Page 230 - Basic Structured Grid Generation

Unstructured grid generation 219

takes the value unity at the ith vertex and zero at the other two. Over a particular triangle
with vertices numbered 1, 2, 3 in anti-clockwise manner, having known cartesian
co-ordinates (x 1 ,y 1 ), (x 2 ,y 2 ), (x 3 ,y 3 ), respectively, it is easy to show that a shape
function linear in x and y and taking the value 1 at the vertex 1 and zero at the
vertices 2 and 3 is given by
1
e
N (x, y) = (a 1 + b 1 x + c 1 y), (8.24)
1 e
2A
e
where a 1 = x 2 y 3 −x 3 y 2 , b 1 = y 2 −y 3 , c 1 = x 3 −x 2 ,and A is the area of the triangle.
e
e
Expressions for similar shape functions N and N which take the value 1 at the
2 3
vertices 2, 3, respectively, and zero at the other vertices may be immediately written
down by cyclic permutation of the suffixes in eqn (8.24). Over a typical triangular
element of the triangulation, we can then write the approximating solution as
e
e
e
e
e
e
e
φ = φ N + φ N + φ N , (8.25)
1 1 2 2 3 3
e
e
e
e
where now the coefficients φ , φ , φ represent the values of φ at the vertices.
1 2 3
As an example of a field equation, we take Poisson’s equation
∂ φ ∂ φ
2 2
κ + + Q (x, y) = 0 (8.26)
∂x 2 ∂y 2
where κ is a constant and in heat transfer problems Q(x, y) could be a heat source
function. The boundary conditions are taken to be that φ is specified on a part C φ of
the boundary and the normal derivative ∂φ/∂n (in the direction of the outward normal)
takes the value q (possibly a function of position) on the remaining part C q . This means
that at nodes on that part of the boundary of the triangulation which approximates C φ
the value of the coefficients φ i in eqn (8.23) is effectively specified.
Suppose that N of the nodes are in the interior of the triangulation or on the boundary
C q and that nodes with global vertex numbers N + 1,N + 2,...,m lie on that part of
the boundary which approximates C φ . Then instead of eqn (8.23) we can put
m N N

˜
˜
φ = φ i R i (x, y) + φ i R i (x, y) = φ 0 (x, y) + φ i R i (x, y), (8.27)
i=N+1 i=1 i=1
where in the first term the coefficients φ i are known from the boundary conditions.
This representation of the family of approximating functions has the same form as
eqn (8.20), since the roof functions R i (x, y) take the value zero at the boundary nodes
N + 1,N + 2,... ,m.
The Galerkin approach gives the integral form

∂ φ ∂ φ
2 ˜ 2 ˜
˜
R i L(φ) dx dy = R i κ 2 + 2 + Q (x, y) dx dy = 0,
D D ∂x ∂y
i = 1,... ,N, (8.28)
but to be able to deal with approximating functions with discontinuous slopes across
triangular edges, we need to transform the double integral by integration by parts (the