Shape Functions are used in a method for interpolating between values. It lends itself most readily to computerised methods of interpolating higher order multi-variable problems, but we can start with a simple example, e.g. if we have a function where we know values:
| \(n\) | \(x_{n}\) | \(y_{n}\) |
|---|---|---|
| 0 | 0 | 7 |
| 1 | 1 | 9 |
We can write an expression to interpolate between these values
\[\begin{align} y & = \sum^{n} y_{n} S_{n}(x) \\ y & = 7 \times S_{0}(x) + 9 \times S_{1}(x) \end{align}\]
where \(S_{0}(x)\) and \(S_{1}(x)\) are Shape Functions. The choice of function to use is mostly arbitrary, but will have the properties:
\[\begin{align} S_{n}(x_{m}) & = 1\textrm{ where } n = m \\ & = 0\textrm{ where } n \neq m \end{align}\]
For the above example, we could select Shape Functions:
\[\begin{align} S_{0}(x) & = 1 - x \\ S_{1}(x) & = x \end{align}\]
and then evaluate the expression:
\[ y = 7(1 - x) + 9x \]
Now we can use this expression to interpolate between the values in the table, e.g. \(x = 0.3\)
\[\begin{align} y & = 7(1 - 0.3) + 9 \times 0.3 \\ y & = 7.6 \end{align}\]
Consider a function \(f(x, y)\) where we know values on a grid:
| \(n\) | \(x_{n}\) | \(y_{n}\) | \(F_{n}\) |
|---|---|---|---|
| 0 | 0 | 0 | f(0, 0) |
| 1 | 0 | 1 | f(0, 1) |
| 2 | 1 | 0 | f(1, 0) |
| 3 | 1 | 1 | f(1, 1) |
We can write an expression to interpolate between these values
\[ f(x, y) = \sum_{n=1}^{4} F_{n} S_{n}(x, y) \]
and select shape functions of the form:
\[ S_{n}(x, y) = A_{n}x + B_{n}y + C_{n}xy + D_{n} \]
with coefficients:
| \(n\) | \(A_{n}\) | \(B_{n}\) | \(C_{n}\) | \(D_{n}\) |
|---|---|---|---|---|
| 0 | -1 | -1 | 1 | 1 |
| 1 | 0 | 1 | -1 | 0 |
| 2 | 1 | 0 | -1 | 0 |
| 3 | 0 | 0 | 1 | 0 |