Constant consistency kernel function and its formulation

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.