Mathematical Physics - Volume II - Numerical Methods

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

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Where: v = ( | x − x ′ | ) / h , D is the number of dimensions of the problem (i.e. 1,2 or 3), C is the scaling factor which depends on the number of dimensions and ensures that the consistency conditions 2 and 3 are satisfied:   2 3 , D = 1 , 10 7 π , D = 2 , 1 π , D = 3 . (5.31) The derivatives of the kernel function have the following property ∂ W ( | x − x ′ | , h ) ∂ x ′ k = − ∂ W ( | x − x ′ | , h ) ∂ x k . It is important not to forget the dimensionality of the kernel function W . For instance, in one dimension, W has dimension of length − 1 and the dimension of ∂ W ∂ x ′ is thus length − 2 . The mass, m , should be interpreted as mass per unit area, with the cross-sectional area numerically equal to one. 5.5 Variable Smoothing Length If large deformations occur, particles can move apart. In the case of conventional (Eulerian) SPH, if the smoothing length remains constant, the particle spacing can become so large that particles will no longer interact. On the other hand, in compression, the number of neighbour particles within the support can become large, which can significantly slow down the calculation. In order to avoid these problems, Gingold and Monaghan [28] suggested that it would be preferable to allow h for any particle to be related to its density according to h = G m ρ 1 / n (5.32) where n is the number of dimensions and G is a constant approximately G ≈ 1 . 3. Benz [16] proposed the use of a variable smoothing length with the intent of maintaining a healthy neighbourhood as the continuum deforms. His equation for evolution of h is: h = h 0 ρ 0 ρ 1 / n (5.33) where h 0 and ρ 0 are initial smoothing length and density and n is the number of dimensions of the problem. Another frequently used equation for evolution of h based on conservation of mass is: d h d t = 1 n h ∇ · v . (5.34) According to Monaghan [65] and Price and Monaghan, h should be determined from the summation equations so that it is consistent with the density obtained from the summation, i.e. ρ I = ∑ J m J W ( | x I − x J | , h J ) , where ρ I is either estimated from the SPH summation.