Publikationen

Monografien

  • 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).

  • H. Abels, K. Disser, H.-Chr. Kaiser, A. Mielke, M. Thomas, eds., Issue on Partial Differential Equations in Fluids and Solids, 14 of Discrete and Continuous Dynamical Systems -- Series S, American Institute of Mathematical Sciences, Springfield, 2021, 292 pages, (Collection Published).

Artikel in Referierten Journalen

  • D. Peschka, A. Zafferi, L. Heltai, M. Thomas, Variational approach to fluid-structure interaction via GENERIC, Journal of Non-Equilibrium Thermodynamics, (2022), published online on 11.02.2022, DOI 10.1515/jnet-2021-0081 .
    Abstract
    We present a framework to systematically derive variational formulations for fluid-structure interaction problems based on thermodynamical driving functionals and geometric structures in different coordinate systems by suitable transformations within this formulation. Our approach provides a promising basis to construct structure-preserving discretization strategies.

  • D. Peschka, L. Heltai, Model hierarchies and higher-order discretisation of time-dependent thin-film free boundary problems with dynamic contact angle, Journal of Computational Physics, 464 (2022), pp. 111325/1--111325/22, DOI 10.1016/j.jcp.2022.111325 .
    Abstract
    We present a mathematical and numerical framework for the physical problem of thin-film fluid flows over planar surfaces including dynamic contact angles. In particular, we provide algorithmic details and an implementation of higher-order spatial and temporal discretisation of the underlying free boundary problem using the finite element method. The corresponding partial differential equation is based on a thermodynamic consistent energetic variational formulation of the problem using the free energy and viscous dissipation in the bulk, on the surface, and at the moving contact line. Model hierarchies for limits of strong and weak contact line dissipation are established, implemented and studied. We analyze the performance of the numerical algorithm and investigate the impact of the dynamic contact angle on the evolution of two benchmark problems: gravity-driven sliding droplets and the instability of a ridge.

  • A. Zafferi, D. Peschka, M. Thomas, GENERIC framework for reactive fluid flows, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, (2022), pp. 1--70 (published online on 09.05.2022), DOI 10.1002/zamm.202100254 .
    Abstract
    We describe reactive fluid flows in terms of the formalism General Equation for Non-Equilibrium Reversible-Irreversible Coupling also known as GENERIC. Together with the formalism, we present the thermodynamical and mechanical foundations for the treatment of fluid flows using continuous fields and present a clear relation and transformation between a Lagrangian and an Eulerian formulation of the corresponding systems of partial differential equations. We bring the abstract framework to life by providing many physically relevant examples for reactive compressive fluid flows.

  • A.K. Giri, P. Malgaretti, D. Peschka, M. Sega, Hydrodynamics at the moving contact line, Physical Review Fluids, 7 (2022), pp. L102001/1--L102001/9 (published online on 10.10.2022), DOI 10.1103/PhysRevFluids.7.L102001 .
    Abstract
    By removing the smearing effect of capillary waves in molecular dynamics simulations we are able to provide a microscopic picture of the region around the moving contact line (MCL) at an unprecedented resolution. On this basis, we show that the continuum character of the velocity field is unaffected by molecular layering down to below the molecular scale. The solution of the continuum Stokes problem with MCL and Navier-slip matches very well the molecular dynamics data and is consistent with a slip-length of 42 Å and small contact line dissipation. This is consistent with observations of the local force balance near the liquid-solid interface.

  • 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 .
    Abstract
    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. 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, DOI 10.3934/dcdss.2021067 .
    Abstract
    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.

  • X. Liu, E. Titi, Local well-posedness of strong solutions to the three-dimensional compressible primitive equations, Archive for Rational Mechanics and Analysis, 241 (2021), pp. 729--764, DOI 10.1007/s00205-021-01662-3 .
    Abstract
    This work is devoted to establishing the local-in-time well-posedness of strong solutions to the three-dimensional compressible primitive equations of atmospheric dynamics. It is shown that strong solutions exist, are unique, and depend continuously on the initial data, for a short time in two cases: with gravity but without vacuum, and with vacuum but without gravity.

Beiträge zu Sammelwerken

  • M. Heida, M. Thomas, GENERIC for dissipative solids with bulk-interface interaction, in: Research in the Mathematics of Materials Science, A. Schlömerkemper, ed., 31 of Association for Women in Mathematics Series, Springer, Cham, 2022, pp. 333--364, DOI 10.1007/978-3-031-04496-0_15 .
    Abstract
    The modeling framework of GENERIC was originally introduced by Grmela and Öttinger for thermodynamically closed systems. It is phrased with the aid of the energy and entropy as driving functionals for reversible and dissipative processes and suitable geometric structures. Based on the definition functional derivatives we propose a GENERIC framework for systems with bulk-interface interaction and apply it to discuss the GENERIC structure of models for delamination processes.

  • S. Bartels, M. Milicevic, M. Thomas, S. Tornquist, N. Weber, Approximation schemes for materials with discontinuities, in: Non-standard Discretisation Methods in Solid Mechanics, J. Schröder, P. Wriggers, eds., 98 of Lecture Notes in Applied and Computational Mechanics, Springer, Cham, 2022, pp. 505--565, DOI 10.1007/978-3-030-92672-4_17 .
    Abstract
    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.

  • 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 .
    Abstract
    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 (the complement 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.

Preprints, Reports, Technical Reports

  • L. Schmeller, D. Peschka, Gradient flows for coupling order parameters and mechanics, Preprint no. 2909, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2909 .
    Abstract, PDF (7667 kByte)
    We construct a formal gradient flow structure for phase-field evolution coupled to mechanics in Lagrangian coordinates, present common ways to couple the evolution and provide an incremental minimization strategy. While the usual presentation of continuum mechanics is intentionally very brief, the focus of this paper is on an extensible functional analytical framework and a discretization approach that preserves an appropriate variational structure as much as possible. As examples, we first present phase separation and swelling of gels and then the approach of stationary states of multiphase systems with surface tension and show the robustness of the general approach.

  • R. Shiri, L. Schmeller, R. Seemann, D. Peschka, B. Wagner, On the spinodal dewetting of thin liquid bilayers, Preprint no. 2861, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2861 .
    Abstract, PDF (12 MByte)
    We investigate the spinodal dewetting of a thin liquid polystyrene (PS) film on a liquid polymethylmethacrylate (PMMA) subtrate. Following the evolution of the corrugations of the PS film via in situ measurements by atomic force microscopy (AFM) and those of the PS-PMMA interface via ex situ imaging, we provide a direct and detailed comparison of the experimentally determined spinodal wavelengths with the predictions from linear stability analysis of a thin-film continuum model for the bilayer system. The impact of rough interfaces and fluctuations is studied theoretically by investigating the impact of different choices of initial data on the unstable wavelength and on the rupture time. The key factor is the mode selection by initial data perturbed with correlated colored noise in the linearly unstable regime, which becomes relevant only for liquid bilayers to such an extent. By numerically solving the mathematical model, we further address the impact of nonlinear effects on rupture times and on the morphological evolution of the interfaces in comparison with experimental results.

  • X. Liu, M. Thomas, E. Titi, Well-posedness of Hibler's dynamical sea-ice model, Preprint no. 2833, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2833 .
    Abstract, PDF (324 kByte)
    This paper establishes the local-in-time well-posedness of solutions to an approximating system constructed by mildly regularizing the dynamical sea ice model of it W.D. Hibler, Journal of Physical Oceanography, 1979. Our choice of regularization has been carefully designed, prompted by physical considerations, to retain the original coupled hyperbolic-parabolic character of Hibler's model. Various regularized versions of this model have been used widely for the numerical simulation of the circulation and thickness of the Arctic ice cover. However, due to the singularity in the ice rheology, the notion of solutions to the original model is unclear. Instead, an approximating system, which captures current numerical study, is proposed. The well-posedness theory of such a system provides a first-step groundwork in both numerical study and future analytical study.

Vorträge, Poster

  • D. Peschka, Discretization of compressible Stokes flow using Hamiltonian and Onsager structures, Workshop on Numerical Methods and Analysis in CFD, July 5 - 8, 2022, WIAS Berlin, July 5, 2022.

  • D. Peschka, Gradient flows coupling order parameters and mechanics (online talk), Colloquium of the SPP 2171 (Online Event), Westfälische Wilhelms-Universität Münster, October 21, 2022.

  • D. Peschka, Steering the pattern formation of dewetting liquids (online talk), SFB 910 Symposium ``Pattern formation and coherent structure in dissipative systems'' (Online Event), Technische Universität Berlin, January 14, 2022.

  • D. Peschka, Variational approach for wetting flows on solid and soft substrates (online talk), Colloquium of the SPP 2171 (Online Event), Westfälische Wilhelms-Universität Münster, January 28, 2022.

  • S. Tornquist, Dynamic phase-field fracture in viscoelastic materials using a first-order formulation, 92th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2022), DFG Priority Program 2256 ``Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials'', August 15 - 19, 2022, Rheinisch-Westfälische Technische Hochschule Aachen, August 16, 2022.

  • A. Zafferi, Analysis of a reactive-diffusive porous media model for rock dehydration processes, 92th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2022), Session 14 ``Applied Analysis'', August 15 - 19, 2022, Rheinisch-Westfälische Technische Hochschule Aachen, August 17, 2022.

  • M. Thomas, First-order formulation for dynamic phase-field fracture in visco-elastic materials, PHAse field MEthods in applied sciences (PHAME 2022), May 23 - 27, 2022, Istituto Nazionale di Alta Matematica, Roma, Italy, May 25, 2022.

  • M. Thomas, First-order formulation for dynamic phase-field fracture in visco-elastic materials, Beyond Elasticity: Advances and Research Challenges, May 16 - 20, 2022, Centre International de Rencontres Mathématiques, Marseille, France, May 16, 2022.

  • M. Thomas, First-order formulation for dynamic phase-field fracture in visco-elastic materials, Jahrestreffen des SPP 2256, September 28 - 30, 2022, Universität Regensburg, September 30, 2022.

  • M.H. Farshbaf Shaker, D. Peschka , M. Thomas, B. Wagner, Variational methods for viscoelastic flows and gelation, MATH+ Day 2021 (Online Event), Technische Universität Berlin, November 5, 2021.

  • 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.

  • X. Liu, Second law of thermodynamics and bounded entropy solutions in the compressible Navier--Stokes system, Summer School 2021 ``Wave Phenomena: Analysis and Numerics'' (Hybrid Event), September 27 - 30, 2021.

  • D. Peschka, Dynamic contact angles and their numerical discretization for gradient systems (online talk), 15th International Conference on Free Boundary Problems: Theory and Applications 2021 (Online Event), September 13 - 17, 2021, WIAS Berlin, September 13, 2021.

  • D. Peschka, Energy-based variational approach to moving interfaces and contact lines (online talk), SIAM Conference on Mathematical Aspects of Materials Science (MS21), Minisymposium 46 ``Multi-phase Flow and Dynamics of Interfaces: Analysis and Numerics'' (Online Event), May 17 - 28, 2021, Basque Center for Applied Mathematics, Bilbao, Spain, May 19, 2021.

  • D. Peschka, Mathematical modeling and simulation of substrate-flow interaction using generalized gradient flows, Conference ``Dynamic Wetting of Flexible, Adaptive, and Switchable Substrates'' of the SPP 2171, Freiburg, November 8 - 10, 2021.

  • D. Peschka, Mathematical modeling and simulation of substrate-flow interaction using generalized gradient flows, SPP 2171 Conference ``Dynamic Wetting of Flexible, Adaptive, and Switchable Substrates'', November 8 - 10, 2021, Albert-Ludwigs-Universität Freiburg, November 10, 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.

  • D. Peschka, Thin-film problems with dynamic contact angle (online talk), 8th European Congress of Mathematics (8ECM), Minisymposium ID 43 ``Higher-order Evolution Equations'' (Online Event), June 20 - 26, 2021, Portoroč, Slovenia, June 23, 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), MSDI4: ``Modeling and Analysis of Geological Fluid Flows'', March 1 - 3, 2021, Freie Universität Berlin, March 2, 2021.

  • A. Zafferi, Dynamics of rock dehydration on multiple scales, CRC 1114 Conference 2021 (Online Event), March 1 - 3, 2021.

  • A. Zafferi, Thermodynamics of reaction-diffusion-induced rock dehydration processes (online talk), 16thJoint European Thermodynamics Conference (Hybrid Event), June 14 - 18, 2021, Charles University Prague, Czech Republic, June 16, 2021.

  • M. Thomas, Convergence analysis for fully discretized damage and phase-field fracture models (online talk), 15th International Conference on Free Boundary Problems: Theory and Applications 2021 (FBP 2021, Online Event), Minisymposium ``Phase Field Models'', September 13 - 17, 2021, WIAS, Berlin, September 14, 2021.

  • M. Thomas, GENERIC structures with bulk-interface interaction (online talk), 16th Joint European Thermodynamics Conference (Hybrid Event), June 14 - 18, 2021, Charles University Prague, Czech Republic, June 17, 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, Thematic Einstein Semester on Energy-based Mathematical Methods for Reactive Multiphase Flows, Technische Universität Berlin, WIAS Berlin, January 21, 2021.

  • X. Liu, Second law of thermodynamics and bounded entropy solutions in the compressible Navier--Stokes system (online talk), 15th International Conference on Free Boundary Problems: Theory and Applications 2021 (Online Event), September 13 - 17, 2021, WIAS Berlin, September 17, 2021.