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MPM basics
The material point method is effectively a finite-element scheme where the integration (material points) points may move independent of the mesh. It is a Lagrangian method - however it appears to act like an Eulerian scheme in that the grid often stays fixed.
The key idea is that the Material Points (MP) hold all of the infomation about the body(ies) in the system, and that the grid/mesh is used to solve the governing equations.
As such multiple rounds of mapping data to the grid (P2G particle-to-grid) and back (G2P grid-to-particle) is needed.
For an explicit scheme this may be implemented as:
- Reset grid
- Map momentum & mass to grid (P2G)
- Calculate grid velocities & Apply boundary conditions
- Update stress (USF)
- Map forces to grid
- Solve for acceleration F = MA, and update velocity v += dT * F/M
- Map velocity & acceleration back to particles (G2P)
- Update particle velocity & location
One option of updating stress is to do it after the first velocity map step. This tends to cause energy accumulation over time - however it is fast.
By updating the stress after moving the MPs (at the end of the step) USL schemes tend to dissipate energy.
MUSL does a second round of momentum mapping at the end of the step and uses the newly mapped velocities to calculate stress - MUSL schemes tend to perform better than USL but are expensive due to the double mapping.