18 Agglomeration Multigrid and FMG

Multigrid accelerates convergence by treating error components on spatial scales appropriate to a hierarchy of control volumes. NeuralFlow uses worker-owned agglomerated coarse grids. The active coarse solver rediscretizes the governing equations on each level and reuses the coupled residual/Jacobian infrastructure rather than constructing a separate scalar pressure equation.

18.1 Hierarchy notation

Let level denote the original fine grid and levels progressively coarser agglomerations. A fine cell belongs to one parent aggregate on the next coarse level. Denote state restriction by and correction prolongation by .

The nonlinear equation on level is (18.1) where is obtained by rediscretizing the active NeuralFlow equations on that level’s geometry and boundary representation.

18.2 Correction multigrid

For a linearized fine-grid correction problem a correction cycle performs:

1. pre-smoothing on level ;

2. residual evaluation ;

3. restriction to the coarse level;

4. approximate coarse solution;

5. prolongation of the coarse correction;

6. fine-state/correction update;

7. post-smoothing.

The smoother is not required to solve the level system accurately. Its purpose is primarily to damp error components that are oscillatory on the current grid.

18.3 Nonlinear FAS

For a nonlinear equation, Full Approximation Storage (FAS) transfers a full approximation, not only an error correction. If the coarse FAS equation can be written as (18.2) with (18.3) The correction reconciles the coarse rediscretization with the restricted fine nonlinear operator. Mixing a correction-only transfer with a full FAS state without the corresponding equation transformation is mathematically inconsistent.

18.4 State restriction

For flow initialization and FAS, state restriction must preserve the meaning of the governing state. In compressible flow, a safe conservative restriction is based on volume-integrated conservative quantities, (18.4) followed by reconstruction of primitive variables from the restricted conservative state. Independently averaging both pressure and temperature can violate total-energy conservation.

For the constant-density incompressible formulation, the stored state in Eq. (4.1) determines which components are restricted. Turbulence, species, and optional particle variables must be transferred with the same equation layout active on the target level.

18.5 Correction prolongation versus full-state prolongation

Ordinary correction multigrid prolongates a correction, Full-multigrid initialization instead requires a complete admissible state on the next finer level, (18.5) where must preserve positivity/bounds and the thermodynamic relation between primitive and conservative variables. A correction interpolator is not automatically a safe full-state interpolator.

18.6 Nested full-multigrid initialization

A full-multigrid (FMG) initializer uses nested iteration:

1. start from a valid initialized fine-grid state;

2. build/validate the hierarchy;

3. restrict the complete state to the coarsest usable level;

4. solve or apply FAS cycles on the coarsest level;

5. prolongate a complete state to the next finer level;

6. perform level/FAS work involving that level and all required coarser levels;

7. continue to level 0;

8. perform fine-grid cleanup and restore normal solver settings.

For levels , the nested-order backbone is with FAS cycles allowed to descend again to the coarsest level from each active finer stage. Simply performing one ordinary V-cycle from the fine grid is not FMG initialization.

18.7 Level stopping criteria

An initialization level should stop because useful convergence has been achieved or because a defined work limit has been reached. A relative criterion is (18.6) combined with a maximum cycle/iteration count. The initialization target need not be the same as the final production convergence criterion; the objective is a physically admissible starting field that substantially reduces fine-grid startup cost.

18.8 Coarse-level physics and turbulence

Rediscretization means that active boundary conditions, fluxes, sources, equation count, and thermodynamic relations have to remain meaningful on the coarse geometry. Turbulence is especially sensitive because wall distance, normals, , , turbulent viscosity, and wall-function quantities must refer to the current level consistently.

A robust FMG initializer may deliberately freeze or stage selected physics if a coarse representation is not valid. Such staging must be explicit and followed by a fine-grid rebuild of turbulence, wall, boundary, and property data before returning control to the normal solver.

18.9 Parallel consistency

Every worker owns its local hierarchy. A multigrid cycle must keep workers synchronized through restriction/prolongation interface exchanges and convergence reductions. If one worker detects an invalid state, the failure has to be reduced collectively before any worker destroys a level or returns to the main event loop.

The fine-grid partition itself is not replaced by FMG. Successful completion must leave level 0 active with valid primitive/conservative consistency and ready linear-solver objects.

18.10 18.10 Availability and case compatibility

Agglomeration multigrid is available for supported compressible and constant-density cases and reuses the active physical model on each retained coarse level. It should be treated as a convergence accelerator, not as a different physical model. Some model combinations restrict hierarchy use; the GUI readiness/availability state should therefore be checked before relying on AggMG or FMG for a particular case.

Do not assume a fixed speed-up from enabling multigrid. Effectiveness depends on mesh size and quality, stiffness, smoother strength, hierarchy quality and the dominant error wavelength. Judge AggMG by residual reduction per wall-clock time and by convergence of the engineering quantities of interest.

18.11 Current hierarchy construction and cycle controls

The GUI provides three agglomeration formulations: Nonlinear FAS, Implicit linear correction and a legacy compatibility formulation. Cycle choices include V, W, F and the ongoing-solve FMG schedule. Correction transfer can use parent-cell injection, inverse-distance interpolation or localized corrected interpolation. These choices change convergence behavior and cost; they do not alter the fine-grid physical equations.

Hierarchy construction provides a default grouping method, an alternative grouping method and a strict best-admissible comparison. Optional creation-time refinement can improve local aggregate quality, and the coarse-grid quality policy can warn only, automatically protect/repair, or reject unresolved coarse levels. These controls primarily affect hierarchy setup cost, memory and multigrid robustness.

These geometric agglomeration controls are distinct from the matrix-based AAMG linear solver described in Chapter 9. Homogeneous VOF cases currently do not use the AggMG hierarchy; use the standard coupled solver/linear-solver controls for those cases.