äärelliselementtimenetelmää

Äärelliselementtimenetelmä, often abbreviated as FEM or FDM, is a numerical technique used to find approximate solutions to boundary value problems governed by partial differential equations. It is widely employed in engineering and computational science to solve problems involving stress analysis, heat transfer, fluid flow, and electromagnetics. The core idea of FEM is to discretize a complex continuous domain into a finite number of smaller, simpler subdomains called finite elements.

Within each element, the unknown variables, such as displacement or temperature, are approximated by simple functions, typically polynomials. These element-wise approximations are then assembled to form a global system of algebraic equations that represents the entire domain. The solution to this system yields the approximate values of the unknown variables at discrete points called nodes, which are located at the boundaries of the elements.

The accuracy of the FEM solution depends on several factors, including the size and shape of the elements and the order of the approximation functions used. Smaller elements and higher-order polynomials generally lead to more accurate results but also increase computational cost. The method is particularly powerful for problems with complex geometries, irregular boundaries, and varying material properties, where analytical solutions are often impossible to obtain. FEM has become an indispensable tool for design and analysis in various industries.