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

Abstract

It is well known that rate-independent systems involving nonconvex energy func-2010 Mathematics Subject Classification. 49J27 49J40 35Q74 65M12 74C05 74H15 .

Existence of parameterized BV-solutions for rate-independent systems with discontinuous loads

“…Thanks to Proposition 2.3 the right hand side is uniformly bounded with respect to ε ≥ 0 and the partitions of [0, T ] and we have shown estimate (24). In order to prove (25) observe that ∂R(0) ⊂ Z * can be identified with a subset of V * that is bounded with respect to the V * -norm, see [7…”

“…We prove the existence of (parameterized) balanced viscosity solutions via a vanishing viscosity analysis (Theorem 4.4) and study the compactness of the solution set (Proposition 5.2). The analysis is carried out in the semilinear rate-independent setting introduced in [18], compare also [13,Example 3.8.4], [7,9]. For a more detailed presentation of the arguments let Z, V be Hilbert spaces and X a Banach space such that Z ⊂ ⊂ V ⊂ X (compact and continuous embeddings, respectively).…”

“…It follows from (10) that ∂R(0) ⊂ V * and bounded in V * -norm (see for instance [7]). For ε > 0, let…”

Irreversibility and alternate minimization in phase field fracture: a viscosity approach

“…The theoretical investigation of the relationship between alternate minimization schemes for phase field models of fracture and rate-independent processes has instead only recently started with the works [2,3,6,33] (see also [31,38] for the application of alternating algorithms in different physical frameworks). Here, we describe the algorithm adopted in [33].…”

A priori error estimates for a linearized fracture control problem

“…For the considered quasi-static fracture model, the time evolution is not smooth, in general, and thus typical convergence estimates, for the forward evolution of the fracture, provide only weak convergence in space-time, see, e.g., Bartels et al (2018). It is subject to future research to see, if concepts such as BV-solutions, see, e.g., Knees (2019), provide a suitable framework for an analogous error analysis with respect to the time discretization.…”