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Weighted energy-dissipation functionals for gradient flows

Alexander MielkeUlisse Stefanelli — 2011

ESAIM: Control, Optimisation and Calculus of Variations

We investigate a global-in-time variational approach to abstract evolution by means of the functionals proposed by Mielke and Ortiz [ (2008) 494–516]. In particular, we focus on gradient flows in Hilbert spaces. The main result is the convergence of minimizers and approximate minimizers of these functionals to the unique solution of the gradient flow. Sharp convergence rates are provided and the convergence analysis is combined with time-discretization. Applications of the...

A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems

Michael OrtizAlexander Mielke — 2008

ESAIM: Control, Optimisation and Calculus of Variations

This 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...

Linearized plasticity is the evolutionary Γ -limit of finite plasticity

Alexander MielkeUlisse Stefanelli — 2013

Journal of the European Mathematical Society

We provide a rigorous justification of the classical linearization approach in plasticity. By taking the small-deformations limit, we prove via Γ -convergence for rate-independent processes that energetic solutions of the quasi-static finite-strain elastoplasticity system converge to the unique strong solution of linearized elastoplasticity.

A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems

Alexander MielkeMichael Ortiz — 2007

ESAIM: Control, Optimisation and Calculus of Variations

This work is concerned with the reformulation of evolutionary problems in a weak form enabling consideration of solutions that may exhibit . This reformulation is accomplished by expressing the evolutionary problem in variational form, , by identifying a functional whose minimizers represent 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 ...

Numerical approaches to rate-independent processes and applications in inelasticity

Alexander MielkeTomáš Roubíček — 2009

ESAIM: Mathematical Modelling and Numerical Analysis

A conceptual numerical strategy for rate-independent processes in the energetic formulation is proposed and its convergence is proved under various rather mild data qualifications. The novelty is that we obtain convergence of subsequences of space-time discretizations even in case where the limit problem does not have a unique solution and we need no additional assumptions on higher regularity of the limit solution. The variety of general perspectives thus obtained is illustrated on several...

Weighted energy-dissipation functionals for gradient flows

Alexander MielkeUlisse Stefanelli — 2011

ESAIM: Control, Optimisation and Calculus of Variations

We investigate a global-in-time variational approach to abstract evolution by means of the functionals proposed by Mielke and Ortiz [ (2008) 494–516]. In particular, we focus on gradient flows in Hilbert spaces. The main result is the convergence of minimizers and approximate minimizers of these functionals to the unique solution of the gradient flow. Sharp convergence rates are provided and the convergence analysis is combined with time-discretization. Applications of the...

BV solutions and viscosity approximations of rate-independent systems

Alexander MielkeRiccarda RossiGiuseppe Savaré — 2012

ESAIM: Control, Optimisation and Calculus of Variations

In 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 that is a viscous regularization of a given rate-independent dissipation...

A metric approach to a class of doubly nonlinear evolution equations and applications

Riccarda RossiAlexander MielkeGiuseppe Savaré — 2008

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

This paper deals with the analysis of a class of doubly nonlinear evolution equations in the framework of a general metric space. We propose for such equations a suitable metric formulation (which in fact extends the notion of for gradient flows in metric spaces, see [5]), and prove the existence of solutions for the related Cauchy problem by means of an approximation scheme by time discretization. Then, we apply our results to obtain the existence of solutions to abstract doubly nonlinear equations...

BV solutions and viscosity approximations of rate-independent systems

Alexander MielkeRiccarda RossiGiuseppe Savaré — 2012

ESAIM: Control, Optimisation and Calculus of Variations

In 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 that is a viscous regularization...

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