On the Shape Differentiability of Objectives: A Lagrangian Approach and the Brinkman Problem

• Main Authors: José Rodrigo González Granada, Joachim Gwinner, Victor A. Kovtunenko
• Format: Article
• Language: English
• Published: MDPI AG 2018-10-01
• Series: Axioms
• Subjects: constrained optimization; variational inequality; Lagrangian; geometry-dependent objective function; shape derivative; Delfour–Zolésio theorem; divergence-free Brinkman flow
• Online Access: https://www.mdpi.com/2075-1680/7/4/76

This paper establishes the shape derivative of geometry-dependent objective functions for use in constrained variational problems. Using a Lagrangian approach, our differentiablity result is based on the theorem of Delfour⁻Zolésio on directional derivatives with respect to a parameter of shape perturbation. As the key issue of the paper, we analyze the bijection under the kinematic transport of geometries that is needed for function spaces and feasible sets involved in variational problems. Our abstract theoretical result is applied to the Brinkman flow problem under incompressibility and mixed Dirichlet⁻Neumann boundary conditions, and provides an analytic formula of the shape derivative based on the velocity method.