Optimizing Fracture Propagation Using a Phase-Field Approach
Within this proposal, 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-self-penetration 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 analyze the resulting MPCC with respect to it's necessary and sufficient optimality conditions by means of a regularization of the lower-level problem and passage to the limit with respect to the regularization parameter. Moreover, we will consider SQP-type algorithms for the solution of this MPCC in function space and investigate its properties. Additionally, we will consider the discretization by finite elements and show the convergence of the discrete approximations to the continuous limit.
The simultaneous consideration of the inexactness due to discretization and regularization error will allow us to construct and analyze an efficient inexact SQP-type solver for the MPCC under consideration.
Robert Haller-Dintelmann, Hannes Meinlschmidt, Winnifried Wollner: Higher Regularity for Solutions to Elliptic Systems in Divergence Form Subject to Mixed Boundary Conditions, Annali di Matematica Pura ed Applicata, 2018 (SPP1962-047).
Bernhard Endtmayer, Ulrich Langer, Ira Neitzel, Thomas Wick, Winnifried Wollner: Multigoal-oriented Optimal Control Problems with Nonlinear PDE Constraints (SPP1962-108, 03/2019, [bib])
Ira Neitzel, Thomas Wick, Winnifried Wollner: An Optimal Control Problem Governed by a Regularized Phase-field Fracture Propagation Model. Part II The Regularization Limit (SPP1962-091, 10/2018, [bib])
Masoumeh Mohammadi, Winnifried Wollner: A Priori Error Estimates for a Linearized Fracture Control Problem (SPP1962-090, 10/2018, [bib])
Veronika Karl, Ira Neitzel, Daniel Wachsmuth: A Lagrange Multiplier Method for Semilinear Elliptic State Constrained Optimal Control Problems (SPP1962-087, 10/2018, [bib])
Robert Haller-Dintelmann, Hannes Meinlschmidt, Winnifried Wollner: Higher Regularity for Solutions to Elliptic Systems in Divergence Form Subject to Mixed Boundary Conditions (SPP1962-047, 01/2018, [bib])
Modeling, problem analysis, algorithm design and convergence analysisThe focus of this area is on the development and analysis of genuinely non-smooth models in the sciences in order to properly capture real-world effects and to avoid comprising smoothing approaches. In simulation and optimization this requires to advance set-valued analysis and the design of robust algorithms for non-smooth problems.
Realization of algorithms, adaptive discretization and model reductionAs the target applications of this SPP involve non-smooth structures and partial differential operators, the discretization of the associated problems and robust error estimation are important issues to be address, and proper model-reduction techniques need to be developed.
Project Related News
Mar 06, 2019 : New preprint submitted
Winnifried Wollner submitted the preprint SPP1962-108, Multigoal-oriented Optimal Control Problems with Nonlinear PDE Constraints
Oct 26, 2018 : New preprint submitted
Winnifried Wollner submitted the preprint SPP1962-091, An Optimal Control Problem Governed by a Regularized Phase-field Fracture Propagation Model. Part II The Regularization Limit
Oct 25, 2018 : New preprint submitted
Winnifried Wollner submitted the preprint SPP1962-090, A Priori Error Estimates for a Linearized Fracture Control Problem
Dec 01, 2017 : Welcome to our new project member
Andreas Hehl joins Project 17