The Hybrid Reproducing Kernel Particle Method For Three-Dimensional Problems

Meshless method is a numerical method developed since the 1990 s.Because its trial function is obtained based on a series of discrete nodes,the method can overcome the dependence of the mesh-based numerical method on the mesh,and it has become one of the important methods in the field of scientific and engineering computing.The reproducing kernel particle method(RKPM),which is developed from smooth particle method,is an important meshless method.When using the RKPM to solve three-dimensional(3D)problems,the computational efficiency of this method is low due to the complicated shape function.In order to improve the computational efficiency,in this thesis,by introducing the idea of dimension splitting,a hybrid reproducing kernel particle method(HRKPM)for solving 3D problems is presented.This method transforms a 3D problem into a series of related two-dimensional(2D)ones to obtain the solution of the original problem,and can improve the computational speed of the RKPM greatly.The HRKPM for 3D potential problems is presented.By introducing the idea of dimension splitting,a 3D potential problem can be transformed into a series of related2 D ones in the dimension splitting direction.Then,the shape functions of these 2D problems are constructed by the RKPM,the essential boundary conditions are imposed by the penalty method,and the discrete equations are obtained from the weak form of the potential problems.Finally,by coupling these discrete equations with the difference method in the dimension splitting direction,the formula of the HRKPM for solving 3D potential problems can be obtained.The numerical examples are given to discuss the influences of scaling parameter of influence domain,penalty factor,node distribution and dimension splitting step number on computational results,and the error analysis and convergence study are carried out.Comparing the RKPM,the HRKPM has higher computational efficiency when solving 3D potential problems.The HRKPM for several kinds of 3D time-dependent problems,such as the transient heat conduction problem,wave equation and advection-diffusion problem,are presented.Different from the potential problem,the governing equations of 3D time-dependent problems contain the first or second derivative of the variable to be solved to time,so it should be discretized in space ans time domains.By introducing the idea of dimension splitting,a 3D time-dependent problem can be transformed into a series of related 2D ones in the dimension splitting direction.Then,the discrete equations of these 2D problems are obtained by using the RKPM,and these discrete equations are coupled by using the difference method in the dimension splitting dierction.Finally,by using the difference method to discretize the time domain,the formulas of the HRKPM for solving 3D time-dependent problems are obtained.In the numerical examples,the influences of scaling parameter of influence domain,penalty factor,node distribution,dimension splitting step number and the time step on the computational results are discussed.Compared with the RKPM,the HRKPM has higher computational efficiency at different times.The HRKPM for 3D elasticity is presented.The three equilibrium equations of elasticity are divided into three sets of equations in which two equilibrium equations are contained.By coupling the discrete equations for solving arbitrary two sets of equations,we can obtain the solution of 3D elasticity.For arbitrary set of equations,a3 D elasticity problem is transformed into a series of related 2D ones in the dimension splitting direction.Then,the discrete equations of these 2D problems are established by using the RKPM.Finally,the discrete equations are coupled by using the difference method in the dimension splitting direction,and the discrete equations of the HRKPM for this set of equilibrium equations can be obtained.Then choosing another set of equilibrium equations arbitrarily,the discrete equations of the HRKPM for this set of equilibrium equations can be obtained in the same way.Combining the discrete equations for these two sets of equilibrium equations,the numerical solution of the original 3D elasticity problem can be obtained.In the numerical examples,the influences of scaling parameter of influence domain,penalty factor and node distribution on the computational results are discussed.Numerical examples show that the HRKPM can improve the computational efficiency of the RKPM for solving 3D elasticity problems.The HRKPM for 3D elastodynamics is presented.The governing equation of 3D elastodynamic includes the first and second derivatives of displacement to time,so it is necessary to discretize the space and time domains.The three equilibrium equations of elastodynamics are divided into three sets of equations in which two equilibrium equations are contained.By coupling the discrete equations for solving arbitrary two sets of equations,we can obtain the solution of 3D elastodynamics.For arbitrary set of equations,a 3D elastodynamics problem is transformed into a series of related 2D ones in the dimension splitting direction.Then,the discrete equations of these 2D problems are established by using the RKPM.The discrete equations are coupled by using the difference method in the dimension splitting direction.Finally,the classical Newmark-β method is used to discretize the time domain,and the discrete equations of the HRKPM for this set of equilibrium equations can be obtained.Then choosing another set of equilibrium equations arbitrarily,the discrete equations of the HRKPM for this set of equilibrium equations can be obtained in the same way.Combining the discrete equations for these two sets of equilibrium equations,the numerical solution of the original 3D elastodynamics can be obtained.In the numerical examples,the influences of scaling parameter of influence domain and the node distribution on the computational results are discussed.Numerical examples show that the HRKPM has higher computational efficiency for solving 3D elastodynamics problems.For the HRKPM presented above for solving several kinds of 3D problems,the corresponding algorithm implementation processes are proposed,and the MATLAB programs are written.From the numerical examples,the advantages of the method in this thesis are shown.