Space-Time Mixed System Formulation of Phase-Field Fracture Optimal Control Problems
研究了相场断裂最优控制问题的时空公式和Galerkin离散化，通过罚函数处理断裂不可逆约束，推导了状态、伴随、切线和伴随Hessian方程，并设计了合适的函数空间和Fréchet导数。
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( $$r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>r</mml:mi> </mml:math> ) discretization in time.