where 2≤τ ≤10 is usually sufficient in order to obtain a converging Uzawa method.
For ˆS (cf. Section 3.3.5) we want to use
Sˆ=ω·diagCAˆ−1B+D. (4.16) Here, ω is a user parameter but can usually be left at its default value ofω = 0.7. In the case of ∆tvirt/∆t ≥0.01 where we do not want to explicitly computeCAˆ−1B+D, we can use different means to estimate the diagonal ofS =CAˆ−1B+D: The default choice in the new AMG method presented here is
sii≈X
k
X
j
dijbjk
ajj
. (4.17)
Other choices include sii ≈X
j
|dijbji/ajj|, sii ≈X
k
X
j
dijbjk/ajj
and sii ≈
X
j
dijbji/ajj
. (4.18) The results of the different options are similar, however the first option shows slightly better convergence rates in many models [116]. The estimation for ˆS can also serve as an indicator as to how good the coarsening based on the pressure-pressure matrixDis, or if the more sophisticated coarsening based on the Schur-ComplementCAˆ−1B+D is needed [98].
Only in very rare cases (mostly for very small virtual time step sizes ∆tvirt) do we need the stabilized interpolation (3.33). One of these cases can be found in [99].
Again, we refer to Section 4.5.7 for an experiment comparing the different techniques.
• can have logically different connectivity patterns.
This gives rise to a number of questions:
• If a point is deleted that was associated to a coarse level point in time step t, this has an immediate impact on the full AMG hierarchy in time step t+ 1.
• Does a point that has been added in time step t+ 1 necessarily become an F-point or can it become a C-F-point in certain situations? If so, do we need to re-built the full AMG hierarchy for that time step?
• What happens if due to the changing neighborhoods, a system becomes reducible in time stept+ 1? A good example for this is when the fluid forms droplets over time: Those necessarily lead to independent subsystems in the linear systems and therefore the full system becomes reducible. This may then influence the convergence dramatically in certain situations which is why we employ means to find those subsystems, see Section 5.2. Therefore, if the neighborhood structure (potentially) changes, we need to re-run our algorithm which means we also have to compute a new AMG setup.
These problems do not exist for discretizations with static meshes. The only variable to consider there is how closely related the two matrices A and B are in terms of coefficient size.
Options to overcome the obstacles for re-using the AMG setup in GFDMs are:
• Keep two separate point clouds: A static one and a moving one. The dynamic one is moved in every time step, just like described above. It can then be used to discretize the continuity equations, but before solving the linear systems, the coefficients need to be projected onto the static point cloud. The idea is similar to what the Material Point Method (MPM) [145] does, where all the physical quantities are mapped back and forth between a background mesh and a moving point cloud. This step introduces an error that would need to be accounted for.
It would also be necessary to move the static point cloud at least in some time steps in order to keep track with moving geometries for example. In fact, this idea introduces difficulties that are similar to those of mesh-based methods and the error introduced by the projection between the point clouds would need to be investigated carefully. For these reasons, this idea is not further pursued.
• Secondly, one may drop the idea of a moving point cloud all together and return back to an Eulerian approach. This is entirely possible with GFDMs, but this approach drops a lot of the strengths of these methods. It does have applications though when looking at transport problems [128].
• An idea that would not interfere with the FPM at all is to solve the problems arising due to the movement of the point cloud completely on the AMG side, especially in the setup. The insertion and deletion of points, and therefore rows in the matrix, could possibly be dealt with using locally adopted AMG
setups. This would reduce the computational effort in the setup phase, because only certain parts of the C/F-Splitting and the coarse level operators had to be rebuilt. To the best of the author’s knowledge though, approaches like this have not been considered so far.
• Another idea that does not alter the FPM algorithm at all would be to use aggre-gation based coarsening in AMG. In this coarsening technique, instead of finding a maximal independent set of coarse level points [138], so called aggregates are formed that consist of multiple points. The interpolation from the coarse level to the fine level within each aggregate is constant. This makes it easy to add or remove single points (rows) from an aggregate, as long as at least one point of the aggregate is still present.
It is unclear though how to handle the changing neighborhoods in this setting as well. In addition to that, convergence rates of aggregative AMG methods are often not as good as for standard coarsening approaches. Although the setup phases for aggregative methods are usually cheaper, we found that in our case this is different and the additional loss in the convergence rate makes aggregation based AMG not usable in our context, cf. Section 5.4.1. Also, since the point cloud is moving, the convergence rate of an AMG solution phase with a re-used setup can be expected to be even worse.
Because of that, aggregation based AMG can be ruled out as well.
• The final idea, which we found to be the most useful one, is to skip the point cloud organization phase for a predefined3 number of time steps. If the point cloud management phase is skipped, then no new points are introduced or deleted and no neighborhood relationships between points are changed. The only change is the position of the points and therefore the coefficients in the stencil. So the two (hydrostatic pressure, for example) matrices A from time step t and B from time step t+ 1 are structurally identical and have the same size, but include different coefficients. In this situation we can use standard setup re-use strategies from AMG, like re-using the full setup from time step t in time step t+ 1. Not only does this strategy save time in the AMG setup, but it also saves time in the FPM algorithm itself, as the point cloud organization is skipped.
On the other hand, the point cloud management is an essential step in the FPM algorithm and cannot be skipped for too many time steps. How many time steps can be carried out without managing the point cloud depends on the particular problem being solved and also on the question the simulation aims to answer.
Generally speaking, the slower the movement of the point cloud or the smaller the time step, the more likely it is that skipping the point cloud management has a negligible impact on the quality of the solution.
Section 4.5.6 compares the benefits of skipping the point cloud management and re-using the AMG setup to the loss in quality of the solution.
3Adaptive strategies are also possible.
Figure 4.1: Geometry for the bifurcated tube model.