Gradient Flows as a Selection Procedure for Equilibria of Nonconvex Energies

Ortner, Christoph

doi:10.1137/050643982

Cited by 7 publications

(6 citation statements)

References 18 publications

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“…While the static questions of -convergence or relaxation are well studied, the related questions for evolutionary systems are treated less systematically, see e.g., [7,8,37]. Only recently, a systematic study for gradient flows was initialized in [36,[38][39][40][41].…”

Section: Introductionmentioning

confidence: 99%

Γ-limits and relaxations for rate-independent evolutionary problems

2007

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This work uses the energetic formulation of rate-independent systems that is based on the stored-energy functionals E and the dissipation distance D. For sequences (E k ) k∈N and (D k ) k∈N we address the question under which conditions the limits q ∞ of solutions q k : [0, T ] → Q satisfy a suitable limit problem with limit functionals E ∞ and D ∞ , which are the corresponding -limits. We derive a sufficient condition, called conditional upper semi-continuity of the stable sets, which is essential to guarantee that q ∞ solves the limit problem. In particular, this condition holds if certain joint recovery sequences exist. Moreover, we show that time-incremental minimization problems can be used to approximate 123 388 A. Mielke et al. the solutions. A first example involves the numerical approximation of functionals using finite-element spaces. A second example shows that the stop and the play operator converge if the yield sets converge in the sense of Mosco. The third example deals with a problem developing microstructure in the limit k → ∞, which in the limit can be described by an effective macroscopic model. Mathematics Subject Classification (2007)

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Section: Introductionmentioning

confidence: 99%

Γ-limits and relaxations for rate-independent evolutionary problems

2007

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“…The limit systems obtained in this paper are gradient flows, but the functionals generating it have a different form. On the other hand, Hamiltonian systems generated by pair interaction potentials have been considered in . Some of the questions required to understand such problems have analogies with the ones considered in this last paper, including questions related with the stability of the dynamics for perturbations with length scales comparable with the distance between particles.…”

Section: Introductionmentioning

confidence: 88%

Friction dominated dynamics of interacting particles locally close to a crystallographic lattice

Bodnar

Velázquez²

2012

Math Methods in App Sciences

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A system of particles, in general d‐dimensional space, that interact by means of pair potentials and adjust their positions according to the gradient flow dynamics induced by the total energy of the system is studied. The case when the range of the interaction is of the same order as the mean interparticle distance is considered. It is also assumed that particles, locally, are located close to some crystallographic lattice. An appropriate system of equations that describes the evolution of macroscopic deformation of the crystallographic lattice, as well as the system that describes the evolution of the main crystallographic directions is derived. Well‐posedness of the derived system is studied as well as the stability of the particle system. Same examples of potentials that yield stable and unstable systems are given. Copyright © 2012 John Wiley & Sons, Ltd.

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“…for example [19,23]). This can be best seen by considering an atomistic body which is clamped at the left-hand end with a small deformation applied to the right-hand end.…”

Section: Model Problem and Qc Approximationmentioning

confidence: 99%

“…In a quite similar spirit, we used an adaptive proximal point algorithm (PPA) (see [13] for an overview), which is essentially a time-discretized version of a gradient flow for E to compute critical points. The novelty of our algorithm is that, motivated by the analysis in [19], we chose the | · | w 1,2 ε -semi-norm as the gradient flow norm. Given an initial condition Y (0) , for the th step of the optimization method, we find a critical point (local minimizer) of…”

Section: Adaptive Algorithmmentioning

confidence: 99%

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Analysis of a quasicontinuum method in one dimension

Ortner¹,

Süli²

2008

ESAIM: M2AN

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Abstract.The quasicontinuum method is a coarse-graining technique for reducing the complexity of atomistic simulations in a static and quasistatic setting. In this paper we aim to give a detailed a priori and a posteriori error analysis for a quasicontinuum method in one dimension. We consider atomistic models with Lennard-Jones type long-range interactions and a QC formulation which incorporates several important aspects of practical QC methods. First, we prove the existence, the local uniqueness and the stability with respect to a discrete W 1,∞ -norm of elastic and fractured atomistic solutions. We use a fixed point argument to prove the existence of a quasicontinuum approximation which satisfies a quasi-optimal a priori error bound. We then reverse the role of exact and approximate solution and prove that, if a computed quasicontinuum solution is stable in a sense that we make precise and has a sufficiently small residual, there exists a 'nearby' exact solution which it approximates, and we give an a posteriori error bound. We stress that, despite the fact that we use linearization techniques in the analysis, our results apply to genuinely nonlinear situations.Mathematics Subject Classification. 70C20, 70-08, 65N15.

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Gradient Flows as a Selection Procedure for Equilibria of Nonconvex Energies

Cited by 7 publications

References 18 publications

Γ-limits and relaxations for rate-independent evolutionary problems

Γ-limits and relaxations for rate-independent evolutionary problems

Friction dominated dynamics of interacting particles locally close to a crystallographic lattice

Analysis of a quasicontinuum method in one dimension

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