Viscous Corrections of the Time Incremental Minimization Scheme and Visco-Energetic Solutions to Rate-Independent Evolution Problems

Abstract

We propose the new notion of Visco-Energetic solutions to rate-independent systems (X, E, d) driven by a time dependent energy E and a dissipation quasi-distance d in a general metric-topological space X.As for the classic Energetic approach, solutions can be obtained by solving a modified time Incremental Minimization Scheme, where at each step the dissipation quasi-distance d is incremented by a viscous correction δ (e.g. proportional to the square of the distance d), which penalizes far distance jumps by inducing a localized version of the stability condition.We prove a general convergence result and a typical characterization by Stability and Energy Balance in a setting comparable to the standard energetic one, thus capable to cover a wide range of applications. The new refined Energy Balance condition compensates the localized stability and provides a careful description of the jump behavior: at every jump the solution follows an optimal transition, which resembles in a suitable variational sense the discrete scheme that has been implemented for the whole construction.

Multiscale analysis of singularly perturbed finite dimensional gradient flows: the minimizing movement approach

“…Adapting to our setting some arguments in [25,Section 7], we show at first that the right continuous representative of the nonincreasing function…”

“…for all 0 ≤ s ≤ t ≤ T . In order to do this, we borrow some ideas from [25,Section 7]. While the " ≤ " inequality above can be obtained by passing to the limit in the considered minimization scheme, the " ≥ " ensues from the stability condition, and requires the additional assumptions (F4)-(F5) on the energy, which are instead not necessary in order to recover compactness (see Section 5.1).…”

From Visco-Energetic to Energetic and Balanced Viscosity Solutions of Rate-Independent Systems

“…3.1. This remarkable property has somehow inspired the approach in [MS16]. There, a new notion of rate-independent evolution has been obtained in the time-continuous limit, as τ ↓ 0, of the perturbed time-incremental minimization scheme…”

“…The analysis carried out in [MS16] in fact covers a more general, topological setting, akin to that of [MM05], with a general viscous correction δ : X × X → [0, ∞) compatible, in a suitable sense, with the metric d: a particular case is in fact δ(u, v) = µ 2 d 2 (u, v) as in (IM µ ). In the simplified metric setting of (X), under the same conditions ensuring the existence of Energetic solutions it is possible to show that the (piecewise constant interpolants of the) discrete solutions arising from (IM µ ) converge, as τ ↓ 0 and µ > 0 is fixed, to a (µ-)Visco-Energetic solution to the rate-independent system (X, E, d).…”

Convergence analysis of time-discretisation schemes for rate-independent systems

“…In comparison to the energy dissipation estimate obtained in [ACFS17] it contains more information and the behavior at jump points can be characterised more precisely. In this context let us finally mention [MS16,RS17]. There, the authors consider (1.6) for fixed η > 0 and at each t k they carry out only one minimization step (i.e.…”