The Interpolating Element-Free Galerkin Method For Nonlinear Large Deformation Problems

Meshless method constructs trial functions based on nodes,which avoids the constraints of elements or grids.It doesn't need remeshing technology when solving the problems with element distortion or mesh movement.Therefore,it is suitable for dealing with complicated problems,such as crack propagation,nonlinear large deformation,high-speed collision and so on.Now meshless method has become one of the research hotspots for scientific and engineering computation.Element-free Galerkin(EFG)method based on moving least-squares(MLS)approximation is one of the most widely studied and applied meshless methods.In the interpolating moving least-squares(IMLS)method based on a nonsingular weight function,the number of undetermined coefficients to obtain the shape function is one less than that of the traditional MLS approximation,and the order of the inverse matrix is one less order.Compared with the interpolating moving least-squares method based on a singular weight function,the IMLS method overcomes the computational difficulties and truncation errors caused by singular weight function,then it can improve the computational efficiency and accuracy.In this dissertation,the shape function is obtained with the IMLS method based on a nonsingular weight function,and then the interpolating element-free Galerkin(IEFG)method for large deformation problems of elasticity,elastoplasticity,viscoelasticity and polymer gels is presented.The IEFG method for elastic large deformation problems is presented.The discretilized equations are established by using Galerkin weak form based on total Lagrange formulation of elastic large deformation problems,the displacement boundary condition is imposed with penalty function method,the discretilized equations are solved with the Newton-Raphson iteration method,then formulae of the IEFG method for elastic large deformation problems are obtained.By analyzing numerical examples,comparing with the EFG method,the IEFG method presented in this dissertation has higher computational accuracy and efficiency.The IEFG method for elastoplastic large deformation problems is presented.The discretilized equations are established by using Galerkin weak form based on total Lagrange formulation of elastoplastic large deformation problems,the displacement boundary condition is also imposed by penalty function method,the elastoplastic constitutive relation based on the incremental form and linear hardening elastoplastic model are used,Mises yield criterion is used for the yield condition,and the discretilized equations are solved by the Newton-Raphson iteration method,then formulae of the IEFG method for elastoplastic large deformation problems are obtained.Again,by analyzing the numerical examples,it is shown that the IEFG method in this dissertation has higher computational accuracy and efficiency.The IEFG method for viscoelastic large deformation problems is presented.The discretilized equations are established by using Galerkin weak form based on total Lagrange formulation of viscoelastic large deformation problems with three-parameter model,then formulae of the IEFG method for viscoelastic large deformation problems are obtained.Numerical examples are given to discuss the influences of the relevant parameters on the numerical solutions,and the advantages of the IEFG method in this paper are shown.The IEFG method for large deformation problems of inhomogeneous swelling of polymer gels is presented.The free energy function of the deformation gradient caused by free swelling and mechanical loading is obtained by multiplicative decomposition,the discretilized equations are established by using Galerkin weak form of the inhomogeneous swelling of polymer gels,then formulae of the IEFG method for large deformation problems of polymer gels are obtained.The evolution of wrinkle model and the evolution of lattice model are studied to show that the IEFG method is more suitable for solving the nonlinear large deformation problems of inhomogeneous swelling of polymer gels.For the IEFG method for large deformation problems proposed in this paper,numerical programs are written by MATLAB.Numerical examples are given to discuss the influences of different weight functions,penalty factors,scale parameters of influence domain and step numbers on the numerical solutions.The numerical results are compared with ones of finite element method(FEM)to illustrate the validity and the advantages of the IEFG method for solving large deformation problems.The interpolating element-free Galerkin method for large deformation problems proposed in this paper can promote the research of meshless methods for large deformation problems or even super-large deformation problems,and promote the application of meshless methods in engineering.