MATLAB拓扑优化88行代码详解
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"MATLAB88行程序 - 一个基础的MATLAB拓扑优化代码，适合初学者学习，基于Sigmund的99行代码优化，加入了密度滤波器并提升了效率。"
MATLAB是一个强大的数学计算软件，广泛应用于科研、工程和教育领域。在结构多学科优化（Structural Multidisciplinary Optimization）中，拓扑优化是一种通过改变设计区域内的材料分布以优化结构性能的方法。MATLAB因其易读易写和强大的数值计算能力，常被用于编写拓扑优化的代码。
本文介绍的MATLAB88行程序，是对Sigmund在2001年提出的99行代码的进一步优化。原始的99行代码已经是一个简洁且功能完整的拓扑优化实现，但新代码在保持可读性的同时，进行了以下改进：
1. **密度滤波器**：在拓扑优化中，密度滤波器用于平滑材料分布，避免出现过于复杂或不连续的结构。通过引入密度滤波，新的代码能生成更加连续和结构合理的优化结果。
2. **效率提升**：通过重新分配数组和向量化循环，代码的运行效率显著提高。在包含7,500个元素的基准测试中，运行速度提高了约100倍。这得益于MATLAB的内向优化和向量化操作，减少了不必要的计算和内存访问。
3. **代码精简**：尽管增加了功能，但代码长度反而减小到88行。这意味着代码的组织更紧凑，更易于理解和维护，对初学者来说更加友好。
这个88行的MATLAB代码不仅是一个实用的工具，还提供了一个教学示例，展示了如何在有限的代码行数内实现高效优化算法。对于想要学习MATLAB编程和拓扑优化的人来说，这是一个宝贵的起点，可以深入理解优化过程中的关键步骤，如问题定义、求解器设置、边界条件处理以及结果后处理等。
在学习和使用这个88行程序时，初学者可以逐步理解每个函数的作用，例如如何定义优化目标和约束，如何更新设计变量，以及如何应用滤波器来改善解决方案的质量。此外，通过对比优化前后的代码，可以学习到代码优化技巧，提高MATLAB编程技能。
MATLAB88行程序是一个简洁而高效的拓扑优化工具，它将理论与实践结合，为初学者提供了宝贵的实践机会，同时也为专业人士提供了一个快速原型开发的模板。
4 E. Andreassen et al.
density variables cannot become zero, and the term γ is
not required.
The density filter transforms the original densities x
e
as
follows:
˜x
e
=
1
i∈N
e
H
ei
i∈N
e
H
ei
x
i
(9)
In the following, the original densities x
e
are referred to as
the design variables. The filtered densities ˜x
e
are referred to
as the physical densities. This terminology is used to stress
the fact that the application of a density filter causes the
original densities x
e
to loose their physical meaning. One
should therefore always present the filtered density field ˜x
e
rather than the original density field x
e
as the solution to the
optimization problem (Sigmund 2007).
In the case where a density filter is applied, the sensitiv-
ities of the objective function c and the material volume V
with respect to the physical densities ˜x
e
are still given by
(5)and(6), provided that the variable x
e
is replaced with ˜x
e
.
The sensitivities with respect to the design variables x
j
are
obtained by means of the chain rule:
∂ψ
∂x
j
=
e∈N
j
∂ψ
∂ ˜x
e
∂ ˜x
e
∂x
j
=
e∈N
j
1
i∈N
e
H
ei
H
je
∂ψ
∂ ˜x
e
(10)
where the function ψ represents either the objective func-
tion c or the material volume V .
3 MATLAB implementation
In this section the 88 line MATLAB code (see Appendix)is
explained. The code is called from the MATLAB prompt by
means of the following line:
top88(nelx,nely,volfrac,penal,rmin,ft)
where nelx and nely are the number of elements in the
horizontal and vertical direction, respectively, volfrac is
the prescribed volume fraction f , penal is the penaliza-
tion power p, rmin is the filter radius r
min
(divided by
the element size), and the additional argument (compared to
the 99 line code) ft specifies whether sensitivity filtering
(ft = 1) or density filtering (ft = 2) should be used.
When sensitivity filtering is chosen, the 88 line code yields
practically
1
the same results as the 99 line code; e.g. the
1
The slight difference which can be observed between the 88-line and
the 99-line code is due to the difference in the SIMP formulation.
optimized MBB beam shown in Fig. 1 of the original paper
by Sigmund (2001) can be reproduced by means of the
following function call:
top88(60,20,0.5,3,1.5,1)
The most obvious differences between the 88 line code
and the 99 line code are the following: (1) the for loops
used to assemble the finite element matrices, to compute
the compliance, and to perform the filtering operation have
been vectorized, (2) the remaining arrays constructed by
means of a for loop are properly preallocated, (3) a maxi-
mum amount of code is moved out of the optimization loop
to ensure that it is only executed once, (4) a distinction is
made between the design variables x and the physical densi-
ties xPhys in order to facilitate the application of a density
filter, and (5) all subroutines have been integrated in the
main program.
The 88 line code consists of three parts: the finite ele-
ment analysis, the sensitivity or density filter, and the
optimization loop. These parts are discussed in detail in
Sections 3.1–3.3. Section 3.4 presents some results obtained
with the 88 line code.
3.1 Finite element analysis
The design domain is assumed to be rectangular and dis-
cretized with square elements. A coarse example mesh
consisting of 12 elements with four nodes per element and
two degrees of freedom (DOFs) per node is presented in
Fig. 2. Both nodes and elements are numbered column-wise
from left to right, and the DOFs 2n − 1and2n corre-
spond to the horizontal and vertical displacement of node
n, respectively. This highly regular mesh can be exploited
in several ways in order to reduce the computational effort
in the optimization loop to a minimum.
The finite element preprocessing part starts with the
definition of the material properties (lines 4–6): E0 is
the Young’s modulus E
0
of the material, Emin is the
Fig. 2 The design domain with 12 elements
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