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25.6.4 Full-Approximation Storage (FAS) Multigrid

FLUENT's approach to forming the multigrid grid hierarchy for FAS is simply to coalesce groups of cells on the finer grid to form coarse grid cells. Coarse grid cells are created by agglomerating the cells surrounding a node, as shown in Figure  25.6.4. Depending on the grid topology, this can result in cells with irregular shapes and variable numbers of faces. The grid levels are, however, simple to construct and are embedded, resulting in simple prolongation and relaxation operators.

Figure 25.6.4: Node Agglomeration to Form Coarse Grid Cells

It is interesting to note that although the coarse grid cells look very irregular, the discretization cannot "see'' the jaggedness in the cell faces. The discretization uses only the area projections of the cell faces and therefore each group of "jagged'' cell faces separating two irregularly-shaped cells is equivalent to a single straight line (in 2D) connecting the endpoints of the jagged segment. (In 3D, the area projections form an irregular, but continuous, geometrical shape.) This optimization decreases the memory requirement and the computation time.

FAS Restriction and Prolongation Operators

FAS requires restriction of both the fine grid solution $\phi$ and its residual (defect) $d$. The restriction operator $R$ used to transfer the solution to the next coarser grid level is formed using a full-approximation scheme [ 41]. That is, the solution for a coarse cell is obtained by taking the volume average of the solution values in the embedded fine grid cells. Residuals for the coarse grid cell are obtained by summing the residuals in the embedded fine grid cells.

The prolongation operator $P$ used to transfer corrections up to the fine level is constructed to simply set the fine grid correction to the associated coarse grid value.

The coarse grid corrections $\psi^H$, which are brought up from the coarse level and applied to the fine level solution, are computed from the difference between the solution calculated on the coarse level $\phi^H$ and the initial solution restricted down to the coarse level $R\phi$. Thus correction of the fine level solution becomes

 \phi^{\rm new} = \phi + P \, \left(\phi^H - R \phi \right) (25.6-29)

FAS Coarse Level Operator

The FAS coarse grid operator $A^H$ is simply that which results from a re-discretization of the governing equations on the coarse level mesh. Since the discretized equations presented in Sections  25.3 and 25.5 place no restrictions on the number of faces that make up a cell, there is no problem in performing this re-discretization on the coarse grids composed of irregularly shaped cells.

There is some loss of accuracy when the finite-volume scheme is used on the irregular coarse grid cells, but the accuracy of the multigrid solution is determined solely by the finest grid and is therefore not affected by the coarse grid discretization.

In order to preserve accuracy of the fine grid solution, the coarse level equations are modified to include source terms [ 160] which insure that corrections computed on the coarse grid $\phi^H$ will be zero if the residuals on the fine grid $d^h$ are zero as well. Thus, the coarse grid equations are formulated as

 A^H \phi^H + d^H = d^H(R \phi) - R d^h (25.6-30)

Here $d^H$ is the coarse grid residual computed from the current coarse grid solution $\phi^H$, and $d^H(R \phi)$ is the coarse grid residual computed from the restricted fine level solution $R \phi$. Initially, these two terms will be the same (because initially we have $\phi^H = R \phi$) and cancel from the equation, leaving

 A^H \phi^H = - R d^h (25.6-31)

So there will be no coarse level correction when the fine grid residual $d^h$ is zero.

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