C++ for Scientists - Technische Universität Dresden

230 CHAPTER 8. FINITE WORLD OF COMPUTERS
8.2.5 Finite Element Integration
In the theory of finite elements [?, ?], a continuous function space is projected onto a finite
function space P k , where the space P k is the space of polynomials up to the total order of k.
For many special cases, finite element integrals can be computed manually and added into
the source code of an application. This results in excellent run time performance but lacks
flexibility. For more general cases, e.g., general coefficients, they must be computed by numerical
integration at run-time. To prevent an ill-conditioned system matrix, orthogonal polynomials
have to be chosen as numerical integration weights. One possible type of polynomial is a
normalized Legendre polynomial [?]. Coefficients for such a polynomial Pk of order k can be
efficiently evaluated by using the recursion procedure:
P0(x) = 1 (8.1)
P1(x) = x
2j − 1
Pk(x) = x Pk−1(x) −
j
j − 1
j Pk−2(x) k ≥ 2
To use arbitrary p-finite elements (polynomial order [?, ?]) the numerical coefficients have to
be calculated either manually and inserted into the source code or determined numerically at
run time.
The polynomial library presented here is then used to store manually pre-calculated integration
tables at compile time (order 1-5). If the user requires higher order finite elements, numerical
coefficients are calculated at run time to any order.
8.3 A Loop and More
One of the important parts in computer science is repetation. A computer was made to do
exactly like this, programmable operations and repetations. To give a simple example, a for
loop is expressed by:
for (long i = 0; i < max counter; ++i)
{}
To give a real application of this concept, integration is used.
� b
a
f(x)dx
Several approximation schemes are also available:
� b
a
� b
a
f(x)dx ≈
f(x)dx ≈
b − a
6
f(a) + f(b)
2
�
f(a) + 4f
· (b − a)
� a + b
2
� �
+ f(b)