Hi everyone,

Does anybody know any paper or article that deals with implicit SPH (Smoothed Particles Hydrodynamics) technique. My question is actually if there is any technique to compute implicitly the smooth kernel W.

Thank you in advance,

Etiene

Implicit SPH in all its glory, with moving particles for computing fluid flow is still and unsolved problem, as far as my information goes. There exists a paper that deals with implicit SPH conduction modeling:

Modeling Transient Heat Transfer Using SPH and Implicit Time Integration Rusty Rook; Mehmet Yildiz; Sadik Dost Num. Heat Transfer, Part B, 51, 1-23.

What do you mean by computing the kernel implicitly ?

Hello,

Thank you for your answer. I read the paper you've recommended and things starts to became clearer.

I'm trying to solve the following equation:

rho * dv/dt = - grad P, (I)

rho is the density, v is velocity, t is time d/dt is the derivative in time, grad is the gradient and P the pressure field.

What I really meant about computing the kernel implicit is that the kernel W(r) is a function of r = xi - xj, which is the relative position. In the beginning I was thinking that a implicit method in time would compute W not in the current time t but in t + 1 instead, which means compute r in time t+1 (unlike the paper you've cited, my particles can move trhough the domain). In this scenario, I would have a more complicated equation implicit in time, for instance, the SPH discretization of Equation (I) could be:

dv/dt = sum(j) { Vj * (P_j(t+1)/rho_j(t+1)^2 + P_i(t+1)/rho_i(t+1)^2) * grad W_ij(t+1) } where W is the kernel, rho is the density and V is the volume of a particle.

Thanks!