Space-time formulation, discretization, and computations for phase-field fracture optimal control problems

TL;DR

This paper develops space-time discretization schemes for phase-field fracture optimal control problems, incorporating regularization and crack irreversibility, and proposes an efficient Newton algorithm with numerical validation.

Contribution

It introduces a novel space-time discretization approach for phase-field fracture optimal control, including a discontinuous Galerkin time discretization and a reduced Newton algorithm.

Findings

Successful implementation of the discretization schemes

Effective handling of regularization and crack irreversibility

Numerical experiments demonstrate algorithm performance

Abstract

The purpose of this work is the development of space-time discretization schemes for phase-field optimal control problems. First, a time discretization of the forward problem is derived using a discontinuous Galerkin formulation. Here, a challenge is to include regularization termsand the crack irreversibility constraint. The optimal control setting is formulated by means of the Lagrangian approach from which the primal part, adjoint, tangent and adjoint Hessian are derived. Herein the overall Newton algorithm is based on a reduced approach by eliminating the state constraint. From the low-order discontinuous Galerkin discretization, adjoint time-stepping schemes are finally obtained. Our algorithmic developments are substantiated and illustrated with some numerical experiments.