Speaker
Description
We present a framework for computing shape derivatives of boundary functionals discretized with unfitted finite element methods. The main idea is to replace the boundary functional by a narrow-band volumetric regularization prior to discretization. This yields an exact Fr'echet derivative of the resulting discrete functional with respect to the discrete level set function, without any mesh-related restrictions.
At the continuous level, we establish Fr'echet differentiability of the regularized boundary functional and show that, as the regularization thickness converges to zero, both the functional and its derivative converge to the classical boundary shape functional and shape derivative.
At the discrete level, we prove optimal-order error estimates with respect to the mesh size for the functional and for its Fr'echet derivative that are independent of the regularization thickness. As a result, the regularized formulation admits consistent and stable discrete gradients suitable for gradient-based shape optimization on unfitted meshes with fixed regularization.
The approach accommodates higher-order level set representations and extends, via standard Lagrangian techniques, to PDE-constrained problems.
Numerical experiments confirm the exactness of the discrete derivative, the predicted convergence rates, and the applicability of the method to geometric optimization problems.