Optimization of rotor shape for constant torque improvement and radial magnetic force minimizat

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数学专业英语第2章课后答案

2.12.比:ratio 比例:proportion 利率:interest rate 速率:speed 除:divide 除法:division 商:quotient 同类量:like quantity 项:term 线段:line segment 角:angle 长度:length 宽:width高度:height 维数:dimension 单位:unit 分数:fraction 百分数:percentage3.(1)一条线段和一个角的比没有意义,他们不是相同类型的量.(2)比较式通过说明一个量是另一个量的多少倍做出的,并且这两个量必须依据相同的单位.(5)为了解一个方程,我们必须移项,直到未知项独自处在方程的一边,这样就可以使它等于另一边的某量.4.(1)Measuring the length of a desk, is actually comparing the length of the desk to that of a ruler.(3)Ratio is different from the measurement, it has no units. The ratio of the length and the width of the same book does not vary when the measurement unit changes.(5)60 percent of students in a school are female students, which mean that 60 students out of every 100 students are female students.2.22.初等几何:elementary geometry 三角学:trigonometry 余弦定理:Law of cosines 勾股定理/毕达哥拉斯定理:Gou-Gu theorem/Pythagoras theorem 角:angle 锐角:acute angle 直角:right angle 同终边的角:conterminal angles 仰角:angle of elevation 俯角:angle of depression 全等:congruence 夹角:included angle 三角形:triangle 三角函数:trigonometric function直角边:leg 斜边:hypotenuse 对边:opposite side 临边:adjacent side 始边:initial side 解三角形:solve a triangle 互相依赖:mutually dependent 表示成:be denoted as 定义为:be defined as3.(1)Trigonometric function of the acute angle shows the mutually dependent relations between each sides and acute angle of the right triangle.(3)If two sides and the included angle of an oblique triangle areknown, then the unknown sides and angles can be found by using the law of cosines.(5)Knowing the length of two sides and the measure of the included angle can determine the shape and size of the triangle. In other words, the two triangles made by these data are congruent.4.(1)如果一个角的顶点在一个笛卡尔坐标系的原点并且它的始边沿着x轴正方向，这个角被称为处于标准位置.(3)仰角和俯角是以一条以水平线为参考位置来测量的,如果正被观测的物体在观测者的上方,那么由水平线和视线所形成的角叫做仰角.如果正被观测的物体在观测者的下方,那么由水平线和视线所形成的的角叫做俯角.(5)如果我们知道一个三角形的两条边的长度和对着其中一条边的角度,我们如何解这个三角形呢?这个问题有一点困难来回答,因为所给的信息可能确定两个三角形,一个三角形或者一个也确定不了.2.32.素数:prime 合数:composite 质因数:prime factor/prime divisor 公倍数:common multiple 正素因子: positive prime divisor 除法算式:division equation 最大公因数:greatest common divisor(G.C.D) 最小公倍数: lowest common multiple(L.C.M) 整除:divide by 整除性:divisibility 过程:process 证明:proof 分类:classification 剩余:remainder辗转相除法:Euclidean algorithm 有限集:finite set 无限的:infinitely 可数的countable 终止:terminate 与矛盾:contrary to3.(1)We need to study by which integers an integer is divisible, that is , what factor it has. Specially, it is sometime required that an integer is expressed as the product of its prime factors.(3)The number 1 is neither a prime nor a composite number;A composite number in addition to being divisible by 1 and itself, can also be divisible by some prime number.(5)The number of the primes bounded above by any given finite integer N can be found by using the method of the sieve Eratosthenes.4.(1)数论中一个重要的问题是哥德巴赫猜想,它是关于偶数作为两个奇素数和的表示.(3)一个数,形如2p-1的素数被称为梅森素数.求出5个这样的数.(5)任意给定的整数m和素数p,p的仅有的正因子是p和1,因此仅有的可能的p和m的正公因子是p和1.因此,我们有结论:如果p是一个素数,m是任意整数,那么p整除m,要么(p,m)=1.2.42.集:set 子集:subset 真子集:proper subset 全集:universe 补集:complement 抽象集:abstract set 并集:union 交集:intersection 元素:element/member 组成:comprise/constitute包含:contain 术语:terminology 概念:concept 上有界:bounded above 上界:upper bound 最小的上界:least upper bound 完备性公理:completeness axiom3.(1)Set theory has become one of the common theoretical foundation and the important tools in many branches of mathematics.(3)Set S itself is the improper subset of S; if set T is a subset of S but not S, then T is called a proper subset of S.(5)The subset T of set S can often be denoted by {x}, that is, T consists of those elements x for which P(x) holds.(7)This example makes the following question become clear, that is, why may two straight lines in the space neither intersect nor parallel.4.(1)设N是所有自然数的集合,如果S是所有偶数的集合,那么它在N中的补集是所有奇数的集合.(3)一个非空集合S称为由上界的,如果存在一个数c具有属性:x<=c对于所有S中的x.这样一个数字c被称为S的上界.(5)从任意两个对象x和y，我们可以形成序列（x，y），它被称为一个有序对，除非x=y，否则它当然不同于（y，x）.如果S和T是任意集合，我们用S*T表示所有有序对（x,y),其中x术语S，y属于T.在R.笛卡尔展示了如何通过实轴和它自己的笛卡尔积来描述平面的点之后，集合S*T被称为S和T的笛卡尔积.2.52.竖直线:vertical line 水平线:horizontal line 数对:pairs of numbers 有序对:ordered pairs 纵坐标:ordinate 横坐标:abscissas 一一对应:one-to-one 对应点:corresponding points圆锥曲线:conic sections 非空图形:non vacuous graph 直立圆锥:right circular cone 定值角:constant angle 母线:generating line 双曲线:hyperbola 抛物线:parabola 椭圆:ellipse退化的:degenerate 非退化的:nondegenerate任意的:arbitrarily 相容的:consistent 在几何上:geometrically 二次方程:quadratic equation 判别式:discriminant 行列式:determinant3.(1)In the planar rectangular coordinate system, one can set up aone-to-one correspondence between points and ordered pairs of numbers and also a one-to-one correspondence between conic sections and quadratic equation.(3)The symbol can be used to denote the set of ordered pairs（x，y）such that the ordinate is equal to the cube of the abscissa.（5）According to the values of the discriminate，the non-degenerate graph of Equation （iii） maybe known to be a parabola， a hyperbolaor an ellipse.4.（1）在例1，我们既用了图形，也用了代数的代入法解一个方程组（其中一个方程式二次的，另一个是线性的）。

树优公司-TOSCA-结构形状优化培训教程-1

• Each surface node may be displaced • The optimization displacement direction is determined based upon the components
geometry • The signed absolute optimization displacement amount is determined by the
Stress reduction of 25 % required!
2012/5/19
Possible Solutions:
(1) Parameter free shape optimization using TOSCA Structure
(2) Change of the Radius of the contour
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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Node Position (Theta = [0? 90癩)
Optimized Design
2012/5/19
• 公司成员大多具有长期的工程经验，是中国最专业的工程设计优化团队之一。
总部位于北京高科技园区，在西安、武汉和香港设有分支机构。
北京树优信息技术有限公司
2012/5/19

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设计、仿真和优化完整解决方案
多学科优化、试验设计结构拓扑形状优化流场网格变形优化
2012/5/19
Optimization displacement direction

Abaqus中Topology和Shape优化指南

Abaqus中Topology和Shape优化指南目录1. 优化模块界面......................................................................................................- 1 -2. 专业术语..............................................................................................................- 1 -3.定义拓扑优化Task(general optimization和condition-based optimization).......- 2 -3.1 General Optimization 参数设置.................................................................- 3 -3.1.1 Basic选项参数..................................................................................- 3 -3.1.2 Density选项参数..............................................................................- 4 -3.1.3 Perturbation选项参数.......................................................................- 5 -3.1.4 Advanced选项参数...........................................................................- 5 -3.2 Condition-based topology Optimization 参数设置....................................- 6 -3.2.1 Basic选项参数..................................................................................- 7 -3.2.2 Advanced选项参数...........................................................................- 7 -4 定义Shape Optimization Task方法....................................................................- 8 -4.1 Basic选项参数............................................................................................- 8 -4.2 Mesh Smoothing Quality选项参数............................................................- 9 -4.3 Mesh Smoothing Quality选项参数..........................................................- 11 -5 定义design response变量方法.........................................................................- 13 -5.1 单个design response定义方法...............................................................- 14 -5.2 combined design response定义方法........................................................- 15 -5.3 design response使用注意事项.................................................................- 17 -5.3.1 定义design response的操作.........................................................- 17 -5.3.2 condition-based topology optimization的design response............- 18 -5.3.3 general topology optimization的design response..........................- 18 -5.3.4 design response for shape optimization...........................................- 21 -6 定义objective function方法..............................................................................- 22 -6.1 目标函数定义...........................................................................................- 23 -6.2 目标函数的运算.......................................................................................- 23 -6.2.1 min运算..........................................................................................- 23 -6.2.2 max运算..........................................................................................- 24 -6.2.3 minimizing the maximum design response......................................- 24 -7 定义Constraints方法........................................................................................- 24 -8 定义Geometric restrictions方法.......................................................................- 25 -8.1 Defining a frozen area................................................................................- 26 -8.2 Specifying minimum and maximum member size....................................- 26 -8.3 maintaining a moldable structure(可拔模结构)........................................- 27 -8.4 maintaining a stampable structure(冲压成型结构)...................................- 28 -8.5 Specifying a symmetric structure...............................................................- 29 -8.6 Applying additional restrictions during a shape optimization...................- 31 -8.7 Combining geometric constraints..............................................................- 31 -9 定义Stop conditions方法..................................................................................- 32 -9.1 Global stop conditions...............................................................................- 32 -9.2 Local stop conditions.................................................................................- 33 -10 Abaqus优化模块支持.......................................................................................- 34 -10.1 Support for analysis types........................................................................- 34 -10.2 Support for geometric nonlinearities.......................................................- 34 -10.3 Support for multiple load cases................................................................- 34 -10.4 Support for acceleration loading..............................................................- 35 -10.5 Support for contact during the optimization............................................- 35 -10.6 Restrictions on an Abaqus model used for topology optimization..........- 35 -10.7 Restrictions on an Abaqus model used for shape optimization...............- 35 -10.8 Support materials in the design area........................................................- 36 -10.8.1 Materials supported by condition-based topology optimization....- 36 -10.8.2 Materials supported by general topology optimization.................- 36 -10.8.3 Material support in shape optimization..........................................- 37 -10.9 支持的单元类型.....................................................................................- 37 -10.9.1 支持的二维实体单元...................................................................- 37 -10.9.2 支持的三维实体单元...................................................................- 38 -10.9.3 支持的对称实体单元...................................................................- 39 -10.9.4 额外支持的单元...........................................................................- 39 -11. Job模块中优化过程的设置............................................................................- 40 -11.1 优化过程的理解.....................................................................................- 40 -11.2 Optimization Process Manager................................................................- 42 -12 拓扑优化理论...................................................................................................- 42 -12.1 General Topology Optimization理论......................................................- 43 -12.1．1 SIMP(Solid Isotropic Material With Penalization Method).......- 43 -12.1.2 RAMP(Rational Approximation of Material Properties)...............- 43 -12.1.3 Gradient-based methods.................................................................- 43 -12.2 General与Condition-based Topology Optimization对比.....................- 44 -13 拓扑优化结果后处理.......................................................................................- 44 -13.1 单元相对密度值.....................................................................................- 44 -13.2 Isosurfaces................................................................................................- 45 -13.3 Extraction.................................................................................................- 47 -14 形貌优化后处理...............................................................................................- 48 -14.1 向量DISP_OPT.....................................................................................- 48 -14.2 场变量DISP_OPT_V AL........................................................................- 48 -14.3 正常分析步中的优化迭代过程中的应力和位移等场变量.................- 49 -14.4 Extracting a surface mesh........................................................................- 49 -15 几何非线性的开与闭对拓扑优化结果的影响...............................................- 50 -16. 形貌优化中的几何约束..................................................................................- 53 -16.1 Demold control(脱模控制)......................................................................- 53 -16.2 Turn control(车床加工控制)...................................................................- 55 -16.3 Drill control(钻孔控制)...........................................................................- 56 -16.4 Planar symmetry(平面对称约束)............................................................- 57 -16.5 Stamp control(锻造控制)........................................................................- 58 -16.6 Growth约束............................................................................................- 58 -16.7 Design direction约束..............................................................................- 59 -16.8 Penetration check(穿越检查)..................................................................- 60 -1. 优化模块界面2. 专业术语① optimization task：对优化任务的一个定义，即定义一个优化Job；② design responses：一个设计响应可以直接从输出数据库中提取，例如模型的体积，另外，对于拓扑优化模块的设计响应不仅可以直接从输出数据库中提取，而且可以计算设计响应，如模型的应变能；③ objective function：目标函数指的是设计响应的函数值或者是一组设计响应的组合，如整个模型的应变能的最小值；④ constraints：约束是一个设计响应的函数值，但不能是多个设计响应组合的函数值；⑤ geometric restriction：A geometric restriction places restrictions on the changes that the Abaqus Topology Optimization Module can make to the topology of the model. Geometrical restrictions include frozen regions from which material cannot be removed and manufacturing constraints, such as restrictions on cavities and undercuts, that would prevent the optimized model from being removed from a mold⑥ stop condition：停止条件是对优化计算收敛的一个指示器，如当在一个指定数量的迭代后一个优化被认为完成了；global stop condition定义了优化迭代的最大数目，local stop condition指定了优化迭代达到所需最小或最大数目；⑦ optimization processes：需要在job模块中创建；⑧ design varible：对于topo优化，优化区域的每个单元的密度即为设计变量；而shape优化，优化区域表面单元的节点的位移即为设计变量；⑨ design cycle：优化过程中的每个迭代成为design cycle；【提示】：I、优化算法总是在满足了约束的基础上才开始最大或最小化目标函数；II、一个优化任务中最多只能包含一个体积约束；【附英文原版】3.定义拓扑优化Task(general optimization和condition-based optimization)3.1 General Optimization 参数设置 3.1.1 Basic选项参数3.1.2 Density选项参数3.1.3 Perturbation选项参数3.1.4 Advanced选项参数在优化计算过程中，拓扑优化模块会自动给优化区域分配一个指定的质量来满足约束和目标函数，在优化结束时，整个优化区域的结构包含了硬单元(hard elements)和软单元(soft elements)，其中软单元对结构的刚度没有任何影响，但是影响着结构的自由度，因此会影响优化计算的速度。

Topology Optimization of Continuum Structures Under Buckling and

Topology Optimization of Continuum Structures Under Buckling andDisplacement ConstraintsBIAN Bing-chuanDepartment of Applied Science and TechnologyTaishan UniversityTaian , ChinaE-mail:bianbingchuan@Abstract—In this paper, the topology optimization model for the continuum structure was constructed. The model had the minimized weight as the objective function subjected to the buckling constraints and displacement constraints. Based on the Taylor expansion and the filtering function, the objective function and the constraints were approximately expressed as an explicit function. The optimization model was translated into a dual programming and solved by the sequence second-order programming. All the corresponding numerical procedures are developed by the PCL toolkit in the MSC.Patran/Nastran software platform. Numerical examples show that this method can solve the topology optimization problem of continuum structures with the buckling and displacement constraints efficiently and give more reasonable structural topologies.Key words- buckling constraints; displacement constraints; ICM method; topology optimization; filtering functionI.I NTRODUCTIONThe intensity, the stiffness and the buckling are very important factors determining the safety of a structure. Structural failure due to buckling of one part is catastrophic. Buckling has attracted more attention in recent years.As the structural static optimization and the dynamical optimization, there are three development phases of the buckling structural optimization. They are cross-sectional, shape, and topology optimizations. The cross-sectional and shape optimization considering buckling constraints were developed by many scholars.The Optimum design with size and shape variables and sensitivity analysis for buckling stability of complex built-up structures and composite material were made byGu yuanxian et al [1]. The plate’s shape and frame’scross-sectional optimization were studied by D. ManickarSUI Yun-kangThe Centre of Numerical Simulation for EngineeringCollege of Mechanical Engineering and AppliedElectronics TechnologyBeijing University of TechnologyBeijing, China-ajah et al [2] using the Evolutional Structural Optimization (ESO) method. The studies by Rong Jian-hua et al [3] reported that the cross-sectional optimum designs of the frame under the maximum bucking critical load using ESO method.The topological optimization studies considering buckling is seldom due to difficulty. The topological optimization of truss studies by Guo Xu[4] reported that the model for the truss structure is constructed, which the minimized weight as the objective function has subjected to the local buckling constraints and global buckling constraints. The optimum designs reported by Zhou Ming [5] are the shell’s topological optimization under linear buckling response using variable-density method.From above we can see that several mechanical properties were difficultly considered and the weight’s upper limit was difficultly confirmed when the object was minimum buckling eigenvalue. In order to overcome these difficulty, based on the independent continuous and mapping idea [6, 7], the continuous independent topological variables are used in this problem. The topology optimization model for the continuum structure is constructed, which the minimized weight as the objective function has subjected to the buckling and displacement constraints.2009 International Conference on Information Technology and Computer ScienceII.F ORMULATION AND SOLUTION OF THE TOPOLOGICALOPTIMIZATION MODELA.The topological optimization model of the continuum structureThe buckling critical load is one structure’s inherent character. It is not alter with the outer load alter. Buckling analytic eigenvalue equation can be represented as)1(,J ,j j j0u G K ˅˄O (1)The symbols K ,G ,j O and u j denote stiffness matrix, thegeometric stiffness matrix, the j-th eigenvalue, and the corresponding eigenvector, respectively.The structure’s displacement which is the key point’s displacement must be limited. So the displacement constraints can be represented asrir u u d (2)Where ir u is the i-th element’s r -th displacement andr u is upper limit of r -th displacement, respectively.In the present study, the filter functions of weight, stiffness matrix, geometric stiffness matrix for one element, are denoted as followsa i i w t t f )(b i i k t t f )( bi i g t t f )((3)Where a and b are constants, respectively. Numerical example demonstrate that the optimal results with most of elements having value 0 or 1can be obtained using the filter functions in which a =1,b =3.3.As a consequence, the variables such as weight, stiffness matrix, geometric stiffness matrix, are identified by the filter functions mentioned previously0)(i i w i w t f w ˈ0)(i i k i t f k k ˈ0)(i i g i t f g g (4)Where 0i w is the weight,0i k is the stiffness matrix,0i g is the geometric stiffness matrix, of the original structure element before topology optimization, respectively. Finally, the topology optimization model for the continuum structure is constructed, which the minimized weight as the objective function has subjected to the buckling and displacement constraints.°°°°¯°°°°® d d d d ¦¦ )1;1;1(10))(())()((s.t.)(min find 1J ,...,j R ,...,r N ,...,i t t f u t f ,t f w t f W E i r N i i k ir j i g i k j N1i ii w N O O t (5)Where W is the total weight of the structure, w i and N are the weight of i -th element and the total number of element,j O is the upper limit of the critical load ,J i s the total number of the buckling mode ,ir u isthe i -th element’s r -th displacement and r u is upper limit of r -th displacement, respectively.B.Explicit approximate function of the constraint 1) Explicit approximate function of the buckling constraint: Since eigenvalue O is implicitly related with topology variable t , eigenvalue can be explicit expression by the first-order Taylor expansion if the first-order derivative of eigenvalue O has been obtained. So to obtain the first-order derivative of eigenvalue O is the most important approach to optimum design.The expression for the eigenvalues is obtained by Eq. (1); i.e.jj j j j Guu Ku u TT / O (6)The stiffness matrix and geometric stiffness matrix can be obtained)(10t K k K ¦ Ni bi i t ˈ)(10t G gG¦ Ni bi it (7)The inverse variable is importedbi i t x /1 (8)From all of above, we can obtain)2//()2/2/(/TT T i j j j i j j j i j i j x x Gu u u g u u k u O O w w ij ij j ij x V V U 6O /)( (9)Where ji j ij U u k u T5.0 i s the mode strain energy for the i -th element under j -th mode .ji j ij V u g u T5.0 is the mode geometric strain energy for the i -th element under j-th mode .i j j j x V Gu u T 5.0 6is the total mode geometric strain energy for the structure under j-th mode. For bucking constraint j j O O d , using first-order Taylor expansion, theapproximate expression of eigenvalue can be obtained ˖¦¦ 66 d Ni jk ij j k ij k j j k i j k ij j k ijNi i V V U x V V Ux 1)()()()()()(1/)()(/)(O O O O x (10)Where superscript k is the number at the k -th iteration. Ifwe define j k ij j k ij k ij V V U A 6 /)()()()(O ,the left-hand of inequation re-written as ˖¦ Ni k i j k ij j k ij i x V V Ux 1)()()(/)(6O iNi ijNi i k ij b k i Ni k i i k ij x dx A t x x A ¦¦¦11)()(1)()()(/(11)The right-hand of inequation re-written as ˖jNi k ij k j j Ni j k ij j k ij k j j e A V V U ¦¦ 1)()(1)()()()(/)()(x x O O O O O 6(12)Then all kinds of buckling constraints can be simplified by the formj i Ni ije x dd ¦ 1(13)2) Explicit approximate function of the displacement constraint: The generalized displacement of one node can be expressed as¦³ Ni iir dvu 1R T V )(˅˄İı(14)Where Vi ıis the stress component of i-th element under the unit virtual load .R i İ is the strain component of i-th element under the unit real load.So the Eq. (14) can be written as follow¦¦³ Ni i i Ni i i r dv u 1VT R 1R T V )()(u F İı˅˄(15)Based on the Mohr’s theorem we can obtain ˖¦¦ N i i i k i i Ni i i r x x u 1VT R )(1VT R )()(u F u F (16)Let ir k iii c x)(VTR /)(u F ˈthe displacement constraintwas expressed as explicit approximate function ˖r i Ni ir Ni i i k ii r x c x x u d ¦¦ 11VT R )()(u F (17)C. Explicit approximate function of the objective function The object function can be expressed as¦N1i i i w w t f W 0)((18)Where the filter function is a i i w t t f )( the filter function can be re-written asb a i i w x t f //1)((19)If we define E b a /ˈthe Eq.(19) is translated into E i i w x t f /1)( ˈ then, the object function is re-written as¦Ni i i x wW 10/E(20)The object function is translated into explicit expression by second-order Taylor expansion}]/)2([]2/)1({[1)(0212)(0i k i i i Ni k i i x x w x x w W ¦E E E E E E (21)Where )(k ix is the initial design variable of i-th element at the k -thiteration.We define 2)(02/)1( E E E k i i i x w A ,1)(0/)2( E E E k i i i x w B The topology optimization model can be re-written as°°°°°¯°°°°°® d d d d ¦¦¦)1(1)1()1(s.t .)(min find112N ,...,i x R ,...,r ux c J ,...,j e x dx B x A WE ii ri Ni ir ji Ni ijN1i i i i i Nx(22)D.Solution of the topological optimization modelIn order to reduce the number of the variable and the model’s scale. The optimization model is translated into a dual programming°¯°®t 0s.t.)(max find ȤȤȤĭ(23)In Eq. (23) ˈ»¼º«¬ª ȘμȤ,),,(min ),(1Șμx ȘμL ĭii x x d d We can get the second-order programming model°°¯°°®t 0s.t.)(21)(min find T 0ȤȤD ȤH D ȤȤȤȤT T ĭ(24)Where ĭ2D ˈĭH The second-order programming model was solved ˈwecan get Ȥ. The optimum values of topological variables *tare accordingly obtained. Then the structural analysis and topological optimization are renewably begun untilsatisfying the following convergence condition which isH'd |/)(|)1()()1(k k k W W W W (25)Where )(k W and )1( k W is the structural weight ofprevious iteration and current iteration, respectively, H theconvergence precision. In the present study, 001.0 H is employed.III.N UMERICAL EXAMPLESA.Example 1Illustrating in Figure.1 (a), the base structure is a plane elastic body with size 80.0u 50.0u 1.0mm. The elastic modulus of structure material is E=1.0u 106MPa, and thePoisson's ratio is J =0.3, and the material density is U =1.0h 10-3kg /mm3. The right-hand points of the structure are clasped. A concentrated force P=9000N along with the up-to-down (-y) direction is located in right-bottom of structure. Quadrilateral element with 4-node is used and number of total elements is 48u 30.The base structure’sweight is 4kg.(a)Base structure(b)Optimal topology configuration subjected tobuckling constraints(c)Optimal topology configurationsubjected to displacement constraints of reference [7](d)Optimal topology configuration subjected to buckling and displacementconstraintsFigure 1.Definition and optimal results of example 1In this example the maximal value of the critical load is 1O ˙1100N and the maximal value of the displacement is 0.5mm.After 47 times iterations the optimal topology configuration depicted in Figure 1 can be get. From figure 1, we can see that the topologyconfiguration subjected to displacement constraints in reference [7] is thin and uniformity. The topologyconfiguration with two constraints is not same as that with single constraint. The middle bar is shortened and the whole structure’s stability is strengthened. B.Example 2Illustrating in Figure.2 (a), the base structure is a solid elastic body with size 120u 45u 8mm. The elastic modulus of structure material is E =7.0u 104MPa, and the Poisson's ratio is J =0.3, and the material density is U =2.7×10-6kg /mm 3. The right-hand and left-hand bottom corner points of the structure are clasped. A distributed force P =12500N along with the up-to-down (-y ) direction is located five nodes in top of surface. Hexahedron element with 8-node is used and number of total elements is 40u 15u 4.The base structure’s weight is 0.117kg.In this example the maximal value of the critical load is1O ˙21000N and the maximal value of the displacement is 0.4mm. After 34 times iterations the optimal topology configuration depicted in Figure 2 can be get.(a) Finite element model(b)Optimal topology configurationsubjected to buckling constraints(c)Optimal topology configuration subjected to displacement constraints (d)Optimal topology configuration subjected to buckling and displacement constraintsFigure 2. Definition and optimal results of example 2IV.C ONCLUSIONS(1) The topological optimization of the continuum structures with two kinds of constraints was solved. The study constraints scope to topological optimization of the continuum structures was expanded. So the structure’s constraints are similar to the technical structure.(2)The buckling and displacement of the continuum structures is one of important aspect of structure design. The studies of topology optimization problem of continuum structures with the buckling and displacement constraints can provide design consult to engineer in conception phase.R EFERENCES[1] Y.X. Gu, G.Z. Zhao, H.W. Zhang, Z. Kang, and R.V. Grandhi.Buckling design optimization of complex built-up structures with shape and size variables. Struct Multidisc Optim 19Springer-Verlag 2000, 183-191.[2] D. Manickarajah, Y.M. Xie, G.P. Steven. Optimization of columns andframes against buckling. Computers and Structures, 75 (2000), 45-54.[3] J.H. Rong, Y.M. Xie, X.Y. Yang .An improved method forevolutionary structural optimization against buckling. Computers and Structures, 79 (2001), 253-263.[4] X. Guo · G.D. Cheng · N. Olhoff. Optimum design of truss topologyunder buckling constraints. Struct Multidisc Optim (2005), DOI 10.1007/s00158-004-0511-z.[5] M.Zhou. Topology Optimization for Shell Structures with LinearBuckling Responses. Computational Mechanics WCCMVI Sept.5-10, 2004.Beijing.China.[6] Sui Yunkang, Yang Deqing, et al. Uniform ICM theory and methodon optimization of structural topology for skeleton and continuum structures. Chinese Journal of Computational Mechanics, 2000, 17(1): 28-33.[7] PENG Xi-rong, SUI Yun-kang. Topological Optimization ofContinuum Structure with Static Displacement and Frequency Constraints by ICM method. Chinese Journal of Computational Mechanics, 2006, 23(4):391-396.。

基于机器视觉的认知康复机器人系统设计

encounter difficulties, firstly, based on machine vision, highlight hint will be carried out
through the interface. If the patient is still unable to complete the task, the robotic arm
image information from time domain to frequency domain, and target detection is
accomplished by comparing the descriptors of reference objects and objects to be tested.
了完整的辅助康复系统，并且针对所用的六自由度机械臂进行了运动学建模和分
析，分析了机械臂的运动和避障策略。
最后，在康复系统样机中，通过模拟病人进行了完整的测试。仿真和实际测
试的结果充分的展示了辅助康复策略的可行性以及该辅助康复机器人系统的高效
性。
关键词：认知康复；机器视觉；傅里叶描述算子；模长位移算法；机械臂
been proved that, to a certain extent, MCI can be slowed down or even cured by human
intervention. However, due to the lack and uneven distribution of related medical
康复过程往往不及时不到位，缺乏持续性和有效性。针对此，本课题设计了一套
基于机器视觉的辅助认知康复机器人系统，用以提高 MCI 等疾病的康复效率。

整体立铣刀圆弧刃前刀面的磨削轨迹算法

机械设计与制造Machinery Design & Manufacture147第6期2021年6月整体立铳刀圆弧刃前刀面的磨削轨迹算法张潇然，罗斌，陈思远，程雪峰（西南交通大学机械工程学院，四川成都610031）摘要:针对圆弧立铳刀磨削中周齿前刀面与端齿前刀面的过渡问题，提出磨削圆弧刃前刀面的砂轮轨迹算法，以此实现周齿与端齿前刀面的光滑连接。

定义了一种切深磨削点轨迹曲线，可以同时约束圆弧前刀面的宽度和前角；定义了圆弧刃在平面中的瞬时前刀面，计算在瞬时前刀面中的砂轮磨削轨迹和姿态,再经过空间坐标变换，得出砂轮实际加工轨迹。

通过C++将算法编写为相应程序，进行仿真和实际加工验证，所得验证结果证明了该方法的正确性和可行性。

关键词:立铳刀;磨削加工;端齿圆弧刃；前刀面中图分类号:TH16;TH161 文献标识码:A 文章编号:1001-3997(2021)06-0147-03The Grinding Algorithm for the Rake Face of the Arc Edge of the Integral End MillZHANG Xiao-ran, LUO Bin, CHEN Si-yuan, CHENG Xue-feng(School of Mechanical Engineering , Southwest Jiaotong University, Sichuan Chengdu 610031, China)Abstract :A iming at the transition problem between the rake f ace of p eripheral f lank and the rake f ace of e nd tooth in the circulararc end mill, proposing a grinding algorithm f or the rake face of the arc edge that can achieve smooth connection between thetwo. Defines a depth-of-depth curve that can simultaneously constrain the width and rake angle of t he arc rake f ace. Defines theinstantaneous rake f ace of the arc edge and calculates the grinding path and attitude of t he grinding wheel in it. After the space coordinate transformation, the actual machining track of t he grinding wheel is obtained. Programming the algorithm into corre sponding p rogram by C++, and p erformming the simulation and p rocessing verification. The obtained results prove the correctnessand f easibility of t he algorithm.Key Words :End Mill ； Grinding ； Arc Edge ； Rake Face1引言圆弧头立铳刀是目前常见的高速切削刀具,具有制造成本低、材料切除率大等特点。

8形状优化

edge domain.
Update: Reparameterize or partition domains. Parameters: Update the morphing parameters. √ retain handles：当在为新创建的或编辑的domain中的 √ partition domains：所有已创建的2D domain（ shell单元或3D domain中的面）将被刨分。
2）改变handle间的距离
3）改变倒圆或孔的半径
4）绘制一新的曲面或线几何
HyperMorph操作来自domain的生成：create
Domain 的操作： Create: Create or update a domain. Organize: Move elements into a domain. Edit edges: Merge, split, or add handles to an
在HyperMorph中，通过Morphing操作来实现网格的变形。Morphing过程包括将模型分成多个domains（域），这些域的形状由handles（控制柄）来控制。
Domains: 区域，模型的多个域 Handles：控制点，即Domains的对应控制点
Domain的分类：
全局 domain (globe domain) 局部 domain (local domain ) 1D
目约 200Mpa
标：质量最小化束：在接头部位的最大Von Mises 应力小于
设计变量：接头的形状变量（使用HyperMorph 定义）
钢轨接头的形状优化
优化方案1
优化方案2
优化方案3=方案1+方案2

2010年11月CAESARII高级培训讲义- 工况编辑器 - 何耀良

z 载荷的形式应该进行独立的分析
z Magnification due to local variations
z 考虑了弯头、三通等管件的局部应力增大现象
z Code committee tradition
z 规范应力是规范委员会的传统惯例。
AECsoft
2010-11-6
CAESAR II 中所有的原始载荷
因此我们需要对工况进行组合编辑
AECsoft
2010-11-6
工况编辑器
推荐工况
AECsoft
原始载荷
组合工况
载荷循环次数工况应力类型
2010-11-6
CAESAR II 工况设置
z 进行一次静态计算后，CAESAR II 保存最后一次运算时的载荷工况设置。
z 注：CAESAR II 不会自动设置偶然工况，用户需要根据实际情况灵活定义。
工况组合方法
z CAESARII中提供了多种工况组合方法，其中最常见的有两类：
z Algebraic（代数合成）: z 用于两个工况之间的减运算（求解EXP）. z 分别求解相关工况的位移后再进行减运算得到位移差. z 通过最后得到的位移量来求解推力、弯矩、应力.
z Scalar（标量合成）: z 用于两个工况之间的加运算（求解OCC）. z 分别求解相关工况的应力后再进行叠加 z 不再单独计算各工况的位移
AECsoft
2010-11-6
组合工况
Used to add or subtract results from previously defined primitive load cases. 通过基础工况的加减获取特定的载荷要求。如在非线性系统中重新计算风和地震载荷。 Necessary for proper EXP and OCC code stress definition. 考虑二次应力及偶然应力的工况组合。 Not used for restraint or equipment load definition, nor for displacement reporting. 有些特定的中间设计工况既不是计算推力也不是计算位移，如弹簧设计工况

面向智能制造的不规则零件排样优化算法

Vol. 27 No. 6June2021第27卷第6期2 0 2 1年6月计算机集成制造系统Computer Integrated Manufacturing SystemsDOI ： 10. 13196/j. cims. 2021. 06. 013面向智能制造的不规则零件排样优化算法高勃X张红艳X朱明皓2+(1-北京交通大学计算机与信息技术学院'匕京100044；2.北京交通大学经济管理学院，北京100044&摘要：以智能工厂应用场景为例，为提高广泛应用于智能制造领域的二维不规则件的排样性能，提出了基于启发式和蚁群的不规则件排样优化算法$首先提取不规则件的几何特征，对零件进行组合操作预处理，使两个或多个不规则零件组合为矩形件或近似矩形件并对其包络矩形，然后利用蚁群学习算法对预处理后的零件进行排样，确定零件排放的最佳位置，不断更新得到最优排样结果。

仿真实验结果表明，综合考虑板材利用率以及耗时情况，所提算法取得了较好的结果能总够满足实际生产的需求$关键词：二维板材；不规则零件；启发式算法；蚁群学习算法；优化排样中图分类号:TP391文献标识码:AOptimization algorithm of irregular parts layout for intelligent manufacturingGAO B o 1 , ZHANG Hongyan 1 , ZHUMinghao 2(1. School of Computer and Information Technology, Beijing Jiaotong University ,Beijing 100044, China ；2. School of Economics and Management , Beijing Jiaotong University , Beijing 100044, China)Abstract ：To improve the performance of two-dimensional irregularly shaped part layout in the field of intelligent manufacturing and smart factories 'an optimization algorithmbased on heuristics and ant colony optimizations wasproposed.Thegeometricfeaturesofirregularlyshapedpartswereextractedtopreprocesscombinatorialoperationof theseparts 'which madetwoormorepartscombineintorectangularorapproximatelyrectangularparts.Thentheant colony learning algorithm was used to find an initial combination of parts.After irregularly shaped parts are nes- ted 'thebestpositionofeachpartwasdeterminedandoptimizediteratively.Theresultsofsimulationexperimentsshowed that the algorithm had achieved satisfactory results in terms of the utilization rate ofboards and time-com-plexity 'which madeareasonablesolutiontobeadoptedforactualproductions.Keywords ：two-dimensional plate ； irregular parts ； heuristic algorithm ； ant colony learning algorithm ； optimized lay outo 引言二维零件排样是实际应用中最常见的排样问题,广泛应用在机械制造、轻工、服装和印刷业等行业中。

螺杆压缩机外文文献翻译、中英文翻译、外文翻译

螺杆压缩机外文文献翻译、中英文翻译、外文翻译英文原文Screw CompressorsN. Stosic I. Smith A. KovacevicScrew CompressorsMathematical Modellingand Performance CalculationWith 99 FiguresABCProf. Nikola StosicProf. Ian K. SmithDr. Ahmed KovacevicCity UniversitySchool of Engineering and Mathematical SciencesNorthampton SquareLondonEC1V 0HBU.K.e-mail:n.stosic@/doc/d6433edf534de518964bcf 84b9d528ea81c72f87.htmli.k.smith@/doc/d6433edf534de51896 4bcf84b9d528ea81c72f87.htmla.kovacevic@/doc/d6433edf534de51 8964bcf84b9d528ea81c72f87.htmlLibrary of Congress Control Number: 2004117305ISBN-10 3-540-24275-9 Springer Berlin Heidelberg New York ISBN-13 978-3-540-24275-8 Springer Berlin Heidelberg New YorkThis work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the GermanCopyright Law.Springer is a part of Springer Science+Business Media/doc/d6433edf534de518964bcf84b9d 528ea81c72f87.html_c Springer-Verlag Berlin Heidelberg 2005Printed in The NetherlandsThe use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.Typesetting: by the authors and TechBooks using a Springer LATEX macro packageCover design: medio, BerlinPrinted on acid-free paper SPIN: 11306856 62/3141/jl 5 4 3 2 1 0PrefaceAlthough the principles of operation of helical screw machines, as compressors or expanders, have been well known for more than 100 years, it is only during the past 30 years thatthese machines have become widely used. The main reasons for the long period before they were adopted were their relatively poor efficiency and the high cost of manufacturing their rotors. Two main developments led to a solution to these difficulties. The first of these was the introduction of the asymmetric rotor profile in 1973. This reduced the blowhole area, which was the main source of internal leakage by approximately 90%, and thereby raised the thermodynamic efficiency of these machines, to roughly the same level as that of traditional reciprocating compressors. The second was the introduction of precise thread milling machine tools at approximately the same time. This made it possible to manufacture items of complex shape, such as the rotors, both accurately and cheaply.From then on, as a result of their ever improving efficiencies, high reliability and compact form, screw compressors have taken an increasing share of the compressor market, especially in the fields of compressed air production, and refrigeration and air conditioning, and today, a substantial proportion of compressors manufactured for industry are of this type.Despite, the now wide usage of screw compressors and the publication of many scientific papers on their development, only a handful of textbooks have been published to date, which give a rigorous exposition of the principles of their operation and none of these are in English.The publication of this volume coincides with the tenth anniversary of the establishment of the Centre for Positive Displacement Compressor Technology at City University, London, where much, if not all, of the material it contains was developed. Its aim is to give an up to date summary of the state of the art. Its availability in a single volume should then help engineers inindustry to replace design procedures based on the simple assumptions of the compression of a fixed mass of ideal gas, by more up to date methods. These are based on computer models, which simulate real compression and expansion processes more reliably, by allowing for leakage, inlet and outlet flow and other losses, VI Preface and the assumption of real fluid properties in the working process. Also, methods are given for developing rotor profiles, based on the mathematical theory of gearing, rather than empirical curve fitting. In addition, some description is included of procedures for the three dimensional modelling of heat and fluid flow through these machines and how interaction between the rotors and the casing produces performance changes, which hitherto could not be calculated. It is shown that only a relatively small number of input parameters is required to describe both the geometry and performance of screw compressors. This makes it easy to control the design process so that modifications can be cross referenced through design software programs, thus saving both computer resources and design time, when compared with traditional design procedures.All the analytical procedures described, have been tried and proven on machines currently in industrial production and have led to improvements in performance and reductions in size and cost, which were hardly considered possible ten years ago. Moreover, in all cases where these were applied, the improved accuracy of the analytical models has led to close agreement between predicted and measured performance which greatly reduced development time and cost. Additionally, the better understanding of the principles of operation brought about by such studies has led to an extension of the areas of application of screw compressors and expanders.It is hoped that this work will stimulate further interest in an area, where, though much progress has been made, significant advances are still possible.London, Nikola StosicFebruary 2005 Ian SmithAhmed KovacevicNotationA Area of passage cross section, oil droplet total surfacea Speed of soundC Rotor centre distance, specific heat capacity, turbulence model constantsd Oil droplet Sauter mean diametere Internal energyf Body forceh Specific enthalpy h = h(θ), convective heat transfer coefficient betweenoil and gasi Unit vectorI Unit tensork Conductivity, kinetic energy of turbulence, time constant m Massm˙ Inlet or exit mass flow rate m˙ = m˙ (θ)p Rotor lead, pressure in the working chamber p = p(θ)P Production of kinetic energy of turbulenceq Source term˙Q Heat transfer rate between the fluid and the compressor surroundin gs˙Q= ˙Q(θ)r Rotor radiuss Distance between the pole and rotor contact points, control volume surfacet TimeT Torque, Temperatureu Displacement of solidU Internal energyW Work outputv Velocityw Fluid velocityV Local volume of the compressor working chamber V = V (θ)˙VVolume flowVIII Notationx Rotor coordinate, dryness fraction, spatial coordinatey Rotor coordinatez Axial coordinateGreek Lettersα Temperature dilatation coefficientΓ Diffusion coefficientε Dissipation of kinetic energy of turbulenceηi Adiabatic efficiencyηt Isothermal efficiencyηv Volumetric efficiencySpecific variableφ Variableλ Lame coefficientμ Viscosityρ Densityσ Prand tl numberθ Rotor angle of rotationζ Compound, local and point resistance coefficientω Angular speed of rotationPrefixesd differentialΔ IncrementSubscriptseff Effectiveg Gasin Inflowf Saturated liquidg Saturated vapourind Indicatorl Leakageoil Oilout Outflowp Previous step in iterative calculations SolidT Turbulentw pitch circle1 main rotor, upstream condition2 gate rotor, downstream conditionContents1Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ………………………. . . . . . . . . . . . . . . 1 1.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. . . . . . . . . 4 1.2 Types of Screw Compressors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . ….. . . . . . . .7 1.2.1 The Oil Injected Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . …... . .71.2.2 The Oil Free Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . ….... .7 1.3 Screw Machine Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .8 1.4 Screw Compressor Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .101.5RecentDevelopments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5.1RotorProfiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 13 1.5.2CompressorDesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2ScrewCompressorGeometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1 The Envelope Method as a Basis for the Profiling of Screw CompressorRotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ………………………….. . . . . ….. . . . . . . . 19 2.2 Screw Compressor Rotor Profile s . . . . . . . . . . . . . . . . . . . . …. . . . . . . . . . . . . . . . . . . ….. . . 20 2.3 Rotor ProfileCalculation . . . . . . . . . . . . . . . . . . . . . . . . . . . …………………………. . . . . .23 2.4 Review of Most Popular Rotor Profiles . . . . . . . . . . . . . . . ………………………….. . . . . . 23 2.4.1 Demonstrator Rotor Profile (“N” Rotor Generated) . . ………………………………….. . 24 2.4.2 SKBK Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . ……………………………... . . . . . . . . .26 2.4.3 Fu Sheng Profile . . . . . . . . . . . . . . . . . . . . . . . . . ………………………………. . . . . . . . .27 2.4.4 “Hyper”Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ………………………………. . .27 2.4.5 “Sigma” Profile . . . . . . . . . . . . . . . . . . . . . . .. . . . . . ………………………………. . . . . .28 2.4.6 “Cyclon” Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . ………………………………. . . . . .28 2.4.7 Symmetric Prof ile . . . . . . . . . . . . . . . . . . . . . . . . . . . ……………………………… . . . . .29 2.4.8 SRM “A” Profile . . . . . . . . . . . . . . . . . . . . . . . . . . ……………………………… . . . . . . .30 2.4.9 SRM “D” Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . ……………………………… . . . . . .31 2.4.10 SRM “G” Profile . . . . . . . . . . . . . . . .. . . . . . . . …………………………….. . . . . . . . . .32 2.4.11 City “N” Rack Generated Rotor Profile . . . . . . . . . . . ………………………………… . . 32 2.4.12 Characteristics of “N” Profile . . . . . . . . . . . . . . . . . . . ………………………………. . . . 34 2.4.13 Blower Rot or Profile . . . . . . . . . . . . . . . . . . . . …………………………….. . . . . . . . . . . 39 X Contents2.5 Identification of Rotor Positionin Compressor Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . …………………………….. . . . . . . .40 2.6 Tools for Rotor Manufacture . . . . . . . . . . . . . . . . . . . . . . …………………………. . . . . . . .45 2.6.1 Hobbing Tools . . . . . . . . . . . . . . . . . . . . . . . . . . ………….…..………………. . . . . . . . . .45 2.6.2 Milling and Grinding Tools . . . . . . . . . . . . . . . . . . . ……………………………….... . . . . 482.6.3 Quantification of ManufacturingImperfections . . . . . ……………………………….... . . 483 Calculation of Screw Compressor Performance . . . . . . . . . . ………………………………. . . 49 3.1 One Dimensional Mathematical Model . . . . . . . . . . . . . . …………………………... . . . . . .49 3.1.1 Conservation Equationsfor Control Volume and Auxiliary Relationships . . . . ............................................... . . 50 3.1.2 Suction and Discharge Ports . . . . . . . . . . . . . . . . . . . ....................................... . . . . 53 3.1.3 Gas Leakages . . . . . . . . . . . . . . . . . . . . . . . . . . .................................... . . . . . . . . . .54 3.1.4 Oil or Liquid Injection . . . . . . . . . . . ...................................... . . . . . . . . . . . . . . . . . 55 3.1.5 Computation of Fluid Properties . . . . . . . . ........................................ . . . . . . . . . . . 57 3.1.6 Solution Procedure for Compressor Thermodynamics . (58)3.2 Compressor Integral Parameters . . . . . . . . . . . . . . . . . . . ………………………….. . . . . . . . 59 3.3 Pressure Forces Actingon Screw Compressor Rotors . . . . . . . . . . . . . . . . . . . . . . ................................... . . . . . . . 61 3.3.1 Calculation of Pressure Radial Forces and Torque . . . . .. (61)3.3.2 Rotor Bending Deflections . . . . . . . . . . . . . . . . . . . . . ……………………………….. . . . 64 3.4 Optimisation of the Screw Compressor Rotor Profile,Compressor Design and Operating Parameters . . . . . . . . . . ……………………………….. . . . 65 3.4.1 OptimisationRationale . . . . . . . . . . . . . . . . . . . . . . . . ……………………………….. . . . 65 3.4.2 Minimisation Method Usedin Screw CompressorOptimisation . . . . . . . . . . . ……………………………………… . . . . . . 67 3.5 Three Dimensional CFD and Structure Analysisof a Screw Compressor . . . . . . . . . . . . . . . . . . . . . . . . . …………………………….. . . . . . . . .71 4 Principles of Screw Compressor Design. . . . . . . . . . . …………………………… . . . . . . . . 77 4.1 Clearance Management. . . . . . . . . . . . . . . . . . . . . . . . ………….….………… . . . . . . . . . .78 4.1.1 Load Sustainability . . . . . . . . . . . . . . . . . . . . . . . . . . . . ………….………………….. . . .79 4.1.2 Compressor Size and Scale . . . . . . . . . . . . . . ………………………………. . . . . . . . . . . 80 4.1.3 RotorConfiguration . . . . . . . . . . . . . . . . . . . . . . . ……………………………... . . . . . . .82 4.2 Calculation Example:5-6-128mm Oil-Flooded Air Compressor . . . . . . . . . . . . . . . ……………………………... . . . 824.2.1 Experimental Verification of the Model . . . . . . . . . . . ………………………………. . . . 845 Examples of Modern Screw Compressor Designs . . . . . . . ……………………………… . . . 89 5.1 Design of an Oil-Free Screw CompressorBased on 3-5 “N” Rotors . . . . . . . . . . . . . . . . . . . . . . . . . . ……………………………. . . . . . . 90 5.2 The Design of Familyof Oil-Flooded Screw Compressors Basedon 4-5 “N” Rotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . …………………………… . . . . . . .93 Contents XI.5.3 Design of Replacement Rotorsfor Oil-FloodedCompressors . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................. . .96 5.4 Design of Refrigeration Compressors . . . . . . . . . . . . . . . .............................. . . . . . . 100 5.4.1 Optimisation of Screw Compressors for Refrigeration . . . (102)5.4.2 Use of New Rotor Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . (103)5.4.3 Rotor Retrofits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ……………………………. . . 103 5.4.4 Motor Cooling Through the Superfeed Port in Semihermetic Compressors . . . . . . . . . . . . . . . . . . . …………………………………… . . . 103 5.4.5 Multirotor Screw Compressors . . . . . . . . . . . . . . . . . …………………………….... . . . . 104 5.5 Multifunctional Screw Machines . . . . . . . . . . . . . . . . . . ……………………….. . . . . . . . . 108 5.5.1 Simultaneous Compression and Expansionon One Pair of Rotors . . . . . . . . . . . . . . . . . . . . . . . . . . ............................................ . 108 5.5.2 Design Characteristics of Multifunctional Screw Rotors .. (109)5.5.3 Balancing Forces on Compressor-Expander Rotors . …………………..……………. . . 1105.5.4 Examples of Multifunctional Screw Machines . . . . . . . . (111)6Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . …………………… . . . . . . . . . 117A Envelope Method of Gearin g . . . . . . . . . . . . . . . . . . . . . . . . ………………………… . . . . . 119B Reynolds TransportTheorem. . . . . . . . . . . . . . . . . . . . . . . …………………………. . . . . . . 123C Estimation of Working Fluid Propertie s . . . . . . . . . . . . . . . …………………………….. . . . 127 Re ferences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ………………… . . . . . . . . . . 133中文译文螺杆压缩机N.斯托西奇史密斯先生A科瓦切维奇螺杆压缩机计算的数学模型和性能尼古拉教授斯托西奇教授伊恩史密斯博士艾哈迈德科瓦切维奇工程科学和数学北安普敦广场伦敦城市大学英国电子邮件：n.stosic@/doc/d6433edf534de518964bcf 84b9d528ea81c72f87.htmli.k.smith@/doc/d6433edf534de51896 4bcf84b9d528ea81c72f87.htmla.kovacevic@/doc/d6433edf534de51 8964bcf84b9d528ea81c72f87.html国会图书馆控制号：2004117305isbn-10 3-540-24275-9 纽约施普林格柏林海德堡isbn-13 978-3-540-24275-8 纽约施普林格柏林海德堡这项工作是受版权保护,我们保留所有权利。

Optimization of compact Soller collimator

Preprint IAE- 6501/14, 2007 Optimization of compact Soller collimatorL.I.OgnevRRC «Kurchatov Institute»AbstractThe opportunity of optimization compact collimator for soft x-ray radiation for wave length 4 nm, corresponding to carbon atoms absorption was investigated at the specified angle of collimation of the beam. As an example of materials for collimator glass and nickel have been chosen. Both absorption at reflection and scattering on roughness were taken into consideration.From the obtained results follows that effect of diffraction becomes significant for minimizing width of the channel up to 20 - 40 µm. Reduction of transmission angle from 0.014 up to 0.007 radian results in increase of factor of attenuation of intensity of the beam ()L absorp incoh ββ+2 for the basic waveguide mode approximately by 5 times. The choice of lighter material for collimator allows to decrease considerably losses at transmission of the basic mode, thus the replacement of nickel with glass decreases losses approximately by 4 times. So in short collimator where angular collimation is achieved on 2 reflections at greater height of roughness of a surface losses are 20% lower, than for more smooth surfaces with 4 beam reflections.During heating hydrogen plasma in tokamak because of degradation of a graphite covering of a wall dust carbon particles probably penetrate into active zone. Research on peripheral part of the active zone with the method of Rayleigh scattering of laser radiation [1] has shown, that the sizes of particles are in the range of 80 nanometers. However research of the center of the active zone is complicated by absorption of laser radiation in plasma. The technique of research by means of soft x-ray radiation in a range of absorption of atoms of carbon 280 eV [2] is supposed to be more selective. However the geometry of thermonuclear installation makes strict demands to possible dimensions of X -ray optics devices. In the present work the opportunity of miniaturization of Soller collimators for formation and research of x-ray beams in the field of 280 eV was investigated.Schematically the ray paths in flat collimator are shown in a Fig. 1.The angular spectrum of the outgoing beam is defined by losses at reflection of beams from collimator walls. From figure it is obvious, that if the ratio between width of the channel d and its length L remains constant d / L = const number of reflections in the geometrical optics approximation is the same and thus losses do not depend on the collimator length. We shall consider, that the factor of reflection can be defined by the formula)())/4(exp()(2ϑλπσϑϑF R R ⋅−=(1)Where σ is root-mean-square height of roughness of collimator walls, λ is a wave length of radiation,R F (θ) is coefficient of the total external reflection of radiation from a surface for a beam with sliding incidence angle θ (Fresnel formula).The typical dependence R F (θ) for reflection of radiation with wave length of λ = 4 nm from glass and nickel surfaces with various values of root-mean-square amplitude of roughness is shown on Fig.2 and 3. Values of optical constants of materials were taken from [2].For restriction of a beam angular halfwidth θ½ on one reflection it is necessary, that R (θ½) ≤ 1/e. If 2 reflections from walls supposed, the condition should be {R(θ½)}2 ≤ 1/e, for 3 reflections condition {R (θ½)}3 ≤ 1/e should be satisfied. For n reflections{R (θ½, n )}n ≤ 1/e.(2)That is, the less is the coefficient of reflection the smaller number of reflections n is required for restriction of angular spectrum of a beam. Thus values of the ratio of the width to length d / L can be obtained.Height of roughness of a surface of the channel, necessary to satisfy the conditions (2) with n reflections at θ½,n < θF , where ))Re(1(εθ−=F isFresnel angle of the total external reflection, can be obtained from a formulan R n F n 1))(ln(4,2/1,2/1+=θπθλσ. (3)For narrow beams the diffraction effects can be expected which results in attenuation of the radiation propagating in the channel collimator even under a zero angle to an axis [3, 4]. The attenuation coefficient of the lower mode due to incoherent scattering on the rough surface at small values of factor of correlation R (z) is proportional to root-mean-square height of roughness [3],)2/exp()2/exp()1(~),(02222ηηξξσεβξd d z d k z V incoh ∫∫∫′∞∞−∞∞−−−′− (4)where 2/12))(1/()(),(z R z R z V ′−′−=′ξξ.Using the approximated approach for wave eigenfunctions, the coefficient of incoherent attenuation can be written approximately asI l dl incoh ⋅+≈))(Re()1(21)(232επσπβ, (5)where l is number of waveguide eigenfunction, I is correlation functional in (2), dependent on correlation properties of a surface of the channel. For exponential autocorrelation function R (z) = exp (-z/l corr ) the values of this correlation functional resulted in Table 1.Table 1l corr [µm] I /2 [µm]0.1 0.7086·10-2 0.5 0.3463·10-1 1.0 0.6918·10-1 5.0 0.3399 10 0.6053 Dependence of factor of attenuation of a beam on of the channel width due to scattering on rough walls resulted in Fig. 4 for glass and Fig. 5 for nickel at various heights of wall roughness. The length of correlation has been taken equal to 0.1 µm.Absorption on walls leads to attenuation of radiation for waveguide mode number l with coefficient[].)1Re()Im()1()(2/3232−−+≈εεπβl d l absorp (6)The approximate formulas (5) and (6) agreed well with results of numerical calculations [3] and explicitly show reverse cubic dependence of factors of attenuation of waveguide modes on width of the channel, earlier noted in numerical calculations [4]. It can be noted also that factor of absorption (4) has the same dependence on parameters of environment as linear approximation of reflection coefficient [5]. Dependence of factor of attenuation of a beam with the wave length λ = 4 nm on width of the channel due to absorption on collimator walls, calculated with these formulas is shown on Fig. 6 for glass and on Fig. 7 for nickel.Let us consider consequences of miniaturization for losses of radiation at passage through collimator at axial orientation of a beam. We shall consider dependence of losses ()L absorp incoh ββ+2on width of the channel d under condition of the constant ratio d / L . That corresponds to constant angular transmission spectrum in geometrical optics approach. It results in approximate equation L = d n / θ½, n .The maximum angular range θ½,n for detection of dust was determined from the expected size of particles of carbon D=60 nm, that is θ½ ~0.1 λ /D ~7·10-3 or 14·10-3 rad.From ratio (2) we determine height of roughness of collimator surfaces which provides attenuation of X-ray radiation with wavelength 4 nm after n=2 and 4 reflections in case of usage of nickel or glass. Results are given in Table 2.Table 2 Nickel n=2 n=4 angle of collimation 0.007 σ =310.7 211.8 0.014 σ =149.7 97.4 Glass n=2 n=4 angle of collimation 0.007 σ =308.1 207.9 0.014 σ =146.9 93.0 Despite of triple difference of the value Re(1 - ε) for glass and nickel, amplitudes of height of the roughness necessary for collimation of the radiation, practically coincide. It is accounted for strong influence of roughness on losses of a mirror component of the reflected beam.These data were used for calculation of losses for the lower wave modes at various values d / l. Results of calculations are shown on Fig. 8 and Fig. 9. From the results for wave length 4 nm follows, that effects of diffraction become significant at reduction of width of the channel up to 20 - 40 µm. Reduction of collimation angle from 0.014 up to 0.007 radian leads to increase of attenuation factor ()L absorp incoh ββ+2 for the basic waveguide modeapproximately by 5 times. The choice of lighter material for collimator allows to decrease considerably minimum losses for transmission of radiation through collimator, so replacement of nickel with glass results approximately by 4 times decrease. Thus in short collimator where angular restriction is achieved on 2 reflections at greater height of surface roughness losses are 20 % smaller.The main results1. The approximate analytical expressions for attenuation of wave modes of the X-ray radiation captured in low absorbing dielectric gap with rough walls were developed.2. Diffraction effects for transmission of X-ray radiation with wavelength of 4nm become significant when narrowing the width of a channel till 20-40µm.3. Choice of a material with smaller nuclear number and the increased roughness of a surface can reduce the minimum losses of the radiation at transmission through the channel at the same angular discrimination of a beam.References1. W.P. West, B.D. Bray, J. Burkart, Measurement of number density and size distribution of dust in DIII-D during normal plasma operation, Plasma Phys. Control. Fusion, 2006, v.48, p. 1661 - 1672.2. I. Diel, J. Friedrich, C. Kunz, S. Di Fonzo, B. R. Müller, W. Jark, Optical constants of float glass, nickel, and carbon from soft X-ray reflectivity measurements, Applied Optics, 1997, Vol. 36, No. 25, p. 6376 - 6382.3. T.A. Bobrova, L.I. Ognev, On the “supercollimation” of x-ray beams in rough interfacial channels, JETP Letters, 1999, v.69, n.10, p. 734-738.4. L.I. Ognev, X-ray diffraction effects in submicron slits and channels, X-ray spectrometry, 2002, vol. 31, n. 3, p. 274-277.5. A.V. Vinogradov, V.F. Kovalev, I.V. Kozhevnikov, V.V. Pustovalov,Zh. Tech. Fiz., 1985, v. 55, p. 244; Sov. Phys. Tech. Phys., 1985, v. 30, 145.Fig. 1Changes of the linear size of collimator with given ratio of width d and length L does not change angular spectrum of a target beam inthe approximation of geometrical optics.0.00.20.40.60.81.04 nml =100 A40 A s=0glass R angleFig. 2 Dependence of reflection coefficient R (θ) on sliding angle of incidence θ of radiation with wavelength of 4 nm on glass surface with various values of height of roughness σ = 0 (solid), 40 Å (dashed) and 100 Å (points). For calculation of Fresnel reflection optical parameters of substance were used from work [2].0.00.20.40.60.81.0)θR ( θnickel100 A40 A s=0angle4 nml =Fig. 3Dependence of reflection coefficient R (θ) from a sliding angle of incidence θ of radiation with wavelength of 4 nm on nickel surface with various values of height of roughness σ = 0 (solid), 40 Å (dashed) and 100 Å (points). For calculation of Fresnel reflection optical parameters of substance were used from work [2].1010010001E-41E-30.010.11102m L b collimator lengthq q q 42GlassL=Nd/ 1/21/2=0.014 1/2=0.007d, mN=2N=4Fig. 4Dependence of attenuation coefficient of the basic waveguide mode on width of the flat glass channel d . The solid line corresponds to attenuation due to absorption in the smooth channel. The dot curve corresponds to attenuation due to scattering on roughness with height σ = 80 Å, dashed line corresponds to σ = 160 Å.0.010.111E-51E-41E-30.010.1110100m m b ,s =160 A80 A absorption4 nml=l =0 modeNickel d, mm-1Fig. 5Dependence of attenuation coefficient of the basic waveguide mode on width of the flat nickel channel d . The solid line corresponds to attenuation due to absorption in the smooth channel. The dot curve corresponds to attenuation due to scattering on roughness with height σ = 80 Å, dashed line corresponds to σ = 160 Å.1010010001E-41E-30.010.11102m L b collimator lengthq q q 42GlassL=Nd/ 1/21/2=0.014 1/2=0.007d, mN=2N=4Fig. 6Dependence of a parameter of attenuation of intensity of radiation()L absorp incoh ββ+2 for the basic waveguide mode l = 0 on width of a flat glass wave guide d . Ratio between collimator length and its width are chosen equal L / d=N / θ½, where θ½ is half-width collimation angle of a X-ray beam, N is maximum number of reflections. Solid lines above correspond to angularrestriction to θ½ =0.007, dashed lines to θ½ =0.014 at double and quadruple reflections.1E-41E-30.010.11102m L b NickelN - number of reflections24q q q collimator lengthL=Nd/ 1/21/2=0.014d, mN=2N=41/2=0.007Fig. 7Dependence of parameter of attenuation of intensity of radiation()L absorp incoh ββ+2 for the basic waveguide mode l = 0 on width of a flat nickel wave guide d . Ratio between collimator length and its width are chosen equal L / d=N / θ½, where θ½ is halfwidth collimation angle of a X-ray beam, N is maximum number of reflections. Solid lines above correspond to angular restriction to θ½ =0.007, dashed lines to θ½ =0.014 at double and quadruple reflections.。

topology[1]

40
No design area (No material)
24
Result of Design Constraint
Influence of Design Domain
12 2 2 1.25 Design Domain 5
1.25
Non-design Domain
1.25
Design Domain
max u s f
design
T
5
Lagrangian
NE 1 NE T T L = ∑ d e K ed e − d e fe + λ ∑ ρ e Ae Le − W 2 e =1 e =1

Total Potential Energy Weight Constraint
new
Shape Design Find the best Ω
Material Design Find the best Dnew
15
Homogenization Design Method
• Shape and Topology Design of Structures is transferred to Material Distribution Design (Bendsoe and Kikuchi, 1988)
min
∑ρ A L
e =1 e e
e
Design Sensitivity
K
P1 P2
∂u ∂K ∂f =− u+ ∂Ae ∂Ae ∂Ae
∂u ∂σ e ∂ De Be ue = De Be e = ∂Ae ∂Ae ∂Ae ∂ ui u ∂u = i • i ∂Ae ui ∂Ae

机械类关于凸轮的中英文翻译

英文原文CamsV arious motions can be produced by the action of a cam against a follower.Mamy timing devices are operated by can action.The purpose of andy cam is to produce a displacement of its follower;a secondary follower is often .used to produce additional displacement in another location.The most popular type is the plate cam.The cylindrical type is used to transmit linear motion to a follower as the cam rotates.Three-dimensional cam are sometimes used;these provide some unusual follower motions,but also make follower design difficult.The camshaft in the automotive engine illustrates a simple but important application of a late cam.The cam assemblies in automatic record players illustrate a somewhat more complex application.Cam profiles are accurately constructed by either praphical or mathematical methods.The transitiom from development drawings to working (shop) drawing can be made in several ways:1.Make a full-scale template.This is desirable from the manufacturing standpoint,but it will not guarantee accurate cam profiles.e radial dimensions.This is fairly accurate,but sometimes produces layout problems in the shop.e coordinate dimensioning.This procedure will ensure accuracy.In selecring one of these methods,one should consider the function of the cam in terms of desired preciseness.Because the cam work outline already determined, therefore the cam structural design mainly was determines the curve outline axial thickness and the cam and the drive shaft connection way. When the work load compares the hour, curve outline axial thickness generally takes for the outline curve biggest radius of vector 1,/10 ～/5; Regarding a stress bigger important situation, must with carry on the design according to the cam contour surface from the contact intensity.When determination cam and drive shaft joint way, should synthesize theconsideration cam the assembling and dismantling, the adjustment and firmly grades the question. Regarding implementing agency more equipment, between its each execution component movement coordination usually determined by the cycle of motion chart, therefore in assembly cam gear time, the cam contour curve initial station (pushes regulation starts) the relative position to have according to the cycle of motion chart to carry on the adjustment, guarantees each execution component to be able according to the pre-set sequence synchronized action. Therefore, requests the cam in the structural design to be able to be opposite to the drive shaft carries on the rotation along the circumference direction, and reliably performs fixedly. The simplest method uses the clamping screw nail fixed cam, or with clamping screw nail pre- fixed, after treats adjusts uses the pin to be fixed again.From structural design: from structure: When design must consider from the guidance and prevented revolves. From movement rule design: Involves many aspects from the movement rule design the questions, besides consideration rigidity impact and flexible impact, but also should maximum speed vmax which has to each kind of movement rule, maximum acceleration amax and its the influence performs the comparison. 1) vmax bigger, then momentum mv is bigger. If from is suddenly prevented, the oversized momentum can cause the enormous impulse, endangers the equipment and the personal safety. Therefore, when is bigger from the quality, in order to reduce the momentum, should choose the vmax value smaller movement rule.2) amax bigger, is bigger. Function in high vice- contact place stress bigger, the organization intensity and the wear resistant request is also higher. Regarding high speed cam, in order to reduce the harm, should choose the amax value smaller movement rule. First states several kind of movements rules vmax, amax, the impact characteristic and the suitable situation following table regarding swings from the cam gear, its movement graph x-coordinate expression cam corner, y-coordinate then separately expresses from, angular speed and angle acceleration. This kind of movement graph has the state of motion and above is same.From structural design: from structure: When design must consider from the guidance and prevented revolves. From movement rule design: The cam gear design basic question 1. cam gears type choice, the definite cam shape, with from maintainsthe high vice- contact from the shape and the movement form and the cam the way 2. from the movement rule design, according to the application situation to from the travelling schedule and the state of motion request, determines from the movement rule. 3. cam gears basic parameter design, determines from the travelling schedule, various movements angle, the cam radius, , the roller radius, the center distance, from the length and so on. 4. cam contours curve design. 5. cam gears bearing capacity computation. 6. cam gears structural design, plan organization assembly drawing and various components shop drawingFromstructural design: from structure: When design must consider from the guidance and prevented revolves. From movement rule design: The cam gear design basic question 1. cam gears type choice, the definite cam shape, with from maintains the high vice- contact from the shape and the movement form and the cam the way 2. from the movement rule design, according to the application situation to fromthe travelling schedule and the movement 1, the cam gear application cam gear is includes the cam the high vice- organization, the cam gear has the structure to be simple, may accurately realize request merit and so on movement rule, thus obtains the widespread application in the industrial production, specially automatic device and in the automatic control device, obtains the widespread application. 2nd, the cam gear classification according to two moves the relative motion characteristic classification between the component (1) the plane cam gear 1) the disk cam; 2) translation cam. (2) space cam gear according to from movement vice- element shape classification (1) apex from; (2) roller from (3) flat base from. Note: Classifies this part of content when the introduction cam gear, should point out each kind of cam gear the good and bad points and its the adaption situation, showed each kind of cam gear the inner link, will build the foundation for the later translation cam and the column cam contour design.3rd, the throwout lever movement rule (1) the cam gear cycle of motion and the basic term terminology push the regulation movement angle: With from pushes the cam corner which the regulation corresponds; Far stops the angle: With from far rests the cam corner which the regulation corresponds; Return trip movement angle: With cam corner which corresponds from the return trip; Nearly stops the angle: With fromnearly rests the cam corner which the regulation corresponds; Cam: Take the cam axle center as the center of a circle, take its outline slightly to diameter r0 as the radius circle; From ravelling schedule: In pushes in the regulation or the return trip from the biggest displacement, indicated with h;: The cam center of rotation with from guides way the bias distance, indicated with e.Types of CamsPlate cams are simple to fabricate.The follower can be moved in various patterns with various rise /fall ratios.Motion should be controlled to avoid abrupt changes in force transmitted from the cam to the follower.One should carefully determine horizontal force components,since these present problems designing the follower assembly guide.Critical reactions occur at points A and B.These reaction values must be computed.The relative vertical position of point A with respect to B needs to be raised if the reaction value at Bis excessive.The position of B should be as close to cam as possible to minimize flexure in the roller-follower support.This type produces reciprocating motion in the follower.Again,dorces need to be determined and dimensions chosen so as to avoid excessive component sizes.A tapered roller follower is frequently employed ;the groove in the periphery of the cam should be shaped to accommodate the follower.This type of cam is expensive to produce.The cylindrical cam has two outstanding features.One is the fact that the cam is positive actiong.N outside forces (such as gravity or spring action ) are needed to hold the follower against the working surface of the cam.The second feature is the fact that the follower can move through a complete cycle in the course of several revolutions of the cam.For example,it is possible to design the cam so the follower could move from a starting position at the left end to the extreme right position in three revolutions( or more),then the starting position in two revolutions.Other variations are possible.A translation cam is illustrated.In the figure shown the cam reciprocates horizontally and the follower moves up and down.A pivoted follower can be used with this type .The translation cam can be made positive by providing a guided plate with an inclined slot for the cam;the slot cam then engage a pin or roller on a guided vertical reciprocated follower.With the latter type ,however,a complete force analysisis a critical phase of the design.In this type,the cam rotates and the follower (ususlly a roller or pin) is guided by a groove cut into the end face of a cylindrical section .Rotation of the cam provides translation of the follower.This type is also positive acting.Production costs for this type of cam are much higher than for a simple plate cam.A constant –diameter cam is illustrated .This is merely a circular plate with the camshaft hole eccentrically located.The amount of eccentricity determines the amount of follower displacement.As the cam rotates,the follower reciprocates.This arrangement is sometimes known as a Scotch yoke mechanism.Follower action is positive ;harmonic motion is produced by this type of arrangement.Types of FollowersIn neneral,the follower is considered to be the part that comes in contact with the cam profile .However,when a seconday follower is used, the motion of the secondary follower is dictated by that of the primary follower.For example ,a roller follower can be reciprocated by acting against the edge of a pivoted follower.The simplest type of follower is the reciprocationg type that merely moves up and down (or in and out ) with the rotation of the cam;the centerline can be either collinear with the cam centerline or offset from it .Contact with the cam can be via a point,a knife edge,a suface ,or a roller.A flat-afced reciprocating follower is shown If a point or surface is employed for contact the high normal force can result in abrasion and excessive wear.If the load being transmitted from the cam to the follower is small,the problem is not serious.For example ,the operation of a small snap-action switch does not produce cam surface wear.Miniature snap-action electrical switches have actuators with various configurations;some of these are in the form of rounded points or thin meta sections.Miniature three-way valves in air circuits have similar actuators.If cams are used to operate mechanical components directly,a roller is much more effective.Cam rollers are commercially available in roller sizes ranging from1/2 in .to 6 in Basic dynamic capacities range from 620 to 60000 ,based on 33.33 rpm and 500hr of minimum life .Correction factors must be used for any other speed or life values.It should be noted that the cam can be lubricated through and oil hole in the end of theshank.Rolling contact with the cam surface minimizes wear problems.Several mounting arrangements are possible with this type of followr .shows the roller follower mounted on a pivoted arm .A pivoted flat-faced follower is shown .As with any flat-faced follower,friction between the follower face and the cam profile must be controlled.Proper lubrication can reduce the effects of friction.汉语翻译:凸轮通过凸轮和从动件的作用，可得到不同的运动。

Optimum angles for a polarimeterpart I

a retarder. A classical system for the measurement of the Stokes vector is that of a quarter-wave plate and a linear polarizer placed in series3 with a detector such as a photodiode or a photomultiplier tube. Pezzaniti and Chipman4 and Morgan et al.5 discuss in detail the application of the almost simultaneous measurement technique of Mueller matrix p0larimetry. An alternative approach has been patented by Azzam et al.6 in which four detectors are placed such that reflections of an incoming light beam between these detectors
Subject terms: polarization analysis and measurement; polarimeter; optimum angles; Mueller matrix.
Optical Engineering 34(6), 1651-1655 (June 1995).
1 Introduction
The influence of polarization on multiplபைடு நூலகம் scattering of light is increasingly being studied. Carswell and Pal1 observed that an anisotropic pattern of multiple scatter is obtained when a linearly polarized laser beam is incident on a scattering medium. Look and Chen2 considered a linearly polarized laser beam incident on a scattering medium. They carried out extensive measurements of the side-scattered radiation for this system. Their data clearly showed the anisotropic and highly polarized nature of this side scattered radiation. Their polarization measurements were, however, restricted to the parallel and perpendicular components of the emerging side-scattered light. For a full characterization of the polarization field, the Stokes vector of the light field must be specified. At least five general techniques are known for the mea-

[IT计算机]山东大学ACM模板_图论

图论1.最短路径 (4)1)Dijkstra之优雅stl (4)2)Dijkstra__模拟数组 (4)3)Dijkstra阵 (5)4)SPFA优化 (6)5)差分约束系统 (7)2.K短路 (7)1)Readme (7)2)K短路_无环_Astar (7)3)K短路_无环_Yen (10)4)K短路_无环_Yen_字典序 (12)5)K短路_有环_Astar (15)6)K短路_有环_Yen (17)7)次短路经&&路径数 (20)3.连通分支 (21)1)图论_SCC (21)2)2-sat (23)3)BCC (25)4.生成树 (27)1)Kruskal (27)2)Prim_MST是否唯一 (28)3)Prim阵 (29)4)度限制MST (30)5)次小生成树 (34)6)次小生成树_阵 (36)7)严格次小生成树 (37)8)K小生成树伪代码 (41)9)MST计数_连通性状压_NOI07 (41)10)曼哈顿MST (43)11)曼哈顿MST_基数排序 (46)12)生成树变MST_sgu206 (49)13)生成树计数 (52)14)最小生成树计数 (53)15)最小树形图 (56)16)图论_最小树形图_double_poj3壹64 (58)5.最大流 (60)1)Edmonds_Karp算法 (60)2)SAP邻接矩阵 (61)3)SAP模拟数组 (62)4)SAP_BFS (63)5)sgu壹85_AC(两条最短路径) (64)6)有上下界的最大流—数组模拟 (67)6.费用流 (69)1)费用流_SPFA_增广 (69)2)费用流_SPFA_消圈 (70)3)ZKW数组模拟 (72)7.割 (73)1)最大权闭合图 (73)2)最大密度子图 (73)3)二分图的最小点权覆盖 (74)4)二分图的最大点权独立集 (75)5)无向图最小割_Stoer-Wagner算法 (75)6)无向图最大割 (76)7)无向图最大割(壹6ms) (76)8.二分图 (78)1)二分图最大匹配Edmonds (78)2)必须边 (79)3)最小路径覆盖（路径不相交） (79)4)二分图最大匹配HK (80)5)KM算法_朴素_O(n4) (81)6)KM算法_slack_O(n3) (82)7)点BCC_二分判定_(2942圆桌骑士) (84)8)二分图多重匹配 (86)9)二分图判定 (88)10)最小路径覆盖（带权） (89)9.一般图匹配 (90)1)带花树_表 (90)2)带花树_阵 (93)10.各种回路 (96)1)CPP_无向图 (96)2)TSP_双调欧几里得 (97)3)哈密顿回路_dirac (98)4)哈密顿回路_竞赛图 (100)5)哈密顿路径_竞赛图 (102)6)哈密顿路径_最优&状压 (102)11.分治树 (104)1)分治树_路径不经过超过K个标记节点的最长路径 (104)2)分治树_路径和不超过K的点对数 (107)3)分治树_树链剖分_Count_hnoi壹036 (109)4)分治树_QTree壹_树链剖分 (113)5)分治树_POJ3237(QTree壹升级)_树链剖分 (117)6)分治树_QTree2_树链剖分 (122)7)Qtree3 (125)8)分治树_QTree3(2)_树链剖分 (128)9)分治树_QTree4_他人的 (130)10)分治树_QTree5_无代码 (135)12.经典问题 (135)1)欧拉回路_递归 (135)2)欧拉回路_非递归 (136)3)同构_树 (137)4)同构_无向图 (140)5)同构_有向图 (141)6)弦图_表 (143)7)弦图_阵 (147)8)最大团_朴素 (149)9)最大团_快速 (149)10)极大团 (150)11)havel定理 (151)12)Topological (151)13)LCA (152)14)LCA2RMQ (154)15)树中两点路径上最大-最小边_Tarjan扩展 (157)16)树上的最长路径 (160)17)floyd最小环 (161)18)支配集_树 (162)19)prufer编码_树的计数 (164)20)独立集_支配集_匹配 (165)21)最小截断 (168)最短路径Dijkstra之优雅stl#include <queue>using namespace std;#define maxn 壹000struct Dijkstra {typedef pair<int, int> T; //first: 权值，second: 索引vector<T> E[maxn]; //边int d[maxn]; //最短的路径int p[maxn]; //父节点priority_queue<T, vector<T>, greater<T> > q;void clearEdge() {for(int i = 0; i < maxn; i ++)E[i].clear();}void addEdge(int i, int j, int val) {E[i].push_back(T(val, j));}void dijkstra(int s) {memset(d, 壹27, sizeof(d));memset(p, 255, sizeof(p));while(!q.empty()) q.pop();int u, du, v, dv;d[s] = 0;p[s] = s;q.push(T(0, s));while (!q.empty()) {u = q.top().second;du = q.top().first;q.pop();if (d[u] != du) continue;for (vector<T>::iterator it=E[u].begin();it!=E[u].end(); it++){ v = it->second;dv = du + it->first;if (d[v] > dv) {d[v] = dv;p[v] = u;q.push(T(dv, v));}}}}};Dijkstra__模拟数组typedef pair<int,int> T;struct Nod {int b, val, next;void init(int b, int val, int next) {th(b); th(val); th(next);}};struct Dijkstra {Nod buf[maxm]; int len; //资源int E[maxn], n; //图int d[maxn]; //最短距离void init(int n) {th(n);memset(E, 255, sizeof(E));len = 0;}void addEdge(int a, int b, int val) {buf[len].init(b, val, E[a]);E[a] = len ++;}void solve(int s) {static priority_queue<T, vector<T>, greater<T> > q;while(!q.empty()) q.pop();memset(d, 63, sizeof(d));d[s] = 0;q.push(T(0, s));int u, du, v, dv;while(!q.empty()) {u = q.top().second;du = q.top().first;q.pop();if(du != d[u]) continue;for(int i = E[u]; i != -壹; i = buf[i].next) {v = buf[i].b;dv = du + buf[i].val;if(dv < d[v]) {d[v] = dv;q.push(T(dv, v));}}}}};Dijkstra阵//Dijkstra邻接矩阵，不用heap！#define maxn 壹壹0const int inf = 0x3f3f3f3f;struct Dijkstra {int E[maxn][maxn], n; //图,须手动传入！int d[maxn], p[maxn]; //最短路径，父亲void init(int n) {this->n = n;memset(E, 63, sizeof(E));}void solve(int s) {static bool vis[maxn];memset(vis, 0, sizeof(vis));memset(d, 63, sizeof(d));memset(p, 255, sizeof(p));d[s] = 0;while(壹) {int u = -壹;for(int i = 0; i < n; i ++) {if(!vis[i] && (u==-壹||d[i]<d[u])) {u = i;}}if(u == -壹 || d[u]==inf) break;vis[u] = true;for(int v = 0; v < n; v ++) {if(d[u]+E[u][v] < d[v]) {d[v] = d[u]+E[u][v];p[v] = u;}}}}} dij;SPFA优化/*** 以下程序加上了vis优化，但没有加slf和lll优化（似乎效果不是很明显）* 下面是这两个优化的教程，不难实现----------------------------------SPFA的两个优化该日志由 zkw 发表于 2009-02-壹3 09:03:06SPFA 与堆优化的 Dijkstra 的速度之争不是一天两天了，不过从这次 USACO 月赛题来看，SPFA 用在分层图上会比较慢。

Topology_optimization_of_continuum_structures_A_review_

Topology optimization of continuum structures:A review*Hans A EschenauerResearch Center for Multidisciplinary Analyses and Applied Structural Optimization,FOMAAS,University of Siegen,D-57068Siegen,Germany;esch@fb5.uni-siegen.deNiels OlhoffInstitute of Mechanical Engineering,Aalborg University,DK-9220Aalborg East,Denmark;no@ime.auc.dkIt is of great importance for the development of new products toﬁnd the best possible topol-ogy or layout for given design objectives and constraints at a very early stage of the designprocess͑the conceptual and project deﬁnition phase͒.Thus,over the last decade,substantialefforts of fundamental research have been devoted to the development of efﬁcient and reliableprocedures for solution of such problems.During this period,the researchers have beenmainly occupied with two different kinds of topology design processes;the Material or Mi-crostructure Technique and the Geometrical or Macrostructure Technique.It is the objective ofthis review paper to present an overview of the developments within these two types of tech-niques with special emphasis on optimum topology and layout design of linearly elastic2Dand3D continuum structures.Starting from the mathematical-physical concepts of topologyand layout optimization,several methods are presented and the applicability is illustrated by anumber of examples.New areas of application of topology optimization are discussed at theend of the article.This review article includes425references.͓DOI:10.1115/1.1388075͔Keywords:Mathematical-Physical Fundamentals,Deﬁnitions,Formulations,Material Models—Microstructure Techniques,Homogenization,Perimeter,and Filtering Techniques—Macrostructure Tech-niques,Approach by Growing and Degenerating a Structure(Material Removal),Approach by InsertingHoles—New Applications of Topology OptimizationCONTENTS1.INTRODUCTION (332)1.1Structures,materials,optimization:A multidisciplinary task (332)1.2Survey of topology optimization of continuumstructures (333)2.MATHEMATICAL-PHYSICAL FUNDAMENTALS (335)2.1Deﬁnition and terms of topology (335)2.2Classiﬁcation of topology optimization (335)2.3Energy principles (336)2.4Problem formulations of shape and topologyoptimization (338)2.5Material models (340)3.MICROSTRUCTURE APPROACHES ANDHOMOGENIZATION TECHNIQUES (343)3.1Mathematically based homogenization techniques (343)3.2Layered2D microstructure:Smear-out technique (344)3.3Layered3D microstructures:Quasiconvexiﬁcation (349)3.4Discussion and examples (352)4.OTHER APPROACHES TO ACHIEVE WELL-POSEDPROBLEM FORMULATIONS:PERIMETER METHODAND FILTERING TECHNIQUES (354)4.1Perimeter method (355)4.2Local constraint on gradient of material density (356)4.3Filtering techniques (356)4.4Discussion of methods (357)5.MACROSTRUCTURE APPROACHES (357)5.1Techniques by degenerating and/or growing a structure (358)5.2Techniques by inserting holes (363)6.FURTHER APPROACHES–NEW APPLICATIONS (372)6.1Recent developments (372)6.2New applications (374)7.CONCLUSIONS (380)ACKNOWLEDGMENT (380)REFERENCES (381)Transmitted by Associate Editor FG Pfeiffer*Dedicated to our friend and colleague,Professor Ernest Hinton,PhD(1946–1999),in fond memoryASME Reprint No AMR$38.00Appl Mech Rev vol54,no4,July2001©2001American Society of Mechanical Engineers3311INTRODUCTION1.1Structures,materials,optimization:A multidisciplinary taskTwo scientists established not only the classical theory of elasticity,but they also laid the foundation for the increas-ingly importantﬁeld of structural optimization.Theﬁrst con-cepts of seeking the optimal shapes of structural elements are contained in the works of Galilei͓37͔.Thus,in his book, Discorsi,Galileo Galilei͑1564–1642͒was theﬁrst to per-form systematic investigations into the fracture process of brittle bodies.In this context,he described the inﬂuence of the shape of a body͑hollow bodies,bones,blades of grass͒on its strength,thus posing and answering questions address-ing the‘‘Theory of bodies with equal strength.’’On the other hand,Robert Hooke͑1653–1703͒formulated the fundamen-tal law of linear theory of elasticity:Strain͑change of length͒and stress͑load͒are proportional to each other.Based on these considerations one could assume the theory of elastic-ity and to a wider extent continuum mechanics to be aﬁeld of science whose problems might be considered as being solved to a large extent.This,however,would be a funda-mental error.The previous years have witnessed increasing challenges in terms of the design of ever more complex me-chanical systems and components as well as of extremely lightweight constructions,a fact that has led,among others, to the development of advanced materials and hence to the demand for increasingly precise calculation methods.The substantial and still undiminished importance of structural mechanics is due to the fact that questions towardﬁnding an optimal design in terms of load bearing capacity,reliability, accuracy,costs,etc,have to be answered already in an early stage of the design process͑concept phase͒.In this context, research into theﬁelds of material laws,advanced materials, contact mechanics,damage mechanics,etc,proved to be of particular importance for solving various problems.This de-velopment naturally includes computer science and technol-ogy,the enormously fast development of which has facili-tated,over the previous decades,the programming and thus the availability of more sophisticated software systems for treating large-scale,highly nonlinear systems.The development and construction of products,especially in industrial practice frequently raises the question of which measures must be taken to improve the quality and reliability in a well-aimed manner without exceeding a certain cost limit.In this respect,a new area in the scope of Computer Aided Engineering has emerged,namely the optimization of structures,commonly called Structural Optimization.It of-fers to the engineers of the development,calculation,and design departments a tool which,by means of mathematical algorithms,allows to determine better,possibly optimal,de-signs in terms of admissible structural responses͑deforma-tions,stresses,eigenfrequencies,etc͒,manufacturing,and the interaction of all structural components.Hence,structural optimization has become a multidisciplinaryﬁeld of re-search.Its foundations,however,date back to one of the last universal scholars of modern times,Gottfried Wilhelm Leib-niz͑1646–1716͒,whose works in theﬁelds of mathematics and natural sciences can be seen as the basis of any analytic procedure and highlight the tremendous importance of coher-ent scientiﬁc thinking͑the latter being an important precon-dition of structural optimization͒.He laid the foundation of differential calculus,and he also built theﬁrst mechanical computer.Without these achievements,modern optimization calculations would not be possible to a larger extent.In this respect,it is of utmost importance to mention Leonard Euler ͑1707–1783͒who has played a most signiﬁcant scientiﬁc role.One of his many achievements is his development of the theory of extremals which provided the basis for the de-velopment of the calculus of variations.With this method Jakob Bernoulli͑1655–1705͒determined the‘‘curve of the shortest falling time’’͑Brachistochrone͒and Sir Isaac New-ton͑1643–1727͒the body of revolution with the smallest resistance.By formulating the principle of the smallest ef-fect,and by developing an integral principle Lagrange ͑1736–1813͒and Hamilton͑1805–1865͒contributed toward the completion of variational calculus as one of the funda-mentals for several types of optimization problems.Euler, Lagrange͓60͔,Clausen͓186͔,and de Saint Venant per-formed initial investigations into the determination of the optimum shape of one-dimensional load bearing structures under arbitrary loads.Typical examples for these pursuits are the problems of optimal design of columns,torsion bars and cantilever beams for which optimum cross-sections were de-termined by means of variational calculus.To achieve this, optimality criteria are derived as necessary conditions;in the case of unconstrained problems Euler equations are used. Constraints are considered by applying the Lagrangian mul-tiplier method.As regards the optimum design of arches and trusses,an important place is held by the works of Le´vy͓64͔. For the history of mechanical principles,see Szabo͓106͔.Thus we can state that structural mechanics in the widest sense is hardly a subject for specialists any more.Living as well as artiﬁcial structures appear in overwhelming variety; research into them must be supported by broad knowledge and by establishing analogies.Scientiﬁc progress usually can be only achieved today by experts of different disciplines working together.Although there still exist tendencies of iso-lation today,the interdisciplinary exchange of information meanwhile has considerably improved.One very important reason is the development of advanced materials͑ceramics, plastics,composites etc͒which have great impact on the development of new,highly complex constructions and structures.In order to account for the manifold phenomena of mate-rials several theories ofﬁnite elasticity,plasticity,viscoelas-ticity,and viscoplasticity have developed independent of each other.In this context,the works of Truesdell and Noll set a milestone within the theory of material behavior͑see, among others,Truesdell͓110,111͔,Truesdell and Noll͓406͔, Noll͓324͔,Prager͓82͔,Krawietz͓56͔͒.Owing to the increasing demands on the efﬁciency,reli-ability,and shortened development cycle of a product,it has become inevitable to solve problems by computer-aided pro-cedures.Substantial progress has been achieved in computa-tional analysis of structures and components,especially by332Eschenauer and Olhoff:Topology optimization of continuum structures Appl Mech Rev vol54,no4,July2001means of the versatileﬁnite element method͑FEM͒.In many applications,an algorithm-based optimization of the compo-nent dimensions has already become general use,however, the development of applicable methods and strategies is still in progress for generating best-possible initial layouts for components.An overview of the different procedures is given in͓13,28,31,38,390͔.In recent years,substantial efforts have been made in the development of topology optimization procedures,and there are several different strategies whose use is in most cases highly problem dependent.Topology strategies are to deter-mine an optimal topology according to the deﬁned optimiza-tion problem independently of the designer.They shall sup-port the interactive work in the design process,since an isolated optimization calculation often does not yield an op-timal result.Thus,it is important to include the designer’s creativity especially in those cases where essential demands cannot be modelled sufﬁciently in the optimization process. Creativity should not be underestimated particularly in com-plex design processes,and it is also important in topology optimization.Michell͓305͔developed a design theory for the topology of thin-bar structures that are optimal with regard to weight. The bars in these structures are all perpendicular to each other and form an optimal arrangement in terms of either maximum tensile or compressive stresses.Very important subsequent generalizations were made by Prager͓83,348͔, and Rozvany and Prager͓368͔,who solved a range of dif-ferent topology optimization problems by analytical proce-dures based on optimality criteria.For an overview,see also Rozvany et al͓370,371͔.1.2Survey of topology optimizationof continuum structuresTopology Optimization is often referred to as layout optimi-zation͑or generalized shape optimization͒in the literature ͑cf,Olhoff and Taylor͓332͔,Kirsch͓264͔,Bedsøe et al ͓146͔,Rozvany et al͓90͔,Bendsøe and Mota Soares͓10͔,Cherkaev͓21͔,Rozvany and Olhoff͓91͔͒and these labels will be used interchangeably in this review.The importance of this type of optimization lies in the fact that the choice of the appropriate topology of a structure in the conceptual phase is generally the most decisive factor for the efﬁciency of a novel product.Moreover,usual sizing and shape opti-mization cannot change the structural topology during the solution process,so a solution obtained by one of these methods will have the same topology as that of the initial design.Topology or layout optimization is therefore most valuable as preprocessing tools for sizing and shape optimi-zation͑Fleury͓218͔,Bremicker͓158͔,Olhoff et al͓327͔͒.Two types of topology optimization exist͑discrete or con-tinuous͒,depending on the type of a structure.For inherently discrete structures,the optimum topology or layout design problem consists in determining the optimum number,posi-tions,and mutual connectivity of the structural members. This area of research has been active for several decades and has been largely developed by Prager and Rozvany.For an up-to-date account of the area of layout optimization of dis-crete structures,the reader is referred to eg,the comprehen-sive review paper by Rozvany et al͓92,367͔,the monographby Bendsøe͓9͔and the proceedings by Eschenauer and Ol-hoff͓28͔,Olhoff and Rozvany͓75͔,Gutkowski and Mroz ͓44͔,Rozvany͓89͔,Bloebaum͓16͔,the annual Proceedings of the ASME Design Automation Conferences,among oth-ers,Gilmore et al͓41͔,and Topping and Papadrakis͓109͔.The present review paper is dedicated to topology optimi-zation of continuum structures.This research has been ex-tremely active since the publication of the papers by Bendsøeand Kikuchi͓150͔and Bendsøe͓142͔.Examples of morerecent publications that provide an overview of the subjectare:Atrek͓130͔,Kirsch͓264͔,Eschenauer and Schumacher ͓210,211͔,Duysinx͓24͔,Olhoff͓325͔,Cherkaev and Kohn ͓22͔,Haber and Bendsøe͓236͔,Bendsøe͓143,144͔,Hassani and Hinton͓252͔,Maute et al͓303͔,and Olhoff and Es-chenauer͓328͔.In topology optimization of continuumstructures,the shape of external as well as internal bound-aries and the number of inner holes are optimized simulta-neously with respect to a predeﬁned design objective.It isassumed that the loading is prescribed and that a givenamount of structural material is speciﬁed within a given2Dor3D design domain with given boundary conditions.Thereare several research activities going on throughout the worldconcentrating on these problems,and different solution pro-cedures have been developed.Very roughly,one can distin-guish between two classes of approaches,the so-calledMaterial-or Micro-approaches vs,the Geometrical or Macro-approaches.Section2of this paper brieﬂy outlines the basic conceptsand mathematical-physical fundamentals of the problem.Firstly,the term topology is discussed and deﬁned math-ematically.Then the conceptual processes of topology opti-mization are presented for the two main types of solutiontechniques just mentioned,the Microstructure͑Material͒ap-proaches and the Macrostructure͑Geometrical͒approaches.Subsequently,a brief overview is presented of the basicequations of elasticity and the variational and energy prin-ciples that constitute the mathematical-physical foundationfor topology optimization,and two typical formulations forsuch a problem are outlined,a variational formulation and amathematical programming formulation.Finally,we presentsome periodic,perforated microstructured material models ofvariable material density which constitute the basis for theso-called Microstructure͑Material͒approaches of topologyoptimization.Now,as was originally pointed out by Lurie͑see Ref.͓66͔for references͒a topology optimization problem is not well-posed if the design space is not closed in an appropriatesense,and a regularization of the formulation of the problemis then needed͓180,181,290,266,267͔.Mathematical indica-tions of the need for regularization are generation of anisot-ropy in the design and the impossibility of satisfying secondorder necessary conditions for optimality in certain subre-gions of the structural domain͑Olhoff et al͓329͔,Lurieet al͓290͔,Cheng͓176͔͒.Numerically,the need manifestsitself by lack of convergence or by dependence of the topol-Appl Mech Rev vol54,no4,July2001Eschenauer and Olhoff:Topology optimization of continuum structures333ogy on the size of the appliedﬁnite element mesh͑Cheng and Olhoff͓180,181͔,Olhoff et al͓329͔,and Cheng͓176͔͒.There are two paths out of this dilemma:one can either extend the design space to include solutions with microstruc-ture in the problem formulation,or restrict the space of ad-missible solutions in the formulation͑Niordson͓320,321͔, Bendsøe͓141͔͒.Section3deals with the former path which encounters an approach termed relaxation in which materials with periodic, perforated microstructure of continuously varying volume density and orientation are included as admissible designs ͑see Olhoff et al͓329͔,Cheng and Olhoff͓181͔,Kohn and Strang͓266,267͔,Avellaneda͓134͔,Mlejnek͓312͔,Lipton ͓280,281͔͒,and where their effective mechanical properties are determined via some sort of homogenization technique, eg,mathematically based homogenization͑Bourgat͓157͔, Bensoussan et al͓12͔,Sanchez-Palencia͓93͔͒,a smear-out method͑Olhoff et al͓329͔,Cheng and Olhoff͓181͔,Thom-sen͓108͔͒or by quasiconvexiﬁcation͑Gibiansky and Cher-kaev͓40͔,Cherkaev and Palais͓185͔,Buttazzo and Del Maso͓165͔,Allaire͓122͔͒.Section4is devoted to the path out of the above men-tioned dilemma which implies introduction of an appropriate restriction in the problem formulation that renders the topol-ogy optimization problem well-posed͑Haber et al͓237–239͔,Ambrosio and Buttazzo͓128͔,Fernandes et al͓34͔, Petersson͓80͔,Petersson and Sigmund͓345͔͒.The ap-proaches of this kind generally provide a means to control the complexity of the topology design and include the Pe-rimeter method͑Haber et al͓237–239͔͒by which a bound constraint on the perimeter or surface area of the solid do-main͑of2D or3D designs,respectively͒restricts solutions to be entirely composed of purely solid and void domains. The same can be achieved in a much simpler way by use of ﬁltering techniques known from image processing͑Sigmund ͓97,377,378͔͒,and this approach is also discussed in Section4.Section5deals with so-called Geometrical or Macro-approaches to topology optimization of continuum struc-tures,and these approaches are all based on constitutive laws for usual solid,isotropic materials.Among these techniques, the variable thickness sheet model for prediction of topology wasﬁrst suggested by Rossow and Taylor,1973͓365͔.Here, the admissible domain for topology optimization is divided into a large number of smaller sub-areas,the thicknesses of which are deﬁned as design variables and are then optimized subject to minimum compliance.The Shape-method devel-oped by Atrek and Kodali͓133͔and Atrek͓4,131͔is based on precisely the same idea and combined with a technique of cutting away elements͑sub-areas of the structure͒with thick-nesses that end up being equal to the prescribed lower limit value.The same idea is again found in Mattheck͓69͔and Mattheck et al͓298͔and implemented in the CAO͑Com-puter Aided Optimization͒-/SKO͑Soft Kill Option-͒Method where Young’s Modulus of the material plays the role as variable thickness and understressed elements are cut away such that a fully stressed design may result.EvolutionaryStructural Optimization͑ESO͒is a further numerical methodof topology optimization which is developed by Xie andSteven͓412͔and Querin et al͓350–352͔,and integratedwithﬁnite element analysis.Bidirectional ESO͑BESO͒is anextension to this method and can begin with a minimumamount of material in contrast to ESO,which uses an ini-tially oversized structure,see Young͓120,417͔,Young et al ͓418͔.The same is valid for the Material Density Functions method by Yang and Chuang͓416͔.A novel topology opti-mization method,called Metamorphic Development͑MD͒,for both trusses and continuum structures and also for com-bined truss/continuum structures has been developed by Liuet al͓285–287͔at the Engineering Design Centre of the Uni-versity of Cambridge.In the last part of Section5,a further important macro-structure approach is presented which uses an iterative posi-tioning and hierarchically structured shape optimization ofnew holes,so-called bubbles.This means that the boundariesof the structure are considered to be variable,and that theshape optimization of new bubbles and of the other variableboundaries of the component is carried out as a parameteroptimization problem͑Eschenauer et al͓208,209͔,Es-chenauer and Wahl͓214͔,Eschenauer and Schumacher ͓212͔,Rosen and Grosse͓364͔,Schumacher͓95͔,Thierauf ͓404͔͒.Following this idea,Garreau et al͓230͔recently pro-posed a similar yet modiﬁed approach by using so-called topological gradients,which provides information on the possible advantage of the occurrence of a small hole in the body.Cea et al͓172͔developed a topological optimization algorithm based on aﬁxed point method using the topologi-cal gradient.Sokolowski and Zochowski͓99͔gave some mathematical justiﬁcations to the topological gradient in the case of free boundary conditions on the hole and generalized it to various cost ing domain truncation and an adaptation of Lagrange’s method,Garreau et al͓230͔exhibit the topological gradient for a large class of problems,bound-ary conditions for the hole,and cost functions.Section6presents an overview of different kinds of prob-lems,design objectives,and constraints that can be currentlyhandled in the area of topological optimization of structures.While the problems dealt with in initial papers on topology optimization concerned stiffness maximization͑minimiza-tion of compliance͒for a single case of loading,subsequent extensions include,eg,handling of multiple load cases; bimaterial structures;plate and shell bending problems; eigenfrequency optimization;buckling eigenvalue optimiza-tion problems;and stress minimization problems.In Section 6,we also demonstrate by way of examples that in very recent years,new avenues have been opened for the application of topology optimization in Biomechanics͑Re-iter and Rammerstorfer͓359͔,Reiter͓358͔,Tanaka et al ͓399͔,Pettermann et al͓346͔,Folgado and Rodrigues͓222͔, Pedersen and Bendsøe͓78͔͒,as well as in areas of design of materials for prescribed mechanical properties͑Sigmund ͓377,378͔͒and design of compliant mechanisms͑Sigmund ͓379,380͔͒.Section7,ﬁnally,presents the conclusions of this article.334Eschenauer and Olhoff:Topology optimization of continuum structures Appl Mech Rev vol54,no4,July20012MATHEMATICAL-PHYSICAL FUNDAMENTALS2.1Deﬁnition and terms of topologyPrior to the treatment of the actual topic of the present re-view article,topology optimization,the term topology as a subﬁeld of geometry shall be explained and deﬁned.Etymo-logically,the word is derived from the Greek noun topos which means location,place,space or domain.Mathemati-cally speaking,topology is concerned with objects that are deformable in a so-called rubber-like manner,ie,it can be shown that Euler’s Polyhedron Rule maintains its validity in the three-dimensional space if objects like tetrahedrons, cubes,octahedrons etc.are deformed in an arbitrary manner. In the1950s and1960s,important papers on topology were contributed by Alexandroff͓1͔,Bourbaki͓19͔,Franz͓36͔, Hilton and Wylie͓48͔,Hocking and Young͓49͔,Kelley͓53͔, Koethe͓55͔,Pontryagin͓81͔,and Schubert͓94͔.All subsets of R3͑including straight lines,sets of points etc͒are called topological domains.From a mathematical point of view,all distortions are transformations or reversibly unique map-pings.As topological transformations or topological map-pings we deﬁne those transformations of one topological do-main into another that neither destroy existing nor generate new neighborhood relations.Two topological domains are termed topologically equivalent if there exists a topological mapping of one of the domains into the other one͑Fig.1͒. Hence,a topological property of a domain is a characteristic maintained at all topological mappings ie,it is invariant.Topology is therefore considered as the invariants theory of topology domains.The term topological mapping can be reduced to the term continuous mapping,where mapping in a topology domain is called continuous if it does not violate any existing neighborhood relations.In a general manner, topological transformations can be formulated as a continu-ous transformations whose reverse transformation is also continuous.The latter case is also called homomorphism,ie, the transformations are reversibly unique͑bijective͒and con-tinuous͑for further information see͓304͔͒.Based on the above-given terms and deﬁnitions of topol-ogy,a link shall now be established between topology and optimization.For that purpose,the term topology class is to be introduced that describes certain objects to be topologi-cally equivalent.A topology class is generally deﬁned by the degree of connection of domains.Domains that belong to one topology class are topologically equivalent͑Fig.2a͒.A second topology class is deﬁned by the degree to which the domains are connected͑Fig.2b͒.A further topology class is termed n-fold connected,if(nϪ1)cuts from one boundary to another are required to transform a given,multiply con-nected domain into a simply connected domain͑Fig.2c͒.Up to this point,topology has only been considered as a mathematical deﬁnition.In the following,the deﬁnitions of structural or design optimization as presented in the subse-quent sections shall be adapted to topology optimization.As indicated in Fig.2a,the neighborhood relations of the single elements that establish a domain remain unviolated in the classical shape optimization of a component;the map-ping rules of homomorphisms are valid.Topology optimiza-tion,however,changes the neighborhood relation,ie,a trans-formation into a different topology class is performed.From a mathematical point of view the position and shape of the new domain is of no importance͑Fig.2b͒.At any rate,both positioning and shape inﬂuence the structural mechanical be-havior of a component,and it is therefore usually intended to improve the component by both topology and shape optimization.The set-oriented topology͓23,51,52͔describes those properties of geometrical shapes that remain unchanged even if the domain is subjected to distortions that are large enough to eliminate all metric and projective properties.The topo-logical properties are the most general qualities of a domain.In the classical shape optimization of structural compo-nents,interrelations between the elements that constitute a domain are maintained,and the isomorphous mapping laws are valid.Topology optimization,ie,improving transforma-tions into other topology classes,modiﬁes these interrela-tions.2.2Classiﬁcation of topology optimization2.2.1Conceptual processesThe topology of a structure,ie,the arrangement of material or the positioning of structural elements in the structure,is crucial for its optimality.Traditionally,the topology of a de-sign is in most cases chosen either intuitively or inspired by already existing designs͑Current-Design-World-State͓͒204͔. However,there is a signiﬁcant necessity of and interest in improving the quality of products byﬁnding their best pos-sible topology in a very early stage of the designprocess.Fig.1Topologicalmapping/transformation Fig.2Topological properties of two-dimensional domains Appl Mech Rev vol54,no4,July2001Eschenauer and Olhoff:Topology optimization of continuum structures335。

Zemax课堂3 (优化)

Operands
ZEMAX提供了200多种操作数(用4个大写字母缩写组成)，如EFFL，可以控制包括系统参数、像差、MTF、圆内能量集中度、光线约束、边界约束条件、玻璃材料的范围等。
Operands分类
• • • • • • • • 一阶光学性能 1. EFFL 透镜单元的有效焦距 2. AXCL 透镜单元的轴向色差 3. LACL 透镜单元的垂轴色差 4. PIMH 规定波长的近轴像高 5. PMAG 近轴放大率 6. AMAG 角放大率 7. ENPP 透镜单元入瞳位置
Optimization过程
Damped Least Squares(DLS)
• DLS算法是所有光学设计软件中的基本优化算法。假定Merit function定义为如下形式：
式中W为操作数的权重的绝对值， V为当前值， T 是目标值，下标i是操作数的号码 (row number in the spreadsheet). 目标：找出x使MF的值最小。
Overall Weight:一般设置为1。 Ignore Lateral Color：对不同的波长按不同的参考点计算。用于按波长来分光分色系统设计中，如棱镜或者光谱仪等。
Weight
操作数的权重：绝大部分为正。 1）<0，相当于权重为无穷大； 2）=0，不考虑这个操作数； 3）>0，使MF最小。
Defining complex operands
ZEMAX缺省的MF不能满足用户需要时，用户可以自己构建评价函数，有两种方法： • 对缺省的merit function进行重新定义； • 用operands手动编辑merit function，
• • • • •
衍射能级 1． DENC 衍射包围圆能量 2． DENF 衍射能量 3． GENC 几何包围圆能量 4． XENC

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