Mesh Generation for hp-Type Finite Element Analysis of Plates

2nd Symposium on Trends in Unstructured Mesh Generation, University of Colorado, Boulder, August 1999

MESHING
RESEARCH
CORNER

Informationsverarbeitung im Konstruktiven Ingenieurbau, Universit,t Stuttgart, Pfaffenwaldring 7, 70550 Stuttgart, Germany
stefan.holzer@po.uni-stuttgart.de

Abstract
It is possible to obtain exponential convergence in error norm in the numerical analysis of plate bending problems based on Reissner-Mindlin theory when p- extension is performed on a mesh which takes proper care of nonsmooth influences in the exact solution. Emphasis is on good a priori mesh design.

Proper mesh design startes with coarse meshing. This means in essence that the required mesh density is mainly governed by the edge size of the original structure. Curved edges will be mapped by blending function techniques and therefore pose no extra problem.

We discuss how to generate a mesh that is "purely quadrilateral and as coarse as possible". The basic strategy is advancing front meshing, combining two triangular elements into one quadrilateral "on the fly". In order to control the meshing algorithm, an additional uniform fine triangular background mesh ("carelessly" created) is used which provides the required local mesh densities. A smooth density distribution is generated by solving an auxiliary problem on the background mesh. The auxiliary problem is a Dirichlet problem of the Laplacian where the boundary data correspond to the required edge size on the boundary. This problem is solved by a fast finite element analysis using constant strain triangle elements. In addition to the local mesh density, this auxiliary problems provides mesh density gradients as well, which are essential in determining mesh gradation towards areas of small element size, such as around small holes or other tiny features.

In addition to serving as a finite element mesh for solving the auxiliary mesh density problem, the background mesh serves at the same time as a data evaluation mesh for the result presentation and superposition of the original plate bending problem (primary problem). Therefore, the (small) amount of computer time required for generating this mesh is not lost labour, but the background mesh serves a dual purpose.

Point singularities require additional point refinements. This part of the problem is standard.

However, in addition to point singularities, the Reissner-Mindlin theory is characterized by boundary layer effects, most significantly a boundary layer in the shear force on soft simply supported and free edges. Anisotropic mesh refinement is essential for regaining exponential convergence under these circumstances. Furthermore, combined point and anisotropic mesh refinements are necessary in corners of the domain. We discuss a simple, yet efficient and effective strategy for introducing all these refinement features after the coarse meshing.

We give ample computational evidence that the meshes generated by the algorithm proposed are indeed suitable for gaining faster-than-algebraic convergence of the global error in energy norm. In p-elements, element distortion is much less a source of quality degradation than in low-order elements.

The lecture will include a live presentation of the techniques described, demonstrating the high efficiency of the approach.