Finitélelemmódszer

Finitélelemmódszer, often translated as Finite Element Method (FEM), is a numerical technique used to find approximate solutions to boundary value problems for partial differential equations. It is widely employed in engineering and computational mechanics for solving problems involving stress analysis, heat transfer, fluid flow, and electromagnetism. The core idea of FEM is to discretize a complex domain into a finite number of smaller, simpler subdomains called finite elements. These elements are typically simple shapes like triangles or quadrilaterals in two dimensions, or tetrahedra or hexahedra in three dimensions.

Within each finite element, the unknown function (which represents a physical quantity like displacement or temperature) is approximated using a set of simple basis functions, often polynomials. The coefficients of these basis functions are the nodal values, representing the unknown function at specific points called nodes, which are usually located at the vertices or edges of the elements. By assembling the equations derived from these local approximations across all elements, a global system of algebraic equations is formed. This system is then solved numerically to obtain the approximate values of the unknown function at the nodes. The accuracy of the solution generally increases with the refinement of the mesh, meaning a larger number of smaller elements. FEM is a powerful tool because it can handle complex geometries and various boundary conditions that are often intractable with analytical methods.