A C1-mapping based on finite elements on quadrilateral and hexahedral meshes

TL;DR

This paper introduces an algorithm for constructing smooth, conforming mappings for higher-continuity finite element meshes on quadrilateral and hexahedral grids, accommodating adaptive refinement and efficient data storage.

Contribution

It presents a novel algorithm for generating $H^2$-conforming mappings on quadrilateral and hexahedral meshes, including variants for orthogonal edges and adaptive mesh refinement.

Findings

Algorithm successfully constructs $H^2$-conforming mappings

Handles adaptive mesh refinement with nonconforming edges

Provides efficient data storage strategies

Abstract

Finite elements of higher continuity, say conforming in $H^2$ instead of $H^1$, require a mapping from reference cells to mesh cells which is continuously differentiable across cell interfaces. In this article, we propose an algorithm to obtain such mappings given a topologically regular mesh in the standard format of vertex coordinates and a description of the boundary. A variant of the algorithm with orthogonal edges in each vertex is proposed. We introduce necessary modifications in the case of adaptive mesh refinement with nonconforming edges. Furthermore, we discuss efficient storage of the necessary data.