On numerical study of constrained coupled shape optimization problems based on a new shape derivative method

In this article, we deal with a numerical method for the approximation of a class of coupled shape optimization problems, which consist in minimizing an appropriate general volume cost functional subjected to coupled boundary value problems by means of a Neumann boundary transmission condition. We show the existence of the shape derivative of the cost functional and express it by means of support functions, using a new formula of shape derivative on a family of convex domains. This allows us to avoid the disadvantages related to the classical shape derivative method using vectors field. Then the numerical discretization is performed using the dual reciprocity boundary element method in order to avert the remeshing task required for the finite element method. Finally, we give some numerical results, based on the gradient method, showing the efficiency of the proposed approach.

Automated summary

In this article, a numerical method for approximating coupled shape optimization problems is presented. The method involves minimizing a general volume cost functional subject to coupled boundary value problems using a Neumann boundary transmission condition. The existence of the shape derivative of the cost functional is shown and expressed using support functions based on a new formula for shape derivative on a family of convex domains. The dual reciprocity boundary element method is employed for numerical discretization to avoid the remeshing task required for the finite element method. Numerical results based on the gradient method demonstrate the efficiency of the proposed approach.