### Browsing by Author "Mielke, Alexander"

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- ItemAnalysis and Numerics for Rate-Independent Processes(Zürich : EMS Publ. House, 2007) Francfort, Gilles; Mielke, Alexander; Roubicek, Tomas[no abstract available]
- ItemAnalytical and numerical methods for finite-strain elastoplasticity(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Gürses, Ercan; Mainik, Andreas; Miehe, Christian; Mielke, AlexanderAn important class of finite-strain elastoplasticity is based on the multiplicative decomposition of the strain tensor $F=F_el F_pl$ and hence leads to complex geometric nonlinearities. This survey describes recent advances on the analytical treatment of time-incremental minimization problems with or without regularizing terms involving strain gradients. For a regularization controlling all of $nabla F_pl$ we provide an existence theory for the time-continuous rate-independent evolution problem, which is based on a recently developed energetic formulation for rate-independent systems in abstract topological spaces. In systems without gradient regularization one encounters the formation of microstructures, which can be described by sequential laminates or more general gradient Young measures. We provide a mathematical framework for the evolution of such microstructure and discuss algorithms for solving the associated space-time discretizations. We outline in a finite-step-sized incremental setting of standard dissipative materials details of relaxation-induced microstructure development for strain softening von Mises plasticity and single-slip crystal plasticity. The numerical implementations are based on simplified assumptions concerning the complexity of the microstructures.
- ItemAnalytical and Statistical Approaches to Fluid Models(Oberwolfach-Walke : Mathematisches Forschungsinstitut Oberwolfach, 2000) Mielke, Alexander; Titi, Edriss S.[no abstract available]
- ItemAn approach to nonlinear viscoelasticity via metric gradient flows(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Mielke, Alexander; Ortner, Christoph; Şengül, YaseminWe formulate quasistatic nonlinear finite-strain viscoelasticity of rate-type as a gradient system. Our focus is on nonlinear dissipation functionals and distances that are related to metrics on weak diffeomorphisms and that ensure time-dependent frame-indifference of the viscoelastic stress. In the multidimensional case we discuss which dissipation distances allow for the solution of the time-incremental problem. Because of the missing compactness the limit of vanishing timesteps can only be obtained by proving some kind of strong convergence. We show that this is possible in the one-dimensional case by using a suitably generalized convexity in the sense of geodesic convexity of gradient flows. For a general class of distances we derive discrete evolutionary variational inequalities and are able to pass to the time-continuous in some case in a specific case.
- ItemAveraging of time-periodic dissipation potentials in rate-independent processes : dedicated to Tomáš Roubícek on the occasion of his sixtieth birthday(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Heida, Martin; Mielke, AlexanderWe study the existence and well-posedness of rate-independent systems (or hysteresis operators) with a dissipation potential that oscillates in time with period. In particular, for the case of quadratic energies in a Hilbert space, we study the averaging limit → 0 and show that the effective dissipation potential is given by the minimum of all friction thresholds in one period, more precisely as the intersection of all the characteristic domains. We show that the rates of the process do not converge weakly, hence our analysis uses the notion of energetic solutions and relies on a detailed estimates to obtain a suitable qui-continuity of the solutions in the limit → 0.
- ItemBalanced viscosity (BV) solutions to infinite-dimensional rate-independent systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Mielke, Alexander; Rossi, Riccarda; Savaré, GiuseppeBalanced Viscosity solutions to rate-independent systems arise as limits of regularized rate-independent flows by adding a superlinear vanishing-viscosity dissipation. We address the main issue of proving the existence of such limits for infinite-dimensional systems and of characterizing them by a couple of variational properties that combine a local stability condition and a balanced energy-dissipation identity. A careful description of the jump behavior of the solutions, of their differentiability properties, and of their equivalent representation by time rescaling is also presented. Our techniques rely on a suitable chain-rule inequality for functions of bounded variation in Banach spaces, on refined lower semicontinuity-compactness arguments, and on new BV-estimates that are of independent interest.
- ItemBalanced-Viscosity solutions for multi-rate systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Mielke, Alexander; Rossi, Riccarda; Savaré, GiuseppeSeveral mechanical systems are modeled by the static momentum balance for the displacement u coupled with a rate-independent flow rule for some internal variable z. We consider a class of abstract systems of ODEs which have the same structure, albeit in a finite-dimensional setting, and regularize both the static equation and the rate-independent flow rule by adding viscous dissipation terms with coefficients α and , where 0<<1 and α>0 is a fixed parameter. Therefore for α different from 1 the variables u and z have different relaxation rates. We address the vanishing-viscosity analysis as tends to 0 in the viscous system. We prove that, up to a subsequence, (reparameterized) viscous solutions converge to a parameterized curve yielding a Balanced Viscosity solution to the original rate-independent system and providing an accurate description of the system behavior at jumps. We also give a reformulation of the notion of Balanced Viscosity solution in terms of a system of subdifferential inclusions, showing that the viscosity in u and the one in z are involved in the jump dynamics in different ways, according to whether α >1, α=1, or 0<α<1.
- ItemBalanced-Viscosity solutions to infinite-dimensional multi-rate systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Mielke, Alexander; Rossi, RiccardaWe consider generalized gradient systems with rate-independent and rate-dependent dissipation potentials. We provide a general framework for performing a vanishing-viscosity limit leading to the notion of parametrized and true Balanced-Viscosity solutions that include a precise description of the jump behavior developing in this limit. Distinguishing an elastic variable $u$ having a viscous damping with relaxation time $eps^alpha$ and an internal variable $z$ with relaxation time $eps$ we obtain different limits for the three cases $alpha in (0,1)$, $alpha=1$ and $alpha>1$. An application to a delamination problem shows that the theory is general enough to treat nontrivial models in continuum mechanics.
- ItemBlow-up versus boundedness in a nonlocal and nonlinear Fokker-Planck equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Dreyer, Wolfgang; Huth, Robert; Mielke, Alexander; Rehberg, Joachim; Winkler, MichaelLiteraturverz.
- ItemBV solutions and viscosity approximations of rate-independent systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Mielke, Alexander; Rossi, Riccarda; Savaré, GiuseppeIn the nonconvex case solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional and a dissipation potential which is a viscous regularization of a given rate-independent dissipation potential. The resulting definition of `BV solutions' involves, in a nontrivial way, both the rate-independent and the viscous dissipation potential, which play a crucial role in the description of the associated jump trajectories. We shall prove a general convergence result for the time-continuous and for the time-discretized viscous approximations and establish various properties of the limiting $BV$ solutions. In particular, we shall provide a careful description of the jumps and compare the new notion of solutions with the related concepts of energetic and local solutions to rate-independent systems.
- ItemCalculation of ultrashort pulse propagation based on rational approximations for medium dispersion(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Amiranashvili, Shalva; Bandelow, Uwe; Mielke, AlexanderUltrashort optical pulses contain only a few optical cycles and exhibit broad spectra. Their carrier frequency is therefore not well defined and their description in terms of the standard slowly varying envelope approximation becomes questionable. Existing modeling approaches can be divided in two classes, namely generalized envelope equations, that stem from the nonlinear Schrödinger equation, and non-envelope equations which treat the field directly. Based on fundamental physical rules we will present an approach that effectively interpolates between these classes and provides a suitable setting for accurate and highly efficient numerical treatment of pulse propagation along nonlinear and dispersive optical media.
- ItemA class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Mielke, Alexander; Ortiz, MichaelThis work is concerned with the reformulation of evolutionary problems in a weak form enabling consideration of solutions that may exhibit evolving microstructures. This reformulation is accomplished by expressing the evolutionary problem in variational form, i.e., by identifying a functional whose minimizers represent entire trajectories of the system. The particular class of functionals under consideration is derived by first defining a sequence of time-discretized minimum problems and subsequently formally passing to the limit of continuous time. The resulting functionals may be regarded as elliptic regularizations of the original evolutionary problem. We find that the $Gamma$-limits of interest are highly degenerate and provide limited information regarding the limiting trajectories of the system. Instead we seek to characterize the minimizing trajectories directly. The special class of problems characterized by a rate-independent dissipation functional is amenable to a particularly illuminating analysis. For these systems it is possible to derive a priori bounds that are independent of the regularizing parameter, whence it is possible to extract convergent subsequences and find the limiting trajectories. Under general assumptions on the functionals, we show that all such limits satisfy the energetic formulation (S) & (E) for rate-independent systems. Moreover, we show that the accumulation points of the regularized solutions solve the associated limiting energetic formulation.
- ItemCoarse-graining via EDP-convergence for linear fast-slow reaction systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Mielke, Alexander; Stephan, ArturWe consider linear reaction systems with slow and fast reactions, which can be interpreted as master equations or Kolmogorov forward equations for Markov processes on a finite state space. We investigate their limit behavior if the fast reaction rates tend to infinity, which leads to a coarse-grained model where the fast reactions create microscopically equilibrated clusters, while the exchange mass between the clusters occurs on the slow time scale. Assuming detailed balance the reaction system can be written as a gradient flow with respect to the relative entropy. Focusing on the physically relevant cosh-type gradient structure we show how an effective limit gradient structure can be rigorously derived and that the coarse-grained equation again has a cosh-type gradient structure. We obtain the strongest version of convergence in the sense of the Energy-Dissipation Principle (EDP), namely EDP-convergence with tilting.
- ItemCoexistence of Hamiltonian-like and dissipative dynamics in chains of coupled phase oscillators with skew-symmetric coupling(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Burylko, Oleksandr; Mielke, Alexander; Wolfrum, Matthias; Yanchuk, SerhiyWe consider rings of coupled phase oscillators with anisotropic coupling. When the coupling is skew-symmetric, i.e. when the anisotropy is balanced in a specific way, the system shows robustly a coexistence of Hamiltonian-like and dissipative regions in the phase space. We relate this phenomenon to the time-reversibility property of the system. The geometry of low-dimensional systems up to five oscillators is described in detail. In particular, we show that the boundary between the dissipative and Hamiltonian-like regions consists of families of heteroclinic connections. For larger chains with skew-symmetric coupling, some sufficient conditions for the coexistence are provided, and in the limit of N oscillators, we formally derive an amplitude equation for solutions in the neighborhood of the synchronous solution. It has the form of a nonlinear Schrödinger equation and describes the Hamiltonian-like region existing around the synchronous state similarly to the case of finite rings.
- ItemComplete damage evolution based on energies and stresses(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Mielke, AlexanderThe rate-independent damage model recently developed in Bouchitté, Mielke, Roubícek ``A complete-damage problem at small strains" allows for complete damage, such that the deformation is no longer well-defined. The evolution can be described in terms of energy densities and stresses. Using concepts of parametrized Gamma convergence, we generalize the theory to convex, but non-quadratic elastic energies by providing Gamma convergence of energetic solutions from partial to complete damage under rather general conditions
- ItemComplete damage in elastic and viscoelastic media and its energetics(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Mielke, Alexander; Roubíček, Tomáš; Zeman, JanA model for the evolution of damage that allows for complete disintegration is addressed. Small strains and a linear response function are assumed. The ``flow rule'' for the damage parameter is rate-independent. The stored energy involves the gradient of the damage variable, which determines an internal length-scale. Quasi-static fully rate-independent evolution is considered as well as rate-dependent evolution including viscous/inertial effects. Illustrative 2-dimensional computer simulations are presented, too.
- ItemA complete-damage problem at small strains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Bouchitté, Guy; Mielke, Alexander; Roubíček, TomášThe complete damage of a linearly-responding material that can thus completely disintegrate is addressed at small strains under time-varying Dirichlet boundary conditions as a rate-independent evolution problem in multidimensional situations. The stored energy involves the gradient of the damage variable. This variable as well as the stress and energies are shown to be well defined even under complete damage, in contrast to displacement and strain. Existence of an energetic solution is proved, in particular, by detailed investigating the $Gamma$-limit of the stored energy and its dependence on boundary conditions. Eventually, the theory is illustrated on a one-dimensional example.
- ItemContinuum descriptions for the dynamics in discrete lattices : derivation and justification(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Giannoulis, Johannes; Herrmann, Michael; Mielke, AlexanderThe passage from microscopic systems to macroscopic ones is studied by starting from spatially discrete lattice systems and deriving several continuum limits. The lattice system is an infinite-dimensional Hamiltonian system displaying a variety of different dynamical behavior. Depending on the initial conditions one sees quite different behavior like macroscopic elastic deformations associated with acoustic waves or like propagation of optical pulses. We show how on a formal level different macroscopic systems can be derived such as the Korteweg-de Vries equation, the nonlinear Schroedinger equation, Whitham's modulation equation, the three-wave interaction model, or the energy transport equation using the Wigner measure. We also address the question how the microscopic Hamiltonian and the Lagrangian structures transfer to similar structures on the macroscopic level. Finally we discuss rigorous analytical convergence results of the microscopic system to the macroscopic one by either weak-convergence methods or by quantitative error bounds.
- ItemConvergence of solutions of kinetic variational inequalities in the rate-independent quasi-static limit(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Mielke, Alexander; Petrov, Adrien; Martins, João A.C.This paper discusses the convergence of kinetic variational inequalities to rate-independent quasi-static variational inequalities. Mathematical formulations as well as existence and uniqueness results for kinetic and rate-independent quasi-static problems are provided. Sharp a priori estimates for the kinetic problem are derived that imply that the kinetic solutions converge to the rate-independent ones, when the size of initial perturbations and the rate of application of the forces tends to 0. An application to three-dimensional elastic-plastic systems with hardening is given.
- ItemConvergence to equilibrium in energy-reaction-diffusion systems using vector-valued functional inequalities : dedicated to Peter Markowich on the occasion of his sixtieth birthday(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Mielke, Alexander; Mittnenzweig, MarkusWe discuss how the recently developed energy-dissipation methods for reaction-diffusion systems can be generalized to the non-isothermal case. For this we use concave entropies in terms of the densities of the species and the internal energy, with the important feature, that the equilibrium densities may depend on the internal energy. Using the log-Sobolev estimate and variants for lower-order entropies as well as estimates for the entropy production of the nonlinear reactions we give two methods to estimate the relative entropy by the total entropy production, namely a somewhat restrictive convexity method, which provides explicit decay rates, and a very general, but weaker compactness method.