 # Finite-Element-Method (FEM)FEM the all-round talent of numerics

The Finite-Element-Method (FEM) is a physically motivated numerical method with which partial differential equations can be solved approximately.

## Partial differential equations

With partial differential equations, the stresses of the material points of a solid body are formulated, among other things, as a function of the movements of the points surrounding them. They can be used to develop powerful mathematical models that realistically depict the deformation behaviour of solids. If the deformation behaviour is to be investigated under consideration of the mass inertia, temporal derivatives concerning the displacement field are to be approximated as well as spatial derivatives.

## Discretization in space and time

In many finite element programs, FEM is only used for spatial approximation. the Newmark-beta-method or the Generalized-alpha-method is often used for temporal approximation . Both are collocation methods which are assigned to the class of finite difference methods. They are used to evaluate the differential equations of structural mechanics at discrete times. At this point it is noted, that FEM can also be used in the time domain. However, the application of the space-time finite element method leads to very large systems of equations which have to be solved numerically. Therefore, mixed methods are often used, where only the spatial approximation determines the number of free values.  In this context, the finite element method should be in the first place, so that reference is made to the relevant literature for collocation procedures.

## Approximation of geometry and physics

According to FEM, the shapes of complex structures are approximated with a variety of geometric primitives, called finite elements. Particularly suitable are cube-shaped elements and tetrahedra. In addition to the geometry, the spatial courses of the movements and the resulting stresses are approximated with polynomial approaches. The degrees of freedom of the approaches are assigned to the finite element nodes, which are distributed regularly, lying on the edges of the finite elements, ensure the complete structure of geometric bodies and the continuity in the displacement processes.

## Approximate solutions

The quality of approximate solutions for the movements of structures can be improved either by refining the finite element mesh or by selecting higher polynomial approaches in the finite elements. Related to the structural mechanics, finally it should be noted, that the underlying differential equations, with the application of FEM, are solved in weak form not at the single material points but over the whole spatial area, following the Lagrange formalism.