Meshing

Gmsh

Three-dimensional finite element mesh generator with a built-in CAD engine and post-processing facilities. Provides fast, reliable meshing for complex 2D and 3D geometries.

C++ / Python

CAD + Meshing

OpenCASCADE

Open Source (GPL)

Supported Element Types

Line (1D)

Order: 1st–5th

Triangle

Order: 1st–5th

Quadrilateral

Order: 1st–2nd

Tetrahedron

Order: 1st–5th

Hexahedron

Order: 1st–2nd

Prism

Order: 1st–2nd

Pyramid

Order: 1st–2nd

Point (0D)

Order: —

Key Features

CAD Engine

Built-in constructive solid geometry (CSG) kernel for creating 2D and 3D models. Boolean operations (union, intersection, difference) on geometric primitives. Supports STEP, IGES, BREP, and STL import/export via OpenCASCADE.

Mesh Generation

Generate structured, unstructured, and hybrid meshes. Supports tetrahedra, hexahedra, prisms, pyramids, and arbitrary polyhedra. Algorithms include Delaunay triangulation, advancing front, and transfinite interpolation.

Mesh Quality & Adaptivity

Mesh quality optimisation with Laplacian smoothing and topological operations. Adaptive mesh refinement (h-refinement, p-refinement) based on error estimators. Anisotropic meshing for boundary layers.

Scripting & API

Full Python and C++ API for programmatic mesh generation. Gmsh scripting language (.geo files) for parametric model definitions. Integration with FEniCS, deal.II, and GetDP for solver coupling.

Mesh Quality Metrics

Gmsh evaluates element quality using shape metrics. For triangles, the standard metric is:

Where A is the triangle area and l_i are edge lengths. η = 1 for an equilateral triangle, η → 0 for degenerate elements.

Example: Geometry Script

plate_with_hole.geo

// Rectangle with circular hole — Gmsh .geo script
SetFactory("OpenCASCADE");

// Define rectangle
Rectangle(1) = {0, 0, 0, 10, 5};

// Define circle (hole)
Disk(2) = {5, 2.5, 0, 1.0};

// Boolean subtraction
BooleanDifference(3) = { Surface{1}; Delete; }{ Surface{2}; Delete; };

// Set mesh size
MeshSize{ PointsOf{ Surface{3}; } } = 0.3;

// Generate 2D mesh
Mesh 2;

Resources: Visit gmsh.info for official documentation. VELSTROM tutorials for meshing geospatial terrain models are under development.