Smoothed Particle Hydrodynamics
⟨ ∇ ⋅ μ ∇ u ( x ) ⟩ = ∑ i = 1 N m i ρ i ( μ ( x ) + μ ( x i ) ) ( u ( x ) − u ( x i ) ) | x − x i | 2 ( x − x i ) ⋅ ∇ w h ( | x − x i | ) ( 13 ) {\displaystyle \langle \nabla \cdot \mu \nabla u(x)\rangle =\sum _{i=1}^{N}{\frac {m_{i}}{\rho _{i}}}{\frac {(\mu (x)+\mu (x_{i}))(u(x)-u(x_{i}))}{|x-x_{i}|^{2}}}(x-x_{i})\cdot \nabla w_{h}(|x-x_{i}|)\qquad \qquad {\rm {(13)}}}
または、
⟨ ∇ ⋅ μ ∇ u ( x ) ⟩ = 2 ∑ i = 1 N m i ρ i μ ( x i ) ( u ( x ) − u ( x i ) ) | x − x i | 2 ( x − x i ) ⋅ ∇ w h ( | x − x i | ) ( 14 ) {\displaystyle \langle \nabla \cdot \mu \nabla u(x)\rangle =2\sum _{i=1}^{N}{\frac {m_{i}}{\rho _{i}}}{\frac {\mu (x_{i})(u(x)-u(x_{i}))}{|x-x_{i}|^{2}}}(x-x_{i})\cdot \nabla w_{h}(|x-x_{i}|)\qquad \qquad {\rm {(14)}}}
と定義される。