Articles in Refereed Journals

  • L. Schmeller, D. Peschka, Gradient flows for coupling order parameters and mechanics, SIAM Journal on Applied Mathematics, 83 (2023), pp. 225--253, DOI 10.1137/22M148478X .
    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.

  • D. Peschka, Partial and complete wetting of thin films with dynamic contact angle, Physics of Fluids, 35 (2023), pp. 041705/1--041705/6, DOI 10.1063/5.0146538 .
    The wetting of thin films depends critically on the sign of the spreading coefficient S. We discuss the cases S<0, S=0, and S>0 for transient models with contact line dissipation and find that the use of a dynamic contact angle solves problems for S>0 that models might otherwise have. For initial data with a non-zero slope and S>0, we show that there exists a finite time at which the contact angle of the thin film goes to zero. Then, a molecular precursor emerges from the thin film and moves outward at a constant velocity.

  • R. Shiri, L. Schmeller, R. Seemann, D. Peschka, B. Wagner, Impact of noise on spinodal dewetting of liquid-liquid films, communications physics, 6 (2023), pp. 109/1--109/11, DOI 10.1038/s42005-023-01208-x .
    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.

  • L. Giacomelli, M. Gnann, D. Peschka, Droplet motion with contact-line friction: Long-time asymptotics in complete wetting, Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences, 479 (2023), pp. 20230090/1--20230090/23, DOI 10.1098/rspa.2023.0090 .
    We consider the thin-film equation for a class of free boundary conditions modelling friction at the contact line, as introduced by E and Ren. Our analysis focuses on formal long-time asymptotics of solutions in the perfect wetting regime. In particular, through the analysis of quasi-self-similar solutions, we characterize the profile and the spreading rate of solutions depending on the strength of friction at the contact line, as well as their (global or local) corrections, which are due to the dynamical nature of the free boundary conditions. These results are complemented with full transient numerical solutions of the free boundary problem.

  • D. Peschka, A. Zafferi, L. Heltai, M. Thomas, Variational approach to fluid-structure interaction via GENERIC, Journal of Non-Equilibrium Thermodynamics, 47 (2022), pp. 217--226, DOI 10.1515/jnet-2021-0081 .
    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/ .
    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, published online on 09.05.2022, DOI 10.1002/zamm.202100254 .
    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, Resolving the microscopic hydrodynamics at the moving contact line, Physical Review Fluids, 7 (2022), pp. L102001/1--L102001/9, DOI 10.1103/PhysRevFluids.7.L102001 .
    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.

  • X. Liu, M. Thomas, E. Titi, Well-posedness of Hibler's dynamical sea-ice model, Journal of Nonlinear Science, 32 (2022), pp. 49/1--49/31, DOI 10.1007/s00332-022-09803-y .
    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.

Contributions to Collected Editions

  • 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 .
    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 .
    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.

Preprints, Reports, Technical Reports

  • M. Thomas, S. Tornquist, Ch. Wieners, Approximating dynamic phase-field fracture in viscoelastic materials with a first-order formulation for velocity and stress, Preprint no. 3002, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3002 .
    Abstract, PDF (493 kByte)
    We investigate a model for dynamic fracture in viscoelastic materials at small strains. While the sharp crack interface is approximated with a phase-field method, we consider a viscous evolution with a quadratic dissipation potential for the phase-field variable. A non-smooth constraint enforces a unidirectional evolution of the phase-field, i.e. material cannot heal. The viscoelastic equation of motion is transformed into a first order formulation and coupled in a nonlinear way to the non-smooth evolution law of the phase field. The system is fully discretized in space and time with a discontinuous Galerkin approach for the first-order formulation. Based on this, existence of discrete solutions is shown and, as the step size in space and time tends to zero, their convergence to a suitable notion of weak solution of the system is discussed.

  • A. Zafferi, K. Huber, D. Peschka, J. Vrijmoed, T. John, M. Thomas, A porous-media model for reactive fluid-rock interaction in a dehydrating rock, Preprint no. 2999, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.2999 .
    Abstract, PDF (3206 kByte)
    We study the GENERIC structure of models for reactive two-phase flows and their connection to a porous-media model for reactive fluid-rock interaction used in Geosciences. For this we discuss the equilibration of fast dissipative processes in the GENERIC framework. Mathematical properties of the porous-media model and first results on its mathematical analysis are provided. The mathematical assumptions imposed for the analysis are critically validated with the thermodynamical rock data sets.

  • L. Schmeller, D. Peschka, Sharp-interface limits of Cahn--Hilliard models and mechanics with moving contact lines, Preprint no. 2990, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.2990 .
    Abstract, PDF (5979 kByte)
    We construct gradient structures for free boundary problems with moving capillary interfaces with nonlinear (hyper)elasticity and study the impact of moving contact lines. In this context, we numerically analyze how phase-field models converge to certain sharp-interface models when the interface thickness tends to zero. In particular, we study the scaling of the Cahn--Hilliard mobility with certain powers of the interfacial thickness. In the presence of interfaces, it is known that the intended sharp-interface limit holds only for a particular range of powers However, in the presence of moving contact lines we show that some scalings that are valid for interfaces produce significant errors and the effective range of valid powers of the interfacial thickness in the mobility reduces.

Talks, Poster

  • D. Peschka, Moving contact lines for sliding droplets, 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2023), Session 11 ``Interfacial Flows'', May 30 - June 2, 2023, Technische Universität Dresden, June 1, 2023.

  • D. Peschka, Multiscale limits of thin-film models with moving support, In search of model structures for non-equilibrium systems, April 24 - 28, 2023, Westfälische Wilhelms-Universität Münster, April 27, 2023.

  • D. Peschka, Multiscale limits of thin-film models with moving support, Kolloquium des SFB 1114, Freie Universität Berlin, April 20, 2023.

  • D. Peschka, Sharp-interface limit of models with mechanics and contact lines, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Session 00247 ``Interfaces and Free Boundaries in Fluid Mechanics and Materials Science'', August 20 - 25, 2023, Waseda University, Tokyo, Japan, August 24, 2023.

  • M. Thomas, Approximating dynamic phase-field fracture with a first-order formulation for velocity and stress, Annual Workshop of the GAMM Activity Group on Analysis of PDEs, September 18 - 20, 2023, Katholische Universität Eichstätt-Ingolstadt, September 20, 2023.

  • M. Thomas, Nonlinear fracture dynamics: Modeling, analysis, approximation, and applications, Jahrestreffen des SPP 2256, September 27 - 29, 2023, Universität Regensburg, September 29, 2023.

  • M. Thomas, Approximating dynamic phase-field fracture with a first-order formulation for velocity and stress, Nonlinear PDEs: Recent Trends in the Analysis of Continuum Mechanics, July 17 - 21, 2023, Universität Bonn, Hausdorff School for Advanced Studies in Mathematics, July 17, 2023.

  • M. Thomas, Approximating dynamic phase-field fracture with a first-order formulation for velocity and stress, Seminar für Angewandte Mathematik, Technische Universität Dresden, June 5, 2023.

  • M. Thomas, Nonlinear fracture dynamics: Modeling, analysis, approximation, and applications, Presentation of project proposals in SPP 2256 ``Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials'', Bad Honnef, March 27, 2023.

  • 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, Rome, 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.