Page 229 - Basic Structured Grid Generation

218 Basic Structured Grid Generation

For further information, standard texts on finite element methods such as Hinton and
Owen (1979), Taylor and Hughes (1981), and George (1991) may be consulted.
Suppose that the field equation for a field quantity φ, subject to certain boundary
conditions, in a planar domain D (assumed simply connected here) is

L(φ) = 0 (8.19)
where L is a second order partial differential operator. Suppose also that the domain
has been triangulated, with n triangles and m points (nodes).
In the Method of Weighted Residuals, we seek an approximate solution for φ from
the family of functions
N

˜
φ = f 0 (x, y) + φ i f i (x, y), (8.20)
1
where usually the function f 0 is chosen to satisfy the given boundary conditions and the
f i s satisfy homogeneous boundary conditions in order to make φ satisfy the boundary
˜
conditions for any choice of the constant coefficients φ i . In addition, we choose a set
of N weighting functions W i (x, y) and impose the conditions

W i L(φ) dx dy = 0, i = 1, 2,...,N, (8.21)
˜
D
˜
to make L(φ) as close to zero as possible in some sense over the domain D.These
equations will generate a set of N equations for the N unknown constants φ i and hence
the best (in some sense) approximate solution from our original set.
In the special case of the Galerkin Method, the weighting functions are taken to be
identical to the approximating functions, which gives the set of equations

f i L(φ) dx dy = 0, i = 1, 2,... ,N, (8.22)
˜
D
˜
with φ given by eqn (8.20).
In the particular form of the finite-element method which is appropriate here for a
plane triangulated region, we look for approximate solutions in the form
m

˜
φ = φ i R i (x, y), (8.23)
i=1
where the φ i s are constant coefficients to be determined and the known R i (x, y) func-
tions are called roof functions, which are continuous and piecewise-linear in both x
and y, such that R i takes the value unity at the ith node and zero at all other nodes.
Each node of the triangulation is assigned a global number between 1 and m.More-
over, a node constitutes one of three vertices of a triangular element, and is assigned
a local number between 1 and 3, these numbers being ordered in an anti-clockwise
sense. This assigning of numbers, together with the numbering of triangular elements
and the listing of cartesian co-ordinates of all nodes, is carried out in the Delaunay
grid generation program delaunay1.f listed below.
e
Roof functions may be built up from linear shape functions N (x, y) defined over
i
e
each triangular element. Here the suffix refers to the local number of a vertex; N (x, y)
i