A Class Of Infinitely Differentiable Functions In SHP Method

Smoothed Particle Hydrodynamics(SPH) method belongs to particle method that independent of grid completely, and now it has been successfully applied to many fields like science and engineering. The advantages of SPH method are generalized as follows. Firstly, it simulate liquid convection directly to the movement of particles in the region; secondly, there is no grid and this eliminates the trouble caused by mesh generation; thirdly, it can eliminate the numerical divergence at free interface; lastly, it can easily simulate flow problems with large deformation, managed to avoid the inconvenience caused by grid distortion and reconstruction.This paper firstly gives some profile to the basic idea of the SPH method. We introduce several common kernel function of SPH method. The smoothness of these kernels is not very good when in high-order derivative, which leads to the decrease of the precision of calculation. In this paper, we develop a new class of kernel functions by using the Dirac function based on the polished function, and called ε-Kernel. We also give examples by using ε-Kernel and other kernel functions to solve ordinary differential equations or partial differential equations to verify availability and efficient of ε-Kernel.The main ingredients in this thesis are described as following.The basic idea of SPH is showed. It indicates SPH method is a non-mesh, adaptive and stable Lagrangian solution. We state the basic equation of SPH method in detail. There are two steps in SPH method, one is the integral representation of function, the other is particle approximation of function. We make some introductions of the integral representation of all derivatives in the region of field functions, this show that the derivative’s value of function f(x) at a point x can be converted to the convolution of f(x) and the derivative of the corresponding order of kernel function W that be employed in SPH method. We also introduce the derivative discrete technology of SPH formula, which give a good illustration to the derivative methods of function in the process of nuclear and particle approximation with SPH method.The construction method of kernel function is given in details and several common kernel functions, such as Gaussian kernel and B-spline kernel, are showed. By smoothing Heaviside function, we develop a new form of weight function, which is called ε-kernel, and noted Wε(R). The function Wε(R) can satisfies all the conditions that a kernel function should satisfy in SPH method. The techniques of Tikhonov regularization is introduced as well, the techniques can provide a good theoretical basis for the solution of numerical examples using SPH method in following paper.Based on the knowledge above, we compare the errors of three different kernel functions in solving differential equations and obtain that ε-kernel has more excellent accuracy, and so it has better practical value. In two-dimensional case, we try to use ε-kernel to solve elliptic equation, the results also verify its availability. We verify the availability of ε-kernel function by comparing the analytical and the numerical solutions of partial differential equations defined in one-dimensional or two-dimensional case. It testifies the SPH method from the perspective of the application and makes a good pavement for solving more complicated hydrodynamics problems in future.