Educational article
Struct Multidisc Optim 21, 120–127
SpringerVerlag 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 ±lter and 35 lines for the ±nite element
code. In fact, excluding comment lines and lines associ
ated with output and ±nite element analysis, it is shown
that only 49 Matlab input lines are required for solving
a wellposed 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 downloaded from the website
http://www.topopt.dtu.dk
.
Key words
topology optimization, education, optimal
ity criteria, worldwide web, Matlab code
1
Introduction
The Matlab code presented in this paper is intended
for engineering education. Students and newcomers to
the ±eld of topology optimization can download the
code from the webpage
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 loadcases, alternative meshindepend
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, DK2800 Lyngby, Denmark
email:
[email protected]
fraction and then the program shows the correct optimal
topology for comparison.
Intheliterature,onecan±ndamultitudeofapproaches
for the solving of topology optimization problems. In the
original paper Bendsøe and Kikuchi (1988) a socalled
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 de±nite lengthscale
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.
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 Spring '08
 Staff
 Operations Research, Optimization, Mathematical optimization, topology optimization, Shape optimization, O. Sigmund

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