Finite strain r-adaption based on a fully variational framework

Abstract

A novel r-adaptive finite element strategy based on a fully variational framework is presented. Provided the underlying physical problem is characterized by means of a minimization principle, the proposed method seeks, for a fixed number of nodes, for the best finite element interpolation depending on the nodal positions with respect to the deformed (x) as well as the undeformed (X) configuration, cf. [1]. The existence of a minimization problem does not represent a very strong restriction, since for many physical problems such as standard dissipative media an incremental potential can also be recast, cf. [2]. While minimizing the potential considered by fixing the nodes within the undeformed configuration corresponds to classical NEWTONian mechanics, a variation with respect to (X) is associated with ESHELBY mechanics, cf. [3]. However, in contrast to the simplicity of the concept, its numerical implementation is far away from being straightforward. According to [4], the resulting system of equations is highly singular and hence, standard optimization strategies cannot be applied. In this paper, a viscous regularization is used. This approach is designed to render the minimization problem well-posed while leaving its solutions unchanged. Obviously, relocating the nodes within the undeformed configuration by fixing the triangulation (the connectivity) may lead to strong topological constraints. As a consequence, an energy based re-meshing strategy is advocated. Contrary to classical mesh-improvement methods based on geometrical quality measures, the novel concepts identifies local energy minimizers. That is, the energy of the new triangulation is always lower than that of the initial discretization. The performance of the resulting finite element model is demonstrated by fully three-dimensional examples.