Details
| Original language | English |
|---|---|
| Pages (from-to) | 1222–1248 |
| Number of pages | 27 |
| Journal | Journal of Optimization Theory and Applications |
| Volume | 199 |
| Issue number | 3 |
| Early online date | 27 Jul 2023 |
| Publication status | Published - Dec 2023 |
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) discretization in time.
Keywords
- Mixed-in-time system, Optimal control, Penalization, Phase-field fracture propagation, Reduced optimization approach
ASJC Scopus subject areas
- Mathematics(all)
- Control and Optimization
- Decision Sciences(all)
- Management Science and Operations Research
- Mathematics(all)
- Applied Mathematics
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In: Journal of Optimization Theory and Applications, Vol. 199, No. 3, 12.2023, p. 1222–1248.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Space-Time Mixed System Formulation of Phase-Field Fracture Optimal Control Problems
AU - Khimin, Denis
AU - Steinbach, Marc Christian
AU - Wick, Thomas
N1 - Funding Information: Submitted to the editors Jan 25, 2023. 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 - 2023/12
Y1 - 2023/12
N2 - 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) discretization in time.
AB - 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) discretization in time.
KW - Mixed-in-time system
KW - Optimal control
KW - Penalization
KW - Phase-field fracture propagation
KW - Reduced optimization approach
UR - http://www.scopus.com/inward/record.url?scp=85165986595&partnerID=8YFLogxK
U2 - 10.1007/s10957-023-02272-7
DO - 10.1007/s10957-023-02272-7
M3 - Article
AN - SCOPUS:85165986595
VL - 199
SP - 1222
EP - 1248
JO - Journal of Optimization Theory and Applications
JF - Journal of Optimization Theory and Applications
SN - 0022-3239
IS - 3
ER -