Towards better understanding of the Smoothed Particle Hydrodynamic Method (original) (raw)

2003, Cranfield University

AI-generated Abstract

Numerous approaches have been proposed for solving partial differential equations; all these methods have their own advantages and disadvantages depending on the problems being treated. In recent years there has been much development of particle methods for mechanical problems. In this thesis, several problem areas of the SPH formulation were examined. Firstly, a new approach based on 'Hamilton's variational principle' is used to derive the equations of motion in the SPH form. Secondly, the application of a complex Von Neumann analysis to SPH method reveals the existence of a number of physical mechanisms accountable for the stability of the method. Finally, the notion of the amplification matrix is used to detect how numerical errors propagate permits the identification of the mechanisms responsible for the delimitation of the domain of numerical stability. By doing so, we were able to erect a link between the physics and the numerics that govern the SPH formulation. It has been now about three decades since the first paper by Lucy using the SPH method to test the fission hypothesis. The extension of the SPH method to gas dynamic and to hydrodynamic problems can be attributed largely to the work of Gingold and Monaghan and Benz. At the beginning of the 90's, W. Benz, L. Libersky et al and Johnson et al extended SPH method to simulate problems of solid mechanics including impacts, penetrations and large deformations. The stability analysis of the SPH method was pioneered by Mas-Gallic and Raviart, Swegle et al, D. Balsara, Dilisio et al, D. Hicks and L. Liebrock and Ben Moussa and Vila. The main advantage of the SPH method over Eulerian methods is that it is a Lagrangian gridless one. The originality of this method lies in the manner the space derivatives are evaluated. The objectives of this research project are multiple and can be regarded as theoretical and practical aim. The primary objective is to give the SPH method certain truthfulness. To accomplish this objective, the Hamilton's principle is used to derive the equations of motion in the SPH form. Hamilton's variational principle is the most prominent one so far and processes the unifying nature throughout all physics. The author, by using Hamilton's principle intends to furnish firm mathematical and physical evidence to sustain the SPH method as complete and respected numerical tool such as other classical numerical methods (PIC, finite volume and finite element, etc…). The second objective of this research concerns the boundary condition treatment in particle method and in SPH particularly. Boundary condition implementation is a very subtle yet difficult issue; many authors mentioned the 'Ghost particles' approach for boundary condition treatment. In our knowledge, there is no available or published complete description of the meaning of 'Ghost particles' notion. To remedy this situation, in this research as the first step, a best understanding of the concept of 'Ghost Particles' is derived theoretically from the equations of motion and in the second step a clear depiction of the algorithm is given.