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

TL;DR

This paper derives a shape derivative formula for a non-smooth, non-convex optimization problem related to cohesive crack modeling, enabling gradient-based shape optimization in fracture mechanics.

Contribution

It introduces a novel shape derivative formula for penalty-constrained, nonsmooth nonconvex problems in fracture mechanics using Lavrentiev regularization.

Findings

Derived explicit shape derivative formula involving primal and adjoint states.

Provided numerical examples demonstrating crack shape optimization in 2D.

Enabled gradient-based algorithms for crack shape identification.

Abstract

A class of non-smooth and non-convex optimization problems with penalty constraints linked to variational inequalities (VI) 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.