Mathematical Physics - Volume II - Numerical Methods

Chapter 5. Review of Development of the Smooth Particle Hydrodynamics (SPH) Method

174

The requirement defined by Equation (5.42) can be interpreted as the ability of the interpo lation scheme to approximate a constant field (zero order consistency). However if this condition is not satisfied an exact approximation of a constant field can be achieved by reorganising equation (5.41).

m J ρ J

m J ρ J

x ′

J W ( | x ′

I − x ′

x J W ( | x I − x J | , h )

∑ J

J | , h ) − ∑ J

c =

=

m J ρ J

W ( | x I − x J | , h )

∑ J

(5.43)

W ( | x ′

I − x ′

W ( | x I − x J | , h )

J | , h )

m J ρ J

m J ρ J

= ∑ J

− ∑ J

x ′

x ′

m J ρ J

m J ρ J

J

J

W ( | x I − x J | , h )

W ( | x I − x J | , h )

∑ J

∑ J

This leads to set of new shape functions, also known as Shepard functions ˜ W = W ( | x I − x J | , h ) ∑ J m J ρ J W ( | x I − x J | , h ) which have the partition of unity property [7].

Similarly, an interpolation technique should not affect isotropy of space (domain of interpolation D ). One way of demonstrating this is to prove that the interpolated space is independent of infinitesimal rotational transformations. The same holds for the SPH approximation. If the relative rotation of the two coordinate systems is defined by C the rotation transformation tensor than the equivalent transformation of a position vector can be expressed as: x ′ = C · x (5.44) where C is an orthonormal tensor. For infinitesimal rotations this transformation can also be stated as: x ′ = x − ∆ ϕ × x (5.45) where ∆ ϕ is an infinitesimal rotation vector. If an SPH approximation is to maintain the isotropy of space then the approximation has to satisfy the following condition: ⟨ x ′ ⟩ ≡ ⟨ C · x ⟩ . (5.46) For the case of infinitesimal rotations, the approximation of the product ⟨ C · x ⟩ can be replaced by the product of applications ⟨ C ⟩· ⟨ x ⟩ with the difference proportional to the second and higher order infinitesimals. This allows for the equation (5.46) to restated as ⟨ x ′ ⟩ = ⟨ C ⟩· ⟨ x ⟩ . (5.47) In order to preserve the properties of approximation ⟨ x ⟩ the rotational matrix has to be approximately exactly, i.e. ⟨ C ⟩ = C ⇒ ⟨ x ′ ⟩ = C · ⟨ x ⟩ . To consider this condition one can start by rewriting x ′ = x − ∆ ϕ × x = x − ∇ ( ∆ ϕ × x ) · x = = x − ϕ x x = ( I − ϕ x ) · x , (5.48)