Lecture of G. Allaire, topological optimization based on the level set method

G. ALLAIRE , Centre de Mathématiques Appliquée, Ecole Polytechnique

The optimization of mechanical structures is an important problem for many applications and there has been a lot of progress in this domain recently. In addition to classical methods of boundary variation (which dates back to Hadamard), a new optimization method has appeared which is called topological optimization, and is based on homogenization theory. This last method has a low computational cost because it captures shape implicitly through a fixed mesh. However, it is mainly restricted to linearized elasticity. Even more recently, a new method has been developed, called the level-set method, which uses the Osher and Sethian formalism in order to describe the boundary of a shape with a level set function. This method combines, as much as possible, the advantages of the method of interface variation and of homogenization, and allows topological optimization. The principle of this method is to carry the level set function (that is to say, the edge of the shape) with a velocity that decreases the objective function. Following the boundary variation method, we compute this velocity by differentiating the objective functions with respect to the shape. Following the homogenization method, we use a fixed mesh which contains both the shape and voids (or vacuum) represented by a very weak material. We consider a two or three dimensional linearized elasticity mechanical model and general regular objective functions ( compliance or a least-squares criterion ). Numerical examples show that it is useful to create additional holes. For this purpose, the notion of topological gradients coupled with the level set method is used.

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