Extending h adaptivity with refinement patterns

TL;DR

The work addresses the challenge of efficiently achieving high-fidelity solutions with $h$-adaptive FEM in complex geometries by introducing refinement patterns. It proposes a formal framework where a library of refinement patterns guides element subdivision, with affine and $L^2$-driven transforms ensuring consistency across sides and neighboring elements. The contributions include a detailed methodology for defining, loading, and using patterns within NeoPZ, a set of pattern-tools (RefPatternEquality, GetCompatibleRefPatterns, PerfectMatchRefPattern, RefineDirectional), and a pattern database, demonstrated through diverse geometry-driven examples. This approach improves mesh quality and adaptability while enabling scalable, pattern-driven refinements in practical simulations such as aerospace, fracture mechanics, and structural engineering.

Abstract

This contribution introduces the idea of refinement patterns for the generation of optimal meshes in the context of the Finite Element Method. The main idea is to generate a library of possible patterns on which elements can be refined and use this library to inform an h adaptive code on how to handle complex refinements in regions of interest. There are no restrictions on the type of elements that can be refined, and the patterns can be generated for any element type. The main advantage of this approach is that it allows for the generation of optimal meshes in a systematic way where, even if a certain pattern is not available, it can easily be included through a simple text file with nodes and sub-elements. The contribution presents a detailed methodology for incorporating refinement patterns into h adaptive Finite Element Method codes and demonstrates the effectiveness of the approach through mesh refinement of problems with complex geometries.

Figures (18)

• Figure 1: Topology of a triangular element composed of seven sides: three zero-dimensional sides (points) with local indices 0, 1, and 2; three one-dimensional sides (edges) with local indices 3, 4 and 5; and one two-dimensional side (face) with local index 6.
• Figure 2: Illustration of two neighboring elements through the side with global node indices 1 and 4.
• Figure 3: Circular data structure of neighboring elements. Black represents neighboring data structures between nodes and red represents neighboring data structures between edges.
• Figure 4: Description of a simple refinement pattern on a tetrahedron and the resulting elements.
• Figure 5: YF-17 aircraft mesh. (a) Perspective view and (b) rear view.