Mesh (Discretization)

Mesh generation (discretization) is the process of dividing a continuous geometric domain into a finite set of nodes and elements so that a partial differential equation can be approximated by a system of algebraic equations. In structural FEA, the displacement field is interpolated within each element from nodal values using shape functions; the element stiffness matrix is formed by integrating the strain-displacement matrix (B) and the material constitutive matrix (D) over the element volume. A node is a discrete point where elements connect and where degrees of freedom (translations, rotations, or temperatures) are computed. Common element types include 1D line/beam elements, 2D triangles (3-node CST, 6-node LST) and quadrilaterals (4-node, 8-node), and 3D tetrahedra (4-node, 10-node) and hexahedra (8-node, 20-node). Mesh density controls accuracy: smaller elements reduce discretization error but increase the number of equations and solution time. Refinement strategies include h-refinement (reduce element size h), p-refinement (increase polynomial order p), and r-refinement (move nodes without changing connectivity). A structured mesh has regular, ordered connectivity (typically quads or hexes) and is efficient for simple geometries; an unstructured mesh has irregular connectivity (triangles or tetrahedra) and can fill complex geometry automatically. Mesh quality metrics include aspect ratio, skewness, Jacobian ratio, and warping. Poor-quality elements cause ill-conditioned stiffness matrices, inaccurate stress gradients, and solver convergence problems. A mesh convergence study refines the mesh until a monitored result (e.g., peak von Mises stress or tip deflection) changes by less than a chosen tolerance, confirming that the solution is mesh-independent.