Mesh design and reliability assurance in hybrid-Trefftz p-element approach

Abstract

This paper is concerned with the reliability of a uniform p-extension process based on the hybrid-Trefftz (HT) finite element model. In this model, introduced more than fifteen years ago, the assumed displacement field of the element a priori satisfies the governing differential equations of the problem, while the interelement continuity and the boundary conditions are enforced in an average weighted residual sense. One of the most important advantages of this concept is the existence, inside each element, of a large internal zone of super convergence where the errors are currently one to two orders of magnitude lower than those in the narrow perturbed zone along its boundary. The final aim of this paper is to present a very simple and efficient way of producing reliable results (displacement, internal forces) in the form of contour lines or other suitable graphical representation. To take full advantage of the HT elements, only the results of a regular grid of internal sampling points in the super convergent zone of the elements are used along with a post-processing approach known as ‘krigeing’. The assessment of reliability is based on the control of undesired displacement and traction jumps along the element interfaces rather than checking the smallness of any kind of global error measure. Special attention is paid to singularities associated with angular corners. In contrast to the traditional HT singularity calculation, involving the use of very accurate but costly to implement special purpose functions, use is made of a local mesh refinement. For a practical application of the HT p-element approach, general guidelines for prior design of the element meshes and the choice of the grids of internal sampling points are presented. The efficiency of the approach is illustrated on a series of examples of thin (Kirchhoff) plates in bending. In the last part of the paper some further possible improvements of the basic approach presented are briefly discussed. They include in particular the automatic identification of corner singularities and the quantitative reliability assessment of the smoothed results at nodes of the FE mesh.