Meshless Method Based On The Kernel Approximation

The meshless method based on the kernel approximation is a pure Lagrangian computational method,in which the typical one is the Smoothed Particle Hydrodynamics (SPH) method.SPH uses the kernel function to approximate the physical quantities.However,the approximation effect will decrease near the boundaries.The idea of Reproducing Kernel Particle Method(RKPM) is to reconstruct a new kernel function by means of correcting function.By multiplied the correcting function,the kernel approximation of RKPM can improve the approximation accuracy and computational stability at the internal and boundary points.Although the corrective function is proposed to exactly reproduce the polynomial functions,it is found that RKPM can also benefit to reproduce the derivatives. In this thesis,the proof will be given.By the theoretical analysis and numerical simulation of the mathematical and physical models,the improved results and mechanisms of RKPM will be presented in this thesis.Another two corrective methods,Corrective Smoothed Particle Method(CSPM) and Modified Smoothed Particle Hydrodynamics(MSPH),are also briefly introduced.\ The comparison among these two methods and RKPM shows the relations and differences.In this thesis,the SPH and RKPM methods in cylindrical and spherical coordinate systems are also deduced.In the deduction,the calculation form of the kernel function and the formulas of kernel approximation are proposed.By the numerical simulation of the mathematical model,the effective results of RKPM compared with SPH are still tenable in the cylindrical and spherical coordinate systems.