maillages

Maillages, or meshes, are discretizations of a continuous geometric domain into a finite collection of elements connected at nodes. They are used to approximate spatial fields such as temperature, pressure, velocity, or displacement and to solve partial differential equations numerically, most commonly by the finite element method (FEM) or finite volume method (FVM).

In two dimensions, common elements are triangles and quadrilaterals; in three dimensions, tetrahedra, hexahedra (brick elements), prisms, and pyramids are used. Meshes can be structured, with regular connectivity, or unstructured, with irregular connectivity. Hybrid meshes that combine different element types are also common to balance accuracy and efficiency.

Mesh generation and quality control aim to produce discretizations that accurately represent the geometry while controlling numerical error and computation cost. Techniques include Delaunay triangulation, advancing-front methods, and octree-based approaches. Quality metrics such as minimum angle, aspect ratio, and element skewness guide refinement and smoothing. Adaptive mesh refinement adjusts the mesh locally where the solution exhibits sharp gradients or complex features.

Applications span engineering and science, including computational fluid dynamics, structural analysis, heat conduction, electromagnetics, and geophysical modeling. Beyond numerical simulation, the concept of maillages appears in network design and urban planning to model connectivity, such as road or communication networks, and in mesh networking, where devices form a decentralized, self-organizing topology.

Key challenges include meshing complex geometries and moving boundaries, preserving mesh quality during deformation, handling singularities, and generating large-scale meshes efficiently in parallel.