Mathematical Physics - Volume II - Numerical Methods

5.8 Derivation of Normalised Corrected Gradient SPH formula 173

proposed by Vignjevic [97], is given below. The derivation is based on the homogeneity and isotropy of space. These are properties of space, which have as a consequence conservation of linear and angular momentum, see Landau [45]. The mixed correction insures that homogeneity and isotropy of space are preserved in the process of spatial discretisation. An interpolation technique should not affect homogeneity of space. One way of demonstrating this is to prove that the interpolation of the solution space itself is invariant with respect to translational transformation. Let x I , I = 1 , . . . , n be a finite set of n points in D ∈ R 3 . Kernel interpolation/approx imation of, for instance, a vector field F ( x ) defined on D consists in finding, for given F ( x 1 ) , . . . , F ( x n ) , we write the general expression for the SPH interpolation of a vector field on D in the ( x 1 , x 2 , x 3 ) coordinate system: ⟨ F ( x ) ⟩ = ∑ J m J ρ J F ( x J ) W ( | x − x J | , h ) . (5.36) If the field to be interpolated is the domain D then F = x , x ∈ D and Equation (5.36) becomes: ⟨ x ⟩ = ∑ J m J ρ J x J W ( | x − x J | , h ) (5.37) A different coordinate system ( x ′ 1 , x ′ 2 , x ′ 3 ) , translated by a vector c relative to ( x 1 , x 2 , x 3 ) where the approximation ⟨ x ⟩ can be expressed as: D x ′ E = ∑ J m J ρ J x ′ J W ( | x ′ − x ′ J | , h ) = ∑ J m J ρ J x J W ( | x − x J | , h ) . (5.38) Note that the translation results in x = x ′ + c (5.39) and that x ′ − x ′ J = x − x J ⇒ | x ′ − x ′ J | = | x − x J | ⇒ W | x ′ − x ′ J | , h = W ( | x − x J | , h ) , where x ′ J are the particle position vectors in the new coordinate system. If the interpolation is independent of the translation of coordinate axes constant c has to be approximated exactly, i.e. ⟨ x ⟩ = ⟨ x ′ + c ⟩ = ⟨ x ′ ⟩ + ⟨ c ⟩ = ⟨ x ′ ⟩ + c . (5.40) By substituting Equations (5.37) and (5.38), the approximations for both x I and x ′ I , into Equation (5.40) one obtains: ∑ J m J ρ J x J W ( | x I − x J | , h ) = ∑ J m J ρ J x ′ J W ( | x I − x J | , h )+ c ∑ J m J ρ J W ( | x I − x J | , h ) (5.41) By comparison of Equation (5.39) and Equation (5.41) it is clear that the discretised space will only be homogeneous, i.e. result in the equivalent approximations, if the following condition is satisfied: ∑ J m J ρ J W ( | x I − x J | , h ) = 1 . (5.42)