Chain-Based Representations for Solid and Physical Modeling

TL;DR

This paper introduces a novel matrix-based representation of (co)chain complexes in computational meshes, enabling a deeper understanding of topology-preserving refinements through multilinear transformations.

Contribution

It presents a new, general representation of (co)chain complexes using the Hasse matrix, applicable to various cell complexes and mesh refinements.

Findings

Hasse matrix provides a compact representation of (co)chain complexes.

Mesh refinements correspond to multilinear transformations of the Hasse matrix.

The approach is applicable to diverse cell complexes without restrictions.

Abstract

In this paper we show that the (co)chain complex associated with a decomposition of the computational domain, commonly called a mesh in computational science and engineering, can be represented by a block-bidiagonal matrix that we call the Hasse matrix. Moreover, we show that topology-preserving mesh refinements, produced by the action of (the simplest) Euler operators, can be reduced to multilinear transformations of the Hasse matrix representing the complex. Our main result is a new representation of the (co)chain complex underlying field computations, a representation that provides new insights into the transformations induced by local mesh refinements. Our approach is based on first principles and is general in that it applies to most representational domains that can be characterized as cell complexes, without any restrictions on their type, dimension, codimension, orientability, manifoldness, connectedness.