FiniteElement

Finite element method (FEM) is a numerical technique for obtaining approximate solutions to boundary value problems for partial differential equations. It subdivides a complex domain into smaller, simple pieces called finite elements, such as triangles, quadrilaterals, tetrahedra, or hexahedra, which are connected at nodes. The unknown field is approximated by piecewise polynomial basis functions defined on the elements, and the global solution is assembled from the element contributions.

The method relies on a variational (weak) formulation of the governing equations. By multiplying the equations by test functions and integrating by parts, one obtains a system of equations in terms of nodal values. Discretization replaces the continuous field with a finite-dimensional approximation, producing a sparse system: for static problems, K u = f, and for dynamics, M u'' + C u' + K u = f, with appropriate boundary conditions.

Key concepts include mesh generation, degrees of freedom associated with nodes, and shape (basis) functions. Assembly combines element matrices into a global stiffness, mass, and damping matrix. Numerical integration, often Gaussian quadrature, and isoparametric mapping are used to handle curved geometries. The Galerkin approach is common, and many problems employ linear or nonlinear solvers and, for time-dependent cases, various time integration schemes. Adaptivity and error estimation guide mesh refinement.

FEM is widely used in engineering and physics for structural analysis, heat conduction and diffusion, acoustics, electromagnetics, and incompressible fluid flow. It supports 2D and 3D domains, linear and nonlinear materials, static and dynamic analyses, and steady-state or transient problems. A range of software libraries and commercial packages provides FEM capabilities, from mesh generation and assembly to solving large sparse systems.