Variable-node element families for mesh connection and adaptive mesh computation

TLDR

Variable‑node finite element families are introduced to extend standard elements by allowing arbitrary numbers of nodes on edges, enabling slope discontinuities in two‑dimensional domains. The elements preserve linear interpolation between neighboring nodes, automatically generate shape functions on a master domain, satisfy the patch test with 2×2 Gauss integration, and relax the 1‑irregular node rule for adaptive mesh refinement. These elements provide flexibility for resolving nonmatching mesh problems such as mesh connection and adaptive refinement, and demonstrate improved accuracy and efficiency in several numerical examples.

Abstract

Variable-node finite element families, termed (4 + k + l + m + n)-node elements with an arbitrary number of nodes (k, l, m, and n) on each of their edges, are developed based on the generic point interpolation with special bases having slope discontinuities in two-dimensional domains. They retain the linear interpolation between any two neighboring nodes, and passes the standard patch test when subdomain-wise <TEX>$2{\times}2$</TEX> Gauss integration is employed. Their shape functions are automatically generated on the master domain of elements although a certain number of nodes are inserted on their edges. The elements can provide a flexibility to resolve nonmatching mesh problems like mesh connection and adaptive mesh refinement. In the case of adaptive mesh refinement problem, so-called "1-irregular node rule" working as a constraint in performing mesh adaptation is relaxed by adopting the variable-node elements. Through several examples, we show the performance of the variable-node finite elements in terms of accuracy and efficiency.