Incompressible SPH problems
Hello all,
Now, I'm doing research about SPH method for incompressible flow for my master degree. I use the weakly incompressible and incompressible SPH method. In incompressible SPH, which use projection method, there are 3 main steps as below 1. calculate intermediate velocity (v*) without pressure gradient term 2. calculate pressure implicitly by solve Pressure Poisson Eq. (PPE) D(1/rho G(p)) = D(v*)/dt D : divergence operator G : gradient operator dt : time step 3. correct the intermediate velocity by pressure gradient to get velocity in the next step I have the problem in 2nd step. I have build the subroutine for this step and then I tried to test the code by simulate Poiseuille Flow. The flow is driven by body force not pressure gradient. From analytical, we know that pressure gradient is zero in flow direction and orthogonal to flow direction. Homogenous Neumann BC for pressure is applied at wall and ghost particles is used to model the wall. The result system of linear equation is singular and only have unique solution up to constant plus there is compatibility eqn for consistent system of linear equation i.e. integral volume of D(v*) must be equal to zero. The problem is, in my code, integral volume of D(v*) not equal to zero. I think integral volume of D(v*) = 0 have physical explanation that mass of the system must be conserved. Please give me some help. Is there something wrong with my code? Thanks for your attention. 
Did you get to have an answer to your problem?

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That post is twomonth old, however I think the error was in the BC.s that do not fulfill the compatibility contraint. 
I think use of dummy particles instead of ghost particles gives better results for ISPH, for details you can refer to Lee's paper, A comparison of weakly and incompressible SPH, (I can't remember the exact name of the paper).
I did not understand your problem about Div(u*), but if you meant that you expected to have Div(u*)=0, you are wrong. u* is not a solution to the continuity eqn. It is merely a prediction for velocity field for the next time step. When you project this velocity field on the Divergence free space(that is by solving pressure poisson eqn), you will have the correct velocity field. (correction step). 
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