Shape Derivative for Penalty-Constrained Nonsmooth–Nonconvex Optimization: Cohesive Crack Problem

Victor A. Kovtunenko () and Karl Kunisch ()
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Victor A. Kovtunenko: Karl-Franzens University of Graz
Karl Kunisch: Karl-Franzens University of Graz

Journal of Optimization Theory and Applications, 2022, vol. 194, issue 2, No 10, 597-635

Abstract: Abstract A class of non-smooth and non-convex optimization problems with penalty constraints linked to variational inequalities is studied with respect to its shape differentiability. The specific problem stemming from quasi-brittle fracture describes an elastic body with a Barenblatt cohesive crack under the inequality condition of non-penetration at the crack faces. Based on the Lagrange approach and using smooth penalization with the Lavrentiev regularization, a formula for the shape derivative is derived. The explicit formula contains both primal and adjoint states and is useful for finding descent directions for a gradient algorithm to identify an optimal crack shape from a boundary measurement. Numerical examples of destructive testing are presented in 2D.

Keywords: Shape optimization; Optimal control; Variational inequality; Penalization; Lagrange method; Lavrentiev regularization; Free discontinuity problem; Non-penetrating crack; Quasi-brittle fracture; Destructive physical analysis; 35R37; 49J40; 49Q10; 74RXX (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10957-022-02041-y

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