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

Denis Khimin (), Marc Christian Steinbach () and Thomas Wick ()
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Denis Khimin: Leibniz Universität Hannover
Marc Christian Steinbach: Leibniz Universität Hannover
Thomas Wick: Leibniz Universität Hannover

Journal of Optimization Theory and Applications, 2023, vol. 199, issue 3, No 14, 1222-1248

Abstract: Abstract 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.

Keywords: Phase-field fracture propagation; Optimal control; Reduced optimization approach; Mixed-in-time system; Penalization; 74R10; 65N30; 49M15; 49K20; 35Q74 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10957-023-02272-7

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