Shape differentiability of Lagrangians and application to Stokes problem

TL;DR

This paper develops a new approach for shape sensitivity analysis of constrained minimization problems, extending existing methods to a broader class and applying it to the Stokes problem with boundary conditions.

Contribution

It introduces primal-dual shape differentiability analysis that relaxes previous assumptions, providing explicit formulas and applying them to fluid dynamics problems.

Findings

Extended shape differentiability results for convex constrained problems.

Explicit formulas for shape derivatives in the primal-dual framework.

Application to Stokes problem with mixed boundary conditions.

Abstract

A class of convex constrained minimization problems over polyhedral cones for geometry-dependent quadratic objective functions is considered in a functional analysis framework. Shape differentiability of the primal minimization problem needs a bijective property for mapping of the primal cone. This restrictive assumption is relaxed to bijection of the dual cone within the Lagrangian formulation as a primal-dual minimax problem. In this paper, we give results on primal-dual shape sensitivity analysis that extends the class of shape-differentiable problems supported by explicit formula of the shape derivative. We apply the results to the Stokes problem under mixed Dirichlet-Neumann boundary conditions subject to the divergence-free constraint.