TABLE OF CONTENTS In Chapter 2, the basic ideas and essential formulations of the SPH method have been presented. It has been shown that the smoothing function plays a very important role in the SPH approximations, as it determines the accuracy of the function representation and efficiency of the computation.
In this chapter, a generalized approach to construct the smoothing functions for the SPH method is introduced. The approach uses the integral form of function representation with the help of the Taylor series expansion. A set of conditions are derived systematically, which can then be utilized to construct both analytical smoothing functions and point-dependent smoothing functions that can only be given in numerical forms. These conditions not only ensure the consistency in the SPH approximations, but also describe the compact support requirements for the smoothing function. Examples of SPH smoothing functions constructed include many existing ones used so far in the SPH literature, and a new quartic smoothing function derived recently by Liu, Liu and Lam (2002). The new quartic smoothing function is then applied to the simulation of a one-dimensional shock problem and a two-dimensional heat conduction problem. Particle inconsistency problem in the discretized form of particle approximation is also discussed with an approach for the consistency restoration.
One of the central issues for the meshfree methods is how to effectively perform function approximation based on a set of nodes scattered in an arbitrary manner without using a predefined mesh or grid that provides the connectivity of the...
