11.4 Adaptive Grid Methods and Local Grid Refinement
355
necessary to compute a new coarse grid solution; this solution is not the one
that would be computed on the coarse grid covering the entire domain but
a smoothed version of the fine grid solution. To see what is needed, suppose
that:
is a discretization of the problem on a grid of size h; L: represents the operator.
To force the solution to be a smoothed version of the fine-grid solution in the
region which has been refined we replace the coarse-grid problem by:
&h(42h) =
{ &h($h), in the refined region;
(11.14)
Q2h,
in the remainder of the domain,
where $h is the smoothed fine-grid solution (i.e. its representation on the
coarse grid). The solution is then iterated between the coarse and fine grids
until the iteration error is small enough; about four iterations usually suffice.
Since the solution on each grid does not need to be iterated t o final tolerance
each time, this method costs only a little more than the passive method.
In another kind of active method (Muzaferija, 1994; Muzaferija and Gosman, 1997) the grids are combined into a single global grid, including the
refined grid as well as the non-refined part of the original grid. This requires
a solution method that allows CVs with arbitrary numbers of faces. CVs at an
interface between refined and non-refined regions have more faces and neighbors than regular CVs, see Fig. 11.10. For the global conservation property of
the FV method to be retained, the face of a non-refined CV on the refinement
boundary has to be treated as two (in 2D, and four in 3D) separate sub-faces,
each common to two CVs. In the discretization, the sub-faces are treated exactly like any other face between two CVs. The computer code needs to have
a data structure that can handle this situation and the solver needs to be
able to handle the irregular matrix structure that results. Conjugate gradient
type solvers are a good choice; multigrid solvers with Gauss-Seidel smoothers
can also be used with some limitations. The data structure can be optimized
by storing cell-face and cell-volume related values in separate arrays. For a
simple discretization scheme like the one described in Chap. 8, this is easily
done: each cell face is common to two CVs so, for each face, one needs to
store pointers t o the nodes of neighbor CVs, surface vector components, and
the matrix coefficients. The computer code for solving the flow problem is
then the same for locally refined and standard grids; only the pre-processor
needs adaptation to enable it to handle the data for locally refined grids.
If Chimera grids are used, the code needs no changes, but the interpolation
coefficients and the nodes involved in interpolation need be redefined after
each refinement.
As many levels of grid refinement as necessary can be used; usually a t least
three levels are required but as many as eight have been used. The advantage
of the adaptive grid method is that, because the finest grid occupies only a
355
necessary to compute a new coarse grid solution; this solution is not the one
that would be computed on the coarse grid covering the entire domain but
a smoothed version of the fine grid solution. To see what is needed, suppose
that:
is a discretization of the problem on a grid of size h; L: represents the operator.
To force the solution to be a smoothed version of the fine-grid solution in the
region which has been refined we replace the coarse-grid problem by:
&h(42h) =
{ &h($h), in the refined region;
(11.14)
Q2h,
in the remainder of the domain,
where $h is the smoothed fine-grid solution (i.e. its representation on the
coarse grid). The solution is then iterated between the coarse and fine grids
until the iteration error is small enough; about four iterations usually suffice.
Since the solution on each grid does not need to be iterated t o final tolerance
each time, this method costs only a little more than the passive method.
In another kind of active method (Muzaferija, 1994; Muzaferija and Gosman, 1997) the grids are combined into a single global grid, including the
refined grid as well as the non-refined part of the original grid. This requires
a solution method that allows CVs with arbitrary numbers of faces. CVs at an
interface between refined and non-refined regions have more faces and neighbors than regular CVs, see Fig. 11.10. For the global conservation property of
the FV method to be retained, the face of a non-refined CV on the refinement
boundary has to be treated as two (in 2D, and four in 3D) separate sub-faces,
each common to two CVs. In the discretization, the sub-faces are treated exactly like any other face between two CVs. The computer code needs to have
a data structure that can handle this situation and the solver needs to be
able to handle the irregular matrix structure that results. Conjugate gradient
type solvers are a good choice; multigrid solvers with Gauss-Seidel smoothers
can also be used with some limitations. The data structure can be optimized
by storing cell-face and cell-volume related values in separate arrays. For a
simple discretization scheme like the one described in Chap. 8, this is easily
done: each cell face is common to two CVs so, for each face, one needs to
store pointers t o the nodes of neighbor CVs, surface vector components, and
the matrix coefficients. The computer code for solving the flow problem is
then the same for locally refined and standard grids; only the pre-processor
needs adaptation to enable it to handle the data for locally refined grids.
If Chimera grids are used, the code needs no changes, but the interpolation
coefficients and the nodes involved in interpolation need be redefined after
each refinement.
As many levels of grid refinement as necessary can be used; usually a t least
three levels are required but as many as eight have been used. The advantage
of the adaptive grid method is that, because the finest grid occupies only a