# CompScienceMeshes.jl

Meshes, charts, and differential geometry for finite and boundary element solvers.

## Installation

In addition to the dependencies declared in REQUIRE, this package relies for some of its functionality on gmsh. Make sure gmsh is installed and on the system path if you require this functionality.

## Introduction

This package provides the geometric framework to facilitated the construction of finite element spaces and the assembly of matrices stemming from the discretisation of local (differential) and global (integral) operators on those finite element spaces.

The package roughly contains three components:

• Meshes: allowing for the (almost) linear construction of connectivity matrices. A default implementation is provided but the algorithms should be easily extendable to user defined mesh structures. It is very common, for example, that mesh data structures contain problem specific information (local elasticity, permittivity, boundary conditions). User can use those enriched structures if they extend a limited number of functions.

• Charts: a concept designed after the differential geometric concept of a chart on a manifold. It allows for the construction of points in Euclidian space from a set of parameters and the other way around.

• Neighborhoods: a concept designed after the derivative of a chart as a map from the parametrising vector space to the tangent space of a point of the manifold. It allows querying for tangents, normal, and the Jacobian determinant for use in integration routines.

## Mesh Interface

This package introduces a minimalistic mesh interface and a standard implementation CompScienceMeshes.Mesh. The interface is defined by the semantics of the following functions:

### Functions to query a mesh for its characteristics

CompScienceMeshes.dimensionMethod
dim = dimension(mesh)

Returns the dimension of the mesh. Note that this is the dimension of the cells, not of the surrounding space.

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dimension(simplex)

Return the manifold dimension of the simplex.

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CompScienceMeshes.numverticesFunction
numvertices(mesh)

Returns the number of vertices in the mesh.

Note: this is the number of vertices in the vertex buffer and might include floatin vertices or vertices not appearing in any cell. In other words the following is not necessarily true:

    numvertices(mesh) == numcells(skeleton(mesh,0))
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### Functions to iterate over the mesh' cells and the underlying point set

CompScienceMeshes.skeletonFunction
skeleton(mesh, dim)

Returns a mesh comprising the dim-dimensional sub cells of mesh. For example to retrieve the edges of a given surface mesh,

edges = skelton(mesh, 1)
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skeleton(pred, mesh, dim)

Like skeleton(mesh, dim), but only cells for which pred(cell) returns true are withheld.

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CompScienceMeshes.connectivityFunction
connectivity(faces, cells, op=sign)

Create a sparse matrix D of size numcells(cells) by numcells(faces) that contiains the connectivity info of the mesh. In particular D[m,k] is op(r) where r is the local index of face k in cell m. The sign of r is positive or negative depending on the relative orientation of face k in cell m.

For op=sign, the matrix returned is the classic connectivity matrix, i.e. the graph version of the exterior derivative.

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