Space-Time Mixed System Formulation of Phase-Field Fracture Optimal Control Problems

In this work, space-time formulations and Galerkin discretizations for phase-field fracture optimal control problems are considered. The fracture irreversibility constraint is formulated on the time-continuous level and is regularized by means of penalization. The optimization scheme is formulated in terms of the reduced approach and then solved with a Newton method. To this end, the state, adjoint, tangent, and adjoint Hessian equations are derived. The key focus is on the design of appropriate function spaces and the rigorous justification of all Fr & eacute;chet derivatives that require fourth-order regularizations. Therein, a second-order time derivative on the phase-field variable appears, which is reformulated as a mixed first-order-in-time system. These derivations are carefully established for all four equations. Finally, the corresponding time-stepping schemes are derived by employing a dG(r) discretization in time.

Automated summary

This work investigates the space-time formulations and Galerkin discretizations for phase-field fracture optimal control problems. The fracture irreversibility constraint is regularized on the time-continuous level using penalization. The optimization scheme is solved using a Newton method and the state, adjoint, tangent, and adjoint Hessian equations are derived. The key focus is on designing appropriate function spaces and rigorously justifying all Fréchet derivatives.