Though this article on wikipedia:

https://en.wikipedia.org/wiki/Smoothed-particle_hydrodynamics

Particle is a abtract term, not a physical thing. It’s a techqiue to compute the distribution of matters.

How to calculate the density

{\displaystyle \rho _{i}=\rho ({\boldsymbol {r}}_{i})=\sum _{j}m_{j}W_{ij},}

What is the difference between cubic and quintic splines?

Today, I want to explore the comparison between cubic and quintic splines for trajectory generation: Cubic splines are great for basic tasks where smooth motion is not a requirement. Quintic splines are good for applications that require more precise and smooth control.

Quintic splines

https://pysph.readthedocs.io/en/latest/reference/kernels.html#pysph.base.kernels.QuinticSpline

Firstly:

q = r/h

for $q \in [0, 1]$:

W(q)=\sigma_5[(3-q)^5 - 6(2-q)^5 + 15(1-q)^5]

for $q \in (1, 2]$:

W(q)=\sigma_5[(3-q)^5 - 6(2-q)^5]

for $q \in (2, 3]$:

W(q)=\sigma_5(3-q)^5

for $q \in (3, \infty]$:

W(q)=0

where $\sigma_5$ is a dimensional normalizing factor for the quintic spline function given by:

\sigma_5 = {1\over 120 \pi h^3}

How to calculate the pressure for a particle

I got to known how to calculate the pressure at a particle position!

p_i = c^2(\rho_i - \rho_{0})

where:

c = 10v_{max}

the max of velocity of particle has to be estimated!

How to caculate the pressure force

{dv_i \over dt} = - {\sum_j}m_j{({p_i\over\rho_i^2}+ {p_j\over\rho_j^2})}\nabla{W_{ij}} + g

and,

a_i = {dv_i \over dt}

How to calculate the $\nabla{Wij}$

for $q \ge 3$: $\nabla{Wij}=0$

else for $q \ge 2$

How to caculate viscosity?

we use Artificial viscosity

while $v_{ij} * {r_{ij}} \ge 0$:

\Pi_{ij} = 0

else:

\Pi_{ij} = {\displaystyle {-\alpha \bar c_{ij}\phi_{ij} + \beta\phi_{ij}^2} \over {\bar \rho_{ij}}}

where:

\phi_{ij} = {\displaystyle h \mathbf v_{ij} \cdot \mathbf r_{ij}  \over { \Vert \mathbf r_{ij} \Vert ^2 + \eta_h^2 } }

📃 Source Code

📝 Git logs

commit 9f2d0ee20c21f42ae322a1053821d3b5865b7be6
Author: singhi <zhangxinghai79@gmail.com>
Date:   Mon Mar 31 23:54:48 2025 +0800

    hei