General concepts » Polytopes

Understanding the geometric elements in Rodin.

Introduction

In finite element analysis, a polytope is a geometric primitive that forms the building block of a mesh. A mesh is a collection of polytopes of various dimensions that together partition a domain .

In Rodin, polytopes are represented by the Polytope class and its associated Polytope::Type enumeration.

Polytope Dimensions

Polytopes exist in different dimensions, each with a specific role in the mesh:

Dimension	Name	Role	Example
0	Vertex (Point)	Mesh nodes	Corner of a triangle
1	Edge (Segment)	Connections between vertices	Side of a triangle, boundary in 2D
2	Face	Surface elements	Triangle, Quadrilateral
3	Cell	Volume elements	Tetrahedron, Hexahedron, Wedge

The cells are the highest-dimensional polytopes in a mesh and correspond to the elements over which the finite element approximation is constructed. In a 2D mesh, the cells are faces (triangles or quadrilaterals); in 3D, the cells are volumes.

Available Polytope Types

The Polytope::Type enumeration defines the available element types:

Type	Dimension	Vertices	Description
`Polytope::Type::Point`	0	1	A single vertex
`Polytope::Type::Segment`	1	2	A line segment (edge)
`Polytope::Type::Triangle`	2	3	A triangular element
`Polytope::Type::Quadrilateral`	2	4	A quadrilateral element
`Polytope::Type::Tetrahedron`	3	4	A tetrahedral element
`Polytope::Type::Hexahedron`	3	8	A hexahedral (brick) element
`Polytope::Type::Wedge`	3	6	A prismatic (wedge) element

Using Polytopes for Mesh Generation

The polytope type determines the element shapes when generating uniform meshes via Mesh::UniformGrid():

#include <Rodin/Geometry.h>
using namespace Rodin::Geometry;

Mesh mesh;

// 2D meshes
mesh = mesh.UniformGrid(Polytope::Type::Triangle, {16, 16});
mesh = mesh.UniformGrid(Polytope::Type::Quadrilateral, {16, 16});

// 3D meshes
mesh = mesh.UniformGrid(Polytope::Type::Tetrahedron, {8, 8, 8});
mesh = mesh.UniformGrid(Polytope::Type::Hexahedron, {8, 8, 8});
mesh = mesh.UniformGrid(Polytope::Type::Wedge, {8, 8, 8});

Sub-Entities

Every polytope has sub-entities of lower dimension. For example, a triangle has 3 edges and 3 vertices; a tetrahedron has 4 faces, 6 edges, and 4 vertices. These relationships are captured by the Connectivity class.

To access sub-entity relationships, first compute the connectivity:

mesh.getConnectivity().compute(1, 2); // edges → cells
mesh.getConnectivity().compute(0, 2); // vertices → cells

Iterating Over Polytopes

You can iterate over polytopes of a given dimension using the mesh iterator interface:

// Iterate over cells (highest dimension)
for (auto it = mesh.getCell(); !it.end(); ++it)
{
  auto idx = it->getIndex();
  auto attr = it->getAttribute();
  // ... process cell ...
}

// Iterate over vertices
for (auto it = mesh.getVertex(); !it.end(); ++it)
{
  auto coords = it->getCoordinates();
  // ... process vertex ...
}

Choosing the Right Element Type

Problem	Recommended Type	Reason
General 2D PDEs	`Triangle`	Flexible, good for complex geometries
Structured 2D grids	`Quadrilateral`	Better accuracy on rectangular domains
General 3D PDEs	`Tetrahedron`	Handles complex 3D geometries
Structured 3D grids	`Hexahedron`	Higher accuracy per DOF