SPH Summary

• Set initial conditions \(v_i\)

• Compute values
• • - Density: \(\rho_i = \sum_j m_j \, W_h^{poly6}(\|p_i-p_j\|)\)
  • - Pressure: \(\mathrm{p}_i = s\,(\rho_i-\rho_0)\)

• Compute forces
• • - \(F_{weight}=m_i\;g\)
  • - \(F_{pressure}=-\frac{m_i}{\rho_i} \, \sum_{j}^{j\neq i} m_j \, \frac{\mathrm{p}_i+\mathrm{p}_j}{2\,\rho_j} \; \nabla W_h^{spiky}(\|p_i-p_j\|)\)
  • - \(F_{viscosity}=m_i\,\nu \, \sum_{j}^{j\neq i} m_j \frac{(\mathrm{v}_j-\mathrm{v}_i)}{\rho_j}\;\triangle W_h^{spiky}(\|p_i-p_j\|)\)

• Time integration: \(v_i^{k+1}=v_i^k + \Delta t\;\left(F_{weight}+F_{pressure}+F_{viscosity}\right)/m_i\)