next up previous contents
Next: Interpolation of the velocity Up: Post-processing Previous: Post-processing

Interpolation of scalars in 2D

  The scalar unknowns are positioned in the centroids of the cells. In order to interpolate these values to the vertices a weighted mean value of the four surrounding cells is used. Figure 11.1 sketches a typical example.

  
Figure 11.1: Vertex i,j with four surrounding cells and mapping of quadrilateral formed by centroids on to a square.

In this figure point i, j is the vertex in which the interpolated values must be computed. This point is part of 4 cells with centroids 11, 21, 12 and 22. In order to compute the interpolated value, the quadrilateral spanned by the 4 centroids is mapped onto a unit square (0, 1) (0,1) by a bilinear mapping as is usual in finite elements.
So

 

with
The value of the scalar in (x, y) is computed by

 

To evaluate (11.2) it is necessary to know the value of in point (x, y). This value can be computed from (11.1) by solving this system of non-linear equation with a Newton-Raphson method.
Define

 

The Newton-Raphson method can be written as:

 

Since Newton is a fast converging process, the maximal number of iterations is restricted to 5. At this moment the iterations is stopped if .
From (11.3) it follows that:

 

and

  

With respect to the boundary points it is not longer possible to use an interpolation. In that case an extrapolation is used. Figure 11.2 shows the four points that are used to compute the value at an under boundary.

  
Figure 11.2: Cells that are used to extrapolate the scalar value at the under boundary i,j.



Tatiana Tijanova
Wed Mar 26 10:36:42 MET 1997