Let be a regular open bounded domain of (or ). Let us consider a Partial Differential Equation (PDE) problem defined in , written in its variational formulation:
We now consider a small perturbation of the domain, e.g. by the insertion of a crack , where represents the point where the crack is inserted, is a straight crack containing the origin of the domain, and is a unit vector normal to the crack. Finally, represents the size of the perturbation, assumed to be small. Let be the perturbed domain. We can consider the same PDE problem as before, but on the perturbed domain:
We can rewrite the cost function as a function of by considering the following map:
(2.3) |
The topological sensitivity theory provides an asymptotic expansion of when tends to zero. It takes the general form:
Then to minimize the criterion , one has to insert small holes (or cracks) at points where the topological gradient is the most negative, in order to make the cost function decrease quickly (see the asymptotic expansion (2.4)).