Adaptivity and Mesh Generation in 2-D Finite Element Analysis

Abstract

Abstract A methodology for the automatic mesh generation of triangular and quadrilateral finite element discretizations for two-dimensional elasticity problems is proposed. The methodology is based on: i) an h-adaptive process with powerful mesh generator facilities capable of achieving meshes of specified density, ii) a general stress recovery technique developed for determining the element solutions at the nodes, and iii) an a posteriori error estimation. The h-adaptive process used is based on a complete mesh regeneration procedure which is guided by specified mesh requirements such as geometry definitions, boundary conditions, and space node functions to achieve an optimal refinement. This optimality condition is, as established by Zienkiewicz and Zhu, the mesh refinement with the least number of elements that yields a uniform strain energy norm error distribution in all elements. In the stress recovery process, the nodal values are assumed to belong to a polynomial expansion of the same complete order in the interpolation function basis used, which is valid over all elements adjoining a particular node. A least-squares fit of superconvergent sampling points existing in the path is used to obtain the recovered nodal point parameters for each element. These parameters are averaged to all elements adjoining the node of interest. The technique is simple and cost-effective, and the recovered nodal values of derivatives are superconvergent at the Gauss integration points, which are used as sampling points for quadrilateral elements. This condition is also achieved when centroid and mid-side points are used for triangular elements. The error estimation is done evaluating differences between the post-processed stress gradients and those from the finite element solutions. The energy error norm associated with stress field differences and the finite element predicted strain energy gives an effective error estimate which can be used for comparison with the process tolerance. The technique has been implemented and allows for a fully automatic numerical analysis under a specified global energy error norm. Numerical tests conducted with various planar element formulations illustrate that the proposed technique converges in fewer steps than with previous methods of adaptive mesh refinement.