• M. Liero, S. Reichelt, G. Schneider, F. Theil, M. Thomas, eds., Analysis of Evolutionary and Complex Systems: Issue on the Occasion of Alexander Mielke's 60th Birthday, 14 of Discrete and Continuous Dynamical Systems -- Series S, American Institute of Mathematical Sciences, Springfield, 2021, 453 pages, (Collection Published).

Artikel in Referierten Journalen

  • E. Ipocoana, A. Zafferi, Further regularity and uniqueness results for a non-isothermal Cahn--Hilliard equation, Communications on Pure and Applied Analysis, 20 (2021), pp. 763--782, DOI 10.3934/cpaa.2020289 .
    The aim of this paper is to establish new regularity results for a non-isothermal Cahn--Hilliard system in the two-dimensional setting. The main achievement is a crucial L estimate for the temperature, obtained by a suitable Moser iteration scheme. Our results in particular allow us to get a new simplified version of the uniqueness proof for the considered model.

  • M.H. Farshbaf Shaker, M. Gugat, H. Heitsch, R. Henrion, Optimal Neumann boundary control of a vibrating string with uncertain initial data and probabilistic terminal constraints, SIAM Journal on Control and Optimization, 58 (2020), pp. 2288--2311, DOI 10.1137/19M1269944 .
    In optimal control problems, often initial data are required that are not known exactly in practice. In order to take into account this uncertainty, we consider optimal control problems for a system with an uncertain initial state. A finite terminal time is given. On account of the uncertainty of the initial state, it is not possible to prescribe an exact terminal state. Instead, we are looking for controls that steer the system into a given neighborhood of the desired terminal state with sufficiently high probability. This neighborhood is described in terms of an inequality for the terminal energy. The probabilistic constraint in the considered optimal control problem leads to optimal controls that are robust against the inevitable uncertainties of the initial state. We show the existence of such optimal controls. Numerical examples with optimal Neumann control of the wave equation are presented.

  • R. Rossi, U. Stefanelli, M. Thomas, Rate-independent evolution of sets, Discrete and Continuous Dynamical Systems -- Series S, 14 (2021), pp. 89--119 (published online in March 2020), DOI 10.3934/dcdss.2020304 .
    The goal of this work is to analyze a model for the rate-independent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of a given time-dependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of volume changes. In the mathematical modeling of this process, we distinguish the adhesive case, in which the constraint that the (complement of) the `external load' contains the evolving sets is penalized by a term contributing to the driving energy functional, from the brittle case, enforcing this constraint. The existence of Energetic solutions for the adhesive system is proved by passing to the limit in the associated time-incremental minimization scheme. In the brittle case, this time-discretization procedure gives rise to evolving sets satisfying the stability condition, but it remains an open problem to additionally deduce energy-dissipation balance in the time-continuous limit. This can be obtained under some suitable quantification of data. The properties of the brittle evolution law are illustrated by numerical examples in two space dimensions.

  • P. Colli, M.H. Farshbaf Shaker, K. Shirakawa, N. Yamazaki, Optimal control for shape memory alloys of the one-dimensional Frémond model, Numerical Functional Analysis and Optimization. An International Journal, 41 (2020), pp. 1421-1471, DOI 10.1080/01630563.2020.1774892 .
    In this paper, we consider optimal control problems for the one-dimensional Frémond model for shape memory alloys. This model is constructed in terms of basic functionals like free energy and pseudo-potential of dissipation. The state problem is expressed by a system of partial differential equations involving the balance equations for energy and momentum. We prove the existence of an optimal control that minimizes the cost functional for a nonlinear and nonsmooth state problem. Moreover, we show the necessary condition of the optimal pair by using optimal control problems for approximating systems.

  • M. Thomas, C. Bilgen, K. Weinberg, Analysis and simulations for a phase-field fracture model at finite strains based on modified invariants, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 100 (2020), pp. e201900288/1--e201900288/51, DOI 10.1002/zamm.201900288 .
    Phase-field models have already been proven to predict complex fracture patterns in two and three dimensions for brittle fracture at small strains. In this paper we discuss a model for phase-field fracture at finite deformations in more detail. Among the identification of crack location and projection of crack growth the numerical stability is one of the main challenges in solid mechanics. We here present a phase-field model at finite strains, which takes into account the anisotropy of damage by applying an anisotropic split and the modified invariants of the right Cauchy-Green strain tensor. We introduce a suitable weak notion of solution that also allows for a spatial and temporal discretization of the model. In this framework we study the existence of solutions and we show that the time-discrete solutions converge in a weak sense to a solution of the time-continuous formulation of the model. Numerical examples in two and three space dimensions are carried out in the range of validity of the analytical results.

Beiträge zu Sammelwerken

  • R. Rossi, U. Stefanelli, M. Thomas, Rate-independent evolution of sets, in: Analysis of Evolutionary and Complex Systems: Issue on the Occasion of Alexander Mielke's 60th Birthday, M. Liero, S. Reichelt, G. Schneider, F. Theil, M. Thomas, eds., 14 of Discrete and Continuous Dynamical Systems -- Series S, American Institute of Mathematical Sciences, Springfield, 2021, pp. 89--119, DOI 10.3934/dcdss.2020304 .

Preprints, Reports, Technical Reports

  • 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 .
    Abstract, PDF (891 kByte)
    Damage and fracture phenomena are related to the evolution of discontinuities both in space and in time. This contribution deals with methods from mathematical and numerical analysis to handle these: Suitable mathematical formulations and time-discrete schemes for problems with discontinuities in time are presented. For the treatment of problems with discontinuities in space, the focus lies on FE-methods for minimization problems in the space of functions of bounded variation. The developed methods are used to introduce fully discrete schemes for a rate-independent damage model and for the viscous approximation of a model for dynamic phase-field fracture. Convergence of the schemes is discussed.

  • M. Thomas, S. Tornquist, Discrete approximation of dynamic phase-field fracture in visco-elastic materials, Preprint no. 2798, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2798 .
    Abstract, PDF (646 kByte)
    This contribution deals with the analysis of models for phase-field fracture in visco-elastic materials with dynamic effects. The evolution of damage is handled in two different ways: As a viscous evolution with a quadratic dissipation potential and as a rate-independent law with a positively 1-homogeneous dissipation potential. Both evolution laws encode a non-smooth constraint that ensures the unidirectionality of damage, so that the material cannot heal. Suitable notions of solutions are introduced in both settings. Existence of solutions is obtained using a discrete approximation scheme both in space and time. Based on the convexity properties of the energy functional and on the regularity of the displacements thanks to their viscous evolution, also improved regularity results with respect to time are obtained for the internal variable: It is shown that the damage variable is continuous in time with values in the state space that guarantees finite values of the energy functional.

  • M.H. Farshbaf Shaker, M. Thomas, Analysis of a compressible Stokes-flow with degenerating and singular viscosity, Preprint no. 2786, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2786 .
    Abstract, PDF (730 kByte)
    In this paper we show the existence of a weak solution for a compressible single-phase Stokes flow with mass transport accounting for the degeneracy and the singular behavior of a density-dependent viscosity. The analysis is based on an implicit time-discrete scheme and a Galerkin-approximation in space. Convergence of the discrete solutions is obtained thanks to a diffusive regularization of p-Laplacian type in the transport equation that allows for refined compactness arguments on subdomains.

  • D. Peschka, M. Rosenau, Two-phase flows for sedimentation of suspensions, Preprint no. 2743, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2743 .
    Abstract, PDF (12 MByte)
    We present a two-phase flow model that arises from energetic-variational arguments and study its implication for the sedimentation of buoyant particles in a viscous fluid inside a Hele--Shaw cell and also compare corresponding simulation results to experiments. Based on a minimal dissipation argument, we provide a simplified 1D model applicable to sedimentation and study its properties and the numerical discretization. We also explore different aspects of its numerical discretization in 2D. The focus is on different possible stabilization techniques and their impact on the qualitative behavior of solutions. We use experimental data to verify some first qualitative model predictions and discuss these experiments for different stages of batch sedimentation.

  • 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 .
    Abstract, PDF (3444 kByte)
    This work provides a convergence analysis of a time-discrete scheme coupled with a finite-element approximation in space for a model for partial, rate-independent damage featuring a gradient regularization as well as a non-smooth constraint to account for the unidirectionality of the damage evolution. The numerical algorithm to solve the coupled problem of quasistatic small strain linear elasticity with rate-independent gradient damage is based on a Variable ADMM-method to approximate the nonsmooth contribution. Space-discretization is based on P1 finite elements and the algorithm directly couples the time-step size with the spatial grid size h. For a wide class of gradient regularizations, which allows both for Sobolev functions of integrability exponent r ∈ (1, ∞) and for BV-functions, it is shown that solutions obtained with the algorithm approximate as h → 0 a semistable energetic solution of the original problem. The latter is characterized by a minimality property for the displacements, a semistability inequality for the damage variable and an energy dissipation estimate. Numerical benchmark experiments confirm the stability of the method.

Vorträge, Poster

  • X. Liu, Justification of the primitive equations (online talk), Global Scientist Interdisciplinary online Forum 2021, Southern University of Science and Technology, Shenzhen, China, January 9, 2021.

  • D. Peschka, Self-similar spreading with dynamic contact angles (online talk), 91st Annual Meeting of the International Association of Applied Mathematics and Mechanics (Online Event), Section S11 ``Interfacial Flows'', March 15 - 19, 2021, Universität Kassel, March 17, 2021.

  • S. Tornquist, Analysis of dynamic phase-field fracture (online talk), 9th BMS Student Conference (Online Event), March 3 - 5, 2021, Berlin Mathematical School, March 4, 2021.

  • A. Zafferi, Coupling of thermoviscoelastic solids and reactive flows via GENERIC (online talk), CRC 1114 Conference 2021 (Online Event), March 1 - 3, 2021, Freie Universität Berlin, March 2, 2021.

  • A. Zafferi, Dynamics of rock dehydration on multiple scales (online poster session), CRC 1114 Conference 2021 (online participation), March 1 - 3, 2021.

  • P.-É. Druet, The free energy of incompressible fluid mixtures: An asymptotic study (online talk), TES-Seminar on Energy-based Mathematical Methods and Thermodynamics, Technische Universität Berlin WIAS Berlin, January 21, 2021.

  • D. Peschka, Mathematical modeling and simulation of flows and the interaction with a substrate using energetic variational methods, Vortrag im Rahmen des SFB1194, Technische Universität Darmstadt, January 22, 2020.

  • D. Peschka, Modeling and simulation of wetting and dewetting with dynamic contact angles, Wetting Dynamics 2020, Bonn, September 28 - 30, 2020.

  • D. Peschka, L. Heltai, Fluid-structure interaction (online talks), Thematic Einstein Semester: Student Compact Course ``Variational Methods for Fluids and Solids'' (Online Event), October 12 - 23, 2020, WIAS Berlin.

  • D. Peschka , Variational modeling of bulk and interface effects in fluid dynamics, SPP 2171 Advanced School ``Introduction to Wetting Dynamics'', February 17 - 21, 2020, Westfälische Wilhelms-Universität Münster, February 18, 2020.

  • S. Tornquist, Temporal regularity of solutions to a dynamic phase-field fracture model in visco-elastic materials (online talk), MA4M: Mathematical Analysis for Mechanics (Online Event), November 23 - 25, 2020, WIAS Berlin, November 23, 2020.

  • S. Tornquist, Dynamic phase-field fracture in visco-elastic materials (online talk), Thematic Einstein Semester: Student Compact Course ``Variational Methods for Fluids and Solids'' (Online Event), October 12 - 23, 2020, WIAS Berlin, October 14, 2020.

  • A. Zafferi, K. Huber, C09 - Dynamics of rock dehydration (online talk), SCCS Days 2020 of the Collaborative Research Center - CRC 1114 ``Scaling Cascades in Complex Systems'', December 2 - 4, 2020, Freie Universität Berlin, December 3, 2020.

  • A. Zafferi, K. Huber, Dynamics of rock dehydration, SCCS Days 2020 of the Collaborative Research Center - CRC 1114 "Scaling Cascades in Complex Systems" (online participation), December 2 - 4, 2020.

  • A. Zafferi, Lagrangian--Eulerian reduction of GENERIC systems (online talk), Thematic Einstein Semester: Student Compact Course ``Variational Methods for Fluids and Solids'' (Online Event), October 12 - 23, 2020, WIAS Berlin, October 12, 2020.

  • M.H. Farshbaf Shaker, D. Peschka, M. Thomas, Modeling and analysis of suspension flows (online poster session), MATH+ Day 2020 (Online Event), November 6, 2020.

  • M. Thomas, Modeling and analysis of flows of concentrated suspensions (online talk), Kolloquium des Graduiertenkollegs (Online Event), Universität Regensburg, July 10, 2020.

  • M. Thomas, Nonlinear fracture dynamics: modeling, analysis, approximation, and applications, Vorstellung der Projektanträge im SPP 2256, Bad Honnef, January 30, 2020.

  • M. Thomas, Thermodynamical modelling via energy and entropy functionals (online talks), Thematic Einstein Semester: Student Compact Course ``Variational Methods for Fluids and Solids'' (Online Event), October 12 - 23, 2020, WIAS Berlin, October 14, 2020.

  • M. Thomas, Weierstraß-Gruppe "Volumen-Grenzschicht-Prozesse", Sitzung des Wissenschaftlichen Beirats, WIAS Berlin, September 18, 2020.

  • P. Farrell, D. Peschka, Challenges in drift-diffusion semiconductor simulations, Finite Volumes for Complex Applications IX (Online Event), Belgien, Norway, June 15 - 19, 2020.

  • D. Peschka, Variational methods for viscoelastic flows and gelation (online talk), MATH+ Cluster Days 2020 (Online Event), September 21 - 22, 2020, Technische Universität Berlin, September 21, 2020.