SPP1748

DFG SPP 1748 - Zuverlässige Simulationstechniken in der Festkörpermechanik - Entwicklung nicht konventioneller Diskretisierungsverfahren, mechanische und mathematische Analyse

Zuverlässigkeit effizienter Approximationen von Funktionen beschränkter Variation in der Beschreibung von Materialunstetigkeiten

Projektleitung
Marita Thomas (WIAS), Sören Bartels (Uni Freiburg)
Mitarbeiter
Jakob Keck (Uni Freiburg), Sven Tornquist (WIAS)
Dieses Projekt wird von der Deutschen Forschungsgemeinschaft (DFG) innerhalb des Schwerpunktprogramms "DFG SPP 1748 - Zuverlässige Simulationstechniken in der Festkörpermechanik" finanziell unterstützt. Der Förderungszeitraum läuft von Juli 2018 bis Dezember 2020.

Hintergrund

Funktionen beschränkter Variation bieten einen attraktiven Rahmen, um Materialunstetigkeiten wie Schädigung oder Risse zu beschreiben. In der jüngsten Vergangenheit wurden geeignete Lösungsformulierungen und Existenztheorien für entsprechende Evolutionsmodelle begründet. Dazu wurden numerische Methoden für die Diskretisierung und das iterative Lösen variationeller Probleme, die die Totalvariationsnorm beinhalten, entwickelt und analysiert.

Projektbezogene Vorarbeit

In einer ersten Förderungs­phase wurden abstrakte Existenz­resultate, gekoppelte raten-abhängige/raten-unabhängige Systeme, Delaminierungs­prozesse in Visko­elastodynamik, und Phasenfeld-Beschreibungen von Schädigungs­evolutionen analytisch untersucht.

Algorithmische Beiträge zur iterativen Lösung von Modellproblemen von Funktionen beschränkter Variation, zur adaptiven Approximation von unstetigen Funktionen, basierend auf berechenbaren a posteriori Fehler­abschätzungen, und für die konvergente finite Elemente Simulation eines BV-regularisierten Schädigungsmodells wurden geleistet.

Ziele

Das Vorhaben ist, diese Ergebnisse zu kombinieren, zu verfeinern und auf komplexe Schadens- und Rissmodelle zu erweitern. Die anvisierten Modelle sollen Material­unstetigkeiten entweder direkt in BV oder als Skalierungs­grenzwert erfassen.

Geplant ist die Herleitung von a priori und a posteriori Fehler­abschätzungen, und die Entwicklung adaptiver Approximations­schemata unter Berücksichtigung der Existenztheorie und Gamma-Konvergenz. Diese sollen implementiert und auf spezifische Benchmark-Probleme in der Mechanik angewendet werden. Besonderes Augenmerk wird dabei auf die Zuverlässigkeit von effizienten Methoden gelegt, z.B. die Konvergenz von geeigneten Zeit­diskretisierungs­verfahren und adaptiven Approximations­schemata unter Ausnutzung der Existenztheorie und Gamma-Konvergenz.

Gäste und Kollaborationen

Projektbezogene Ereignisse

  • 30. November - 1. Dezember 2020: >>cancelled due to COVID-19<< Workshop on Material Discontinuities, organized by S. Bartels and M. Thomas at U Freiburg.
  • 25.-26. Mai 2020: >>cancelled due to COVID-19<< Workshop on Material Discontinuities, organized by S. Bartels and M. Thomas at U Freiburg.
  • 16.-20. März 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.
  • 24.-28. September 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.
  • 05.-09. Juli 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.

Projektbezogene Veröffentlichungen

  • 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 .

Vorträge und Poster

  • 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.