The Interpolating Element-Free Galerkin Method For Elastoplasticity And Elastic Large Deformation Problems

Meshless method constructs approximation functions or interpolating functions based on nodes information,which avoids or reduces the dependence of the traditional mesh-based numerical methods on elements or meshes.The method does not need remesh technology in computational process involving mesh distortion and mesh movement,and can overcome the disadvantages,such as low computational accuracy,difficulty of adaptive analysis and limitations of analyzing some complicated problems,of mesh-based numerical methods.It is more suitable for solving complicated problems,such as crack growth,nonlinear large deformation,fluid-solid coupling,high-speed collision and metal stamping forming.The element-free Galerkin(EFG)method is one of meshless methods studied and applied most widely.In this paper,by using the improved interpolating moving least-squares(IMLS)method to form the shape function,the formulae of the interpolating element-free Galerkin(IEFG)method for the 3D elastoplasticity and 2D elastic large deformation problems are obtained.The shape function based on the IMLS method can satisfy the property of Kronecker ? function,then the meshless method based on the IMLS method can apply the displacement boundary conditions directly.The unknown coefficients of the interpolating function are one less than that of the interpolating moving least-squares method obtained by Lancaster,which results in higher computational efficiency and accuracy.By using the IMLS method to form the shape function,using the Galerkin weak form of 3D elastoplasticity problems to obtain the discretized equations of the incremental form,the formulae of the IEFG method for the 3D elastoplasticity problems are obtained.The Mises criterion is used to determine whether the material enters the plastic state.The increment tangent stiffness matrix method is used to solve the final eqautions.And the boundary conditions are applied directly.By using the IMLS method to form the shape function,using the Galerkin weak form of 2D elastic large deformation problems based on the total Lagrange formulation to obtain the discretized equations,the formulae of the IEFG method for 2D elastic large deformation problems are obtained.And Newton-Raphson iterative method is used to solve the final eqautions.For the formulae of the IEFG method presented above,computer code of Matlab is written.From the analysis of numerical examples,the influences of node distributions,scale parameters of influence domains and the loading steps on the numerical results are discussed.Compared the numerical results in this thesis with the ones of the finite element software ABAQUS,the effectiveness of the obtained method is verified.Compared with the ones of the EFG method,it is shown that the IEFG method in this thesis for solving 3D elastoplasticity problems and 2D elastic large deformation problems has higher computational accuracy and efficiency.The research results in this thesis provide an efficient and high-precision numerical method for solving complicated engineering problems by using meshless methods.