TY - JOUR

T1 - Space-time formulation, discretization, and computational performance studies for phase-field fracture optimal control problems

AU - Khimin, D.

AU - Steinbach, M. C.

AU - Wick, T.

N1 - Funding Information: The first and third author are partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Priority Program 1962 (DFG SPP 1962) within the subproject Optimizing Fracture Propagation using a Phase-Field Approach with the project number 314067056 . The second author is funded by the DFG – SFB1463 – 434502799 .

PY - 2022/12/1

Y1 - 2022/12/1

N2 - The purpose of this work is the development of space-time discretization schemes for phase-field optimal control problems. Specifically in the optimal control minimization problem, a tracking-type cost functional is minimized to steer the crack via the phase-field variable into a desired pattern. To achieve such optimal solutions, Neumann type boundary conditions need to be determined. First, a time discretization of the forward problem is derived using a discontinuous Galerkin formulation. Here, a challenge is to include regularization terms and 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, namely the displacement and phase-field unknowns, but keeping the control variable as the only unknown. From the low-order discontinuous Galerkin discretization, adjoint time-stepping schemes are finally obtained. Both our formulation and algorithmic developments are substantiated and illustrated with six numerical experiments.

AB - The purpose of this work is the development of space-time discretization schemes for phase-field optimal control problems. Specifically in the optimal control minimization problem, a tracking-type cost functional is minimized to steer the crack via the phase-field variable into a desired pattern. To achieve such optimal solutions, Neumann type boundary conditions need to be determined. First, a time discretization of the forward problem is derived using a discontinuous Galerkin formulation. Here, a challenge is to include regularization terms and 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, namely the displacement and phase-field unknowns, but keeping the control variable as the only unknown. From the low-order discontinuous Galerkin discretization, adjoint time-stepping schemes are finally obtained. Both our formulation and algorithmic developments are substantiated and illustrated with six numerical experiments.

KW - Galerkin discretization

KW - Optimal control

KW - Phase-field fracture

KW - Reduced optimization approach

KW - Space-time

UR - http://www.scopus.com/inward/record.url?scp=85137165048&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2203.14643

DO - 10.48550/arXiv.2203.14643

M3 - Article

AN - SCOPUS:85137165048

VL - 470

JO - Journal of computational physics

JF - Journal of computational physics

SN - 0021-9991

M1 - 111554

ER -