Directional differentiability for shape optimization with variational inequalities as constraints

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 variat...

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Bibliographic Details
Published inESAIM. Control, optimisation and calculus of variations Vol. 29; p. 64
Main Authors Kovtunenko, Victor A., Kunisch, Karl
Format Journal Article
LanguageEnglish
Published 2023
Online AccessGet full text
ISSN1292-8119
1262-3377
DOI10.1051/cocv/2023056

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Summary: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.
ISSN:1292-8119
1262-3377
DOI:10.1051/cocv/2023056