Page 231 - Basic Structured Grid Generation

220 Basic Structured Grid Generation

Divergence Theorem, or Green’s formula) to

∂R i ∂φ ∂R i ∂φ ∂φ
˜ ˜ ˜
− κ + − R i Q dx dy + κR i ds = 0, (8.29)
D ∂x ∂x ∂y ∂y C ∂n
where the contour integral is taken anti-clockwise around the boundary C of D.Ifwe
replace ∂φ/∂n by q on C q ,and R i by zero on C φ , we obtain
˜

∂R i ∂φ ∂R i ∂φ
˜ ˜
κ + dx dy = R i Q dx dy + κR i q ds, i = 1,. ..,N.
D ∂x ∂x ∂y ∂y D C q
(8.30)
Using eqn (8.27), this becomes, for i = 1,...,N,

N
∂R i ∂R j ∂R i ∂R j
κ + dx dy φ j
D ∂x ∂x ∂y ∂y
j=1

˜ ˜
∂R i ∂φ 0 ∂R i ∂φ 0
= R i Q dx dy + κR i q ds − κ + dx dy.
D C q D ∂x ∂x ∂y ∂y
(8.31)
The contribution of a triangular element to the terms

∂R i ∂R j ∂R i ∂R j
κ + dx dy
D ∂x ∂x ∂y ∂y
in eqn (8.31) is then, using eqn (8.25),
3 e e e e 3
i
∂N ∂N j ∂N ∂N j e e e
i
κ + dx dy φ = K φ , say, (8.32)
ij j
j
e ∂x ∂x ∂y ∂y
j=1 j=1
where K e is clearly symmetric and can be regarded as the element stiffness matrix.
ij
By eqn (8.24) we can write, in the present case,
κ κ

e
K = (b i b j + c i c j ) dx dy = (b i b j + c i c j ). (8.33)
ij e 2 e
e (2A ) 4A
For a given triangulation, reducing eqns (8.31) to a global matrix equation
N

K ij φ j = F i (8.34)
j=1
involves first the evaluation of the stiffness matrices for all elements and then assem-
bling and adding them into a large N ×N matrix K ij . This is usually a straightforward
task once a connectivity table has been established listing the global and local ver-
tex numbers for each element in the triangulation. This enables global numbers to be
made to correspond to the row (and column) numbers of K ij so that the 3 × 3entries
K e can be added in the correct places. The ‘force’ terms F i can be calculated from
ij
the RHS of eqn (8.31) using the values of Q and q at appropriate nodes. Finally a