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

Abstract

The aim of this contribution is a comparison of different mapping techniques usually applied in the field of hierarchical adaptive FE-codes. The calculation of mechanical field variables for the modified mesh is an important but sensitive aspect of adaptation approaches of the spatial discretization. Regarding non-linear boundary value problems 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. The kind of mapping of the field variables affects the convergence behaviour as well as the costs of an adaptive FEM-calculation in a non-negligible manner. In order to improve the stability as well as the efficiency of the adaptive process a comparison of different mapping algorithms is presented and evaluated. Within this context, the mapping methods taken into account are -an element-oriented extrapolation procedure using special shape functions, -a patch-oriented transfer approach and, -the allocation of nodal history-dependent state (field) variable data using a supplementary integration of the material law at the nodes of the elements.

A Robust Numerical Procedure for the Thermomechanical Flow Simulation of Friction Stir Welding Process Using an Adaptive Element-Free Galerkin Method

“…The definition of n on a discretized contact surface can be found in the literature [41]. Substituting (50) into (48) yields…”

“…An adaptive solution is then completed by mapping the solution variables between the old and new spatial discretization. Apparently, the element or patch-based remap procedures [50] are not suitable for the meshfree method. In this study, we introduce a second-order accurate projection operation based on the meshfree convex approximation [29] to transfer the solution valuables.…”

Lie-group interpolation and variational recovery for internal variables

“…They also note the desirable properties of the L 2 minimization for mapping, such as that it requires solving a linear system of equations with a corresponding matrix that is positive-definite and sparse. Bucher et al [5] derive transfer operators that extrapolate the internal variables at integration points to nodes in the source mesh using serendipity interpolation functions.…”

“…The transfer of field data from one mesh to another is a need that arises frequently within the context of mesh adaption in the finite element method [5,14,26,27,31,33,34]. Fields that are available at the nodes may be directly mapped by using the corresponding interpolation functions.…”

Efficient 3D data transfer operators based on numerical integration

“…), by moving least squares approximation (as in ), by the superconvergent patch recovery technique or by alternative techniques (e.g. ). Finally, after the transfer of the nodal values with the ETM, the value at the quadrature points of the new mesh is interpolated from the nodal values of the new mesh.…”