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 Fréchet 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($$r$$
r
) discretization in time.
The purpose of this work are computational demonstations for a newly developed space-time phase-field fracture optimal control framework. The optimal control solution algorithm is a Newton algorithm, which is obtained with the reduced approach by eliminating the state constraint. Due to the crack irreversibility constraint, a rate-independent problem arises, which is treated by penalization and for which we utilize a space-time approach. Therein, we deal with the state, adjoint, tangent, and adjoint Hessian equations. Our fully discretized space-time optimization algorithm is presented and extensively tested with various numerical experiments.
In this work, we undertake additional computational performance studies of a recently developed space-time phase-field fracture optimal control framework. Therein, the phase-field forward problem is formulated in a monolithic fashion. The optimal control problem is formulated with the help of the reduced approach in which the state variable is represented with a solution operator applied to the control. To this end, a Newton algorithm in the control variable is formulated for which auxiliary equations must be solved. Two numerical experiments demonstrate the capabilities of our framework.
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