Page 214 - Introduction to Computational Fluid Dynamics

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6.3 UNSTRUCTURED MESHES
where AP occupies the diagonal positions and the neighbouring coefficients occupy 11:10 193
the off-diagonal positions, forming a pentadiagonal matrix (in the 2D case). It is
this special feature of the structured grids that permitted employment of the ADI
solution method.
The arbitrary [A] matrix formed on unstructured grids is called a sparse matrix.
For such matrices, rapidly convergent methods such as conjugate-gradient (CG)
and generalised minimal residual (GMRES) are available [3]. These methods are
particularly attractive when the number of elements and, hence, the number of
equations requiring simultaneous solutions are large. Description of these methods
is considered beyond the scope of the present book. However, the diagonally
dominant position occupied by the AP coefficient in our equations still permits
employment of the simple point-by-point GS procedure. Thus, the equations can
be solved by a simple routine as follows:

DO1N=1,NE
SUM = SU(N)
DO2K=1, NK(N)
NEBOR = NHERE(N, K)
2 SUM = SUM + AE(N, K) * FI(NEBOR)
FI(N) = SUM / (AP(N) + SP(N))
1 CONTINUE

where NK(N) stores the number of neighbours of node N, NHERE (N, K) stores
the element number of the kth neighbouring node of N, and source term SU (N)
and AP (N) and SP (N) have already been calculated.

6.3.13 Overall Calculation Procedure

The important features of the overall calculation are described through the proce-
dural steps that follow.

Preliminaries
1.Read element and vertex files. Determine neighbouring elements of each node
N to form NHERE (N , K). This is done by searching the shared vertices between
neighbouring elements. Note that there will be no neighbouring elements when
a boundary face is encountered. At such a face, a boundary node is created
and such nodes are identified with numbers NE + 1, NE + 2, etc., where NE
are the total number of elements read from the element file. The coordinates of
interior nodes are calculated using Equation 6.48 and of boundary nodes using
Equation 6.107.
2.Tag the boundary nodes with identification numbers for inflow, symmetry, wall,
and exit boundaries. Note that here boundary nodes rather than near-boundary
cells are tagged. This is unlike the practice on structured grids.