P16

SPH-Based Fluids

P17

Fluid Dynamics

We model fluid dynamics by applying three forces on particle i.
- Gravity
- Fluid Pressure
- Fluid Viscosity

P18

$$ \mathbf{F} _ \mathbf{i}^ \mathbf{gravity} = m _i \mathbf{g} $$

P19

✅ 要解决的问题：怎么计算压强？怎么把压强转化为力？

Pressure is related to the density
- First compute the density of Particle i:
$$ \rho _ i = \sum _ j m _ j W _ {ij} $$
- Convert it into pressure (some empirical function):
$$ P_i=k((\frac{\rho _i}{\rho _\mathrm{constant } } )^7-1) $$

✅ 密度到压强的计算是一个经验公式。

P20

P21

$$ \mathbf{F} _i^{pressure}=-V_i\nabla _iP^{smooth} $$

✅ 体积为粒子在空间中占有的体积，体积越大受到的压力越大、$\nabla$代表压强的差。

To compute this pressure gradient, we assume that the pressure is also smoothly represented:

$$ P_i^{smooth}= \sum _ j V_jP_j W_{ij} $$

✅ 假设空间是一个压强场、粒子是空间中的采样。$P^{smooth}$是通过周粒子$P$的插值得到的采样点压强。

$$ \mathbf{F} _ i^{pressure} = - V _ i \sum _ j V _ j P _ j \nabla _ i W _ {ij} $$

P22

Viscosity effect means: particles should move together in the same velocity.
In other words, minimize the difference between the particle velocity and the velocities of its neighbors.

✅ Viscosity (粘滞)类似于 damping (阻尼)，但有些区别，后者的目标是让粒子的运动停下来，前者的目的是让所有粒子的运动整齐划一，即速度差趋于0.

P23

Mathematically, it means:
$$ \mathbf{F} _i^{vis \cos ity}=-\nu m_i\Delta _i\mathbf{V} ^{smooth} $$

✅ $V$：粘滞系数， $\nabla V$：速度的 Laplacian.注意速度是3D矢量。

To compute this Laplacian, we assume that the velocity is also smoothly represented:

$$ \mathbf{V} _i^{smooth}= \sum_jV_j \mathbf{v} _ j W _ {ij} $$

$$ \mathbf{F} _i^{vis \cos ity}=-\nu m_i\sum _jV_j\mathbf{v} _j\Delta _iW _{ij} $$

✅ smooth会产生粘滞的效果。

P24