Let be a bounded open set of . We assume in this section that contains a perfectly insulating crack . We impose a flux on the boundary of , and we want to find such that the solution of
(2.7) |
A topological gradient approach has been introduced in [8], and consists of defining a Dirichlet and a Neumann problem, as we have an over-determination in the boundary conditions:
It is clear that for the actual crack , the two solution and are equal. The idea is then to consider and minimize the following cost function
The topological asymptotic expansion of this cost function is detailed in [8].