Homogenization and Related Topics

Transcription

1 Homogenization and Related Topics 1st Berlin Dresden Prague Würzburg Workshop Technische Universität Dresden Department of Mathematics 22 June, Webpage: sneukamm/bdpw-workshop/

2 1 General Information Objectives This workshop is the first of a series of Berlin-Dresden-Prague-Würzburg -meetings, which we plan to organize on a regular basis. The goal of these workshops is to bring together researchers (in particular young scientists) from the above mentioned regions interested in the analysis, modeling and simulation of materials, multiscale and evolution problems, continuum mechanics and related fields. The first meeting has a focus on homogenization problems with applications to continuum mechanics. Organizers Martin Kružík (Czech Academy of Sciences, Prague), Alexander Mielke (WIAS Berlin), Stefan Neukamm (TU Dresden) and Anja Schlömerkemper (U of Würzburg). Location of the Lectures The workshop will be hosted at TU Dresden. All lectures take place at Dinner Departement of Mathematics Willers-Bau (Zellescher Weg 12-14) Room C207 The social dinner (no-host) takes place at the Lebanese restaurant Sindbad. It is located at Hübnerstraße 13, Dresden and in walking distance (about 10 minutes) of the main station, see For more information see Support The organizers gratefully acknowledge financial support by the DFG in the context of TU Dresden s Institutional Strategy The Synergetic University. Contact Prof. Dr. Stefan Neukamm stefan.neukamm@tu-dresden.de, phone: Visiting address: TU Dresden, Department of Mathematics, Institute of Scientific Computing Willersbau, room B208 Zellescher Weg Dresden 2

3 2 Program 10:30 11:25 registration & welcome coffee 11:25 11:30 opening remarks 11:30 12:00 Oliver Sander (TU Dresden) 12:00 12:30 Barbora Benešová (U Würzburg) Discretization methods for oriented materials Non-interpenetration for thin films 12:30 13:00 Sina Reichelt (WIAS Berlin) 13:00 13:45 lunch (buffet) 13:45 14:15 Mathias Schäffner (TU Dresden) 14:15 14:45 Martin Heida (WIAS Berlin) 14:45 15:15 Jan Valdman (AS CR Prague) 15:15 15:45 coffee 15:45 16:15 Hanuš Seiner (AS CR Prague) 16:15 16:45 Jan Zeman (TU Prague) 16:45 closing ca. 18:00 Error estimates for elliptic equations with not exactly periodic coefficients Stochastic homogenization of discrete energies with degenerate growth Stochastic homogenization of rate independent systems Computations of some elasto-plasto-damage models Homogenized properties of microstructured and phase-transforming materials Fourier spectral methods in image-based homogenization of composites with complex microstructure social dinner (no-host) 3

4 3 Abstracts Discretization methods for oriented materials Oliver Sander Department of Mathematics, TU Dresden, Germany Materials such as ferromagnets, liquid crystals, and granular media involve orientation degrees of freedom. Mathematical descriptions of such materials involve fields of nonlinear objects such as unit vectors, rotations matrices, or unitary matrices. Classical numerical methods like the finite element method cannot be applied in such situations, because linear and polynomial interpolation is not defined for such nonlinear objects. Instead, a variety of heuristic approaches is used in the literature, which are difficult to analyze rigorously. We present nonlinear generalizations of the finite element method that allow to treat problems with orientation degrees of freedom in a mathematically sound way. This allows to show solvability of the discrete problems, makes the construction of efficient solvers easier, and allows to obtain reliable bounds for the finite element approximation error. We use the technique to calculate stable configurations of chiral magnetic skyrmions, and wrinkling patterns of a thin elastic polyimide film. Non-interpenetration for thin films Barbora Benešová Institute for Mathematics, University of Würzburg, Germany Within this talk, we consider the dimension-reduction procedure in the membrane regime in order to obtain thin-film models of elastic solids. One of the crucial requirements on such models is that penetration of matter is prohibited. While in the bulk this translates to injectivity of deformations, in the film this is no longer true because a thin film can be folded.thus, we introduce suitable conditions on thin-film deformations that characterize non-interpenetration and show these conditions are enforced by thin-film Γ-limits of bulk energies assuring injectivity. This is joint work with Martin Kružík (Prague). Error estimates for elliptic equations with not exactly periodic coefficients Sina Reichelt WIAS Berlin, Germany The talk addresses error estimates for partial differential equations with coefficients that are not exactly ɛ-periodic, where ɛ > 0 denotes the ratio between the microscopic and the macroscopic length scale. We derive quantitative estimates for linear elliptic equations where the ellipticity may degenerate with order O(ɛ 2γ ). It is shown that for γ = 0 and γ = 1 the error between the original solution and the effective solution is of order O(ɛ 1/2 ). Therefore suitable test functions are constructed via the periodic unfolding method and a gradient folding operator making only minimal additional assumptions with respect to the macroscopic scale on the given data and the effective solution. If time permits, we generalize the obtained error estimates to parabolic equations. 4

5 Stochastic homogenization of discrete energies with degenerate growth Mathias Schäffner Department of Mathematics, TU Dresden, Germany We present a discrete-to-continuum analysis for lattice systems with random interactions via Γ convergence. In particular, we assume that the interaction potentials satisfy polynomial growth conditions which degenerate and are given in terms of certain weight functions. Under suitable moment conditions on the weight functions and stationarity/ergodicity assumptions for the interaction potentials, we prove that the discrete energy Γ converges almost surely to a deterministic and homogeneous integral functional. This is a joint work with S. Neukamm (TU Dresden) and A. Schlömerkemper (U Wuerzburg). Stochastic homogenization of rate independent systems Martin Heida WIAS Berlin, Germany We study stochastic homogenization problems of the form 0 Ψ ɛ ( t u ɛ ) + DE ɛ (t, u ɛ ) where E ɛ : [0, T ] B ɛ R is a proper, quadratic functional and Ψ ɛ : B ɛ R is proper and 1-homogeneous and B ɛ is an ɛ-dependent Banach space. As usual in homogenization, the index ɛ > 0 is a smallnes parameter and (in general) relates to the scale of the underlying geometry of the pysical system, such as crystaline structure, microscopic cracks etc.... As an application of the abstract results, we study Prandtl-Reuss plasticity and elasticity problems coupled with Coulomb-friction. Computations of some elasto-plasto-damage models Jan Valdman Institute of Information Theory and Automation, Czech Academy of Sciences, Prague, Czech Republic Based on the two recent papers [1], [2] of Roubicek, Valdman, numerical computations of some elasto-plasto-damage models will be discussed. Computations present time dependent (rate dependent or rate independent) 2D simulations using the finite element methods. Plasticity component (perfect plasticity or kinematic hardening) is resolved by the quasi Newton method and the damage component by a quadratic programming with box constraints. At the end of the talk, new simulations of elastic waves will be presented. [1] Tomáš Roubíček, Jan Valdman. Stress-driven solution to rate-independent elasto-plasticity with damage at small strains and its computer implementation. Mathematics and Mechanics of Solids (published online) [2] Tomáš Roubíček, Jan Valdman. Perfect plasticity with damage and healing at small strains, its modelling, analysis, and computer implementation. SIAM Journal on Applied mathematics 76, No. 1, (2016) 5

6 Homogenized properties of microstructured and phase-transforming materials Hanuš Seiner Institute of Thermomechanics, Czech Academy of Sciences, Prague, Czech Republic The talk will give an overview of several modern materials in which the macro-scale properties are governed by microstructures, with the focus laid on micro- and macro-scale elastic properties and their experimental determination by ultrasonic methods. For each material system, a brief summary of the available experimental observations will be given and the main open questions in theoretical relations between the micro- and macro-scale properties will be outlined. The set of described materials will include: shape memory alloys, modulated structures, crystals with embedded phasetransforming particles, ceramic micro-scaffolds, nanoporous media and graphene-based composites. The aim of the talk will be to find possible overlaps between some of the current trends in materials science and homogenization problems interesting from the mathematics point of view. Fourier spectral methods in image-based homogenization of composites with complex microstructure Jan Zeman Department of Mechanics, Czech Technical University in Prague, Czech Republic The Fourier-based homogenization solvers were introduced by Moulinec and Suquet into the field of computational micromechanics of materials in Since then, they have established themselves as a competitive alternative to finite elements in terms of accuracy, efficiency, versatility, and simplicity of implementation. In its basic version, the method works as a fixed-point iterative solution to a periodic Lippman-Schwinger integral equation, whose kernel can be efficiently handled by the Fast Fourier Transform (FFT). Recently, we have interpreted and analyzed FFT-based methods in a Galerkin framework that involves the four standard steps: (i) introducing a weak form of the governing equations, (ii) projecting the weak form to an approximation space of trigonometric polynomials, (iii) applying a numerical quadrature, (iv) solving the ensuing system of linear equations by a suitable iterative solver. Specifically, the basic Moulinec-Suquet scheme is obtained when (i) the weak form involves the gradients gradients of the field variables, (ii) the approximation space is spanned by trigonometric polynomials, (iii) the trapezoidal rule is employed for numerical integration, and step (iv) is performed by the Richardson iteration. The purpose of this talk is twofold: to summarize these developments and to explain how they can be used to develop more efficient FFT-based solvers, considering scalar elliptic problems for simplicity. Specifically, I will focus on a-posteriori error estimation based on duality arguments, related to the steps (i) (iii) above, and on the comparison of iterative solvers, i.e. step (iv). This is a joint work with Jaroslav Vondejc (Technische Universitt Braunschweig), Ivo Marek (Czech Technical University in Prague), and Nachiketa Mishra (Tata Institute for Fundamental Research). 6