Computational Modeling and Visualization of Physical Systems with Python by Jay Wang

Author:Jay Wang
Language: eng
Format: epub
Publisher: Wiley
Published: 2018-10-03T16:00:00+00:00

Figure 7.7: The triangular meshes in a domain (unshaded).

Figure 7.8: The tent functions at nodes 1 and 3.

The tent function is continuous, though not smooth, across the element boundaries (the ridges). Because the basis functions are zero outside their own element boundaries, the value of ϕ1 at a given point (x, y) can be found on one, and one only, basis function of the element containing that point.

This is true for all points except those on the ridges and the exact node point for which the tent function is defined. To avoid double counting, we require that only one term in the sum (7.12) is actually selected: the basis function of the first element determined to contain the point. The value of the tent function at the node is unity by definition.

Likewise, tent functions at other nodes can be built in the same way. For example, the tent function at node 3 (Figure 7.8, bottom) has two basis functions, one each from elements e1 and e2. There are six such tent functions, one for each node. We note that two tent functions overlap each other only if the two nodes are directly connected by an edge. For instance, the two tent function shown in Figure 7.8 overlap because nodes 1 and 3 are connected by the common edge between e1 and e2. In contrast, the tent functions at node 3 and node 5 would not overlap.

Suppose we are given the values of a function at the nodes ui, and we wish to interpolate the function over the whole domain. Then from Eq. (7.9), we just need to run the sum over all the tent functions thus defined.

To see it in action, we show the interpolation of the function f(x, y) = sin(πx) sin(πy) over the unit square in Figure 7.9. The meshes are shown in the left panel, and the function (wireframe) and its interpolations (solid pyramids) in the right panel.

Two different sets of meshes are used. In the first instance, the domain is divided into eight elements, with one internal node and eight boundary nodes. The values of ui = f(xi, yi) are zero at the boundary nodes and 1 at the lone internal node (the center). Of the nine tent functions, only the one at the center contributes. The interpolation is therefore given by this tent function which is a six-sided pyramid. The front and back facets have been sliced off because all node values of elements e4 and e5 are zero. The interpolation is below the actual function over most of the domain, except at the other two edges along the diagonal (between e1/e2 and e7/e8).

The second set (Figure 7.9, second row) increases the number of meshes to 18. Now there are four internal nodes with the same value . Summing up the four tent functions gives us a flat-top pyramid. Again the front and back edges corresponding to elements e6 and e13 are sliced off, but by a smaller amount than before. There are no improvements over the other two edges.

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