Researches On The Complex Variable Reproducing Kernel Particle Methods

In recent years,a class of new numerical methods called meshless methods has been proposed.The meshless methods only need the information at nodes,and don't discretize the domain into a mesh.The advantages of meshless methods are that the pre-processing is simple,and they have high precision.These features make the meshless methods a hot point and the development trend of numerical methods for science and engineering problems.The reproducing particle method(RKPM) is one,which is studied and applied widely,of the meshless methods.But it has great computation cost because of large numbers of nodes selected in the domain of problem.Then in this dissertation,on the basis of the RKPM,the complex variable reproducing kernel particle method (CVRKPM) is developed,and is applied to solve potential problems,transient heat conduction problems,elasticity problems,elastodynamics problems and elastoplasticity problems respectively.At last,the coupled FEM and CVRKPM for analyzing potential and elasticity problems are presented.The main researches of this thesis are as follows.On the basis of the RKPM,the CVRKPM is presented in this paper.The formations of the CVRKPM are obtained in detail.Comparing to the RKPM,the advantage of the CVRKPM is that the correction function of a 2D problem is formed with 1D basis function when the shape function is formed.Then the unknown coefficients of correction function in the CVRKPM are fewer than in the RKPM.And then the computational efficiency of the CVRKPM is greater.The CVRKPM is applied to two-dimensional potential problems,and the CVRKPM for potential problem is presented,and the corresponding formulae are obtained.Compared with the RKPM,under the same precision,the CVRKPM can select fewer nodes,and then has greater computational efficiency.On the other hand, under the same nodes,the CVRKPM has greater precision than the RKPM.On the basis of the steady heat conduction problems,the CVRKPM is applied to two-dimensional transient heat conduction problems.Combining the Galerkin weak form of transient heat conduction problems,the CVRKPM for transient heat conduction problems is investigated.And the corresponding formulae are obtained.The CVRKPM is applied to two-dimensional elasticity.The discrete equation is produced from the weak form of variational equation,the penalty parameters are used to enforce the essential boundary conditions,and then the CVRKPM for elasticity is presented.And the corresponding formulae are obtained.The advantages of this method are higher precision,and that volume closure phenomena can be avoided.The CVRKPM is applied to two-dimensional elastodynamics.The Galerkin weak form of elastodynamics problems is employed to obtain the discretized system equations,and the Newmark time integration method is used for time history analyses. And the penalty method is employed to apply the essential boundary conditions.Then the CVRKPM for elastodynamics is presented.And the corresponding formulae are obtained.On the basis of the CVRKPM for elasticity,when the small deformation is assumed,the incremental complex variable reproducing particle approximation is adopted,the increments of stress and strain are used to characterize the elastoplastic constitutive relationship,the penalty parameters are adopted to revise the energy variation equations to enforce the essential boundary conditions,and the Newton-Raphson iteration techniques are introduced into the numerical implementation, then CVRKPM for elastoplasticity problems is proposed.Numerical examples show that the CVRKPM for elastoplasticity problems has the advantages of good stability and higher convergence speed.As the shape function of the CVRKPM does not satisfy the property of Kronecker Delta function,which makes it difficult to impose the essential boundary condition of the problem.By combining the CVRKPM and finite element method(FEM),a coupled method of FEM and CVRKPM for analyzing two-dimensional elasticity or potential problems is presented.The coupled method not only emerge the essential boundary condition conveniently,but also exploit their advantages while avoiding their disadvantages to have higher computation efficiency.In order to show the efficiency of the CVRKPM in the dissertation,the MATLAB codes of the methods above have been written.Some numerical examples are provided, and the validity and efficiency of these methods are demonstrated.