Mathematical Physics - Volume II - Numerical Methods

5.8 Derivation of Normalised Corrected Gradient SPH formula 175

where ϕ x is a skew-symmetric dyadic: ϕ x =   0 ∆ ϕ z e 2 e 1

  .

− ∆ ϕ z e 1 e 2 ∆ ϕ y e 1 e 3

− ∆ ϕ x e 2 e 3

0

(5.49)

− ∆ ϕ y e 3 e 1 ∆ ϕ x e 3 e 2

0

For small rotations, the rotation transformation tensor is given by: C = I − ϕ x .

(5.50)

The approximation of the rotated coordinates is:

⟨ x ′ ⟩ ≡ ⟨ Cx ⟩ = ⟨ C ⟩⟨ x ⟩ = ⟨ I − ϕ x ⟩⟨ x ⟩ .

(5.51)

This means that the requirement on the interpolation is: I − ϕ x = ⟨ I − ϕ x ⟩

(5.52)

or

ϕ x = ⟨ ϕ x ⟩ .

(5.53)

Expanding this expression leads to: ⟨ ϕ x ⟩ = ∑ J m J ρ J

m J ρ J

∆ ϕ × x J ∇ W ( x I − x J , h ) = ∑ J

( ϕ x x

J ) ∇ W ( x I − x J , h ) =

(5.54)

m J ρ J

= ϕ x ∑ J

x J ⊗ ∇ W ( x I − x J , h ) .

Therefore to preserve space isotropy, i.e. ϕ = ⟨ ϕ ⟩ the following condition has to be satisfied.

mbr ∑ J = 1

m J ρ J

x J ⊗ ∇ W ( x I − x J , h ) = I .

(5.55)

Space Homogeneity

Space Anisotropy

m J ρ J

m J ρ J x J ⊗ ∇

∑ nnbr J = 1

nnbr J = 1

W ( x I − x J , h ) = 1

W ( x I − x J , h ) = I

∑

Condition which has to be satisfied Normalised – Cor rected form

W ( x I − x J , h )

˜ ∇ ˜ W IJ ∇ ˜ W IJ

˜ W IJ =

=

m J ρ J

∇ ˜ W IJ

nnbr ∑ J = 1

− 1

m J ρ J ⊗

nnbr ∑ J = 1

W ( x I − x J , h )

Table 5.1: Corrected forms of the kernel function and its gradient.

The form of the normalised kernel function and the approximation of the first order derivatives which provides first order consistency is given in Table 5.1 below.