TY - JOUR

T1 - An optimal control problem governed by a regularized phase-field fracture propagation model. part II

T2 - The regularization limit

AU - Neitzel, Ira

AU - Wick, Thomas

AU - Wollner, Winnifried

N1 - Funding Information: ∗Received by the editors October 30, 2018; accepted for publication (in revised form) April 1, 2019; published electronically May 16, 2019. http://www.siam.org/journals/sicon/57-3/M122385.html Funding: The authors received financial support from the DFG SPP 1962 through grants NE1941/1-1 and WO1936/4-1, as well as the Austrian Science Fund (FWF) under grant P 29181, Goal-Oriented Error Control for Phase-Field Fracture Coupled to Multiphysics Problems. †Institut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 19b, 53115 Bonn, Germany (neitzel@ins.uni-bonn.de). ‡Leibniz Universität Hannover, Institut für Angewandte Mathematik, Welfengarten 1, 30167 Hannover, Germany (thomas.wick@ifam.uni-hannover.de). §Fachbereich Mathematik, Technische Universität Darmstadt, Dolivostr. 15, 64293 Darmstadt, Germany (wollner@mathematik.tu-darmstadt.de). Funding Information: The authors received financial support from the DFG SPP 1962 through grants NE1941/1-1 and WO1936/4-1, as well as the Austrian Science Fund (FWF) under grant P 29181, Goal-Oriented Error Control for Phase-Field Fracture Coupled to Multiphysics Problems. Publisher Copyright: Copyright © by SIAM. Copyright: Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2019/1

Y1 - 2019/1

N2 - We consider an optimal control problem of tracking type governed by a time-discrete regularized phase-field fracture or damage propagation model. The energy minimization problem describing the fracture process is formulated by the corresponding Euler-Lagrange equations that contain a regularization term that penalizes the violation of the irreversibility condition in the evolution of the fracture. We prove convergence of solutions of the regularized problem when taking the limit with respect to the penalty term and obtain an estimate for the constraint violation in terms of the penalty parameter. To this end, we make use of convexity of the energy functional due to a viscous regularization which corresponds to a time-step restriction in the temporal discretization of the problem. Numerical experiments underline our theoretical findings.

AB - We consider an optimal control problem of tracking type governed by a time-discrete regularized phase-field fracture or damage propagation model. The energy minimization problem describing the fracture process is formulated by the corresponding Euler-Lagrange equations that contain a regularization term that penalizes the violation of the irreversibility condition in the evolution of the fracture. We prove convergence of solutions of the regularized problem when taking the limit with respect to the penalty term and obtain an estimate for the constraint violation in terms of the penalty parameter. To this end, we make use of convexity of the energy functional due to a viscous regularization which corresponds to a time-step restriction in the temporal discretization of the problem. Numerical experiments underline our theoretical findings.

KW - Optimal control

KW - Phase-field

KW - Regularization limit

KW - Regularized fracture model

UR - http://www.scopus.com/inward/record.url?scp=85070688194&partnerID=8YFLogxK

U2 - 10.1137/18m122385x

DO - 10.1137/18m122385x

M3 - Article

AN - SCOPUS:85070688194

VL - 57

SP - 1672

EP - 1690

JO - SIAM Journal on Control and Optimization

JF - SIAM Journal on Control and Optimization

SN - 0363-0129

IS - 3

ER -