Finite Element Method (FEM)

Numerical integration (2)

It has to be emphasized that the coordinate transformation is done for calculation purposes of numerical integration only. The algebraic equations that have to be solved always refer to the wanted degrees of freedom (vertex values) in the global coordinate system.

The transformation of any element of the global coordinate system (x,y) to the master element of the local system  is described by the following, for example:

 

where  is the approximation function for the master element. In order to make this transformation function clearer, we can assume a square master element with linear approximation functions, and the local coordinates satisfy the relation 1 <=   <= 1 (see figure 1 here). This is called a rectangular Lagrange element with vertex count k=4. The approximation functions for a master element of this type can be derived according to Eq. 3-28. If these linear approximation functions are used, the straight line  =1 (local coordinate system) can be describe by the following equations in the global coordinate system:

This means that x and y are well-defined linear functions of  meaning that the transformation maps the straight line =1 also to a straight line in the global coordinate system.


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Hydrodynamics of Floods
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