We propose a set of Scilab routines which perform shape and topology optimization of plane elastic structures. For the moment only compliance minimization is implemented in the single or multiple loads case. The proposed algorithm uses the level set method of S. Osher and J. Sethian. It is based on the work of G. Allaire, F. Jouve, A.-M. Toader, J. Comp. Phys. Vol 194/1, pp.363-393 (2004). It features the possiblity of using the notion of topological gradient to nucleate new holes, as implemented in the work of G. Allaire, F. de Gournay, F. Jouve, A.-M. Toader, Control and Cybernetics 34, pp.59-80 (2005). These routines have been written by G. Allaire, A. Karrman and G. Michailidis. Users are free to use this code as far as they acknowledge their source!

**Warning :**
Although they have been written and tested with great care, these Scilab
routines come with absolutely no warranty. Their authors decline any responsability
linked to their use.

** What is shape optimization ?**
In the context of solid mechanics it is also called structural optimization.
It is the mathematical theory which makes possible the ''automatic'' optimization of shapes
of mechanical structures. By ``automatic'' it is meant that these methods
and algorithms can be implemented on a computer which can analyse and
improve the designs of numerous successive configurations without any
help from the engineer or designer. For more details on this topic we
refer to the book
Conception optimale de structures published by Springer
in the series "Mathématiques et Applications", volume 58, 2007, 280 p.,
91 illus., ISBN-10: 3-540-36710-1, ISBN-13: 978-3-540-36710-9.
There are, of course, many more references on the topic!

** What is Scilab ?**
Scilab is a free software developed and distributed by INRIA.
You can freely download it on the web site
` www.scilab.org`

.
It is very similar in spirit to Matlab.

This Scilab code is inspired from the Fortran code used in the work of G. Allaire, F. Jouve, A.-M. Toader, J. Comp. Phys. Vol 194/1, pp.363-393 (2004). In a fixed computational domain the 2-d structure is parametrized by a level set function. The iterative algorithm of shape optimization alternates finite element analysis of the current structure and deformation of the structure by transport of the level set function. Its main features are the use of an explicit second-order finite difference upwind scheme for the transport Hamilton-Jacobi equation, a re-initialization process at fixed frequency (function mesh00), a regularization of the velocity field, a possibility of using the topological gradient to nucleate new holes (new !), a careful interpolation to project the structure on the fixed mesh (function FEdensity), a monitoring of the decrease of the objective function by varying the number of time steps in the transport Hamilton-Jacobi equation. We borrowed the finite element solver (functions FE and lk) from the famous '' 99 line topology optimization code written in Matlab'' of O. Sigmund (Struct. Multidisc. Optim. 21, pp.120-127, 2001) whose kind permission is greatefully acknowledged. The objective function to be minimized is the sum of three terms: the compliance, the volume and the perimeter. The two last ones are multiplied by fixed non-negative Lagrange multipliers. However, at the end of the manual, we have also given a function that can be added to the code in order to keep the total volume of the structure constant in each iteration (new !).

The code is not optimized for minimal CPU cost: some routines are still not written in a vectorial form (shame on me). It works only in 2-d for square regular meshes. It performs only compliance minimization in the single or multiple loads case. There is no stopping criterion (not hard to add one). There are many small improvements in solving the level set equation which are implemented in the original Fortran code (cited above) and not reproduced in the Scilab code. Let us mention a few: narrow band, fast marching, Russo-Smereka correction for the re-initialization. One day maybe...

I plan to deal with more general objective functions and to add an adjoint state in the process. In theory 3-d is easy to implement, but in practice Scilab is probably too slow...

You may find more informations on this Scilab code in the
report written by
A. Karrman (from Caltech) after his internship at Ecole Polytechnique, as well as in a manual of the code written by G. Michailidis (Phd student at Ecole Polytechnique).
Please send any bug report to
` gregoire.allaire_AT_polytechnique.fr`

and I will try to correct it as soon as possible.

** Thanks. **
This work has partly been supported by many institutions, the support
of which is gratefully acknowledged: Ecole Polytechnique, CNRS, INRIA
Saclay Ile-de-France, the Chair ''Mathematical modelling and numerical
simulation, F-EADS - Ecole Polytechnique - INRIA''.