SPP1748

DFG Priority Programme 1748: Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretization Methods, Mechanical and Mathematical Analysis

Reliability of efficient approximation schemes for material discontinuities described by functions of bounded variation

Project heads
Marita Thomas (WIAS), Sören Bartels (Uni Freiburg)
Staff
Jakob Keck (Uni Freiburg), Sven Tornquist (WIAS)
This project is financially supported by the German Research Foundation (DFG) within the priority programme "Reliable Simulation Techniques in Solid Mechanics" (SPP1748). The funding period is from July 2018 until December 2020.

Background

Spaces of functions of bounded variations provide an attractive framework to describe material discontinuities such as damage and fracture. In the recent past, suitable notions of solution and general existence theories for corresponding evolutionary model problems have been established and numerical methods for discretizing and iteratively solving variational problems involving the total variation norm have been developed and analyzed.

Preliminary work

In a first funding phase, abstract existence results for coupled rate-dependent/rate-independent systems, delamination processes in visco-elastodynamics, and phasefield descriptions of damage evolutions have been investigated analytically.

Algorithmic contributions have been made to the iterative solution of model problems on functions of bounded variation, adaptive approximation of discontinuous functions based on fully computable a posteriori error estimates, and the convergent finite element simulation of a BV-regularized damage model.

Objectives

The aim is now to combine, refine and extend these results to complex models describing damage and fracture evolution. The envisaged models shall capture material discontinuities either in BV directly or as a scaling limit.

It is planned to derive a priori and a posteriori error estimates, construct adaptive approximation schemes intertwined with results based on existence theory and Gamma-convergence, and to implement and apply them to specific benchmark problems in mechanics. Particular attention will be paid to the reliability of efficient methods, e.g., convergence of suitable time-stepping and adaptive approximation schemes, intertwined with results based on existence theory and convergence.

Guests and collaborations

Project-related events

  • November 30 - December 1, 2020: >>cancelled due to COVID-19<< Workshop on Material Discontinuities, organized by S. Bartels and M. Thomas at U Freiburg.
  • May 25-26, 2020: >>cancelled due to COVID-19<< Workshop on Material Discontinuities, organized by S. Bartels and M. Thomas at U Freiburg.
  • March 16-20, 2020: >>cancelled due to COVID-19<< Section S14 Applied Analysis, organized by D. Knees (Kassel) and M. Thomas at the 91st Annual Meeting of the GAMM, Kassel, Germany.
  • September 24-28, 2018: ALEX 2018: Analysis of evolutionary and complex systems, organized by M. Liero, S. Reichelt (both WIAS), G. Schneider (Stuttgart), F. Theil (Warwick), and M. Thomas at WIAS and HU Berlin, Germany.
  • July 05-09, 2018: Special Session 144 Analytic properties and numerical approximation of differential models arising in applications, organized by: E. Bonetti, C. Cavaterra (both U Milano), E. Rocca (U Pavia), and M. Thomas, at the 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Taipei, Taiwan.

Project-related publications

  • M. Thomas, S. Tornquist, Discrete approximation of dynamic phase-field fracture in visco-elastic materials, Discrete and Continuous Dynamical Systems -- Series S, 14 (2021), pp. 3865--3924 (published online on 29.06.2021), DOI 10.3934/dcdss.2021067 .
  • S. Bartels, M. Milicevic, M. Thomas, S. Tornquist, N. Weber, Approximation schemes for materials with discontinuities, Preprint no. 2799, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2799 . (PDF)
  • S. Bartels, M. Milicevic, M. Thomas, N. Weber, Fully discrete approximation of rate-independent damage models with gradient regularization, Preprint no. 2707, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2707 . (PDF)
  • R. Rossi, U. Stefanelli, M. Thomas, Rate-independent evolution of sets, Preprint no. 2578, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2578 . (PDF)
  • G. Lazzaroni, R. Rossi, M. Thomas, R. Toader, Rate-independent damage in thermo-viscoelastic materials with inertia, Journal of Dynamics and Differential Equations, 30 (2018), pp. 1311--1364, DOI 10.1007/s10884-018-9666-y .
  • S. Bartels, M. Milicevic, M. Thomas, Numerical approach to a model for quasistatic damage with spatial $BV$-regularization, in: Proceedings of the INdAM-ISIMM Workshop on Trends on Applications of Mathematics to Mechanics, Rome, Italy, September 2016, E. Rocca, U. Stefanelli, L. Truskinovsky, eds., 27 of Springer INdAM Series, Springer International Publishing, Cham, 2018, pp. 179--203, DOI 10.1007/978-3-319-75940-1_9 .
  • R. Rossi, M. Thomas, From nonlinear to linear elasticity in a coupled rate-dependent/independent system for brittle delamination, in: Proceedings of the INdAM-ISIMM Workshop on Trends on Applications of Mathematics to Mechanics, Rome, Italy, September 2016, E. Rocca, U. Stefanelli, L. Truskinovsky, A. Visintin, eds., 27 of Springer INdAM Series, Springer International Publishing, Cham, 2018, pp. 127--157, DOI 10.1007/978-3-319-75940-1_7 .
  • M. Thomas, C. Bilgen, K. Weinberg, Phase-field fracture at finite strains based on modified invariants: A note on its analysis and simulations, GAMM-Mitteilungen, 40 (2018), pp. 207--237, DOI 10.1002/gamm.201730004 .
  • M. Thomas, A comparison of delamination models: Modeling, properties, and applications, in: Mathematical Analysis of Continuum Mechanics and Industrial Applications II, Proceedings of the International Conference CoMFoS16, P. VAN Meurs, M. Kimura, H. Notsu, eds., 30 of Mathematics for Industry, Springer Nature, Singapore, 2018, pp. 27--38, DOI 10.1007/978-981-10-6283-4_3 .

Talks and posters

  • M. Thomas, Coupling of rate-independent and rate-dependent systems with application to delamination processes in solids, Mathematics for Mechanics, October 29 - November 1, 2019, Czech Academy of Sciences, Prague, Czech Republic, October 31, 2019.
  • M. Thomas, Coupling of rate-independent and rate-dependent systems with application to delamination processes in solids, Seminar ``Applied and Computational Analysis'', University of Cambridge, UK, October 10, 2019.
  • M. Thomas, Analytical and numerical aspects of rate-independent gradient-regularized damage models, ``Dynamics, Equations and Applications'' (DEA), September 16 - 20, 2019, AGH University of Science and Technology, Kraków, Poland, September 19, 2019.
  • M. Thomas, ``Analytical and numerical aspects for the approximation of gradient-regularized damage models'', Thematic Minisymposium MS A3-2-26 ``Phase-Field Models in Simulation and Optimization'', ``9th International Congress on Industrial and Applied Mathematics'' (ICIAM 2019), July 15 - 19, 2019, Universitat de València, Spain, July 17, 2019.
  • M. Thomas, Coupling of rate-independent and rate-dependent systems, Lecture Series at MURPHYS-HSFS 2019 Summer School, June 17 - 19, 2019, Politecnico di Torino, Turin, Italy.

  • S. Tornquist, Towards the analysis of dynamic phase-field fracture, MURPHYS-HSFS 2019 Summer School, June 17 - 21, 2019, Politecnico di Torino, Turin, Italy.

  • S. Tornquist, Towards the analysis of dynamic phase-field fracture, Spring School on Variational Analysis 2019, Paseky nad Jizerou, Czech Republic, May 19 - 25, 2019.
  • M. Thomas, Analysis for the discrete approximation of gradient-regularized damage models, PDE Afternoon, Universität Wien, Austria, April 10, 2019.
  • M. Thomas, Analysis for the discrete approximation of gradient-regularized damage models, Mathematics Seminar Brescia, March 11 - 14, 2019, Università degli Studi di Brescia, Italy, March 13, 2019.
  • M. Thomas, Rate-independent evolution of sets and application to fracture processes, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S14 ``Applied Analysis", February 18 - 22, 2019, Technische Universität Wien, Austria, February 20, 2019.
  • S. Tornquist, Variational problems involving Caccioppoli partitions, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S14 ``Applied Analysis", February 18 - 22, 2019, Technische Universität Wien, Austria, February 19, 2019.
  • M. Thomas, Analysis for the discrete approximation of gradient-regularized damage models, Oberseminar Analysis und Angewandte Mathematik, November 30 - December 3, 2018, Universität Kassel, Institut für Mathematik, December 3, 2018.
  • M. Thomas, Analytical and numerical aspects of damage models, Berlin Dresden Prague Würzburg Workshop ``Mathematics of Continuum Mechanics'', November 29 - 30, 2018, Universität Würzburg, Institut für Mathematik, November 30, 2018.
  • S. Tornquist, Towards the analysis of dynamic phase-field fracture, Berlin Dresden Prague Würzburg Workshop ``Mathematics of Continuum Mechanics'', November 29 - 30, 2018.
  • M. Thomas, Analysis for the discrete approximation of gradient-regularized damage models, Workshop ``Women in Mathematical Materials Science'', November 5 - 6, 2018, Universität Regensburg, Fakultät für Mathematik, November 6, 2018.
  • M. Thomas, Analytical and numerical approach to a class of damage models, The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 75 ``Mathematics and Materials: Models and Applications'', July 5 - 9, 2018, National Taiwan University, Taipeh, Taiwan, Province Of China, July 6, 2018.
  • M. Thomas, Analysis for the discrete approximation of damage and fracture, Applied Analysis Day, June 28 - 29, 2018, Technische Universität Dresden, Chair of Partial Differntial Equations, Germany, June 29, 2018.
  • M. Thomas, Rate-independent evolution of sets & applications to damage and delamination, PDEs Friends, June 21 - 22, 2018, Politecnico di Torino, Dipartimento di Scienze Matematiche ``Giuseppe Luigi Lagrange'', Italy, June 22, 2018.
  • M. Thomas, Phase-field fracture at finite strains based on modified invariants, Special Materials and Complex Systems (SMACS 2018), June 17 - 22, 2018, University of Milan, Department of Mathematics, Gargnano, Italy, June 18, 2018.
  • M. Thomas, Analysis and simulation for a phase-field fracture model at finite strains based on modified invariants, Workshop ``Special Materials and Complex Systems'' (SMACS 2018), June 18 - 22, 2018, University of Milan/University of Pavia, Gargnano, Italy, June 18, 2018.
  • M. Thomas, Analysis and simulation for a phase-field fracture model at finite strains based on modified invariants, Analysis Seminar, University of Brescia, Department of Mathematics, Italy, May 10, 2018.
  • M. Thomas, Analysis and simulation for a phase-field fracture model at finite strains based on modified invariants, 89th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2018), Section DFG Priority Programmes PP1748 ``Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretization Methods, Mechanical and Mathematical Analysis'', March 19 - 23, 2018, Technische Universität München, March 20, 2018.