finitelyelementtimenetelmät

Finitely elementtimenetelmät, often abbreviated as FEM or in English as Finite Element Method, is a numerical technique used for finding approximate solutions to boundary value problems. It is a powerful tool for solving complex engineering and mathematical physics problems that are difficult or impossible to solve analytically. The core idea behind FEM is to divide a large, complex domain into smaller, simpler, and interconnected subdomains called finite elements.

These finite elements are typically simple geometric shapes, such as triangles or quadrilaterals in two dimensions, or tetrahedra or hexahedra in three dimensions. Within each element, the unknown function, which represents the physical phenomenon being studied (e.g., temperature, stress, displacement), is approximated by a simpler function, often a polynomial. These simpler functions are defined by a set of unknown coefficients, usually corresponding to the values of the function at specific points called nodes, which are located at the vertices and sometimes along the edges or faces of the elements.

The governing differential equations that describe the physical problem are then reformulated in a weak or variational form. By applying this weak form to each finite element and assembling the results from all elements, a system of algebraic equations is created. This system of equations relates the unknown nodal values. Solving this system of equations yields the approximate values of the unknown function at the nodes of the discretized domain. From these nodal values, the behavior of the system within each element and across the entire domain can be determined. FEM is widely applied in fields such as structural analysis, heat transfer, fluid dynamics, electromagnetics, and biomechanics.