Directional differentiability for shape optimization with variational inequalities as constraints

¹ Department of Mathematics and Scientific Computing, Karl-Franzens University of Graz, NAWI Graz, Heinrichstr.36, 8010 Graz, Austria
² Lavrent’ev Institute of Hydrodynamics, Siberian Division of Russian Academy of Sciences, 630090 Novosibirsk, Russia
³ Department of Mathematics and Scientific Computing, Karl-Franzens University of Graz, NAWI Graz, Heinrichstr.36, 8010 Graz, Austria
⁴ Radon Institute, Austrian Academy of Sciences, RICAM Linz, Altenbergerstraße 69, 4040 Linz, Austria

^* Corresponding author: victor.kovtunenko@uni-graz.at

Received: 30 January 2023
Accepted: 19 July 2023

Abstract

For equilibrium constrained optimization problems subject to nonlinear state equations, the property of directional differentiability with respect to a parameter is studied. An abstract class of parameter dependent shape optimization problems is investigated with penalty constraints linked to variational inequalities. Based on the Lagrange multiplier approach, on smooth penalties due to Lavrentiev regularization, and on adjoint operators, a shape derivative is obtained. The explicit formula provides a descent direction for the gradient algorithm identifying the shape of the breaking-line from a boundary measurement. A numerical example is presented for a nonlinear Poisson problem modeling Barenblatt’s surface energies and non-penetrating cracks.