Enriching low-order finite elements by interpolation covers

Abstract

This dissertation presents an enriched finite element procedure based on the use of interpolation cover functions for low-order finite elements, namely, the 3-node triangular and 4- node tetrahedral elements. The standard linear finite element shape functions are coupled with interpolation cover functions over patches of elements to enrich the finite element approximation space and obtain higher convergence of the finite element scheme. Throughout the thesis, we describe the fundamentals of the enriched scheme used for analyses of solid mechanics and heat transfer in solids, but these can be directly applied for other solutions. The cover functions not only capture steep variations of a solution variable but also smooth out discontinuous inter-element quantities such as, for example, stress jumps. The scheme gives similar convergence rates as the equivalent higher-order standard finite elements but can be numerically more efficient due to the use of the simple finite element data structures. Since the enriched cover interpolations are compatible, the scheme provides flexibility to use different cover orders for different patches and efficiently increases solution accuracy without any local or global mesh refinement. Adaptive cover interpolation procedures can be used based on an evaluation of the gradient of the computed field variables. This study presents a fully automatic adaptive interpolation procedure where an error indicator is introduced to apply appropriate local cover orders at the finite element mesh nodes to efficiently improve the accuracy of the solution. The power of the adaptivity is illustrated by two- and three-dimensional simulation examples.