Abstract

The reliable computation of shell structures requires a tool to assess and control the quality of the finite element solution. For practical purposes, the quality of the numerical solution must be measured using a quantity of engineering interest rather than in the standard energy norm. however, the assessment of the error in an output of interest is based on a standard energy norm error estimator. The standard error estimator has to be applied quantity. In shells with assumed-strain models, the combination of primal and dual error estimation is performed differently than in the continuum mechanics case. Moreover, a part form the goal-oriented error estimator, the adaptive process requires a remeshing criterion. This work introduces a specific remeshing criterion for goal-oriented adaptivity and tis particularization to the context of shell elements.

Abstract

This paper introduces a new goal-oriented adap- tive technique based on a simple and effective post-process of the finite element approximations. The goal-oriented character of the estimate is achieved by analyzing both the direct problem and an auxiliary problem, denoted as adjoint or dual problem, which is related to the quantity of interest. Thus, the error estimation technique proposed in this paper would fall into the category of recovery-type explicit residual a posteriori error estimates. The procedure is valid for general linear quantities of interest and it is also extended to non-linear ones. The numerical examples demonstrate the efficiency of the proposed approach and discuss: 1) different error representations, 2) assessment of the dispersion error, and 3) different remeshing criteria.

Abstract

Materials such as composites are heterogeneous at the micro-scale, where several constituents with different material properties can be distinguished like elastic inclusions and the elasto-plastic matrix with isotropic hardening. One has to deal with these heterogeneities on the micro-scale and then perform a scale transition to obtain the overall behavior on the macro-scale, which is often referred to as homogenization. The present contribution deals with the combination of numerically inexpensive mean-field and numerically expensive full-field homogenization methods in elasto-plasticity coupled to adaptive finite element method (FEM) which takes into account error generation and error transport at each time step on the macro-scale. The proposed adaptive procedure is driven by a goal-oriented a posteriori error estimator based on duality techniques. The main difficulty of duality techniques in the literature is that the backwards-in-time al gorithm has a high demand on memory capacity since additional memory is required to store the primary solutions computed over all time steps. To this end, several down wind and upwind approximations are introduced for an elasto-plastic primal problem by means of jump terms [1]. Therefore, from a computational point of view, the forwards-in time duality problem is very attractive. A numerical example illustrates the effectiveness of the proposed adaptive approach based on forwards-in-time method in comparison to backwards-in-time method.