# Verlet integration on grids or how to get better stability in hyperelastic simulation

I am using MLS-MPM to simulate both solid and fluids. It works, but the amount of time steps I must do for hyperelastic solids is absurd.

To give you some perspective, I am able to simulate just the fluid at interactive speeds (60 fps) using only one simulation step per frame. The fluid simulation is just extremely stable. The solid simulation on the other hand requires me to do 60 simulation steps per frame, because if the time step is too large the system accumulates energy at a ludicrous speed.

I know that stormer verlet is useful in spring mass systems, because it is significantly more numerically stable and does not explode as fast as forward euler does.

issue is, the physics update in MLS-MPM happens on the grid, so the verlet formula:

$$x_{i+1} = 2x_{i} - (x_{i-1}) + \vec a t^2$$

Doesn't make much sense, since the positions of the grid centers never change. I cannot use implicit integration here because it takes too long per simulation step. So what option do I have to increase the time step for a hyper elastic neo-hookean simulation? Accuracy is less of a problem as preventing the simulation from exploding.

• Since you used the words '60 fps' , I'm assuming that you're doing graphics and not mechanics. If that is the case, and you don't care about the exact results, why not tweak the material properties and get the time-step you want?
– NNN
Jul 4, 2023 at 7:25
• @NNN I have tried it in the [ast but the particles don;t remain together if I make the solid less rigid, which deefeats the purpose of the simulation. Jul 4, 2023 at 8:32
• How about coarsening the solid grid?
– NNN
Jul 4, 2023 at 8:40
• @NNN coarsening, like reducing the resolution? Does that improve numerical stability? Jul 4, 2023 at 8:42
• Yes. Increase $\Delta{x}$ and you increase $\Delta{t}$ - see the wikipedia page: en.wikipedia.org/wiki/…
– NNN
Jul 4, 2023 at 9:25