A priori error estimates for a linearized fracture control problem
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A priori error estimates for a linearized fracture control problem Masoumeh Mohammadi1 · Winnifried Wollner1 Received: 30 October 2019 / Revised: 10 August 2020 / Accepted: 14 October 2020 © The Author(s) 2020
Abstract A control problem for a linearized time-discrete regularized fracture propagation process is considered. The discretization of the problem is done using a conforming finite element method. In contrast to many works on discretization of PDE constrained optimization problems, the particular setting has to cope with the fact that the linearized fracture equation is not necessarily coercive. A quasi-best approximation result will be shown in the case of an invertible, though not necessarily coercive, linearized fracture equation. Based on this a priori error estimates for the control, state, and adjoint variables will be derived. Keywords Optimal control · Linearized fracture model · Finite element method · A priori error estimate Mathematics Subject Classification 65N12 · 65N15 · 65N30 · 49M25 · 74S05
1 Introduction Modeling, predicting, and control of fracture or damage in solid materials are of great technical importance for the safety requirements of structures in various fields of engineering, e.g., automobile components, aerospace, and marine industries. Therefore, developing a comprehensive model of fracture propagation has long been a challenge in physics, mechanics, and material sciences (Lawn 1993; Marder and Fineberg 1996). The classical method for modeling the fracture propagation is to This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 314067056. * Winnifried Wollner [email protected]‑darmstadt.de Masoumeh Mohammadi [email protected]‑darmstadt.de 1
TU-Darmstadt, Fachbereich Mathematik, 64283 Darmstadt, Germany
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consider a sharp interface in order to separate the structure explicitly into a fully broken part and a fully intact one. This approach implies tracking the exact position of the interface to be able to follow the propagation of the fracture. Therefore, in finite element settings for fracture description, the numerical implementation requires handling of the discontinuities. To overcome the problem of explicit interface tracking, the phase-field method, going back to Ambrosio and Tortorelli (1990), is now widely used for the description of fracture phenomena. This method is also attractive because of the ability of simulating the fracture initiation, propagation, merging, and branching. The phase-field approach to model the fracture, as a twophase discontinuous model with a sharp interface, consists of introducing a continuous phase-field variable in order to approximate the sharp fracture discontinuity. The phase-field variable smoothly differentiates between the two phases. In fact, the fracture phase-field represents the smooth transition from the fully destroyed phase to the fully intact part. The fracture propagation is tracked by the evolution of the phase field. In
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