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Next: 3.2 Finite Boxes Up: 3.1 Finite Elements Previous: 3.1.2 Numerical Integration

3.1.3 Shape Functions

The essence of the finite element method, already stated above, is to approximate the unknown by an expression given as

$\displaystyle \bf u$ $\displaystyle \approx$ $\displaystyle \sum_{i=1}^{n}$Ni . ai = $\displaystyle \bf N \cdot a$     (3.24)

where Ni are the interpolating shape functions prescribed in terms of linear independent functions and ai are a set of unknown parameters. Under normal circumstances the variables are chosen to be identical with the values of the unknown function at the element nodes, thus making
ui $\displaystyle \equiv$ ai     (3.25)

Now consider to interpolate a constant function over the simulation domain. It is clear that a constant value of ai specified at all nodes must result in a constant value of $ \bf u$ which immediately implies that

$\displaystyle \sum_{i=1}^{n}$Ni = 1     (3.26)

at all points of the domain. The so defined shape functions are referred to as standard shape functions and are the basics of most finite element programs. Several extensions to the standard shape functions exist and the reader is advised to relevant literature [Sch80].

Figure 3.3: Linear and quadratic shape functions for one-dimensional elements

The used interpolation scheme is illustrated in Fig. 3.3 where the node points have to be multiplied with the shape functions to get the values inside the element. The order of interpolation of the shape function stipulates the accuracy of the element. Thus, there are two strategies to get high quality results (Fig. 3.4):

Figure 3.4: Approximation characteristics of linear and quadratic shape functions in case of h-refinement and p-refinement

The shape function itself can be calculated using a polynomial approach. For linear interpolation on a one-dimensional element with two nodes a polynomial of first order

u($\displaystyle \xi$) = $\displaystyle \alpha_{1}^{}$ + $\displaystyle \alpha_{2}^{}$$\displaystyle \xi$     (3.27)

can be used. To calculate the coefficients $ \alpha_{1}^{}$ and $ \alpha_{2}^{}$ the values at the element nodes can be inserted into (3.27)
$\displaystyle \alpha_{1}^{}$ = u1     (3.28)
$\displaystyle \alpha_{2}^{}$ = - u1 + u2     (3.29)

Substituting these results into (3.27) yields

u($\displaystyle \xi$) = u1(1 - $\displaystyle \xi$) + u2$\displaystyle \xi$ = u1N1($\displaystyle \xi$) + u2N2($\displaystyle \xi$)     (3.30)

The same principle can be used for all other orders of interpolation as well as for different dimensions. For higher orders of interpolation derivatives are additionally used to calculate the coefficients of the polynomial.

For a summary of shape functions in one, two, and three dimensions calculated for different interpolation orders refer to Appendix B.

next up previous
Next: 3.2 Finite Boxes Up: 3.1 Finite Elements Previous: 3.1.2 Numerical Integration
Mustafa Radi