SPH方法中常数一致性核函数的建立及公式化

Constant consistency kernel function and its formulation

施加边界条件的不足是SPH方法的一个棘手问题,因为在结构边界外没有颗粒的存在,使得在边界处核函数的单位特性不能得到满足。施加"伪"颗粒是目前通用的一种方法,但是对于不规则结构和复杂几何边界,确定这些"伪"颗粒非常困难。本文讨论通过使用满足常数一致性的核函数来改善边界的不足。文章首先通过三种方法推导了满足常数一致性条件的核函数及其函数梯度的表达式,发现了两个不同分母式的表达,分析了满足常数一致性的修正核函数的数学特性。开展了二维和三维的算例比较,结果发现使用修正的核函数对边界条件有明显改善,对计算精度和稳定性也有显著提高。

Boundary conditions have been a sore point in the Smoothing Particle Hydrodynamics （SPH） method. The well-known problem originates from the kernel summation deficiency near the boundary, because there is no contributions of particles outside the boundary. Applying ghost particles or virtual particles is a commonly used approach. However, for an irregular structure or a complicated geometry, it would be difficult to determine these ghosts or virtual particles. In this paper we have discussed the application of a corrected constant consistency （or completeness） of kernel function to deal with the boundary deficiency. First of all, the corrected kernel functions with constant consistency （or completeness） are derived from three different corrective approaches, and their derivatives are also derived. The mathematical features and the error analysis of the corrected kernel functions are presented through the comparison with the traditional kernel function. Tests are carried out for both 2D and 3D case of a tension specimen and an impact example. It should be noted that the influences of the denominator differences in the derived formulations are analyzed and a few remarks are given. The improvement of the corrected constant consistency （or completeness） of kernel function is obvious near the boundary as we expect. In addition, the numerical accuracy and stability are improved as well.