 # Finite Element Method (FEM) Transformation of a finite element graph consisting of rectangles to a square unit element (master element), taken from [*1].

## Numerical integration

For purposes of error minimization during the finite element discretization of complex and non-uniform model areas, arbitrary triangles and squares are used as finite elements (see figure here). Please note, that for triangles and also squares it is important to comply with specific rules including side lengths, angle and the ratio to the neighbour elements as well as with criteria such as the Delauny criterion.

These arbitrary element shapes make the calculation of the integrals more difficult, however, as for example for the coefficient matrix from Eq. 3-19 or 3-20. For this reason, a coordinate transformation is done, which maps the elements from a global coordinate system (x,y) to a so-called master element in a local coordinate system . The coordinate transformation for a square is exemplarily displayed in the figure to the right.

The master element is selected with thought of the numerical solving method used. If Gauss integration is used ( this method is used in the program RMA2) for rectangular elements, for example, a square master element of edge length 2 having its midpoint in the origin of the coordinate system and its edges parallel to the axes of the coordinate system is selected (see figure 1 here). The Gauss integration method is presumed to be used further on.

Although the coordinate transformation makes the wanted integral expression more complicated, the following simplifications are achieved by the introduction of a geometrically simple master element that is used in place of the actual elements:

 (1) The approximation functions can be obtained quiet easily and they are the same for each element since they are always applied to the master element. (2) The calculation of the integrals for the element is simplified by a common, simple element geometry. Commonly available numerical integration methods can be used.  