Combining topological and shape derivatives in structural optimization

Two recent methods in shape and topology optimization of structures are combined in order to obtain an efficient optimization algorithm that benefits of advantages from both methods. The level set method, based on the classical shape derivative, is known to easily handle boundary propagation with topological changes. However, in practice it does not allow for the nucleation of new holes (at least in 2-d). The bubble or topological gradient method of Schumacher, Masmoudi, Sokolowski and their co-workers, is precisely designed for introducing new holes in the optimization process. Therefore, the coupling of these two methods yields a robust algorithm which can escape from local minima in a given topological class of shapes. The method we propose is a logical sequel of our previous work [

1

], [

2

] where we proposed a numerical method of shape optimization based on the level set method and on shape differentiation. The novelty is in the coupling and in the robustness of the proposed numerical implementation. Our basic algorithm is to iteratively use the shape gradient or the topological gradient in a gradient-based descent algorithm. The tricks are to carefully monitor the decrease of the objective function (to avoid large changes in shape and topology) and to choose the right ratio of successive iterations in each method.We provide several 2-d and 3-d numerical examples for compliance minimization and mechanism design. The main advantage of our coupled algorithm is to make the resulting optimal design largely independent of the initial guess, although local minima may still exist (even in the class of shapes sharing the same topology). Similar numerical results where discussed in [

3

].