A contribution to error estimation and mapping algorithms for a hierarchical adaptive FE-strategy in finite elastoplasticity

Abstract

The aim of this contribution is the presentation of an adaptive finite element procedure for the solution of geometrically and physically non-linear problems in structural mechanics. Within this context, the attention is mainly directed on the error estimation and hierarchical strategies for mesh refinement and coarsening in the case of finite elasto-plastic deformations. An important but sensitive aspect of adaptation approaches of the space discretization is the calculation of mechanical field variables for the modified mesh. Procedures of mesh refinement and coarsening imply the determination of strains, stresses and internal variables at the nodes and the Gauss points of new elements based on the transfer of the required data from the former mesh. In order to improve the efficiency as well as the convergence behaviour of the adaptive FE process an approach of data transfer primarily related to nodal values is presented. It is characterized by solving the initial value problem not only at the Gauss points but additionally at the nodes of the elements.

A Comparison of Mapping Algorithms for Hierarchical Adaptive FEM in Finite Elasto-Plasticity

“…On the other hand, the additional solution of initial value problems at the nodes undoubtedly involves a non-negligible numerical effort which however accrues only once at the end of each load step if the equilibrium conditions are satisfied. As in [15] this apparent discrepancy has not been discussed in detail we will close this gap in the present paper, and we will show that the complexity of the abovementioned transfer procedure is justified because a better convergence behaviour as well as an improved numerical stability are obtained in comparison with other mapping algorithms. Within this context, additionally -an element-oriented extrapolation procedure using special shape functions, and -a patch-oriented transfer approach have been implemented into the FE-code SPC-PM2AdNl in order to compare the effect of such different mapping strategies on the solution process.…”

“…For details of the calculation of the particular terms of error estimator (1) realized in the FE code SPC-PM2AdNl we refer to [15].…”

“…The material law finally is represented as a system of differential and algebraic equations (DAE) which has to be integrated at each Gauss point to solve the initial value problem. In [15] as well as in the present paper the authors consider exclusively isotropic and kinematic hardening.…”

“…Different strategies for their mapping to the Gauss points of newly created son elements have been developed. In [15] the authors presented a special mapping algorithm as a crucial point of their adaptive approach: In contrast to the majority of FE applications known from the literature the solution of the initial value problem is performed in SPC-PM2AdNl not only at the Gauss points but supplementary at the nodes of the elements. As for FE approaches usually only the displacement field is continuous over the element boundaries, this procedure leads to different values of the field variables at nodes pertaining to several elements.…”

“…near cracks, notches and in contact areas) and/or at the boundary between elastic and plastic zones. In a previous paper (see [15]) the authors presented details of this strategy concerning the description of the internal data structure of the FE code mentioned above, a brief discussion of the material model as well as more comprehensive comments on the error estimation and the algorithms of mesh refinement and coarsening.…”