Solution of One-Time-Step Problems in Elastoplasticity by a Slant Newton Method

Abstract

We discuss a solution algorithm for quasi-static elastoplastic problems with hardening. Such problems can be described by a time dependent variational inequality, where the displacement and the plastic strain fields serve as primal variables. After discretization in time, one variational inequality of the second kind is obtained per time step and can be reformulated as each one minimization problem with a convex energy functional which depends smoothly on the displacement and non-smoothly on the plastic strain. There exists an explicit formula how to minimize the energy functional with respect to the plastic strain for a given displacement. By substitution, the energy functional can be written as a functional depending only on the displacement. The theorem of Moreau from convex analysis states that the energy functional is differentiable with an explicitly computable first derivative. The second derivative of the energy functional does not exist, hence the plastic strain minimizer is not differentiable on the elastoplastic interface, which separates the continuum in elastically and plastically deformed parts. A Newton-like method exploiting slanting functions of the energy functional's first derivative instead of the nonexistent second derivative is applied. Such method is called a slant Newton method for short. The local super-linear convergence of the slant Newton method in the discrete case is shown and sufficient regularity assumptions are formulated, which would guarantee the local super-linear convergence also in the continuous case.

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

“…Using Moreau's theorem, (EP) can be further reduced to a (Fréchet) differentiable problem in the displacement only, cf. [20]. However, the resulting optimality condition is not eligible to Newton differentiation (in the sense of [30]) in infinite dimensions which may result in mesh-dependent convergence of an associated generalized Newton scheme.…”

“…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. e.g.…”

“…Among the few available references we mention [8] for domain decomposition methods leading to a linear rate of convergence. The approach to plasticity problems without contact constraints in [20], however, turns out to be problematic as far as function space convergence of the employed semismooth Newton (SSN) solver is concerned. In fact, due to the lack of a sufficient norm gap between domain and image space of the mapping involved in the underlying nonsmooth system, generalized differentiability in the sense of [30] does not hold true.…”

Discretization and numerical realization of contact problems for elastic‐perfectly plastic bodies. PART II – numerical realization, limit analysis

“…The respective necessary and sufficient optimality conditions lead to a system of equations in R n involving Lipschitz continuous terms. Such problems are frequently solved by semi-smooth Newton methods, see [6,17,27,29] for a general theory and [7,8,19,21,28,30,36] for applications in plasticity. In this paper two methods are proposed.…”

Quasi-Static Small-Strain Plasticity in the Limit of Vanishing Hardening and Its Numerical Approximation

“…Beside justification of the Prandtl-Reuss model as such a limit, the motivation of considering and implementing a small hardening is also numerically justified, e.g. a-posteriori error estimates and convergence of iterative schemes can be proved [3,19,47]. One should nevertheless mention that there are algorithms allowing for a direct treating of the Prandtl-Reuss model without hardening, e.g.…”