Mathematical Physics - Volume II - Numerical Methods

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

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time for data access time. The integer coordinates int x i

int are computed from the particle

position vector x I as:

x i int = f loor

I

N SUB 2 h MAX x i

(5.35)

.

Where N SUB is the number of box subdivisions, typically N SUB = 1000.

5.7 SPH Shortcomings

As mentioned in the introduction, the basic SPH method has been shown to have several problems: - Consistency,

- Tensile instability, - Zero-energy modes.

5.7.1 Consistency The SPH method even in its continuous form is inconsistent within 2 h of the domain boundaries due to the kernel support incompleteness. In its discrete form the method loses its 0 th order consistency not only in the vicinity of boundaries but also over the rest of the domain if particles have an irregular distribution. Meglicki [56] showed that node disorder results in a systematic error. Therefore a proper SPH grid should be as regular as possible and not contain large discrepancies in order to perform most accurate simulation. First order consistency of the method can be achieved in two ways. Firstly, by correct ing the kernel function, and secondly, by correcting the discrete form of the convolution integral of the SPH interpolation. Johnson [40] uses this correction procedure and proposed the Normalised Smoothing Function. Vignjevic [97] also implemented a kernel normali sation and correction to lead to a Corrected Normalised Smooth Particle Hydrodynamics (CNSPH) method which is first order consistent. The full derivation of this correction is given below. In SPH methods based on a corrected kernel, it is no-longer possible to ignore boundary conditions. In basic SPH, free surface boundary conditions are not imposed and are simply ignored as variables tends to zero at boundaries because of the deficiency of neighbour particles. 5.8 Derivation of Normalised Corrected Gradient SPH formula Starting from the conventional SPH method of Gingold and Monaghan, which is not even zero order consistent, a number of researchers worked on development of a first order consistent form of SPH. The result of this effort at Cranfield was named Normalised Corrected SPH (NCSPH) and the same term is used in this text for any other similar version of the method. The approximation of fields using a NCSPH interpolation has been published by Randles and Libersky [78], Vignjevic [97] , Bonet [17]. Bonet used properties of the integrals of motion (linear and angular momentum) to derive Normalisation and Gradient Correction for kernel interpolation. This approach lacks generality and does not provide the insight into the origin and the nature of the problem. A full derivation of the correction