Discretization and Meshing
Discretization breaks a continuous region into finite pieces so a computer can solve physics on it; the mesh is that decomposition.
From continuous to countable
Physical fields — temperature, magnetic flux, stress — vary continuously through space. A computer can only track finitely many numbers, so the region is divided into small cells or elements, and the field is represented by values at nodes. This division is the mesh.
Kinds of mesh
- Structured grids: regular arrays, simple and fast, best for simple geometry.
- Unstructured meshes: triangles or tetrahedra that conform to complex shapes.
- Adaptive meshes: fine where the solution changes rapidly, coarse elsewhere.
Resolution and error
A finer mesh captures more detail but costs more to compute. Discretization error shrinks as the mesh refines, ideally at a known rate. This is why a mesh-convergence study — solving on progressively finer meshes until the answer stops changing meaningfully — is a standard verification step.
Where resolution matters most
In a fusion machine, gradients are steepest near plasma edges, magnet windings, and material interfaces. Meshes are refined there so that steep gradients, such as heat flux onto a wall, are resolved without wasting cells in quiet regions.
Mesh quality
Badly shaped elements — too thin, too skewed — degrade accuracy and can destabilize a solver even when the mesh is fine. Checking element quality is part of preparing any serious simulation, because a beautiful physics model on a poor mesh still gives poor answers.