Mathematical Physics - Volume II - Numerical Methods

5.4 Kernel Function

169

The surface integral is zero if the domain of integration is larger than the compact support of W or if the field variable assumes zero value on the boundary of the body (free surface). If none of these conditions are satisfied, modifications should be made to account for boundary conditions. One should observe that in Equations (5.18), (5.19) and (5.20) the spatial derivatives of the field variables are substituted by the derivatives of the kernel function giving: D ρ ( x ) D t = ρ ( x ) Z Ω v ( x ′ ) ∇ W ( | x − x ′ | , h ) d x ′ , (5.24) D v ( x ) D t = − Z Ω σ ( x ′ ) ρ ( x ′ ) ∇ W ( | x − x ′ | , h ) d x ′ −

σ ( x ) [ ρ ( x )] 2 Z Ω

ρ ( x ′ ) ∇ W ( | x − x ′ | , h ) d x ′

(5.25)

−

D E ( x )

D t

σ ( x ) [ ρ ( x )] 2 Z Ω [ ρ ( x )] 2 Z Ω

ρ ( x ′ ) v ( x ) ∇ W ( | x − x ′ | , h ) d x ′ +

= −

σ ( x ) v ( x )

ρ ( x ′ ) ∇ W ( | x − x ′ | , h ) d x ′ ,

(5.26)

+

The final step is to convert the continuous volume integrals to sums over discrete interpola tion points. Finally, after a few arrangements in order to improve the consistency between all equations, the most common form of the SPH discretised conservation equations are obtained: D ρ I D t = ρ I N ∑ J = 1 m J ρ J ( v J − v I ) ∇ W ( | x I − x J | , h ) , (5.27) D v I D t = − N ∑ J = 1 m J σ J ρ 2 J + σ I ρ 2 I ∇ W ( | x I − x J | , h ) (5.28)

D E I

D t

N ∑ J = 1

σ I ρ I

m J ( v J − v I ) ∇ W ( | x I − x J | , h )

(5.29)

= −

5.4 Kernel Function To perform the spatial discretisation one has to define the kernel function. Numerous possibilities exist, and a large number of kernel function types are discussed in literature, ranging from polynomial to Gaussian. The most common is the B-spline kernel that was proposed by Monaghan [59]:

C h D   0 ,

1 − 3 1 4 ( 2 − v ) 2 v

3 , v < 1

2 + 3

4 v

3 ,

W ( v , h ) =

(5.30)

1 ≤ v ≤ 2 otherwise .