Fractured Meshes

TL;DR

This paper introduces generalized meshes for discretizing PDEs in complex geometries, allowing overlapping elements and flexible adjacency, with a focus on fractured domains and a novel mesh construction algorithm.

Contribution

It presents the concept of generalized meshes, a new mesh type for fractured geometries, and an algorithm to construct virtually inflated meshes for boundary representation.

Findings

Generalized meshes handle non-regular geometries with overlapping elements.

The proposed algorithm constructs two-sided meshes for fractures.

Discrete differential forms are characterized on these meshes.

Abstract

This work introduces ``generalized meshes", a type of meshes suited for the discretization of partial differential equations in non-regular geometries. Generalized meshes extend regular simplicial meshes by allowing for overlapping elements and more flexible adjacency relations. They can have several distinct ``generalized" vertices (or edges, faces) that occupy the same geometric position. These generalized facets are the natural degrees of freedom for classical conforming spaces of discrete differential forms appearing in finite and boundary element applications. Special attention is devoted to the representation of fractured domains and their boundaries. An algorithm is proposed to construct the so-called {\em virtually inflated mesh}, which correspond to a ``two-sided" mesh of a fracture. Discrete $d$-differential forms on the virtually inflated mesh are characterized as the trace space of discrete $d$-differential forms in the surrounding volume.