A convergent adaptive finite element method for the primal problem of elastoplasticity

Abstract

SUMMARYThe boundary value problem representing one time step of the primal formulation of elastoplasticity with positive hardening leads to a variational inequality of the second kind with some nondifferentiable functional. This paper establishes an adaptive finite element algorithm for the solution of this variational inequality that yields the energy reduction and, up to higher order terms, the R-linear convergence of the stresses with respect to the number of loops. Applications include several plasticity models: linear isotropic-kinematic hardening, linear kinematic hardening, and multisurface plasticity as model for nonlinear hardening laws. For perfect plasticity, the adaptive algorithm yields strong convergence of the stresses. Numerical examples confirm an improved linear convergence rate and study the performance of the algorithm in comparison with the more frequently applied maximum refinement rule.

Convergence of adaptive finite element methods in elastoplasticity

“…Proof of the reliability of η uses Jensen inequality as in [4], whereas in [5] using inverse estimates and convex analysis we prove a local discrete efficiency estimate in terms of the discrete energies, and the existence of positive constants ρ E , ρ with ρ E < 1, depending on the regularity of the initial triangulation T 0 and on the material parameters, such that there holds δ +1 ≤ ρ E δ + ρ(osc 2 (f ) + osc 2 (g)) with δ := J(w ) − J(w) and similar definition for δ +1 . The control of the data oscillations osc (f ) and osc (g) with conditions similar to (3) as step (e) of the adaptive algorithm using the inner node property leads finally to the existence of a sequence (α ) ∈N linearly convergent to zero such that there holds |σ − σ | C −1 ;Ω ≤ α .…”

“…Along the same line as [3] this short note states the energy reduction and the R−linear convergence of the stresses for a conforming FE method of the primal problem of plasticity [8]. Proof and details are given elsewhere [5].…”

A duality-based path-following semismooth Newton method for elasto-plastic contact problems

“…Appropriate discretization schemes for plasticity problems with hardening have been investigated extensively in the recent past. Here we only mention [3,10,9,43] for adaptive finite element methods. Concerning numerical solution methods, we refer to the multigrid approach in [47], various generalized Newton methods in finite dimensions [12,20,42,47,48], including the standard return mapping algorithm in [44] as well as interior point strategies, cf.…”

Error Estimates for Space-Time Discretizations of a Rate-Independent Variational Inequality

“…The difficulty here is the nonsmoothness of the subdifferential operator ∂Ψ(•) as well as the nonlinearity of the map q → D q E(t, q). In the linear case this would reduce to classical evolutionary variational inequalities, for which the numerics is well studied; see, e.g., [29,1,2,15,14,13,41,42,51].…”

“…). A posteriori estimates are established in [13], indeed paving the way to the possibility of applying adaptive techniques to elastoplastic problems [1,15,14,51]. Error control for strain gradient plasticity is presented in [20] where, nevertheless, the extra regularity (u, z) ∈…”