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Publications

Articles in Refereed Journals

  • 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 .
    Abstract
    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, published online in March 2020, 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 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.

  • 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, published online on 22.07.2020, DOI 10.1002/zamm.201900288 .
    Abstract
    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.

  • D. Peschka, S. Haefner, L. Marquant, K. Jacobs, A. Münch, B. Wagner, Signatures of slip in dewetting polymer films, Proceedings of the National Academy of Sciences of the United States of America, 116 (2019), pp. 9275--9284, DOI 10.1073/pnas.1820487116 .

  • L. Adam, M. Hintermüller, D. Peschka, Th.M. Surowiec, Optimization of a multiphysics problem in semiconductor laser design, SIAM Journal on Applied Mathematics, 79 (2019), pp. 257--283, DOI 10.1137/18M1179183 .
    Abstract
    A multimaterial topology optimization framework is suggested for the simultaneous optimization of mechanical and optical properties to be used in the development of optoelectronic devices. Based on the physical aspects of the underlying device, a nonlinear multiphysics model for the elastic and optical properties is proposed. Rigorous proofs are provided for the sensitivity of the fundamental mode of the device with respect to the changes in the underlying topology. After proving existence and optimality results, numerical experiments leading to an optimal material distribution for maximizing the strain in a Ge-on-Si microbridge are given. The highly favorable electronic properties of this design are demonstrated by steady-state simulations of the corresponding van Roosbroeck (drift-diffusion) system.

  • P. Farrell, D. Peschka, Nonlinear diffusion, boundary layers and nonsmoothness: Analysis of challenges in drift-diffusion semiconductor simulations, Computers & Mathematics with Applications. An International Journal, 78 (2019), pp. 3731--3747, DOI 10.1016/j.camwa.2019.06.007 .
    Abstract
    We analyze and benchmark the error and the convergence order of finite difference, finite-element as well as Voronoi finite-volume discretization schemes for the drift-diffusion equations describing charge transport in bulk semiconductor devices. Three common challenges, that can corrupt the precision of numerical solutions, will be discussed: boundary layers at Ohmic contacts, discontinuties in the doping profile, and corner singularities in L-shaped domains. The influence on the order of convergence is assessed for each computational challenge and the different discretization schemes. Additionally, we provide an analysis of the inner boundary layer asymptotics near Ohmic contacts to support our observations.

Contributions to Collected Editions

  • D. Peschka, M. Thomas, T. Ahnert, A. Münch, B. Wagner, Gradient structures for flows of concentrated suspensions, in: Topics in Applied Analysis and Optimisation, M. Hintermüller, J.F. Rodrigues, eds., CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, pp. 295--318, DOI 10.1007/978-3-030-33116-0 .
    Abstract
    In this work we investigate a two-phase model for concentrated suspensions. We construct a PDE formulation using a gradient flow structure featuring dissipative coupling between fluid and solid phase as well as different driving forces. Our construction is based on the concept of flow maps that also allows it to account for flows in moving domains with free boundaries. The major difference compared to similar existing approaches is the incorporation of a non-smooth two-homogeneous term to the dissipation potential, which creates a normal pressure even for pure shear flows.

Preprints, Reports, Technical Reports

  • 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.

  • P. Colli, M.H. Farshbaf Shaker, K. Shirakawa, N. Yamazaki, Optimal control for shape memory alloys of the one-dimensional Frémond model, Preprint no. 2737, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2737 .
    Abstract, PDF (700 kByte)
    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.

  • E. Ipocoana, A. Zafferi, Further regularity and uniqueness results for a non-isothermal Cahn--Hilliard equation, Preprint no. 2716, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2716 .
    Abstract, PDF (270 kByte)
    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.

  • 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.

  • 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, Preprint no. 2626, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2626 .
    Abstract, PDF (424 kByte)
    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.

Talks, Poster

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

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

  • M. Thomas, Bulk-interface processes, Sitzung des Wissenschaftlichen Beirats, Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, September 18, 2020.

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

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

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

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

  • M.H. Farshbaf Shaker, D. Peschka, M. Thomas, Modeling and analysis of suspension flows, Visit of the Scientific Advisory Board of MATH+, November 11, 2019.

  • M.H. Farshbaf Shaker, D. Peschka, M. Thomas, Modeling and analysis of suspension flows, 1st MATH+ Day, Berlin, December 13, 2019.

  • D. Peschka, Dynamic contact angles via generalized gradient flows, Modelling of Thin Liquid Films -- Asymptotic Approach vs. Gradient Dynamics, April 28 - May 3, 2019, Banff International Research Station for Mathematical Information and Discovery, Canada, April 30, 2019.

  • D. Peschka, Dynamic contact angles via gradient flows, 694. WE-Heraeus-Seminar ``Wetting on Soft or Microstructured Surfaces'', Bad Honnef, April 10 - 13, 2019.

  • D. Peschka, Gradient formulations with flow maps -- Mathematical and numerical approaches to free boundary problems, Kolloquium des Graduiertenkollegs 2339 ``Interfaces, Complex Structures, and Singular Limits'', Universität Regensburg, May 24, 2019.

  • D. Peschka, Gradient structures for flows of concentrated suspensions - jamming and free boundaries, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S11 ``Interfacial Flows", February 18 - 22, 2019, Technische Universität Wien, Austria, February 20, 2019.

  • D. Peschka, Mathematical modeling and simulation of substrate-flow interaction using generalized gradient flow, Begutachtungskolloquium für die Anträge des SPP 2171 ``Dynamische Benetzung flexibler, adaptiver und schaltbarer Oberflächen'', Mainz, February 7 - 8, 2019.

  • D. Peschka, Mathematical modeling of fluid flows using gradient systems, Seminar in PDE and Applications, Delft University of Technology, Netherlands, May 28, 2019.

  • D. Peschka, Steering pattern formation of viscous flows, DMV-Jahrestagung 2019, Sektion ``Differentialgleichungen und Anwendungen'', September 23 - 26, 2019, KIT - Karlsruher Institut für Technologie, September 23, 2019.

  • D. Peschka, ``Numerical methods for charge transport in semiconductors: FEM vs FV", 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), July 15 - 19, 2019, Valencia, Spain, July 17, 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.

  • A. Zafferi, An approach to multi-phase flows in geosciences, MURPHYS-HSFS 2019 Summer School on Multi-Rate Processes, Slow-Fast Systems and Hysteresis, Turin, Italy, June 17 - 21, 2019.

  • A. Zafferi, Dynamics of rock dehydration on multiple scales, SCCS Days 2019 of the Collaborative Research Center - CRC 1114, Zeuthen, May 20 - 22, 2019.

  • A. Zafferi, Some regularity results for a non-isothermal Cahn-Hilliard model, 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.

  • M. Thomas, Analysis for the discrete approximation of gradient-regularized damage models, Mathematics Seminar Brescia, Università degli Studi di Brescia, Italy, March 13, 2019.

  • M. Thomas, Analysis for the discrete approximation of gradient-regularized damage models, PDE Afternoon, Universität Wien, Austria, April 10, 2019.

  • M. Thomas, Analytical and numerical aspects for the approximation of gradient-regularized damage models, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), Thematic Minisymposium MS A3-2-26 ``Phase-Field Models in Simulation and Optimization'', July 15 - 19, 2019, Valencia, Spain, July 17, 2019.

  • M. Thomas, Analytical and numerical aspects of rate-independent gradient-regularized damage models, Conference ``Dynamics, Equations and Applications (DEA 2019)'', Session D444 ``Topics in the Mathematical Modelling of Solids'', September 16 - 20, 2019, AGH University of Science and Technology, Kraków, Poland, September 19, 2019.

  • M. Thomas, Coupling of rate-independent and rate-dependent systems, MURPHYS-HSFS 2019 Summer School on Multi-Rate Processes, Slow-Fast Systems and Hysteresis, June 17 - 19, 2019, Politecnico di Torino, Turin, Italy.

  • 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, Dynamics of rock dehydration on multiple scales, SCCS Days 2019 of the Collaborative Research Center - CRC 1114, May 20 - 22, 2019, Freie Universität Berlin, Zeuthen, May 21, 2019.

  • M. Thomas, GENERIC structures with bulk-interface interaction, SFB 910 Symposium ``Energy Based Modeling, Simulation and Control'', October 25, 2019, Technische Universität Berlin, October 25, 2019.

  • M. Thomas, Gradient structures for flows of concentrated suspensions, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), Thematic Minisymposium MS ME-7-75 ``Recent Advances in Understanding Suspensions and Granular Media Flow'', July 15 - 19, 2019, Valencia, Spain, July 17, 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, Towards the analysis of dynamic phase-field fracture, Spring School on Variational Analysis 2019, Paseky, Czech Republic, May 19 - 25, 2019.

  • S. Tornquist, Towards the analysis of dynamic phase-field fracture, MURPHYS-HSFS 2019 Summer School on Multi-Rate Processes, Slow-Fast Systems and Hysteresis, Turin, Italy, June 17 - 21, 2019.

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