Civil Engineering Reference
In-Depth Information
3
Programming Finite Element
Computations
3.1
Introduction
In Chapter 2, the finite element spatial discretisation process was described, whereby partial
differential equations can be replaced by matrix equations which take the form of linear
and non-linear algebraic equations, eigenvalue equations, or ordinary differential equations
in the time variable. The present chapter describes how programs can be constructed in
order to formulate and solve these kinds of equations.
Before this, two additional features must be introduced. First, we have so far dealt only
with the simplest shapes of elements, namely lines and rectangles. Obviously if differential
equations are to be solved over regions of general shape, elements must be allowed to
assume general shapes as well. This is accomplished by introducing general triangular,
quadrilateral, tetrahedral and hexahedral elements together with the concept of a coordinate
system local to the element.
Second, we have so far considered only a single element, whereas useful solutions will
normally be obtained by many elements, usually from hundreds to millions in practice,
joined together at the nodes. Also, various types of boundary conditions may be prescribed
which constrain the solution in some way.
Local coordinate systems, multi-element analyses, and incorporation of boundary con-
ditions are all explained in the sections that follow.
3.2 Local coordinates for quadrilateral elements
Figure 3.1 shows two types of plane 4-noded quadrilateral elements. The shape functions
for the rectangle (Figure 3.1(a)) were shown to be given by equation (2.60), namely
N
1
=
(
1
−
y/b)
and so on. If it is attempted to construct similar shape functions in
the “global” coordinates
(x, y)
for the general quadrilateral (Figure 3.1(b)), rather complex
−
x/a)(
1
