finiteelementti

Finiteelementti, commonly referred to as the finite element method (FEM), is a numerical technique for solving boundary value problems governed by partial differential equations. It works by subdividing a complex domain into a mesh of smaller elements and approximating the unknown field with simple polynomials defined locally on each element. A variational (weak) formulation underpins the method, yielding a system of algebraic equations.

The typical workflow includes creating a mesh, choosing element shapes and interpolation (shape) functions, deriving the element equations, assembling a global stiffness matrix and load vector, applying boundary conditions, and solving the resulting linear or nonlinear system. Time-dependent problems add a time-integration step, often using implicit or explicit schemes, with results post-processed to obtain quantities of interest.

Common elements are one-dimensional line elements, two-dimensional triangles and quadrilaterals, and three-dimensional tetrahedra and hexahedra. Elements can be linear or higher-order (quadratic, cubic) with various families (Lagrange, Hermite). Mesh quality, refinement strategies (h-, p-, hp-refinement), and isoparametric mapping influence accuracy. There are also mixed and stabilized formulations for challenging problems.

Applications span structural mechanics, heat and mass transfer, fluid flow, acoustics, electromagnetics, and geophysics. FEM is widely used in engineering design, simulation, and research, often via specialized software packages such as ANSYS, ABAQUS, or COMSOL. While powerful, FEM requires careful discretization, boundary condition specification, and error assessment to ensure convergence and reliability.