TABLE OF CONTENTS In Chapter 1, it is seen that as a meshfree particle method, the smoothed particle hydrodynamics (SPH) shares many similarities with other particle methods and meshfree methods while possessing its unique features. In this chapter, the basic concept and the essential formulations of the SPH method are introduced. The SPH approximations are discussed, which include the strategy of the SPH method, the continuous integral representation (kernel approximation) and the discretized particle approximation. Some tricks to be used in deriving the SPH formulations for complex PDEs are presented. Concepts on the support domain and the influence domain are described. Differences between the SPH method and other particle methods such as the particle-in-cell (PIC) method are discussed. These concepts, tricks and essential formulations discussed in this chapter are very useful in the development of various variants of the SPH method.
As mentioned in detail in Chapter 1, the SPH method was developed for hydrodynamics problems that are basically in the form of partial differential equations (PDE) of field variables such as the density, velocity, energy, etc. Obtaining analytical solutions for such a set of PDEs is not usually possible, except for very few simple cases. Therefore, efforts have been made in seeking for numerical solutions. In doing so, one needs to first discretize the problem domain where the PDEs are defined. Next, a method is needed to provide an approximation for the values of the field functions and their derivatives at any point. The
