gaussian distributionx 336036正态分布Gaussian distributionx 4 2.26ˑ10-41.13ˑ10-5正态分布Gaussian distributionx 50.50.05正态分布Gaussian distributionx 60.120.006正态分布Gaussian distributionx 7406正态分布Gaussian distribution表7㊀不同可靠性方法求解算例3问题的对比Tab.7㊀Comparison of different reliability methods in example 3方法Method 失效概率Failure probability /%相对误差Relative error /%功能函数调用次数Number of performance functionRBF0.301010.65640RBF-LP0.36107.15340RBF-SPCE 0.33311.12840MCS0.3369图4㊀算例3迭代过程中失效概率的变化情况Fig.4㊀Failure probability in the iteration process of example 34㊀汽车侧面碰撞工程算例㊀㊀汽车抗撞击能力对于保证乘客的安全至关重要㊂在所有汽车交通事故中,侧面碰撞是导致乘客死亡的主要因素,仅次于正面碰撞㊂对于车身来说,车身两侧的机械性能相对较弱,汽车侧面碰撞的缓冲面积相对较小,其碰撞变形易对乘客造成严重伤害,针对汽车侧面碰撞问题[29]做可靠性分析具有重要意义㊂汽车侧面碰撞问题如图5所示[30]㊂针对汽车侧面碰撞问题,以B 柱处前门的速度作为性能函数,验证本文所提方法的工程适用性㊂表8给出了不确定变量的统计特性㊂汽车侧面碰撞功能函图5㊀汽车侧面碰撞Fig.5㊀Vehicle side impact数表达式为g X ()=0.489x 3x 7+0.843x 5x 6-0.0432x 9x 10+0.0556x 9x 11+0.000786x 211-0.75(32)表8㊀汽车碰撞问题不确定变量分布Tab.8㊀Distributions of uncertain variablesfor vehicle side impact变量描述Variable description变量Variable 均值Mean 标准差Standard deviation分布类型DistributionB 柱内侧B-pillar innerx 110.05正态分布Gaussian distribution B 柱加固件B-pillar reinforcementx 210.05正态分布Gaussian distribution地板侧面内侧Floor side innerx 310.05正态分布Gaussian distribution横梁Cross membersx 410.05正态分布Gaussian distribution车门防撞梁Door beamx 510.05正态分布Gaussian distribution门带加固件Door belt line reinforcementx 610.05正态分布Gaussian distribution车顶纵梁Roof railx 710.05正态分布Gaussian distributionB 柱内侧材料Material of B-pillar innerx 80.30.006正态分布Gaussian distribution地板侧面内侧材料Material of floor side innerx 90.30.006正态分布Gaussian distribution障碍高度Barrier heightx 10010正态分布Gaussian distribution撞击位置Barrier hitting positionx 11010正态分布Gaussian distribution表9中给出了RBF 模型㊁RBF-SPCE 模型以及MCS 方法在求解汽车侧面碰撞问题的结果,直接采用MCS 方法得到的失效概率为0.0107%㊂由表9可以㊀第45卷第5期赵子达等:径向基函数-稀疏多项式混沌展开混合代理模型可靠性分析方法1115㊀㊀看出,RBF方法在调用12次功能函数的情况下,失效概率为0.0140%,与MCS 方法的相对误差较大,为30.841%㊂而RBF-SPCE 模型在调用10次功能函数的情况下就达到稳定的结果,且计算得到的失效概率为0.0111%,相对误差3.738%远小于RBF 模型㊂图6给出了在初始样本点相同的情况下,随着不断迭代,增加样本点的失效概率变化情况㊂表9㊀不同可靠性方法求解汽车侧面碰撞问题的对比Tab.9㊀Comparison of different reliability methodsfor vehicle side impact方法Method 失效概率Failure probability /%相对误差Relative error /%功能函数调用次数Number of performance functionRBF0.014030.84112RBF-SPCE 0.01113.73810MCS0.0107图6㊀汽车侧面碰撞问题迭代过程中失效概率的变化情况Fig.6㊀Failure probability in the iteration process of vehicle side impact5㊀结论㊀㊀本文以RBF 与SPCE 展开的正交组合为基础,提出一种可有效解决结构可靠性分析问题的混合代理模型㊂1)该方法SPCE 展开增广到RBF 代理模型中,一定程度上弥补了RBF 拟合线性响应能力差的问题,将RBF 的局部拟合能力与SPCE 展开的全局拟合能力有机结合,在降低计算成本的同时提高了代理模型的预测精度㊂2)数值算例表明,与现有方法相比,本文所提方法在处理高维㊁强非线性问题时呈现出较高的计算效率和计算精度㊂3)汽车侧面碰撞的工程算例表明,本文所提方法可准确预测汽车侧面碰撞问题的失效概率,有效减少了有限元分析所需的样本点,表明本文方法具有良好的工程适用性㊂参考文献(References )[1]㊀OUYANG H,LIU J,HAN X,et al.Correlation propagation foruncertainty analysis of structures based on a non-probabilisticellipsoidal model[J].Applied Mathematical Modelling,2020(88):190-207.[2]㊀刘㊀俊,安子军.基于改进的响应面法的双激波套筒活齿传动接触疲劳强度模糊可靠性分析[J].机械强度,2020,42(6):1362-1368.LIU Jun,AN ZiJun.Fuzzy reliability analysis of contact fatiguestrength of double shock sleeve movable teeth transmission based onan improved RSM [J].Journal of Mechanical Strength,2020,42(6):1362-1368(In Chinese).[3]㊀OUYANG H,LIU J,HAN X,et al.Non-probabilistic uncertaininverse problem method considering correlations for structural parameter identification [J ].Structural and MultidisciplinaryOptimization,2021:1-16.[4]㊀白㊀斌,张俊一,周㊀策,等.基于T-S 模糊故障树的工业机器人可靠性分析[J].机械强度,2021,43(6):1348-1358.BAI Bin,ZHANG JunYi,ZHOU Ce,et al.Reliability analysis ofindustrial robot using T-S fuzzy fault tree[J].Journal of MechanicalStrength,2021,43(6):1348-1358(In Chinese).[5]㊀李永欣,常㊀涛,杨立明,等.基于组合代理模型的三角履带轮多工况疲劳优化[J].机械强度,2021,43(5):1088-1094.LI YongXin,CHANG Tao,YANG LiMing,et al.Fatigueoptimization of triangle track wheel under multi-working conditions based on ensemble of surrogate model [J].Journal of MechanicalStrength,2021,43(5):1088-1094(In Chinese).[6]㊀韩忠华.Kriging 模型及代理优化算法研究进展[J].航空学报,2016,37(11):3197-3225.HAN ZhongHua.Kriging surrogate model and its application todesign optimization:a review of recent progress [J ].ActaAeronautica et Astronautica Sinica,2016,37(11):3197-3225(InChinese).[7]㊀陈㊀哲,杨旭锋,程㊀鑫.基于改进Kriging 模型的主动学习可靠性分析方法[J].机械强度,2021,43(1):129-136.CHEN Zhe,YANG XuFeng,CHENG Xin.Active learning methodbased on improved Kriging model for reliability analysis[J].Journalof Mechanical Strength,2021,43(1):129-136(In Chinese).[8]㊀张天龙,曾㊀鹏,李天斌,等.基于主动学习径向基函数的边坡系统可靠度分析[J].岩土力学,2020,41(9):3098-3108.ZHANG TianLong,ZENG Peng,LI TianBin,et al.Systemreliability analyses of slopes based on active-learning radial basisfunction[J].Rock and Soil Mechanics,2020,41(9):3098-3108(In Chinese).[9]㊀刘㊀鑫,陈㊀德,周振华,等.基于概率-区间混合模型的汽车乘员约束系统可靠性优化设计[J].振动与冲击,2021,40(20):240-246.LIU Xin,CHEN De,ZHOU ZhenHua,et al.Reliability-baseddesign optimization for an occupant restraint system based on a probability and interval hybrid model[J].Journal of Vibration andShock,2021,40(20):240-246(In Chinese).[10]㊀FANG H B,HORSTEMEYER M F.Global response approximationwith radial basis functions[J].Engineering Optimization,2006,38(4):407-424.[11]㊀WEI Y,BAI G,SONG L K.A novel reliability analysis approachwith collaborative active learning strategy-based augmented RBF metamodel[J].IEEE ACCESS,2020(8):199603-199617.。

高二英语期末试卷带答案

高二英语期末试卷带答案考试范围：xxx ；考试时间：xxx 分钟；出题人：xxx 姓名：___________班级：___________考号：___________1．答题前填写好自己的姓名、班级、考号等信息 2．请将答案正确填写在答题卡上一、单项选择1.In this country it is required that anyone who_______recently come here______to pass this kind of test.A ．has ；hasB ．has ；hadC ．have ；haveD ．has ；have 2.______ by a crowd of fans, he couldn’t move a bit.A ．SurroundingB ．SurroundedC ．To be surroundedD ．To surround3.here yesterday, he would attend the lecture with us today. A ．Were he B ．If he were C ．Had he been D ．Should he be 4.He suddenly fell ill, ______ prevented him from attending his friend’s wedding.A ．whyB ．whatC ．thatD ．which5. Don’t be discouraged. ______ things as they are and you will enjoy every day of your life.A ．TakeB ．To takeC ．TakingD ．Taken 6.I wonder why he didn’t do it as ______.A ．told himB ．he toldC ．telling toD ．told to7.This is the first time we________a film in the cinema together as a family. A ．see B ．had seen C ．saw D ．have seen8.—How often do you eat out ?—____________, but usually once a week. A ．Have no ideaB．It dependsC．As usualD．Generally speaking9.20.—I’m looking for a house. I’d like ____ with a big garden.—Yes， madam. We have several of this kind.A．that B．it C．this D．One10.There is no________ that students will listen attentively in class, if they are _____ The World Cup in South Africa at nightA．probability ; accustomed toB．possibility ; addicted toC．need ; drunk withD．chance ; familiar with二、单词拼写单词拼写11.Fresh air and exercise (有助于) to good health.12.The origins of the (宇宙) are still a mystery.13.The government has (阐明) its position on equal pay for women.14.Her gentleness has given me a deep （印象）.15.She demanded that she be kept (通知）of developments.16.Most of the factories under (建设)have been designed by the Chinese designers.17.I don’t mind (学到)that skill if it is useful.18.Researchers use the terms east and west very （粗略地）.19.When he came to life, he found himself （包围）by some small people.20.We gave our classroom a ______ (彻底) cleaning before the National Day.句型转换21.A.There is only one bedroom,so Mary and I live together.B.There is only one bedroom.So IitMary.22.Now that he has been so rude to us, I feel like we have to punish him if he doesn’t respect us._______ he has been so rude to us, I feel like ________ him _______ he respects us.23.One such person in our city is Old Wang.Old Wang is _________ _________ _________ in our city.24.A lot of children like this TV program.This TV program _______________________ a lot of children.25.He performed badly in the exam, so he didn’t pass it.He failed ______ _______ the exam as a ______ ____ his poor performance.26.No sooner had he returned home than he was told to go to another country.________ had he returned home ______ he was told to go to another country.27.It happened that I was out when he called on me.Ihappened ______ _______ ______ when he called on me28.More and more people have shown great interest in this incident because of reports of aliens.This incident _______ ______ ______ _______ _______ _____ reports of aliens.29.He concentrated all his attention on his novel, so he didn’t notice me come in._________ _______ his novel, he didn’t notice me come in.30.The paintings are on show in the assembly hall. David donated them to the school.The paintings___David donated to the school are being______in the assembly hall.31.V. 单词拼写（10×1）1. He is seen as a ________ (潜在的) leader of our party.2. We will _____（任命） him as your secretary, for his carefulness.3. Because of _______（采用） the new teaching methods, the students feel at easy to study.4. They have made a _______（灵活的） holiday plan to satisfy everyone.5. It is against my will if I _______（反对） you.6.In summer, fruit _____ （易于）to decay, so you had better not to buy so much at one time.7. People don’t listen to him, for he is always ________（神经兮兮）.8. You should make a _______（具体的） analysis on the problem.9. Some officials should not ______ （滥用）their authority.10. Under the _____（压力） of life, he had to go abroad.根据下列句子及所给汉语注释, 写出空缺处各单词的正确形式。

理论计算研究二维二维BPg-C3N4 异质结的光催化CO2 还原性能

物理化学学报Acta Phys. -Chim. Sin. 2021, 37 (6), 2010027 (1 of 9)Received: October 13, 2020; Revised: November 4, 2020; Accepted: November 4, 2020; Published online: November 12, 2020.*Corresponding authors. Emails: Jftanhaiyan@ (H.T.); zhubicheng1991@ (B.Z.); zly2017@ (L.Z.). Tel.: +86-151******** (H.T.); +86-132******** (B.Z.); +86-157******** (L.Z.).This work was supported by the National Key Research and Development Program of China (2018YFB1502001), the National Natural Science Foundation of China (51872220, 21905219, 51932007, U1905215, 21871217, U1705251), National Postdoctoral Program for Innovative Talents (BX20180231), China Postdoctoral Science Foundation (2020M672432), Hubei Postdoctoral Program for Innovative Research Post.国家重点研发计划(2018YFB1502001), 国家自然科学基金(51872220, 21905219, 51932007, U1905215, 21871217, U1705251), 博士后创新人才支持计划(BX20180231), 中国博士后科学基金(2020M672432)和湖北省博士后创新研究岗位项目资助 © Editorial office of Acta Physico-Chimica Sinica[Article] doi: 10.3866/PKU.WHXB202010027 2D/2D Black Phosphorus/g-C 3N 4 S-Scheme HeterojunctionPhotocatalysts for CO 2 Reduction Investigated using DFT Calculations Xingang Fei 1, Haiyan Tan 2,*, Bei Cheng 1, Bicheng Zhu 1,*, Liuyang Zhang 1,*1 State Key Laboratory of Advanced Technology for Materials Synthesis and Processing, Wuhan University of Technology, Wuhan 430070, China.2 School of Chemistry and Environmental Engineering, Hubei University for Nationalities, Enshi 445000, Hubei Province, China. Abstract: Photocatalytic reduction of CO 2 to hydrocarbon compounds is apromising method for addressing energy shortages and environmental pollution.Considerable efforts have been devoted to exploring valid strategies to enhancephotocatalytic efficiency. Among various modification methods, the hybridization ofdifferent photocatalysts is effective for addressing the shortcomings of a singlephotocatalyst and enhancing its CO 2 reduction performance. In addition, metal-freematerials such as g-C 3N 4 and black phosphorus (BP) are attractive because of theirunique structures and electronic properties. Many experimental results have verifiedthe superior photocatalytic activity of a BP/g-C 3N 4 composite. However, theoretical understanding of the intrinsic mechanism of the activity enhancement is still lacking. Herein, the geometric structures, optical absorption, electronic properties, and CO 2 reduction reaction processes of 2D/2D BP/g-C 3N 4 composite models are investigated using density functional theory calculations. The composite model consists of a monolayer of BP and a tri-s -triazine-based monolayer of g-C 3N 4. Based on the calculated work function, it is inferred that electrons transfer from g-C 3N 4 to BP owing to the higher Fermi level of g-C 3N 4 compared with that of BP . Furthermore, the charge density difference suggests the formation of a built-in electric field at the interface, which is conducive to the separation of photogenerated electron-hole pairs. The optical absorption coefficient demonstrates that the light absorption of the composite is significantly higher than that of its single-component counterpart. Integrated analysis of the band edge potential and interfacial electronic interaction indicates that the migration of photogenerated charge carriers in the BP/g-C 3N 4 hybrid follows the S-scheme photocatalytic mechanism. Under visible-light irradiation, the photogenerated electrons on BP recombine with the photogenerated holes on g-C 3N 4, leaving photogenerated electrons and holes in the conduction band of g-C 3N 4 and the valence band of BP , respectively. Compared with pristine g-C 3N 4, this S-scheme heterojunction allows efficient separation of photogenerated charge carriers while effectively preserving strong redox abilities. Additionally, the possible reaction path for CO 2 reduction on g-C 3N 4 and BP/g-C 3N 4 is discussed by computing the free energy of each step. It was found that CO 2 reduction on the composite occurs most readily on the g-C 3N 4 side. The reaction path on the composite is different from that on g-C 3N 4. The heterojunction reduces the maximum energy barrier for CO 2 reduction from 1.48 to 1.22 eV, following the optimal reaction path. Consequently, the BP/g-C 3N 4 heterojunction is theoretically proven to be an excellent CO 2 reduction photocatalyst. This work is helpful for understanding the effect of BP modification on the photocatalytic activity of g-C 3N 4. It also provides a theoretical basis for the design of other high-performance CO 2 reduction photocatalysts.Key Words: Photocatalysis; CO 2 reduction; Step-scheme heterojunction; Graphitic carbon nitride;Density functional theory. All Rights Reserved.理论计算研究二维/二维BP/g-C3N4异质结的光催化CO2还原性能费新刚1，谭海燕2,*，程蓓1，朱必成1,*，张留洋1,*1武汉理工大学，材料复合新技术国家重点实验室，武汉 4300702湖北民族大学，化学与环境工程学院，湖北恩施 445000摘要：光催化二氧化碳还原成烃类化合物是解决能源短缺和环境污染的重要途径。

01(东北大学学报(自然科学版))MRD与LRB相混合的结构振动控制

收稿日期:2004212222基金项目:国家自然科学基金资助项目(50508008);辽宁省博士启动基金资助项目(20041014)・作者简介:张延年(1976-),男,辽宁葫芦岛人,东北大学博士研究生;刘　斌(1940-),男,辽宁沈阳人,东北大学教授,博士生导师・第27卷第1期2006年1月东北大学学报(自然科学版)Journal of Northeastern University (Natural Science )Vol 127,No.1Jan.2006文章编号:100523026(2006)0120095204MRD 与LRB 相混合的结构振动控制张延年,刘　斌,李　艺,范　鹤(东北大学资源与土木工程学院,辽宁沈阳　110004)摘要:磁流变阻尼器(MRD )利用磁流变液提供可控性是当今最有前途的半主动控制装置,附加MRD 是铅芯橡胶垫(L RB )隔震结构的发展方向・因此提出3种MRD 与L RB 隔震混合方案,建立了MRD 与L RB 隔震混合结构的动力分析模型,推导出运动微分方程,对其进行时程反应分析・以7层框架结构为例,采用瞬时最优控制算法分别对3种混合方案进行地震反应分析,与L RB 隔震结构进行比较,各混合方案的各种地震反应均明显得到更好地控制,而混合方案3的控制效果最显著・关　键　词:振动控制;混合控制;隔震结构;MRD ;L RB ;恢复力模型中图分类号:TU 31113 文献标识码:A铅芯橡胶垫(L RB )由于装置简明、构造简单、造价低,容易在工程上实现,易于维护且无需外界能源支持等优点而引起广泛关注,并成为当前应用开发的热点[1]・然而,它作为被动控制装置存在着无法避免的缺陷,一旦被安装在结构上后,其固有特性就无法改变,因此无法对各种特性的地震激励都有控制效果[2]・而且风载作用、结构物倾覆及地震后隔震系统的残余变形影响等,使其应用受到一定的限制・Spencer 等人发现在隔震系统底部增加一定的阻尼,可减少上部结构的加速度及层间侧移,但增加阻尼过大反而会增大上部结构的响应[3]・磁流变液(MRF )是在1948年由Rabinow 发明的一种智能材料,具有强度高、黏度低、能量需求小、温度稳定性好、对通常在制造过程中引入的杂质不敏感等特点[4]・由其制成的磁流变阻尼器(MRD )是当今最新的半主动控制装置,除性能安全可靠,制造成本较低外[5],还具有体积小、功耗少、耐久性好、机构简单、可靠性强、适用面大、响应速度快、动态范围广、频率响应高、阻尼力大且连续可调等特点,特别是它能根据系统的振动特性产生最佳阻尼力,因而具有广阔的应用前景[6～9],因此附加MRD 是L RB 隔震结构的发展方向[2]・本文作者将MRD 安装在L RB 隔震结构的隔震层及上部结构,形成MRD 与L RB 隔震混合结构,使其在地震作用下的水平位移反应得到很好的控制,可以避免隔震层的位移超过其设计容许值,有效地提高了结构的稳定性和可靠性・1　MRD 与L RB 混合隔震结构的模型建立假定同一层各构件的上下移动量基本相同,采用层间剪切型模型,墙体的质量各集中于各层,整个结构建立在刚性地基上,不考虑基础的提离,不考虑土与结构的相互作用・以n 层MRD 与L RB 隔震混合结构为例,MRD 恢复力模型采用平行板模型[10],建立动力分析模型如图1所示・m b 为隔震层质量,m 1～m n 为上部结构各层质量;k h ,b 为隔震层总水平刚度,k h ,1～k h ,n 为上部结构各层水平刚度;c h ,b 为隔震层水平阻尼,c h ,1～c h ,n 为上部结构各层水平阻尼;c mc ,b ,c mv ,b 为隔震层MRD 提供的库仑阻尼和黏滞阻尼,c mc ,1～c mc ,n ,c mv ,1～c mv ,n 分别为上部结构各层MRD 提供的库仑阻尼和黏滞阻尼・由于隔震层有L RB 存在,所以水平恢复力按双线性模型(见图2)考虑・F y 为隔震层的屈服力;x b y 为隔震层的屈服位移;k R 为天然橡胶垫(RB )的刚度;k s 为铅的初始刚度・由于铅芯的灌入,L RB 的初始刚度k b1较k R 有较大的提高・当铅芯屈服后,L RB 屈服刚度k b2趋于k R ,从而可获得比较饱满的滞回曲线,达到耗能的目的・图1　MRD 与LRB 隔震混合结构的动力分析模型Fig.1　Model of dynamic analysis of hybrid MRDand LRB base 2isolation图2　LRB 双线性恢复力模型Fig.2　Bilinear hysteretic model of LRB(a )—橡胶恢复力特性;(b )—铅阻尼器恢复力特性;(c )—L RB 恢复力特性・2　水平运动微分方程的建立MRD 与L RB 隔震混合结构的水平运动微分方程为M {¨x }+C h { x }+K h {x}=C m +F {¨x g }・(1)式中,{x},{ x },{¨x }分别为MRD 与L RB 隔震混合结构各层水平相对位移,速度和加速度列向量;M ,C h ,K h 分别为MRD 与L RB 隔震混合结构的质量、水平阻尼和水平刚度矩阵;F 为地面地震加速度转换矩阵;{¨x g }为地震加速度输入;C m 为MRD 的总阻尼力向量・如果结构每一层都安装MRD ,问题将容易解决,但是一些情况下是在结构上选择安装MRD ,并不是在每一层间都安装MRD ・假设安装r 个MRD ,则需要引入一个n ×r 控制装置位置矩阵E ,这时的C m 为r 维MRD的总阻尼力向量,可得运动方程为M {¨x }+C h { x }+K h {x}=EC m +F {¨x g }・(2)由于选择安装MRD ,则阻尼系数矩阵也比较复杂,将MRD 所产生的总阻尼力向量C m 分解:C m =C v +U ・(3)式中,C v ,U 分别为MRD 的黏滞阻尼力和库仑阻尼力向量・一般情况下,MRD 都采用同一型号,因此,黏滞阻尼系数均为c v ,则C v =c v V ,(4)其中,V 为各自MRD 活塞与缸体间的相对速度向量,它与各楼层的运动速度向量 X 的关系为V =-E T { x },(5)则C v 为C v =-c v EE T { x },(6)则运动方程为M {¨x }+(C h +c n EE T ){ x }+K h {x}=EU +F {¨x g }・(7)3　工程实例分析以7层MRD 与LRB 隔震混合结构为例,结构主要参数见表1・1～7层的层高为316m ,隔震层等效阻尼比ξ=0116,屈服后刚度k b2为初始刚度k b1的016,屈服位移为0103m ・在隔震结构上安装Load 公司生产的20t 足尺MRD ,其主要性能参数见表2・MRD 与LRB 隔震混合方案分为3种:(1)方案1:在上部结构层间各安装一个MRD ;(2)方案2:在隔震层安装一个MRD ;(3)方案3:在隔震层和上部结构层间各安装一个MRD ・表1　MRD 与LRB 隔震混合结构参数Table 1　Structural parameters of hybrid MRDand LRB base 2isolation 层数隔震层1～6层7层楼层质量/t215240180层间水平刚度/(MN ・m -1)21515515表2　20t 足尺MRD 主要性能参数Table 2　De sign parameters of the 20t large 2scale MRD冲程cm最大阻尼力kN 最大最小力比v =10(cm ・s -1)最大耗电功率W (1A ,22V )线圈缸体直径cm磁流变液动黏系数Pa ・s流体最大屈服应力kPa两极间隙mm有效流体体积cm 3±820010.1223×105020.320.65029069东北大学学报(自然科学版) 第27卷3种方案的地震波均选用El 2Centro (1940205218),加速度峰值为220cm/s 2,采用IOC 算法[11]对混合结构进行地震反应分析・图3～图6分别为方案3的相对加速度、速度、位移、层间剪力的时程曲线・图3　MRD 与LRB 隔震混合结构加速度时程曲线Fig.3　Acceleration 2time history curve MRDandLRB base 2isolation图4　MRD 与LRB 隔震混合结构速度时程曲线Fig.4　Velocity 2time history of hybrid MRD andLRB base 2isolation图5　MRD 与LRB 隔震混合结构位移时程曲线Fig.5　Displacement 2time history curve of hybridMRD and LRB base 2isolation图6　MRD 与LRB 隔震混合结构层间剪力时程曲线Fig.6　Interstory shear 2time history curve of hybridMRD and LRB base 2isolationMRD 与L RB 隔震混合结构的3种混合方案的相对加速度峰值、相对速度峰值、相对位移峰值和层间剪力峰值与L RB 隔震结构分别进行比较・如表3所示,各混合方案的各种地震反应均得到了更好的控制・表3　结果的地震反应比较Table 3　Comparison of seismic re sponse s of different hybrid scheme s方案相对加速度峰值与L RB 隔震结构比较cm ・s -2%相对速度峰值与L RB 隔震结构比较cm ・s -2%相对位移峰值与L RB 隔震结构比较cm %层间剪力峰值与L RB 隔震结构比较kN %L RB 隔震结构323.21—35.38—7.34—2160.26—混合结构方案1286.97减小11.2131.48减小11.02 6.46减小11.991892.39减小12.40混合结构方案2273.53减小15.3729.60减小16.34 6.16减小16.081817.02减小15.89混合结构方案3241.82减小25.1826.26减小25.78 5.51减小24.931632.18减小24.454　结论为了使L RB 隔震技术有更好的适应性,本文提出3种MRD 与L RB 隔震混合方案,建立MRD 与L RB 隔震混合结构的动力分析模型并得出运动微分方程・以7层MRD 与L RB 隔震混合结构为例,采用IOC 算法对MRD 与L RB 隔震混合结构进行地震反应分析・结果表明各混合方案的各种地震反应均得到了更好的控制,而混合方案3的控制效果更加明显・参考文献:[1]刘季,周云・结构抗震控制研究与应用状况(上)[J ]・哈尔滨建筑大学学报,1995,28(4):1-10・(Liu J ,Zhou Y.State of the art and state of the practice in earthquake response control of structures (1)[J ].Journal of U niversity of A rchitect ure and Engi neeri ng ,1995,28(4):1-10.)[2]Ribakov Y ,G luck J.Active control of MDOF structures with supplemental electrorheological fluid dampers [J ].Earthquake Engi neeri ng and St ruct ural Dynamics ,1999,28(2):143-156.[3]Spencer B F ,Dyke S J ,Sain M K.Phenomena logical model for magnet or ecologicaldampers [J ].JournalofEngi neeri ng Mechanics ,A S CE ,1997,123(3):230-238.79第1期张延年等:MRD 与L RB 相混合的结构振动控制[4]Takesue N,Furusho J,K iyota Y.Fast response MR2fluidactuator[J].JS M E International Journal,2004,47(3):783-791.[5]张延年,刘剑平,刘斌・多向地震耦合作用下MRD结构的地震反应分析[J]・东北大学学报(自然科学版),2005,26(9):897-900・(Zhang Y N,Liu J P,Liu B.Seismic responses of MRDconfiguration under coupled action of multi2directionalearthquake[J].Journal of Northeastern U niversity(N at ural Science),2005,26(9):897-900.)[6]Toshihiko S,Tomoya S,Shin M.Design and performanceverification of variable damper using MR fluid[J].A mericanSociety of Mechanical Engi neers,Dynamic Systems andCont rol Division(Publication)DS C,2003,72(2):989-994.[7]Y ang G Q,Spencer J B,J ung H J,et al.Dynamic modelingof large2scale magnetorhrological damper systems for civilengineering applications[J].Journal of Engi neeri ngMechanics,2004,130(9):1107-1114.[8]Xia P Q.An inverse model of MR damper using optimalneural network and system identification[J].Journal ofSound and V ibration,2003,266(5):1009-1023.[9]Atray V S,Roschke P N.Neuro2fuzzy control of railcarvibrations using semiactive dampers[J].Com puter2A i dedCivil and Inf rast ruct ure Engi neeri ng,2004,19(2):81-92.[10]周云,徐龙河,李忠献・磁流体阻尼器半主动控制结构的地震反应分析[J]・土木工程学报,2001,34(5):10-14・(Zhou Y,Xu L H,Li Z X.Seismic responses of semi2controlusing magnetorhrological fluid dampers[J].Chi na CivilEngi neeri ng Journal,2001,34(5):10-14.)[11]Y ang J N,Wu J C,Li Z.Control of seismic2excitedbuildings using active variable stiffness systems[J].Journalof St ruct ure Engi neeri ng,1996,18(8):589-596.Hybrid Control of MRD and L RB for Structural VibrationZHA N G Yan2nian,L IU Bin,L i Yi,FA N He(School of Resources&Civil Engineering,Northeastern University,Shenyang110004,China.Corres pondent:ZHAN G Y an2 nian,E2mail:zhangyannian@)Abstract:The magnetorhrological damper(MRD)that uses MR fluids to provide controllability is one of the most promising semi2active devices to control structural vibration,which is available to be attached to Lead2Rubber Bearing(L RB)so as to form a base2isolated structure.Three kinds of such hybrid schemes of MRD and L RB were thus proposed with dynamic analysis set up.A kinetic differential equation was derived and a dynamic time history analysis was made.Taking a72story frame as example,the seismic res ponse analysis was made for the proposed three kinds of hybrid schemes according to the theory of instantaneous optimal control(IOC)algorithm pared with the single L RB base2isolated structure,all the three hybrid schemes present clearly the effective controllabilities of seismic res ponse especially the scheme3.K ey w ords:vibration control;hybrid control;base2isolated structure;MRD;L RB;restoring model(Received December22,2004)待发表文章摘要预报食物链算法及其在分销网络优化中的应用喻海飞,汪定伟根据人工生命突现集群及其对环境进行动态作用的特点,即人工生命个体间微观的相互作用将在整个人工生命系统中产生突现集群的现象,同时基于食物链也是生命系统中重要而又有广泛存在的重要现象,通过定义各级人工生命的局部活动规则,提出一种具有食物链形式的人工生命算法,并称之为食物链算法・把食物链算法应用于选址2分配问题的求解,优化设计分销网络结构并最小化供应链成本・该算法取得了较好的解,可以作为企业分销渠道设计的参考,也可应用于物流、电子商务中的优化问题・钢铁企业生产与运输费用协调调度问题关　静,唐立新针对钢铁企业生产前存在不可忽略运输的实际,研究了生产与生产前运输费用协调调度问题・由于钢铁企业被调度的工件体积较大及加工前不能等待太长的时间,因此运输车辆有容量限制及工件在机器前的缓冲等待时间有限制・考虑的机器环境为单机环境,单机前有无限的缓冲空间,运输车辆数目无限,调度的目标函数为传统的调度函数加上运输费用・对于不同的目标函数,证明了工件在单机加工前缓冲等待时间有限制的调度问题是强NP难的,对于运输车辆有容量限制问题的可解情况给出了多项式时间算法・89东北大学学报(自然科学版) 第27卷。

国外关于有限元方面的书籍

Adams, V. and Askenazi, A., Building Better Products With Finite Element Analysis , 1998
Ainsworth, M. and Oden, J. T., A Posterior Error Estimation in Finite Element Analysis , 2000
Backstrom, G., Fields of Physics on the PC by Finite Element Analysis, 1994
Backstrom, G., Fields of Physics by Finite Element Analysis, An Introduction, 1998
Baldwin, K., ed., Modern Methods for Automating Finite Element Mesh Generation, 1986, CP
Baran, N. M., Finite Element Analysis on Microcomputers, 1988
Argyris, J. H. and Mlejnek, H. P. Computerdynamik der Tragwerke, Band III Die Methode der Finiten Elemente, 1996
Ashwell, D. G. and Gallagher, R. H., eds. Finite Elements for Thin Shells and Curved Members, 1976
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某混合动力汽车的NVH实验及分析

某混合动力汽车的ＮＶＨ实验及分析ＮＶＨＥｘｐｅｒｉｍｅｎｔａｎｄＡｎａｌｙｓｉｓｏｆａＨｙｂｒｉｄＥｌｅｃｔｒｉｃＶｅｈｉｃｌｅ邱鹏飞　何东伟　崔明阳（同济大学浙江学院机械与汽车工程系，嘉兴　３１４０００）摘　要牶随着新能源汽车的发展，混合动力汽车具有良好的节油环保优势以及驾乘体验，被广大客户所接受。

然而，混合动力汽车在不同车速、不同工况下，会表现出不同的ＮＶＨ相关问题，对驾驶员主观感受有着不同的影响。

文章分析了混动汽车动力总成系统的ＮＶＨ性能，针对某ＨＥＶ汽车ＳＯＣ工况下，由ＥＶ模式进入并联模式时存在明显的金属敲击声问题，分析了激励产生原因，并排除了故障。

关键词牶混合动力汽车　ＮＶＨ　性能分析ＤＯＩ牶１０．１６４１３／ｊ．ｃｎｋｉ．ｉｓｓｎ．１０１７０８０ｘ．２０２２．０６．００９Ａｂｓｔｒａｃｔ牶Ｗｉｔｈｔｈｅｄｅｖｅｌｏｐｍｅｎｔｏｆｎｅｗｅｎｅｒｇｙｖｅｈｉｃｌｅｓ，ｈｙｂｒｉｄｖｅｈｉｃｌｅｓｈａｖｅｇｏｏｄａｄｖａｎｔａｇｅｓｏｆｆｕｅｌｓａｖｉｎｇａｎｄｅｎｖｉｒｏｎｍｅｎｔａｌｐｒｏｔｅｃｔｉｏｎ，ａｓｗｅｌｌａｓｄｒｉｖｉｎｇｅｘｐｅｒｉｅｎｃｅ，ａｎｄａｒｅａｃｃｅｐｔｅｄｂｙｃｕｓｔｏｍｅｒｓ．Ｈｏｗｅｖｅｒ，ｈｙｂｒｉｄｖｅｈｉｃｌｅｓｗｉｌｌｓｈｏｗｄｉｆｆｅｒｅｎｔＮＶＨｒｅｌａｔｅｄｐｒｏｂｌｅｍｓｕｎｄｅｒｄｉｆｆｅｒｅｎｔｓｐｅｅｄｓａｎｄｗｏｒｋｉｎｇｃｏｎｄｉｔｉｏｎｓ，ｗｈｉｃｈｗｉｌｌｈａｖｅｄｉｆｆｅｒｅｎｔｅｆｆｅｃｔｓｏｎｔｈｅｄｒｉｖｅｒｓｓｕｂｊｅｃｔｉｖｅｆｅｅｌｉｎｇｓ．ＴｈｅＮＶＨｐｅｒｆｏｒｍａｎｃｅｏｆｈｙｂｒｉｄｖｅｈｉｃｌｅｐｏｗｅｒｔｒａｉｎｓｙｓｔｅｍｉｓｒｅｓｅａｒｃｈｅｄｉｎｔｈｉｓｐａｐｅｒ．ＵｎｄｅｒｔｈｅＳＯＣｗｏｒｋｉｎｇｃｏｎｄｉｔｉｏｎｏｆａＨＥＶｖｅｈｉｃｌｅ，ｗｈｅｎｔｈｅｅｎｇｉｎｅｉｎｔｅｒｖｅｎｅｓｆｒｏｍＥＶｍｏｄｅｔｏｐａｒａｌｌｅｌｍｏｄｅ，ｔｈｅｒｅｉｓａｎｏｃｃａｓｉｏｎａｌｏｂｖｉｏｕｓｍｅｔａｌｋｎｏｃｋｉｎｇｓｏｕｎｄ．Ｔｈｅｃａｕｓｅｏｆｅｘｃｉｔａｔｉｏｎｉｓａｎａｌｙｚｅｄ，ａｎｄｔｈｅｆａｕｌｔｉｓｅｌｉｍｉｎａｔｅｄ，ｗｈｉｃｈｉｓｖｅｒｉｆｉｅｄｂｙｅｘｐｅｒｉｍｅｎｔｓ．Ｋｅｙｗｏｒｄｓ牶ｈｙｂｒｉｄｅｌｅｃｔｒｉｃｖｅｈｉｃｌｅ　ＮＶＨ　ｐｅｒｆｏｒｍａｎｃｅａｎａｌｙｓｉｓ０　引　言随着双碳目标的推进，对汽车降低污染排放及减少能源消耗提出了很高的要求，油电混动汽车有着较好的燃油经济性，同时解决了续航焦虑，成为受欢迎的新能源车型。

叶轮机械非定常流动及气动弹性计算

中图分类号：V211.3 论文编号：1028701 18-B061 学科分类号：080103博士学位论文叶轮机械非定常流动及气动弹性计算研究生姓名周迪学科、专业流体力学研究方向气动弹性力学指导教师陆志良教授南京航空航天大学研究生院航空宇航学院二О一八年十月Nanjing University of Aeronautics and AstronauticsThe Graduate SchoolCollege of Aerospace EngineeringNumerical investigations of unsteady aerodynamics and aeroelasticity ofturbomachinesA Thesis inFluid MechanicsbyZhou DiAdvised byProf. Lu ZhiliangSubmitted in Partial Fulfillmentof the Requirementsfor the Degree ofDoctor of PhilosophyOctober, 2018南京航空航天大学博士学位论文摘要气动弹性问题是影响叶轮机械特别是航空发动机性能和安全的一个重要因素。

作为一个交叉学科，叶轮机械气动弹性力学涉及与叶片变形和振动相关联的定常/非定常流动特性、颤振机理以及各种气弹现象的数学模型等的研究。

本文基于计算流体力学(CFD)技术自主建立了一个适用于叶轮机械定常/非定常流动、静气动弹性和颤振问题的综合计算分析平台，并针对多种气动弹性问题进行了数值模拟研究。

主要研究内容和学术贡献如下：由于叶轮机械气动弹性与内流空气动力特性密切相关，真实模拟其内部流场是研究的重点之一。

基于数值求解旋转坐标系下的雷诺平均N–S(RANS)方程，首先构造了适合于旋转机械流动的CFD模拟方法。

特别的，针对叶片振动引起的非定常流动问题，采用动网格方法进行模拟，通过一种高效的RBF–TFI方法实现网格动态变形；针对动静叶排干扰引起的非定常流动问题，采用一种叶片约化模拟方法，通过一种基于通量形式的交界面参数传递方法实现转静子通道之间流场信息的交换。

中国杂交水稻之父袁隆平的英语作文

中国杂交水稻之父袁隆平的英语作文1Yuan Longping, known as the Father of Hybrid Rice in China, was a remarkable scientist whose dedication and contribution have left an indelible mark on the world.He spent countless days and nights in the fields, observing the growth of rice with meticulous care. Regardless of the harsh weather or the fatigue from long hours of work, he never wavered in his pursuit. His perseverance was like a sturdy oak, standing firm against all odds.The experiments he conducted were numerous and challenging. Time and again, he faced failures, but his spirit of never giving up was unwavering. It was as if he had an unquenchable fire within him, driving him forward despite the difficulties.His dream was to ensure that people all over the world had enough to eat. For decades, he toiled silently, devoting his life to the research of hybrid rice. His efforts were like a gentle breeze, bringing hope and sustenance to countless hungry souls.Yuan Longping's spirit of selflessness and his unwavering commitment to science will always inspire us. His story is a testament to the power of determination and the pursuit of a noble cause. We will remember him not only for his achievements but also for his great heartand unwavering spirit.2Yuan Longping, known as the Father of Hybrid Rice in China, was a remarkable figure whose dedication and passion for agricultural science left an indelible mark on the world. His journey in the field of hybrid rice research was not an easy one. Facing doubts and setbacks, he remained steadfast in his belief, driven by an unwavering love for agriculture.Time and again, when confronted with challenges and skepticism from others, Yuan Longping did not waver. Instead, he viewed these obstacles as opportunities to prove the potential of his research. Through countless experiments and tireless efforts, he persisted in his pursuit, eventually achieving tremendous success that transformed the landscape of agriculture.Not only did Yuan Longping focus on his own research, but he also placed great emphasis on nurturing young scientific talents. He shared his knowledge and experiences generously, guiding and inspiring a new generation of researchers to carry forward the cause of hybrid rice studies. His mentorship was a source of strength and inspiration for many, fostering a spirit of innovation and perseverance within the research community.Yuan Longping's story is not just about scientific achievements; it is a testament to the power of passion, determination, and the unwavering pursuit of a dream. His life's work has ensured food security for countlesspeople and serves as a shining example for all of us to follow.3Yuan Longping, known as the Father of Hybrid Rice in China, was a remarkable figure whose life was a tapestry of simplicity and dedication.He led a life marked by frugality. Despite his significant contributions to agriculture and the alleviation of global hunger, he remained humble and lived a simple life. He cared deeply about farmers and their hardships, constantly striving to improve rice yields to ensure their livelihoods.In his spare time, Yuan Longping had a passion for playing the violin. This hobby revealed the rich and diverse inner world hidden beneath his unassuming exterior. Through the strings of the violin, he expressed emotions and found solace after long days of research in the fields.He would often visit the fields himself, getting his hands dirty and closely observing the growth of the rice. His commitment was unwavering, and he never sought personal glory or material wealth. His sole focus was on making a difference in the lives of people through his work.Yuan Longping's story is not just one of scientific achievement but also of a person with a heart full of kindness and a spirit of unwavering determination. His life serves as an inspiration for us all to pursue our passions with sincerity and to always think of the greater good.Yuan Longping, the Father of Hybrid Rice in China, was a remarkable figure whose contributions have had an immeasurable impact on global agriculture. One significant moment that showcases his wisdom and courage was during a crucial experiment when the odds seemed stacked against him. The weather conditions were unfavorable, and initial results were far from promising. However, Y uan Longping, with his unwavering determination and acute scientific insight, made a bold decision to adjust the parameters of the experiment. He meticulously analyzed the data, consulted with his team, and worked tirelessly day and night. Against all expectations, his perseverance paid off, and the experiment ultimately achieved a breakthrough that revolutionized the field of rice cultivation.His achievements not only brought about a substantial increase in rice yields in China but also garnered respect and acclaim on the international stage. His innovative techniques and unwavering dedication to improving food security have been widely recognized and appreciated by the global agricultural community. Yuan Longping's work has been a source of inspiration for countless scientists and researchers, demonstrating that with determination, intelligence, and the courage to take risks, even the most challenging problems can be overcome. His legacy will continue to inspire future generations to strive for excellence and make significant contributions to the betterment of humanity.Yuan Longping, known as the Father of Hybrid Rice in China, was a remarkable figure whose contributions have had an immense and lasting impact on society and humanity. His entire life was dedicated to the research and development of hybrid rice, a pursuit that transformed the landscape of global food production.Yuan Longping's research achievements were nothing short of revolutionary. Through his tireless efforts and innovative spirit, he successfully developed high-yield hybrid rice varieties that significantly increased grain output. This not only solved the food problem in China, a country with a large population, but also provided a crucial solution for the world's hunger issues.His perseverance and dedication in the face of numerous challenges are truly inspiring. Despite encountering various difficulties and setbacks during his research, Yuan Longping remained steadfast in his pursuit. He never gave up, constantly exploring and innovating, which ultimately led to his remarkable success.From Yuan Longping's deeds, we draw valuable strength and enlightenment. We learn the importance of persistence and the spirit of never giving up in the face of difficulties. His work ethic and sense of responsibility also teach us to have a broad vision and a heart dedicated to the well-being of humanity.In conclusion, Y uan Longping's life and work will forever be engraved in our memories. His spirit will continue to inspire generations to come, guiding us to strive for the betterment of society and the well-being of mankind.。

2021高中英语外刊阅读(MayⅢ)

2021高中英语外刊阅读(MayⅢ)Father of hybrid rice' ___ 13:07 ___ Hunan province。

___。

known for developing the first hybrid rice strains。

was born on the ninth day of the seventh month in 1930.according to the ___ - ___-fifth of the world's n with less than 9% of the world's total land.Note: No format errors or usly problematic paragraphs were found in the original text.China'___ has been suspended just days before its ___。

___ amounts of milk。

leading to public outcry and the show'___ the negative effects of the fan culture in China。

which has ___ use of social media and the need for n to ___ future.China'___。

"Youth With You 3," was ___ of milk。

The trigger for this event was a viral video that circulated during the five-day Labor Day holiday in early May。

The video showed a group of middle-aged people opening bottles of dairy products。

袁隆平英语作文100字简单

袁隆平英语作文1Yuan Longping, a name that shines brightly in the agricultural field, was a remarkable figure whose dedication and perseverance have left an indelible mark on the world.He was born in a time when food scarcity was a pressing issue. Despite the harsh conditions and numerous challenges, Yuan Longping was determined to find a solution to increase the yield of rice. He embarked on the arduous journey of researching hybrid rice, undeterred by difficulties and setbacks. Days and nights were spent in the fields, observing, experimenting, and analyzing. He faced doubts and criticisms, but his unwavering belief in his mission kept him going.His efforts finally paid off. The successful development of hybrid rice brought about a revolutionary change in agriculture. It not only increased the output of rice significantly but also provided a reliable food source for countless people around the world. Yuan Longping's contribution was not limited to his scientific achievements. His spirit of never giving up, his passion for solving global food problems, and his selfless dedication have inspired generations.Even in his later years, Yuan Longping remained committed to his research, constantly seeking ways to further improve the quality and yieldof rice. His life was a true testament to the power of determination and the pursuit of a noble goal. We will always remember him as a great scientist and a hero who dedicated his life to ensuring that people have enough to eat.2Yuan Longping, a name that resonates deeply within the hearts of countless people, dedicated his life to the pursuit of agricultural advancements. Picture this: under the burning sun, he could be seen tirelessly working in the fields, his figure bent yet resolute. Each step he took on that fertile soil was a step towards a brighter future for agriculture.His belief in the power of technology to transform the agricultural landscape was unwavering. No matter how challenging the circumstances, he remained steadfast in his pursuit. Years passed, and through countless experiments and research, he persisted, driven by a vision of a world where hunger would be eradicated.The days spent observing the growth of rice seedlings, the nights filled with analysis and reflection – all were part of his journey. He would meticulously note every detail, refusing to overlook even the slightest change. His eyes, filled with wisdom and determination, saw beyond the present, envisioning a future where abundant harvests would be the norm.Yuan Longping's efforts were not in vain. His achievements have not only filled our stomachs but also kindled hope in the hearts of peoplearound the world. His spirit, a guiding light, will continue to inspire generations to come, urging us to strive for the betterment of humanity with unwavering dedication and unyielding perseverance.3Yuan Longping was a remarkable scientist who dedicated his entire life to agricultural research, especially in the field of hybrid rice. His story is one filled with innovation and selfless dedication.In the early days, traditional beliefs held that increasing rice production was an insurmountable challenge. However, Y uan Longping refused to accept these limitations. With unwavering determination and a spirit of innovation, he embarked on a journey that would change the course of agriculture. Through countless experiments and tireless efforts, he overcame numerous difficulties and pioneered a new path in hybrid rice research.Yuan Longping's entire life was devoted to this cause. He spent days and nights in the fields, observing, analyzing, and conducting experiments. He did not seek personal glory or wealth but was driven by a single-minded goal - to ensure that people around the world had enough to eat. His selfless dedication knew no bounds as he freely shared his research findings and knowledge, enabling countless farmers to benefit and increase their yields.Even in his later years, when he could have chosen to rest and enjoy a peaceful life, Yuan Longping remained committed to his work. Hispassion and perseverance were infectious, inspiring a new generation of scientists to follow in his footsteps and continue his mission.In conclusion, Yuan Longping's contributions not only alleviated hunger for millions but also left a lasting legacy of innovation and dedication that will inspire generations to come.4Yuan Longping, a remarkable figure in the field of agriculture, demonstrated extraordinary teamwork spirit and leadership throughout his career. Working closely with his team, he embarked on a journey filled with challenges and achievements.One of the notable instances was when they faced the problem of increasing rice yields in adverse conditions. Yuan Longping, with his unwavering determination and in-depth knowledge, led the team in conducting countless experiments and analyses. He encouraged each member to contribute their unique ideas and perspectives, fostering an environment of innovation and collaboration. Under his guidance, they meticulously examined various factors, from soil quality to climate conditions, leaving no stone unturned.Moreover, Yuan Longping was not only a brilliant leader but also a caring mentor. He showed genuine concern for the well-being and growth of each team member. When they encountered setbacks, he would patiently offer encouragement and practical advice, instilling in them the confidenceto persevere. His words of wisdom and support were like a guiding light, leading the team out of the darkness of confusion and towards the path of success.The team's efforts eventually paid off, as they successfully developed new varieties of rice that significantly improved yields and contributed to solving global food shortages. Yuan Longping's leadership and the collective efforts of the team became a shining example of what can be achieved through teamwork and dedication.5Yuan Longping, a name that shines brightly in the annals of history, was a remarkable figure whose contributions have left an indelible mark on our society. His dedication to agricultural science was nothing short of extraordinary.He spent his entire life working tirelessly in the fields, driven by a passion to solve the problem of food scarcity. Through countless experiments and research, he developed high-yield hybrid rice varieties that have fed millions of people around the world.His deeds have inspired numerous young people to devote themselves to agricultural scientific research. They see in him a role model of perseverance and innovation. His spirit of never giving up in the face of difficulties has become a guiding light for many.The selfless dedication of Yuan Longping is not only reflected in hisscientific achievements but also in his concern for the well-being of humanity. He worked not for personal gain but for the benefit of all. His work ethic and sense of responsibility have set an exemplary standard for people from all walks of life.In a world often driven by material pursuits, Yuan Longping's pursuit of truth and his commitment to making the world a better place stand as a powerful reminder of what true greatness is. His story will continue to inspire generations to come, urging us to strive for noble goals and make significant contributions to the betterment of society.。

英语论文引言分析

they are not proposed for dense urban environments and
deep canyons. Besides, only a minor number of real urban
buildings with hybrid ventilation systems exist, where an
the conclusion of the experiment
In the present study a full comparison analysis, taking
into account nominal time constants and air-exchange
efficiencies, is performed for different ven performance of these ventilation systems in buildings
located in urban environment.
the general purpose
Thanks for listening
located in a deep pedestrian canyon. The study was
performed for single-sided and cross-ventilation
configurations and it revealed that the potential of
natural ventilation is seriously decreased in the urban
environment, while high errors may occur when non-

袁隆平的事迹介绍英语作文

袁隆平的事迹介绍英语作文袁隆平的事迹介绍英语作文（精选10篇）在现实生活或工作学习中，大家一定都接触过事迹吧，事迹是对先进集体、先进人物的模范事迹进行综合整理所形成的总结性书面材料。

事迹到底怎么拟定才正确呢？下面是店铺为大家整理的袁隆平的事迹介绍英语作文，仅供参考，欢迎大家阅读。

袁隆平的事迹介绍英语作文篇1Yuan Longping is known as Chinas “father of hybrid rice”. Its said that in China, we eat depending on “Two Ping” ---- Deng Xiaoping, who made the policy of System of Production Responsibility, & Yuan Longping, who invented hybrid rice.Yuan Longping, who was born in September, 1930, graduatedfromAgriculture Department in Southwest Agricultural Institute.He has been working on agriculture education & the research into hybrid rice since he left the institute.In the 1960s, when China was suffering serious famine, he came up with the idea of hybrid rice, which has a high yield.Ten years later, he succeeded in inventing a new species that produced a 20 percent higher yield than common types of rice.Yuan devoted himself to the research into agriculture, & was honored by UNESCO & FAO.Although he is 70 years old, he is still working on the research into agriculture.袁隆平的事迹介绍英语作文篇2How many journeys have they made and how many missions have they carried on their shoulders?He put the shoes deep into the paddy field, let the clear footprints point to the distant future, towards our beautiful newera!Since ancient times, the people have been the trendof promoting the development of history, food for the people and rice for the food. Therefore, the scientist I admire most is Yuan Longping, the father of hybrid rice. In my mind, the image of Yuan Longping is always fixed as an old man in a white shirt, standing in theendless paddy field, working hard. All the year round, he is more like an ordinary farmer than a scientist.The dream of enjoying the cool under the grass and the dream of hybrid rice covering the world are two grand ideas. What is the concept of a large-scale yield of 900 kg / mu of hybrid rice? This is a peak that no one has ever reached in the world, and it is also a new leap forward for Yuan Longping, the father of hybrid rice, to lead Chinese experts to face the world food problem. Yuan Longping once bluntly said that although this road is difficult, the future is bright. If its hard, its hard. New varieties with higher output have been produced, and people all over China and the world can solve the problem of food and clothing. He is happy, and it doesnt matter if he suffers.Yuan Longping has the habit of mixing Chinese and English when he speaks. Its like eating too much hybrid rice and automatically translating after speaking Chinese. People call this Yuans humor. He also speaks Russian, likes sports and music, and has passed the air force examination. How can such a versatile treasure boy not be admired!In2019, the sun is blazing and the land is warm. Yuan Longping, who is 90 years old, is still working hard. He hopes to strive for another ten years to plant our hybrid rice in half of the paddy fields in the world.This is the quality of every scientist with conscience. Manyscientists devote themselves to scientific research all their lives. Maybe they dont have hundreds of billions of dollars, but they are willing to contribute technology to the benefit of all mankind. They work day and night. Who has the time to watch yunyin; They accurately estimate the meaning of life, who has selfishness to care about the length of life; They pick up small grains of sand on the fuzzy soil. Its these wonderful crystal blossoms that give us direction and make the world more wonderful.Every day I fantasize about when I can germinate, thrive, become a green shade, and be grateful to those scientists who have done nothing for us. Because, I firmly believe that I can continue to move forward to the future, pay homage to the new era!袁隆平的事迹介绍英语作文篇3Yuan Longping is known as Chinas “father of hybrid rice”.Its said that in China, we eat depending on “Two Ping” --Deng Xiaoping, who made the policy of System of Production Responsibility, Yuan Longping, who invented hybrid rice. Yuan Longping, who was born in September, 1930, graduatedfromAgriculture Department in Southwest Agricultural Institute.He has been working on agriculture education the research into hybrid rice since he left the institute. In the 1960s, when China was suffering serious famine, he came up with the idea of hybrid rice, which has a high yield. T en years later, he succeeded in inventing a new species that produced a 20 percent higher yield than common types of rice. Yuan devoted himself to the research into agriculture, was honored by UNESCO FAO. Although he is 70 years old, he is still working on the research into agriculture.袁隆平的事迹介绍英语作文篇4There are a lot of great people in China. From ancient times to the present, they are very patriotic. I really admire them. Of all the great Chinese, I think Yuan Longping is the greatest.Yuan Longping is the father of hybrid rice. When it comes to him, most people respect him very much. Yuan Longping listed such a formula: Knowledge + sweat + inspiration + opportunity = success.In 1960, a rare natural and man-made disaster brought about a serious food famine. All the people were thin and hungry... Yuan Longping also experienced the pain of hunger. He witnessed the harsh reality, remembered the old society and the hardships of the common people, so he wanted to cultivate new varieties of rice with his own wisdom.In the experimental field, he found a different rice plant, differentfromordinary rice. The difference between them lies in the huge difference in rice yield. This discovery surprised Yuan Longping. After careful searching, this was the only one in the huge experimental field. After careful observation and statistical analysis, it is proved that this rice is a natural hybrid rice Proved that this kind of rice has obvious heterosis phenomenon, he excitedly made it into a specimen, took a picture and sent it to his assistant, and went outside to look for it.He and his assistant searched for a long time, and finally found six precious natural rice plants at a water edge. The results of the first cross were not ideal, and new problems came again. Rice is monogamous and self pollinated, so it is difficult to get rid of the male flowers one by one.In order to overcome the worlds problems, he bent down in the experimental field every day, taking good care of rice trees,as if he were his own son. When the next batch of rice was successful, he just walked into the field and rushed out excitedly, looking up and laughing: I succeeded! My hybrid rice is successful After repeated cultivation, the yield has jumpedfrom800 Jin / mu to 1600 Jin / mu. By2020, the yield of rice stem is higher than that of Zizania latifolia, which is more than2000Jin / mu.Yuan Longping has made a great contribution to China, which has fed more than one billion people. It has also made due contributions to the world. Now hybrid rice has been popularized all over the world. I will study hard and repay my motherland in the future.袁隆平的'事迹介绍英语作文篇5Yuanlongping is a great scientist who is devoted to agriculture. Let me introduce him.From an early age he was hard-working and was curious about everything, that is why he was given the nick name, “the student who asks questions.”.He studied agriculture in collage and began experiments in crop breeding. He thought that only by crossing different species of rice plant can we solve the food problem. Step by step, Yuan became the leading figure of the rice-growing world. In 1970 he made a breakthrough which is supported by government.Yuan made a contribution in agriculture, not only to China, but to the whole world. As far as I am concerned, he is more than a scientist. He is a hero.袁隆平的事迹介绍英语作文篇6Every time I get a good place, I sit in a comfortable room with my mothers delicious dishes and fragrant bowls of rice. I cant help but think of it - once on TV news, it was alwaysbroadcast: "there is a man in the field, sometimes he watches, mashes or takes a notebook to record something.". Under the scorching sun, he was wearing a loose and coarse white shirt and a pair of gray and yellow faces like an old farmer, which were the ravines left by the wind and frost.When I was a child, I reluctantly watched this "ordinary" farmer, who was also farming. Why couldnt my grandfather be on TV? Is it because hes more handsome? But also helpless, can only accompany grandfather on time every day to watch the only live TV "news broadcast.".After watching the weather forecast, which is very important to my grandfather, its time for me to do my homework. At that time, although I got good grades, I couldnt concentrate on my study, like skipping all the math problems that I couldnt figure out the solutions at a glance.When meeting difficulties and challenges, they all shrink back, and even begin to shake their goals: why should I take the examination of Sun Yat sen University? Why study? In theend, I finished the work according to the call of sleepy and grandparents.Until the fourth and fifth grade, I saw such a "knowledge link - Yuan Longpings simple life" in "morality and society". He won numerous fame and wealth. His farming shirt usually rangesfromten to fifty on the street. Even on the international stage such as the world awards, his most expensive suit is only 500 yuan. Once, Yuan Longping bought a shirt worth more than ten yuan on the street. He couldnt put it down and shut his mouth“ This kind of shirt is light, breathable, durable and cheap. Its perfect for working in the field! "In the illustration, the old man who is smiling and touchingthe sun is the "old farmer" who has seen him appear in the news many times. I began to look for Yuan Longping on the Internet TV of my parents home.Yuan Longpings world is so small that he has only rice in his experimental field. He has been associated with and accompanied with agriculture all his life. Since he began to study hybrid rice in xxx, he devoted most of his life to the rice field without any regrets. Day after day, year after year, but Yuan Longping firmly believes in providing food and clothing for people all over the world, still looks forward to tomorrow optimistically, and works harder in the hot sun, rain and rice fields.I cant help shivering. I have goose bumps and cold sweats all over my body. Yuan Longping has been fighting for others for more than 50 years in the palm of his hand. However, I study for my future, but I shake my heart"Those who engage in research should not be afraid of failure, and those who are afraid of failure should not engage in research.". He has always adhered to his own philosophy, "never at the expense of production in exchange for quality.". Along the way, the yield of 700kg, 900kg, 1000kg per mu... Chinas Super Hybrid Rice has constantly made breakthroughs in leaps and bounds - it is the rickety and thin figure in the field that has been pursuing and surpassing the dream for nearly 60 years! Many people say that China has enough food, but he always thinks: "the world is so big, billions of people want to eat, food security can not be taken lightly.".My face is very hot, and I can feel the blood flow clearly, as if I could roll an egg on my face and it would be hot soon. Im restless. It is Yuan Longping who forgets to eat and sleep, who has gone through failures and tackling key problems again andagain, that has made the breakthrough. However, I am frivolous and impetuous, indulgent and slack, and encounter a trivial stumbling block, but I am at a loss, timid and dare not challenge myselfIn the year of chub mackerels back, Yuan Longping still insists on striving in the front line of scientific research, never forgetting his original intention, and keeping everyone awayfromhunger“ He also cherishes higher dreams: 1) a dream of "enjoying the cool under the grass" (sitting in the cool under the rice as high as the number of trees, the fragrance of rice as big as p eanuts...); ② "The dream of paddy field in saline alkali land" ("there are still many places in the world where food is not enough. Is it not a waste of land that is not suitable for crop production?"). So he devoted himself more diligently to scientific research.My heart trembled, like thousands of needles straight into my body, tears: Yuan Longping has been 90 years old, still trying to realize his dream! And when I am young, why cant I pursue my dream?I finally got to know the "old farmer" and the good intentions of the national TV station. Start no longer afraid to retreat, guard against arrogance and impatience, try to boldly start to challenge yourself, pursue excellence, and bravely move forward!A thousand hectares of fertile land, but three meals a day, ten thousand rooms, sleeping only three feet. Growing Chinas farmland, the heart is filled with the worlds food and clothing. His spirit of striving for dreams has not only changed hybrid rice, but also inspired me to forge ahead bravely on the road of pursuing dreams. He is my idol, Yuan Longping, the father of hybrid rice.袁隆平的事迹介绍英语作文篇7We watched the movie "yuan longping". The film describes Mr Yuan longping hybrid rice planting and successfully, I watched the movie, feel very sigh with emotion, touched me is Mr Yuan longping indomitable spirit.This movie to a foreigner talk to Mr Yuan longping, on Mr Yuan longping to account before some of the things.Mr Yuan longping is very strong, again and again before him, and he is not afraid of difficulties, not back, he was planting hybrid rice when there are many difficulties, but he always remembered his belief that the pursuit of my life is to make all the people awayfromhunger.Mr Yuan longping is not just sitting in the lab, but to study in the field. CCTV reported, the agriculture department assessment team all just released, guided by Mr Yuan longping "y two optimal 2" one hundred mu of super hybrid rice plots on average 926.6 kg per mu, broke the record of large area of rice per mu in our country.Mr Yuan longping represents China perennial spirit, do we stand? We must take action, contribute to the motherland 袁隆平的事迹介绍英语作文篇8Yuan Longping, male, Han nationality, Jiangxi Province, Jiujiang Dean County, was born in September 7, 1930 in Beijing, now living in Hunan Changsha. August 1953 to participate in the work, university degree, no party, researcher. [1] Chinese hybrid rice breeding experts, known as Chinas "father of hybrid rice", Chinese Academy of engineering.Foreign fellow of the National Academy of Sciences, April2006. won the honorary doctorate degreefromMacao University of science and technology in2010. Won the Mahathir prize for Science in2011.Yuan Longping is a former member of the Standing Committee of the twelve CPPCC National Committee, vice chairman of the CPPCC Hunan Provincial Committee, vice chairman of the Hunan Provincial Association for science and technology. Southwestern University College of Agronomy and biotechnology, Hunan Agricultural University Professor, honorary president of China Agricultural University Professor, honorary president, honorary president of the Huaihua Vocational and Technical College, Hunan Biological and Electromechanical Polytechnic, FAO, the worlds chief adviser Hua Renjian Kang Food Association honorary chairman, Heilongjiang Yanshou County Economic Development Consultant .In January 3,2014, the Yuan Longping team to respond to genetically modified rice research, has not yet been used for practice. Norway members of the nomination China famous hybrid rice breeding expert, India geneticists and human rights activists in Pakistan for the2014Nobel peace prize.袁隆平的事迹介绍英语作文篇9Yuan Longping (born September 7,1930) is a Chinese agricultural scientist and educator,known for developing the first hybrid rice varieties in the 1970s.His "hybrid rice" has since been grown in dozens of countries in Africa,America,and Asia -providing a robust food source in high famine risk areas.Mr.Yuan won the State Preeminent Science and Technology Award of China in 20xx,the Wolf Prize in agricultureand the World Food Prize in 20xx.He is currently is DirectorGeneral of the China National Hybrid Rice R&D Center andhas been appointed as Professor at Hunan Agricultural University,Changsha.He is a member of the Chinese Acade my of Engineering,foreign associate of the US National Academy of Sciences (20xx) and the 20xx CPPCC.Mr.Yuan was born in Beijing,China.He loves playing Majong and the Erhu (Chinese violin),swimming and motorcycling.袁隆平的事迹介绍英语作文篇10Yuan Longping is known as Chinas “father of hybrid rice”.Its said that in China, we eat depending on “Two Ping” --Deng Xiaoping, who made the policy of System of Production Responsibility, Yuan Longping, who invented hybrid rice. Yuan Longping, who was born in September, 1930, graduated from Agriculture Department in Southwest Agricultural Institute.He has been working on agriculture education the research into hybrid rice since he left the institute. In the 1960s, when China was suffering serious famine, he came up with the idea of hybrid rice, which has a high yield. T en years later, he succeeded in inventing a new species that produced a 20 percent higher yield than common types of rice. Yuan devoted himself to the research into agriculture, was honored by UNESCO FAO. Although he is 70 years old, he is still working on the research into agriculture.【袁隆平的事迹介绍英语作文（精选10篇）】。

实时混合试验方法研究与应用综述_章红梅(1)

1
引
言
2
常见的抗震试验方法
目前，结构抗震试验的方法主要有三种
［2 ］
：
收稿日期： 2014 － 09 － 03 基金项目：港澳台科技合作专项资助（ 2012DFH70130 ）；中央高校基本科研业务费专项资金；自然科学基金（ 51008226 ）；光华交叉学科基金 * 联系作者，Email： 395693690@ qq． com
DOI:10.15935/ki.jggcs.2015.01.032
摘
要
Overview of Ｒeal - time Hybrid Simulation with Substructure
ZHENG Hongmei1
* ZENG Song1，
DUAN Yuanfeng2
DU Qingfeng3
Abstract
Ｒealtime hybrid simulation is a viable experiment technique to evaluate the performance of struc-
tural systems subjected to earthquake loads． It combines the advantage of structure experiment and numerical analysis，and becomes the hot topic in the field． This paper aimed to explain the theory of realtime hybrid simulation，emphasized on the analysis of integration algorithm，actuator timedelay and compensation ，introduced the current advancement． Some characteristics of hybrid test and advises have been put forward by literature analysis． Keywords realtime hybrid simulation，integration algorithm，delay compensation，overview 杂的非线性段，仅仅利用数值模拟分析很难准确地反映出结构的动力响应，特别是复杂结构和新型结构。结构抗震试验研究就是为了解决这一问题。我国是一个多地震的国家，每年地震的发生对我们国家都会造成较大的人员伤亡和严重的经济损失。地震作用下，建筑结构失效是导致人员伤亡、经济损失最为直接的因素。因此，设计、建造抗震性能良好的建筑结构是减轻震害损失最重要的任务。了解结构抗震性能的主要途径包括数值模拟与［ 1 ］试验分析。对于结构的简单弹性分析，利用有限元软件进行数值模拟能有效地分析结构的地震响应。而事实上，在地震作用下，结构往往会进入到复结构抗震能力试验是检验结构抗震的最直接的方法，而限于试验规模和经费的限制，子结构试验成为一种具有强大适应性的方法。本文归纳整理了当前关于混合试验的研究现状，对混合试验的应用前景和关键技术提出了建议。

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Analysis of Hybrid Field Prc)blems by the Method of Lines with Non.equidistant Discretization
HEINRICH DIESTEL
AND STEPHAN B. WORM
Abstract —Tfte
efficient for dimensional discretizatiom maintain equidistant substantially. computational planar
IEEE TRANSACTIONS ON MICROWAVETHEORYAND TECHNIQUES, VOL. MTT-32, NO. 6, JU~ 1984
633
Shigesawa (S’62-M63) was born in Hyogo, Japan, on January 5, 1939. He received the B. S., M. S., and Ph.D. degrees in electrical engineering from Doshisha University, Kyoto, Japan, in 1961, 1963, and 1969, respectively. Since 1963, he has been with Doshisha University. From 1979 to 1980, he was a Visiting Scholar at the Microwave Research Institute, Polytechnic Institute of New York, Brooklyn, NY. Currently, he is a Professor of the Faculty of Enzineerimz. Doshisha University. His Present research activities involve kcrowa~e and submillimeter-wave tr&mission lines and devices of open structure, fiber optics, and scattering problems of electromagnetic waves. Dr. Shigesawa is a member of the Institute of Electronics and Communication Engineers (IECE) of Japan, the Japan Society of Applied Physics, and the Opticaf Society of America (OSA). Hirosbi
A
SUCCESSFUL cuits presupposes
DESIGN accurate
of planar knowledge
microwave
cir-
In exactly
the
of the char-
the same solution
as in the limiting
of an infinite number of terms in the Fourier series expansions. The relative convergence phenomenon, which is a consequence of the Fourier series truncations, do% not occur with the method of lines. Optimum convergence is always assured, if, the simple condition is satisfied that the stripedges are located at definite positions with respect to the adjacent *’- and $~-lines [5]. It should be noted, however, that the convergence of the propagation constant, the characteristic impedancp or the resonant frequency does not critically depend on the edge parameters, so that the problem of convergence on the whole is not critical. $01.00 @1984 IEEE
Man&cript received November 3, 1983; revised February 6, 1984. This work was supported by Deutsche Forschungsgemeinschaft. The authors are with the Department of Electrical Engineering, Femuniversitaet, Hagen, Federal Republic of Germany.
method ,of
microwave
tines,
which
has been is extended
proved
to be very and two-
A completely method mately formulated
different problem
way is taken of lines
Fig, 1.
L
z
x
is given by the
Cross section of a planar microwave structure.
fact that, in case of extreme differences
in the widths
conductors and the spacings between them, the total number of lines increases considerably. The reason for these drawbacks lies in the inflexibility of the equidistant discretization. In the present paper, it will equidistant fully discretization, which method, in the grid-point be shown that the nonsuccessin
I.
INTRODUCTION
acteristics of the elementary components. In principle, an exact determination of the characteristics of passive components like transmission lines, resonators, and filters is possible by means of complete Fourier series expansions. For numerical evaluation, only a finite number of terms can be taken into account. Hence, this method mulated is characterized problem by the fact that the exactly foris solved approximately.
that are given in the case of is improved with reduced
The semi-analytical
method
of the method of lines is increased
As a consequence,
various problems of physics [2]. An essential extension of this method is given in [3] for the one-dimensional and in [4] for the twodimensional hybrid problem of planar waveguides. It has been shown that this class of waveguides and in a simple manner. an infinite number is obtained ~ of lines, case limiting case of can be solved accurately
By means of an intermediate transformation discretization. effort. The flexibltity
is solved exactly. of lines has been applied
all essential
properties the accuracy
(M’58) was born in Osaka, Japan, on October 20. 1920. He received the B.S. and Ph.D. degrees’ in electrical engineering from Kyoto University, Kyoto, Japan, in 1942 and 1955, respectively. Since 1954, he has been a Professor of Electromc Engineering at Doshisha University, Kyoto, Japan, where he carried out research @ the fields of microwave transmission lines’ and opticrd engineering. Fmm 1957 to 1958, he was a Fnlbright Scholar and a Research Associate at the Microwave Research Instit~te, Polytechnic Institute of Brooklyn, New York. Dr. Takiyama is a member of the Institute of Electronics and Communication Engineers (IECE) of Japan, the Institute of Electricrd Engineers of Japan, and the OpticaJ Society of America (OSA). Kei Takiyama