Multi-fluid PCISPH

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: \[ F_i^{vis} = 2(d+2)\sum_j \frac{m_im_j}{\rho_i\rho_j}\frac{v_{ij}\cdot x_{ij}}{\|x_{ij}\|^2+0.01h^2}\nabla W(x_{ij}, h) \]

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: