Dynamic Mesh (Mesh Deformation)
Theory: spreading displacement into the domain
Overview
I want to simulate flow through an opening and closing valve. How should the mesh move?
The simplest is mesh deformation: move node positions without changing connectivity. Body-surface motion is the boundary condition, and interior node motion is set smoothly. A common, cheap approach diffuses displacement with a Laplace equation. As motion grows, some cell gets crushed and eventually inverts, stopping the run. How far you can go depends on the diffusivity distribution and how many cells lie between body and wall. Beyond that limit, remesh partway or move whole grids as overset methods do.
Displacement diffusion by the Laplace equation
Choosing the diffusivity
Where γ is large, displacement is nearly uniform and cells move without distorting (stiff). Deformation concentrates where γ is small.
ALE and moving grids
Moving grids in flow simulation trace back to the Arbitrary Lagrangian–Eulerian (ALE) method of the 1970s, which let grids move freely between the Lagrangian view (moving with the fluid) and the Eulerian view (fixed in space). With moving grids, the geometric conservation law (GCL) must hold so changing cell volumes don't generate flow by themselves. Violating it disturbs velocity and pressure in a uniform, resting flow just by moving the grid.
Worked examples
Example: moving a body of side 2 upward
Square 10×10 domain, spacing 0.25 (41×41 nodes). A square body of side 2 at the centre moves up by d. The outer wall is fixed, with a gap of 4 (16 cells) from the body's top to the wall. Each cell's area is compared with its original, and the smallest ratio is reported.
| Diffusivity | Min area ratio at d = 1 | Below 20% at | Inverts at | Worst cell |
|---|---|---|---|---|
| Uniform | 0.46 | 1.50 | 1.90 | Body corner |
| 1/r | 0.61 | 2.10 | 2.60 | Near outer wall |
| 1/r² | 0.38 | 1.30 | 1.65 | Near outer wall |
| Diffusivity | Displacement just above body | Displacement midway to wall |
|---|---|---|
| Uniform | 0.90 d | 0.38 d |
| 1/r | 0.98 d | 0.65 d |
| 1/r² | 1.00 d | 0.82 d |
I thought stiffer near the body was always better, but 1/r² inverts earlier than uniform.
The total motion doesn't change, so it must be absorbed somewhere. With uniform diffusivity, deformation gathers right at the body corner. With 1/r, the region around the body moves together and deformation spreads across the gap, extending the limit from 1.9 to 2.6. With 1/r², though, the body region is so stiff that cells midway to the wall still move 0.82 d, and the last few cells by the outer wall absorb everything and collapse. Stiffening exists to protect fine boundary-layer cells near the body; overdone, it only moves the problem elsewhere. Always check where the worst cell is.
Choosing a method
- For small motion (a fraction of the gap), mesh deformation suffices.
- Start with roughly 1/r diffusivity and monitor the worst cell's location and area ratio.
- Remesh when the ratio drops below a threshold (e.g. 0.2).
- Use overset or sliding meshes for large translation or rotation.
- Verify the geometric conservation law by checking a uniform flow stays undisturbed.
“A run that died just before the valve closed”
A check-valve opening/closing simulation died every time the disc approached the seat: the few cells in the gap were crushed and inverted by deformation. Closure, where the gap goes to zero, cannot in principle be represented by deformation alone. Keeping a minimum gap of 0.1 mm, remeshing at that position, and treating gap flow as a separate resistance allowed the full stroke to be computed.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Uniform diffusivity | Cells crush near body | Use about 1/r |
| Too stiff | Cells crush at outer wall | Check worst-cell location |
| Closing the gap to zero | Inversion guaranteed | Keep a minimum gap |
| Ignoring the GCL | Uniform flow disturbed | GCL-consistent scheme |
| Large rotation by deformation | Grid twists | Sliding or overset |
I'd like to learn related topics.
Related pages include overset (chimera) grids, CFD mesh quality, adaptive mesh refinement, inflation layers and ALE FSI.
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