Topological Gradient

The standard geometrical shape optimization method proceed by the application of successive diffeomorphisms close to the identity starting from an initial guessed shape. Consequently, it does not allow for the optimization of the topology which is kept unchanged from one iteration to the other. The topological gradient enables to determine if the inclusion of a small hole of given shape is cost efficient. Such holes can be included at any time during the geometrical shape optimization process. Moreover, as seen thereafter, the topological gradient can be explicitly computed from the primal and adjoint states of the optimization problem that are already computed during standard geometrical shape optimization. Finally let us mention that level set methods can also handle topological changes during the optimization process [2], possibly coupled with the use of topological gradient [1].