A Meshless Radial Point Interpolation Method with Polynomial Basis and Its Applications

Abstract: At the beginning of this thesis, recent developments of the meshless method are overviewed. Several typical meshless methods are reviewed and appraised in term of their discretization scheme. Characteristics, advantages and disadvantages of all kinds of meshless methods are pointed out.The shape function is constructed by using the radial basis functions with polynomial basis functions, the singularity of the system matrix, because of using pure polynomial basis functions, is overcome. The shape function and its derivatives are simple, consequently have lower computational cost. The shape functions possess the Kronecker Delta function property, and the essential boundary conditions can be easily imposed.The meshless local radial point interpolation method have been used to analyze the bending problem of Timoshenko beam. The deflection and rotation are interpolated separately,and the discretized system equations are established based on a local Petrov-Galerkin weak-form. Results obtained show when the present method is applied to classical thin beam, the shear locking can easily be resolved by using the higher order polynomial in the basis function.The fundamental equations of moderately thick plate based on a Galerkin global weak-form are obtained using the energy principle. The three displacement components are interpolated separately. The discretized equations are derived, and the two optimal shape parameters of the multiquadrics (MQ) function are studied. Background integral cells is needed to evaluate the numerical integrations. Numerical results show that the results obtained converge to the exact solutions when field nodes increase.The present method has good stability and faster convergence. The shear locking can be avoided in the bending analyzing of thin plates.In the end of the thesis, based on Pasternak foundation model, the bending problems for the thick plates on an elastic foundation are analyzed by the meshless global weak-form radial point interpolation method . The Galerkin global weak-form equations for isotropic thick plates on the elastic foundation is derived by the radial point interpolation method. Numerical implementation are studied. Several numerical examples for isotropic thick plates on the elastic foundation are presented. Examples show that the analysis for thick plates on the elastic foundation by the meshless radial point interpolation method has a number of advantages, such as the high efficiency and the quite good accuracy and easy to implement…
Key words: radial point interpolation with polynomial basis functions; local Petrov-Galerkin method ; the Galerkin global weak-form; Timoshenko beam ; moderately thick plate; elastic foundation

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