Finite Element Method (FEM)

Numerical integration (3)

Before we go into deeper detail of coordinate transformation, a few terms have to be explained. We recall the approximation for the solution of the wanted variable from Eq. 3-5 for this. It contains approximation functions just as in Eq. 3-36.

  • If an approximation of the same order (k=n) is chosen for the transformation (geometry) and the wanted variable, the finite elements are called isoparametric.
  • If the order of the approximation for the geometry is bigger than the order of the wanted variable (kn), the elements are superparametric, and in the opposite case they are subparametric [*1], [*2].

Only isoparametric finite elements are used in the context of this chapter.

Our knowledge of coordinate transformation that we have now enables us to recognize that

  • a different shape (e.g. triangle or rectangle) or
  • a different order (e.g. linear or square) of the master element

leads to different transformations and thus to different representations of the elements in the global coordinate system. It is important that the transformation does not introduce unwanted gaps or overlapping elements into the finite element graph. In order to set up appropriate requirements for the transformation, we have a look at the elements of the coefficient matrix from Eq. 3-20.


The integrand that not only contains functions but also derivatives with respect to the global coordinates x and y, now has to be expressed in dependence of  and .

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Hydrodynamics of Floods



*1 Helmig (1996)

*2 Reddy et al. (1994)



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