Staff Andreas Hehl, M.Sc.

Address:
Institut für Numerische Simulation
Friedrich-Hirzebruch-Allee 7
53115 Bonn
Phone: +49 228 73-69746
Office: FHA7 1.040
E-Mail: ed tod nnob-inu tod sni ta lheha tod b@foo tod de

Research Projects

Current

Optimizing Fracture Propagation Using a Phase-Field Approach, Part II

Project SPP 1962 Phase 2, DFG priority program 1962, Phase II.

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Participants

Institut für Numerische Simulation der Rheinischen Friedrich-Wilhelms-Universität Bonn: Prof. Dr. Ira Neitzel

Institut für angewandte Mathematik der Leibniz Universtität Hannover: Prof. Dr. Thomas Wick

Fachbereich Mathematik der Technischen Universtität Darmstadt: Prof. Dr. Winnifried Wollner

Description

Within this project, we consider the numerical approximation and solution of control problems governed by a quasi-static brittle fracture propagation model. As a central modeling component, a phase-field formulation for the fracture formation and propagation is considered.

The fracture propagation problem itself can be formulated as a minimization problem with inequality constraints, imposed by multiple relevant side conditions, such as irreversibility of the fracture-growth or non-selfpenetration of the material across the fracture surface. These lead to variational inequalities as first order necessary conditions. Consequently, optimization problems for the control of the fracture process give rise to a mathematical program with complementarity constraints (MPCC) in function spaces.

Within this project, we intend to focus on mathematical challenges, that are also motivated by applications, such as control of the coefficients of the variational inequality, or nonsmooth and/or nonconvex cost functionals in the outer optimization, such as, e.g., maximizing the released energy of the fracture. We will develop first and second order optimality conditions for the resulting MPCC as well as other obstacle-like formulations. Additionally, we will consider the discretization by finite elements and show the convergence of the discrete approximations to the continuous limit. These findings will be substantiated with prototype numerical tests.

Cooperation

DFG Priority Programme SPP 1962, “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization”

FWF-Project P29181, “Goal-Oriented Error Control for Phase-Field Fracture Coupled to Multiphysics Problems”

Completed

Optimizing Fracture Propagation Using a Phase-Field Approach

Project SPP 1962, DFG priority program 1962.

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Publications

  1. Second Order Optimality Conditions for an Optimal Control Problem Governed by a Regularized Phase-Field Fracture Propagation Model. A. Hehl and I. Neitzel. Optimization, 2022. BibTeX DOI
  2. Optimizing fracture propagation using a phase-field approach. A. Hehl, M. Mohammadi, I. Neitzel, and W. Wollner. In to appear. 2020. BibTeX