Shape and Topology Optimization by the Level Set Method

Numerical methods of shape and topology optimization based on the level set representation and on shape di erentiation make possible topology changes during the optimization process. But they do not solve the inherent problem of illposedness of shape optimization which manifests itself in the existence of many local minima, usually having di erent topologies. The reason is that the level set method can easily remove holes but can not create new holes in the middle of a shape. In practice, this e ect can be checked by varying the initialization which yields di erent optimal shapes with di erent topologies. This absence of a nucleation mechanism is an inconvenient mostly in 2-d: in 3-d, it is less important since holes can appear by pinching two boundaries. In [1] we have proposed, as a remedy, to couple our previous method with the topological gradient method (cf. [5][6][7][13]). Roughly speaking it amounts to decide whether or not it is favorable to nucleate a small hole in a given shape. Creating a hole changes the topology and is thus one way of escaping local minima. Our coupled method of topological and shape gradients in the level set framework is therefore much less prone to nding local, non global, optimal shapes. For most of our 2-d numerical examples of compliance minimization, the expected global minimum is attained from the trivial full domain initialization.