Ole Sigmund的99行拓扑优化Matlab程序

Educational article Struct Multidisc Optim 21, 120–127  Springer-Verlag 2001 A 99 line topology optimization code written in Matlab O. Sigmund Abstract The paper presents a compact Matlab im- plementation of a topology optimization code for com- pliance minimization of statically loaded structures. The total number of Matlab input lines is 99 including opti- mizer and Finite Element subroutine. The 99 lines are divided into 36 lines for the main program, 12 lines for the Optimality Criteria based optimizer, 16 lines for a mesh- independency filter and 35 lines for the finite element code. In fact, excluding comment lines and lines associ- ated with output and finite element analysis, it is shown that only 49 Matlab input lines are required for solving a well-posed topology optimization problem. By adding three additional lines, the program can solve problems with multiple load cases. The code is intended for edu- cational purposes. The complete Matlab code is given in the Appendix and can be down-loaded from the web-site http://www.topopt.dtu.dk. Key words topology optimization, education, optimal- ity criteria, world-wide web, Matlab code 1 Introduction The Matlab code presented in this paper is intended for engineering education. Students and newcomers to the field of topology optimization can down-load the code from the web-page http://www.topopt.dtu.dk. The code may be used in courses in structural optimiza- tion where students may be assigned to do extensions such as multiple load-cases, alternative mesh-independ- ency schemes, passive areas, etc. Another possibility is to use the program to develop students’ intuition for optimal design. Advanced students may be asked to guess the op- timal topology for given boundary condition and volume Received October 22, 1999 O. Sigmund Department of Solid Mechanics, Building 404, Technical Uni- versity of Denmark, DK-2800 Lyngby, Denmark e-mail: sigmund@fam.dtu.dk fraction and then the program shows the correct optimal topology for comparison. In the literature, one canfindamultitude of approaches for the solving of topology optimization problems. In the original paper Bendsøe and Kikuchi (1988) a so-called microstructure or homogenization based approach was used, based on studies of existence of solutions. The homogenization based approach has been adopted in many papers but has the disadvantage that the deter- mination and evaluation of optimal microstructures and their orientations is cumbersome if not unresolved (for noncompliance problems) and furthermore, the resulting structures cannot be built since no definite length-scale is associated with the microstructures. However, the ho- mogenization approach to topology optimization is still important in the sense that it can provide bounds on the theoretical performance of structures. An alternative approach to topology optimization is the so-called “power-law approach” or SIMP approach (Solid Isotropic Material with Penalization) (Bendsøe 1989; Zhou and Rozvany 1991; Mlejnek 1992). Here, ma- terial properties are assumed constant within each elem- ent used to discretize the design domain and the variables are the element relative densities. The material proper- ties are modelled as the relative material density raised to some power times the material properties of solid ma- terial. This approach has been criticized since it was ar- gued that no physical material exists with properties de- scribed by the power-law interpolation. However, a recent paper by Bendsøe and Sigmund (1999) proved that the power-law approach is physically permissible as long as simple conditions on the power are satisfied (e.g. p ≥ 3 for Poisson’s ratio equal to 13 ). To ensure existence of so- lutions, the power-law approach must be combined with a perimeter constraint, a gradient constraint or with fil- tering techniques (see Sigmund and Petersson 1998, for an overview). The power-law approach to topology op- timization has been applied to problems with multiple constraints, multiple physics and multiple materials. Whereas the solution of the above mentioned ap- proaches is based on mathematical programming tech- niques and continuous design variables, a number of pa- pers have appeared on solving the topology optimization problem as an integer problem. Beckers (1999) success- 121 fully solved large-scale compliance minimization prob- lems using a dual-approach but other approaches based on genetic algorithms or other semi-random approaches require thousands of function evaluations even for small number of elements and must be considered impractical. Apart from above mentioned approaches, which all solve well defined problems (e.g. minimization of com- pliance) a number of heuristic or intuition based ap- proaches have been shown to decrease compliance or other objective functions. Among these methods are so- called evolutionary design methods (see e.g. Xie and Steven 1997; Baumgartner et al. 1992). Apart from be- ing very easy to understand and implement (at least for the compliance minimization case), the main moti- vation for the evolutionary approaches seems to be that mathematically based or continuous variable approaches “involve some complex calculus operations and mathe- matical programming” (citation from Li et al. 1999) and they contain “mathematical methods of some complex- ity” (citation from Zhao et al. 1998) whereas the evo- lutionary approach “takes advantage of powerful com- puting technology and intuitive concepts of evolution processes in nature” (citation from Li et al. 1999). Two things can be argued against this. First, the evolutionary approaches become complicated themselves, once more complex objectives than compliance minimization are considered and second, as shown in this paper, the “math- ematically based” approaches for compliance minimiza- tion are simple to implement as well and are compu- tationally equally efficient. Furthermore, mathematical programming based methods can easily be extended to other non-compliance objectives such as non-self-adjoint and multiphysics problems and to problems with multiple constraints (e.g. Sigmund 1999). Extensions of the evolu- tionary approach to such cases seem more questionable. The complete Matlab code is given in the Appendix. The remainder of the paper consists of definition and discussion of the optimization problem (Sect. 2), com- ments about the Matlab implementation (Sect. 3) fol- lowed by a discussion of extensions (Sect. 4) and a conclu- sion (Sect. 5). 2 The topology optimization problem A number of simplifications are introduced to simplify the Matlab code. First, the design domain is assumed to be rectangular and discretized by square finite elements. In this way, the numbering of elements and nodes is simple (column by column starting in the upper left corner) and the aspect ratio of the structure is given by the ratio of elements in the horizontal (nelx) and the vertical direc- tion (nely).1 1 Names in type-writer style refer to Matlab variable names that differ from the obvious (see the Matlab code in the Appendix) A topology optimization problem based on the power- law approach, where the objective is to minimize compli- ance can be written as min x : c(x) =UTKU= N∑ e=1 (xe) p uTe k0 ue subject to : V (x) V0 = f : KU= F : 000< xmin ≤ x≤ 111   , (1) where U and F are the global displacement and force vectors, respectively, K is the global stiffness matrix, ue and ke are the element displacement vector and stiffness matrix, respectively, x is the vector of design variables, xmin is a vector of minimum relative densities (non-zero to avoid singularity), N (= nelx×nely) is the number of elements used to discretize the design domain, p is the penalization power (typically p = 3), V (x) and V0 is the material volume and design domain volume, respectively and f (volfrac) is the prescribed volume fraction. The optimization problem (1) could be solved using several different approaches such as Optimality Criteria (OC) methods, Sequential Linear Programming (SLP) methods or the Method of Moving Asymptotes (MMA by Svanberg 1987) and others. For simplicity, we will here use a standard OC-method. Following Bendsøe (1995) a heuristic updating scheme for the design variables can be formulated as xnewe =   max(xmin, xe−m) if xeB η e ≤ max(xmin, xe−m) , xeB η e if max(xmin, xe−m)< xeBηe