PCISPH

Here is the original paper: "Predictive-Corrective Incompressible SPH"

1.Introduction

In the context of SPH, two different strategies has been purued to model incompressibility:

• using a stiff equation of state (EOS)
  • standard SPH: low stiffness, not real
  • WCSPH: high stiffness, small time steps
• solving a pressure Poisson equation
  • ISPH: slow

This paper proposed an incompressible SPH method featuring the advantages of WCSPH and ISPH:

• PCISPH: low computational cost, large time steps

2. PCISPH Model

2.1 Basic SPH / WCSPH Algorithm

The desity $\rho_i$ of particle $i$ at location $x_i$: $$ \rho_i = m_i\sum_j W(x_{ij}, h) $$

The pressure $p_i$ of particle $i$ is: $$ p_i = \frac{k\rho_{0i}}{\gamma}\left(\left(\frac{\rho_i}{\rho_{0i}}\right)^\gamma - 1\right) $$

The pressure force is: $$ F_i^{p} = -\sum_{j}m_im_j \left(\frac{p_i}{\rho_i^2} + \frac{p_j}{\rho_j^2}\right)\nabla W(x_{ij}, h) $$

The viscosity force is: $$ F_i^{vis} = 2\nu(d+2)\sum_j \frac{m_im_j}{\rho_i\rho_j}\frac{v_{ij}\cdot x_{ij}}{\|x_{ij}\|^2+0.01h^2}\nabla W(x_{ij}, h) $$

2.2 PCISPH Algorithm

Here is the entire algorithm:

, where: $$ \rho_{err, i}^* = \rho_{i}^* - \rho_{0i} $$

$$ \tilde{p}_i = \delta \rho_{err, i}^* $$

To avoid falsified values, we choose a fixed $\delta$ here, which is evaluated for a prototype particle with a filled neighborhood:

$$ \delta = \frac{1}{\beta\left(\sum_j\nabla W_{ij}\cdot \sum_j\nabla W_{ij} + \sum_j \left(\nabla W_{ij}\cdot \nabla W_{ij}\right) \right)} $$

$$ \beta = \Delta t^2m^2\frac{2}{\rho_{0i}^2} $$

3. Result

Here is the results of 2-phase PCISPH with different viscosity coeffcient: