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- ItemRigorous derivation of a plate theory in linear elastoplasticity via [Gamma]-convergence(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Liero, Matthias; Roche, ThomasThis paper deals with dimension reduction in linearized elastoplasticity in the rate-independent case. The reference configuration of the elastoplastic body is given by a two-dimensional middle surface and a small but positive thickness. We derive a limiting model for the case in which the thickness of the plate tends to 0. This model contains membrane and plate deformations which are coupled via plastic strains. The convergence analysis is based on an abstract Gamma convergence theory for rate-independent evolution formulated in the framework of energetic solutions. This concept is based on an energy-storage functional and a dissipation functional, such that the notion of solution is phrased in terms of a stability condition and an energy balance.
- ItemThe weighted energy-dissipation principle and evolutionary [Gamma]-convergence for doubly nonlinear problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Liero, Matthias; Melchionna, StefanoWe consider a family of doubly nonlinear evolution equations that is given by families of convex dissipation potentials, nonconvex energy functionals, and external forces parametrized by a small parameter ε. For each of these problems, we introduce the so-called weighted energy-dissipation (WED) functional, whose minimizer correspond to solutions of an elliptic-in-time regularization of the target problems with regularization parameter δ. We investigate the relation between the Γ-convergence of the WED functionals and evolutionary Γ-convergence of the associated systems. More precisely, we deal with the limits δ→0, ε→0, as well as δ+ ε→0 either in the sense of Γ-convergence of functionals or in the sense of evolutionary Γ-convergence of functional-driven evolution problems, or both. Additionally, we provide some quantitative estimates on the rate of convergence for the limit ε→0, in the case of quadratic dissipation potentials and uniformly λ-convex energy functionals. Finally, we discuss a homogenization problem as an example of application.
- ItemOn gradient structures for Markov chains and the passage to Wasserstein gradient flows(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Disser, Karoline; Liero, MatthiasWe study the approximation of Wasserstein gradient structures by their finite-dimensional analog. We show that simple finite-volume discretizations of the linear Fokker-Planck equation exhibit the recently established entropic gradient-flow structure for reversible Markov chains. Then we reprove the convergence of the discrete scheme in the limit of vanishing mesh size using only the involved gradient-flow structures. In particular, we make no use of the linearity of the equations nor of the fact that the Fokker-Planck equation is of second order.
- ItemHomogenization of Cahn-Hilliard-type equations via evolutionary Gamma-convergence(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Liero, Matthias; Reichelt, SinaIn this paper we discuss two approaches to evolutionary Gamma-convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ Gamma-convergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savare 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004. We apply both approaches to rigorously derive homogenization limits for Cahn-Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals Gamma-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches.
- ItemOptimal transport in competition with reaction: The Hellinger-Kantorovich distance and geodesic curves(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Liero, Matthias; Mielke, Alexander; Savaré, GiuseppeWe discuss a new notion of distance on the space of finite and nonnegative measures on Omega C Rd, which we call Hellinger-Kantorovich distance. It can be seen as an infconvolution of the well-known Kantorovich-Wasserstein distance and the Hellinger-Kakutani distance. The new distance is based on a dynamical formulation given by an Onsager operator that is the sum of a Wasserstein diffusion part and an additional reaction part describing the generation and absorption of mass. We present a full characterization of the distance and some of its properties. In particular, the distance can be equivalently described by an optimal transport problem on the cone space over the underlying space Omega. We give a construction of geodesic curves and discuss examples and their general properties.
- ItemRate independent Kurzweil processes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Krejčí, Pavel; Liero, MatthiasThe Kurzweil integral technique is applied to a class of rate independent processes with convex energy and discontinuous inputs. We prove existence, uniqueness, and continuous data dependence of solutions in $BV$ spaces. It is shown that in the context of elastoplasticity, the Kurzweil solutions coincide with natural limits of viscous regularizations when the viscosity coefficient tends to zero. The discontinuities produce an additional positive dissipation term, which is not homogeneous of degree one.
- ItemThermistor systems of p(x)-Laplace-type with discontinuous exponents via entropy solutions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Bulicek, Miroslav; Glitzky, Annegret; Liero, MatthiasWe show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two difficulties appear: (i) the elliptic term in the current-flow equation is of p(x)-Laplacian-type with discontinuous exponent p, which limits the use of standard methods, and (ii) in the heat equation, we have to deal with an a priori L1 term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder's fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem.
- ItemHybrid finite-volume/finite-element schemes for p(x)-Laplace thermistor models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Fuhrmann, Jürgen; Glitzky, Annegret; Liero, MatthiasWe introduce an empirical PDE model for the electrothermal description of organic semiconductor devices by means of current and heat flow. The current flow equation is of p(x)-Laplace type, where the piecewise constant exponent p(x) takes the non-Ohmic behavior of the organic layers into account. Moreover, the electrical conductivity contains an Arrhenius-type temperature law. We present a hybrid finite-volume/finite-element discretization scheme for the coupled system, discuss a favorite discretization of the p(x)-Laplacian at hetero interfaces, and explain how path following methods are applied to simulate S-shaped current-voltage relations resulting from the interplay of self-heating and heat flow.
- ItemAnalysis of p(x)-Laplace thermistor models describing the electrothermal behavior of organic semiconductor devices(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Glitzky, Annegret; Liero, MatthiasWe study a stationary thermistor model describing the electrothermal behavior of organic semiconductor devices featuring non-Ohmic current-voltage laws and selfheating effects. The coupled system consists of the current-flow equation for the electrostatic potential and the heat equation with Joule heating term as source. The self-heating in the device is modeled by an Arrhenius-like temperature dependency of the electrical conductivity. Moreover, the non-Ohmic electrical behavior is modeled by a power law such that the electrical conductivity depends nonlinearly on the electric field. Notably, we allow for functional substructures with different power laws, which gives rise to a p(x)-Laplace-type problem with piecewise constant exponent. We prove the existence and boundedness of solutions in the two-dimensional case. The crucial point is to establish the higher integrability of the gradient of the electrostatic potential to tackle the Joule heating term. The proof of the improved regularity is based on Caccioppoli-type estimates, Poincaré inequalities, and a Gehring-type Lemma for the p(x)-Laplacian. Finally, Schauders fixed-point theorem is used to show the existence of solutions.
- ItemWIAS-TeSCA - Two-dimensional semi-conductor analysis package(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Gajewski, Herbert; Liero, Matthias; Nürnberg, Reiner; Stephan, HolgerWIAS-TeSCA (Two- and three-dimensional semiconductor analysis package) is a simulation tool for the numerical simulation of charge transfer processes in semiconductor structures, especially in semiconductor lasers. It is based on the drift-diffusion model and considers a multitude of additional physical effects, like optical radiation, temperature influences and the kinetics of deep impurities. Its efficiency is based on the analytic study of the strongly nonlinear system of partial differential equations – the van Roosbroeck system – which describes the electron and hole currents. Very efficient numerical procedures for both the stationary and transient simulation have been implemented. WIAS-TeSCA has been successfully used in the research and industrial development of new electronic and optoelectronic semiconductor devices such as transistors, diodes, sensors, detectors and lasers and has already proved its worth many times in the planning and optimization of these devices. It covers a broad spectrum of applications, from heterobipolar transistor (mobile telephone systems, computer networks) through high-voltage transistors (power electronics) and semiconductor laser diodes (fiber optic communication systems, medical technology) to radiation detectors (space research, high energy physics). WIAS-TeSCA is an efficient simulation tool for analyzing and designing modern semiconductor devices with a broad range of performance that has proved successful in solving many practical problems. Particularly, it offers the possibility to calculate self-consistently the interplay of electronic, optical and thermic effects.
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