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- ItemSpectrum and amplitude equations for scalar delay-differential equations with large delay(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Yanchuk, Serhiy; Lücken, Leonhard; Wolfrum, Matthias; Mielke, AlexanderThe subject of the paper are scalar delay-differential equations with large delay. Firstly, we describe the asymptotic properties of the spectrum of linear equations. Using these properties, we classify possible types of destabilization of steady states. In the limit of large delay, this classification is similar to the one for parabolic partial differential equations. We present a derivation and error estimates for amplitude equations, which describe universally the local behavior of scalar delay-differential equations close to the destabilization threshold.
- ItemOn the Darwin--Howie--Whelan equations for the scattering of fast electrons described by the Schrödinger equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Koprucki, Thomas; Maltsi, Anieza; Mielke, AlexanderThe Darwin-Howie-Whelan equations are commonly used to describe and simulate the scattering of fast electrons in transmission electron microscopy. They are a system of infinitely many envelope functions, derived from the Schrödinger equation. However, for the simulation of images only a finite set of envelope functions is used, leading to a system of ordinary differential equations in thickness direction of the specimen. We study the mathematical structure of this system and provide error estimates to evaluate the accuracy of special approximations, like the two-beam and the systematic-row approximation.
- ItemFree energy, free entropy, and a gradient structure for thermoplasticity(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Mielke, AlexanderIn the modeling of solids the free energy, the energy, and the entropy play a central role. We show that the free entropy, which is defined as the negative of the free energy divided by the temperature, is similarly important. The derivatives of the free energy are suitable thermodynamical driving forces for reversible (i.e. Hamiltonian) parts of the dynamics, while for the dissipative parts the derivatives of the free entropy are the correct driving forces. This difference does not matter for isothermal cases nor for local materials, but it is relevant in the non-isothermal case if the densities also depend on gradients, as is the case in gradient thermoplasticity. Using the total entropy as a driving functional, we develop gradient structures for quasistatic thermoplasticity, which again features the role of the free entropy. The big advantage of the gradient structure is the possibility of deriving time-incremental minimization procedures, where the entropy-production potential minus the total entropy is minimized with respect to the internal variables and the temperature. We also highlight that the usage of an auxiliary temperature as an integrating factor in Yang/Stainier/Ortiz "A variational formulation of the coupled thermomechanical boundary-value problem for general dissipative solids" (J. Mech. Physics Solids, 54, 401-424, 2006) serves exactly the purpose to transform the reversible driving forces, obtained from the free energy, into the needed irreversible driving forces, which should have been derived from the free entropy. This reconfirms the fact that only the usage of the free entropy as driving functional for dissipative processes allows us to derive a proper variational formulation.
- ItemFrom discrete visco-elasticity to continuum rate-independent plasticity : rigorous results(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Mielke, Alexander; Truskinovsky, LevWe show that continuum models for ideal plasticity can be obtained as a rigorous mathematical limit starting from a discrete microscopic model describing a visco-elastic crystal lattice with quenched disorder. The constitutive structure changes as a result of two concurrent limiting procedures: the vanishing-viscosity limit and the discrete to continuum limit. In the course of these limits a non-convex elastic problem transforms into a convex elastic problem while the quadratic rate-dependent dissipation of visco-elastic solid transforms into a singular rate-independent dissipation of an ideally plastic solid. In order to emphasize ideas we employ in our proofs the simplest prototypical system describing transformational plasticity of shape-memory alloys. The approach, however, is sufficiently general and can be used for similar reductions in the cases of more general plasticity and damage models.
- ItemTraveling fronts in a reaction-diffusion equation with a memory term(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Mielke, Alexander; Reichelt, SinaBased on a recent work on traveling waves in spatially nonlocal reaction-diffusion equations, we investigate the existence of traveling fronts in reaction-diffusion equations with a memory term. We will explain how such memory terms can arise from reduction of reaction-diffusion systems if the diffusion constants of the other species can be neglected. In particular, we show that two-scale homogenization of spatially periodic systems can induce spatially homogeneous systems with temporal memory. The existence of fronts is proved using comparison principles as well as a reformulation trick involving an auxiliary speed that allows us to transform memory terms into spatially nonlocal terms. Deriving explicit bounds and monotonicity properties of the wave speed of the arising traveling front, we are able to establish the existence of true traveling fronts for the original problem with memory. Our results are supplemented by numerical simulations.
- ItemRelating a rate-independent system and a gradient system for the case of one-homogeneous potentials(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Mielke, AlexanderWe consider a non-negative and one-homogeneous energy functional $mathcal J$ on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-inpendent system given in terms of the time-dependent functional $mathcal E(t,u)=t mathcal J(u)$ and the norm as a dissipation distance. The relation between the two flows is given via a solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constant-speed intervals for the solutins of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.
- ItemOn an evolutionary model for complete damage based on energies and stresses(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Mielke, AlexanderA recent model allows for complete damage, such that the deformation is not well-defined. The evolution can be described in terms of energy densities and stresses. We introduce the notion of weak energetic solution and show how the existence theory can be generalized to convex, but non-quadratic elastic energies.
- ItemVariational approaches and methods for dissipative material models with multiple scales(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Mielke, AlexanderIn a first part we consider evolutionary systems given as generalized gradient systems and discuss various variational principles that can be used to construct solutions for a given system or to derive the limit dynamics for multiscale problems. These multiscale limits are formulated in the theory of evolutionary Gamma-convergence. On the one hand we consider the a family of viscous gradient system with quadratic dissipation potentials and a wiggly energy landscape that converge to a rate-independent system. On the other hand we show how the concept of Balanced-Viscosity solution arise as in the vanishing-viscosity limit. As applications we discuss, first, the evolution of laminate microstructures in finite-strain elastoplasticity and, second, a two-phase model for shape-memory materials, where H-measures are used to construct the mutual recovery sequences needed in the existence theory.
- ItemDecay to equilibrium for energy-reaction-diffusion systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Haskovec, Jan; Hittmeir, Sabine; Markowich, Peter; Mielke, AlexanderWe derive thermodynamically consistent models of reaction-diffusion equations coupled to a heat equation. While the total energy is conserved, the total entropy serves as a driving functional such that the full coupled system is a gradient flow. The novelty of the approach is the Onsager structure, which is the dual form of a gradient system, and the formulation in terms of the densities and the internal energy. In these variables it is possible to assume that the entropy density is strictly concave such that there is a unique maximizer (thermodynamical equilibrium) given linear constraints on the total energy and suitable density constraints. We consider two particular systems of this type, namely, a diffusion-reaction bipolar energy transport system, and a drift-diffusion-reaction energy transport system with confining potential. We prove corresponding entropy-entropy production inequalities with explicitly calculable constants and establish the convergence to thermodynamical equilibrium, at first in entropy and further in L1 using Cziszàr-Kullback-Pinsker type inequalities.
- ItemNon-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Mielke, Alexander; Patterson, Robert I.A.; Peletier, Mark A.; Renger, D.R. MichielWe study stochastic interacting particle systems that model chemical reaction networks on the microscopic scale, converging to the macroscopic Reaction Rate Equation. One abstraction level higher, we also study the ensemble of such particle systems, converging to the corresponding Liouville transport equation. For both systems, we calculate the corresponding large deviations and show that under the condition of detailed balance, the large deviations enables us to derive a non-linear relation between thermodynamic fluxes and free energy driving force.