Mathematical Physics - Volume II - Numerical Methods

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

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The resulting dispersion relation for the current configuration is expressed as follows, the reader is referred to Randles et al. [78][79] and Rabczuk et al. [73] for a detailed explanation:

ρ " ∑ J ∈ S

W ′ ( j ∆ x ) sin ( κ ∆ x ) # 2

¯ C στ

ω 2 =

−

σ ρ   ∑ J ∈ S

2  

(5.77)

W ′′ ( j ∆ x ) [ 1 − cos ( κ j ∆ x )] − " ∑ J ∈ S

W ′ ( j ∆ x ) sin ( κ j ∆ x ) #

= −

Upon inspection of equation (5.77) it is immediately clear that the three terms contained herein yield three different conditions for stability: 1) When the material is unstable, in other words when ¯ C στ vanishes, which corresponds to the material instability of the continuum ( ¯ ω 2 = ( ¯ C κ 2 ) / ( ρ 0 ) , ¯ C ≤ 0, equation (5.11), Reveles [81]). This implies (5.77) could have two possible solutions: ω = ± i √ x , the negative solution would yield Condition 2 outlined below. 2) At the cut-off wave number κ = π ∆ x , j = − 1, this is the onset of stability for the particle equations in current configuration for an equally spaced particle arrange ment. In this case, the first term in the right hand side of equation (5.77) vanishes. Again, two possible solutions exist: ω = ± i √ x , the negative solution would yield Condition 2. 3) when σ > 0 and ¯ C στ̸ = 0, which is the tensile instability identified by Swegle. The second term inside the brackets on the right hand side is always positive, hence, if only ¯ C στ ρ " ∑ J ∈ S W ′ ( j ∆ x ) sin ( κ ∆ x ) # 2 and ∑ J ∈ S W ′ ( j ∆ x ) sin ( κ ∆ x ) ! 2 existed, the particle equation would be unconditionally stable (i.e. the only pos sible solution for ω is a positive real). However, if ∑ J ∈ S W ′ ( j ∆ x ) [ 1 − cos ( κ j ∆ x )] is sufficiently positive and σ > 0, the product of what is in brackets in equation (5.77) and σ would yield a negative value, hence ω = ± i √ x and again, the negative solution would yield Condition 2. This condition is given by Swegle et al. [87][88] as σ W ′′ > 0 which defines the onset of tensile instability of the SPH equations with nodal integration. Note that stability condition 1) is desirable as it represents the stability of continuum equations. Conditions 2) and 3) are the result of the type of discretisation carried out in SPH. From this analysis it is clear why some special smoothing functions can reduce or eliminate the tensile instability altogether: as long as the smoothing function is carefully selected, the second derivative might yield a negative value which can restore stability in the particle equation. For the cubic spline, (widely employed for SPH approximations) the value of the second derivative of the smoothing function at a distance u = 2 / 3 (Figure 5.4), from particle I ispositive. Therefore, the onset of tensile instability is defined by σ W ′′ > 0 Swegle [87].