99行MATLAB代码实现的拓扑优化教程

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"99line topology optimization 是一个用于教育目的的Matlab代码，旨在帮助学生和初学者了解拓扑优化领域。该代码可以在http://www.topopt.dtu.dk网站上下载，适用于结构优化课程，让学生进行扩展练习，如处理多个荷载情况、实现不同的网格独立性方案或引入被动区域等。此外，它还可以用来培养学生的最优设计直觉，让高级学生预测给定边界条件和体积下的最优结构形态。" 本文介绍了一个简洁的Matlab实现的拓扑优化代码，主要用于静态加载结构的刚度最小化问题。整个Matlab输入代码仅99行，包括优化器和有限元子程序。这99行代码分为36行主程序、12行基于最优性准则的优化器、16行网格独立性过滤器以及35行有限元代码。如果不算注释行和与输出及有限元分析相关的行，解决一个有良好定义的拓扑优化问题只需要49行Matlab代码。通过增加3行额外代码，该程序可以处理具有多个荷载工况的问题。关键词：拓扑优化、教育、最优性准则、互联网、Matlab代码这个99行的Matlab拓扑优化代码是一个理想的教育资源，它简化了复杂的优化算法，使初学者能够更容易地理解和应用。拓扑优化是工程设计中的一个重要工具，它通过寻找结构材料分布的最优配置来最大化性能，同时最小化重量或成本。在这个代码中，使用了基于最优性准则的优化器，这是一种常见的优化策略，可以根据设计变量的梯度信息更新设计。此外，网格独立性过滤器对于确保优化结果不依赖于特定的网格划分至关重要，因为不同的网格可能会导致不同的优化结果。有限元代码则负责将结构模型离散化为可求解的元素，从而在计算过程中实现对结构行为的近似。通过这个简单的代码，学生可以深入了解拓扑优化的基本原理，同时可以通过修改和扩展代码来提高他们的编程技能和理解力。这不仅提供了理论学习的基础，还鼓励实践操作，有助于培养未来的工程师具备解决实际问题的能力。因此，无论是在课堂上还是个人研究中，这个99行的Matlab代码都是一个宝贵的资源。

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

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 can ﬁnd a multitude 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

). 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-

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