High order transition elements: The xNy-element concept -- Part I: Statics

TL;DR

This paper introduces xNy-elements, a new compatible transition element for finite element analysis that enables coupling different mesh refinements and element types, improving local mesh refinement efficiency.

Contribution

It proposes the xNy-element, extending the pNh-element concept to allow coupling of different element sizes, shapes, and polynomial orders in FEM.

Findings

Demonstrates promising convergence behavior

Numerical analysis confirms effective local mesh refinement

Achieves improved accuracy with adaptive hp-refinement

Abstract

Advanced transition elements are of utmost importance in many applications of the finite element method (FEM) where a local mesh refinement is required. Considering problems that exhibit singularities in the solution, an adaptive hp-refinement procedure must be applied. Even today, this is a very demanding task especially if only quadrilateral/hexahedral elements are deployed and consequently the hanging nodes problem is encountered. These element types, are, however, favored in computational mechanics due to the improved accuracy compared to triangular/tetrahedral elements. Therefore, we propose a compatible transition element - xNy-element - which provides the capability of coupling different element types. The adjacent elements can exhibit different element sizes, shape function types, and polynomial orders. Thus, it is possible to combine independently refined h- and p-meshes. The approach is based on the transfinite mapping concept and constitutes an extension/generalization of the pNh-element concept. By means of several numerical examples, the convergence behavior is investigated in detail, and the asymptotic rates of convergence are determined numerically. Overall, it is found that the proposed approach provides very promising results for local mesh refinement procedures.