Mathematical Physics - Volume II - Numerical Methods

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

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These requirements, formulated by Lucy [43], ensure that the kernel function reduces to the Dirac delta function when h tends to zero:

W ( | x − x ′ | , h ) = δ ( | x − x ′ | , h ) .

lim h → 0

(5.4)

And therefore, it follows that:

f ( x ) ⟩ = f ( x ) .

lim h → 0 ⟨

(5.5)

If the function f ( x ) is only known at N discrete points, the integral of equation (5.1) can be approximated by a summation: f I = f ( x I ) ≈ ⟨ f ( x I ) ⟩ = Z Ω f ( x ′ ) W ( x − x ′ ) d Ω ≈ ≈ N ∑ J = 1 m J ρ J f ( x J ) W ( | x I − x J | , h ) . (5.6) In the above equation, the subscript I and J denote particle number, m J and ρ J the mass and the density of particle J , N the number of neighbours of particle I (number of particles that interact with particle I , i.e. the support of the kernel), m J ρ J is the volume associated to the point or particle J and W IJ = W ( | x I − x J | , h ) . In SPH literature, the term particles is misleading as in fact these particles have to be thought of as interpolation points rather than mass elements. Equation (5.6) constitutes the basis of SPH method. The value of a variable at a particle, denoted by superscript I , is calculated by summing the contributions from a set of neighbouring particles (Figure 5.1), denoted by superscript J and for which the kernel function is not zero: ⟨ f ( x I ) ⟩ = ∑ J m J ρ J f ( x J ) W ( | x I − x J | , h ) . (5.7)

Figure 5.1: Set of neighbouring particles.