Mathematical Physics - Volume II - Numerical Methods

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

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5.9.1 Stability analysis of conventional (Eulerian) SPH For the analysis of the SPH momentum equation in current configuration, the following 1 − D SPH momentum equation with nodal integration is considered: m I ¨ u I = − ∑ J ∈ S V J W ′ I ( x I − x J , h ) σ J . (5.60) Where: V J is the current volume of particle J , σ J is the Cauchy stress of particle J and

∂ W I ( x I − x ′ , h ) ∂ x ′

W ′

I ( x I − x J , h ) =

.

x ′ = x J

In order to introduce the displacement perturbation on the right hand side of (5.60), the current volume V J was expressed in terms of the initial particle mass and density as:

m ρ J

V J =

(5.61)

where the current density in (5.61) was defined as ρ = J − 1 ρ 0 . (5.62) The substitution of equation (5.62) and equation (5.61) into equation (5.60) and using the fact that in 1 D J = F , yields: m I ¨ u I = − ∑ J ∈ S m J ρ 0 W ′ I ( x I − x J , h ) F J σ J . (5.63) In equation (5.63), the deformation gradient is expressed with respect to the current configuration, hence: Equation (5.63) is linearised using perturbations ¯ u = u + ˜ u , ¯ x = x + ˜ x , ¯ F = F + ˜ F and ¯ σ = σ + ˜ σ as follows: m I ¨¯ u I = − ∑ J ∈ S m J ρ 0 W ′ I ( ¯ x I − ¯ x J , h ) ¯ F J ¯ σ J , (5.65) m I ¨¯ u I = − ∑ J ∈ S m J ρ 0 W ′ I ( ¯ x I − ¯ x J , h ) ( σ + ˜ σ )( F + ˜ F ) , (5.66) which yields m I ¨¯ u I = − ∑ J ∈ S m J ρ 0 W ′ I ( ¯ x I − ¯ x J , h ) ( ¯ x I − ¯ x J , h )( σ J F J + σ J ˜ F J + ˜ σ J F J ) (5.67) in (5.67) the product ˜ σ ˜ F was neglected. F J = ∂ x ∂ X J = ∂ u ∂ X J + 1 = ∂ u ∂ x ∂ x ∂ X J + 1 = 1 1 − ∂ u ∂ x J . (5.64)