Consistent High Order Element-free Galerkin Method And Crack Propagation Analysis

Element-free Galerkin(EFG)method gradually developed in the past twenty years is a meshfree method.In EFG,construction of approximation functions does not depend on a mesh.In addition,it is convenient to construct high order approximation functions and the shape functions possess high order smoothness.Due to these merits,EFG method shows significant advantages in dealing with crack propagation,adaptive analysis,computation of plates and shells,large deformation analysis,etc.However,the shape functions of the EFG method are non-polynomial rational functions and thus accurate evaluation of the weak form by a numerical integration is quite difficult.Quite a number of integration points have to be employed to ensure the stability of the method.This leads to low computational efficiency.Furthermore,the numerical integration is not accurate enough to let the method exactly pass patch tests which are very important in the guarantee of convergence.To a large extent,this drawback impedes the application of the EFG method in industry.It is significant to study how to rationally reduce the number of integration points,meanwhile,to ensure the accuracy and thus to significantlyimprove the computational efficiency of the element-free Galerkin method.This is exactly the major issue this thesis devotes to study.The present work in this thesisis based on the "consistency framework for nodal derivatives" proposed by Duan et al.in 2012 for meshfree methods.Using the corrected nodal derivatives which are determined by the consistency framework to compute the stiffness matrix is able to reduce the number of integration points and to improve the computational efficiency.However,the introduction of the corrected nodal derivatives to the stiffness matrix is lack of theoretical basis.Based on the Hu-Washizu three-field variational principle,the present thesis re-derive "consistency framework for nodal derivatives" and the corrected nodal derivatives naturally exist in the weak form.This lays a theoretical basis for the method,and names it as consistent element-free Galerkin method(CEFG).For two dimensional problems,the present thesis establishes quadratic and cubic CEFG methods.They remarkably reduce the number of integration points and improve computational accuracy and convergence.This results a significant increase in the computational efficiency.Moreover,the present thesis extends this method to three-dimensional problems by developing the quadratically consistent 4-point(QC4)integration scheme and the computational accuracy,convergence and efficiency of the three-dimensional element-free Galerkin method are significantly improved.The present thesis also develops a one-point integration scheme for consistent element-free Galerkin method.The scheme employs only one evaluation point in each sub-domain for integration and is able to reproduce a linear strain field without hourglass modes.Development of the scheme is based on the proposed consistency framework to correct nodal derivatives and the introduction of the Taylor expansion technique.The proposed one-point integration scheme is able to exactly pass linear and quadratic patch tests and thus is named as quadratically consistent 1-point integration method(QC1).Compared with other one point integration methods,the proposed QC1 method shows significant superiorities in the computational accuracy,convergence,stability and efficiency.Numerical simulation of crack propagation is an important application field of the element-free Galerkin method.The present thesis extends the proposed consistent high order element-free Galerkin method to crack problems(i.e.discontinuous problems).Phantom nodes are used to describe the strong discontinuity of the displacement at cracks.The algorithm to introduce the phantom nodes and the method to evaluate the domain integral in cracked"element" are proposed.Compared with the standard high order element-free Galerkin method and the low order consistent element-free method,the proposed consistent high order meshfree method remarkably improves the computational accuracy of stress intensity factors and is able to accurately predict crack paths.Further,the present thesis adaptively adds computational nodes near crack tip.as crack propagates by taking full advantage of the convenience of the EFG method in local refinement of computational nodes.The adaptive simulation of crack propagation is achieved and this significantly reduces the number of nodes as well as computational scales.In addition,the present thesis also studies accurate enforcement of essential boundary conditions in the element-free Galerkin method.The methods coupling finite elements and weight functions to enforce essential boundary conditions are investigated respectively,and some improvements are presented.