Nodal Integration Of High Order Meshfree Methods And Mechanical Analysis For Thin Plates And Shells

Meshfree methods,such as the element-free Galerkin(EFG)method,developed in the past two decades possess several appealing advantages.For example,construction of meshfree approximation does not depend on a mesh(topological connectivity of nodes).Also,it is convenient to construct high order smooth approximation in meshfree methods.Due to these reasons,they have been widely used in many fields such as large deformation analysis,adaptive analysis and dynamic fracture,etc.However,due to the non-polynomial character of meshfree approximation functions,it is difficult to integrate the Galerkin weak form accurately.High order Gauss integration method with plenty of quadrature points in each background cell is required to result a stable method.This apparently leads to low computational efficiency and limits the applications of meshfree methods in real industrial analysis.The work done by this thesis can be divided into two parts including solids and thin plates and shells.On one hand,aiming at the difficulty of domain integration for solids,this thesis derives the nodal derivative correction equations based on Hu-Washizu three-field variational principle.Nodal integration schemes which satisfies high order consistency for two-dimensional and three-dimensional meshfree methods are developed by introducing Taylor expansion into the nodal derivative correction equations.The proposed integration schemes can reproduce linear strain fields exactly and show much better accuracy,efficiency and stability than those of standard Gauss integration and the existing stabilized conforming nodal integration(SCNI)which can only reproduce constant strain fields exactly.In comparison with the trilinear finite element method(FEM),i.e.eight-node hexahedron element,the proposed method also shows better numerical performance in accuracy,convergence and even in computational efficiency.On the other hand,as an important class of engineering components,thin plates and shells are widely used in many industries such as aerospace engineering,civil engineering,marine engineering,mechanical engineering and chemical engineering,etc.Due to the real industrial value,the mechanical behaviors of thin plates and shells under different loads are always a hotspot in computational mechanics.In mathematics,the government equations of thin plates and shells are high order partial differential equations.To solve these equations numerically,the approximation functions at least should possess C1 continuity.However,in the FEM which is widely applied in engineering analysis,the nodal shape functions based on the Lagrange interpolation only possess C0 continuity.Therefore,FEM cannot be applied to thin plates and shells straightforwardly.In contrast,the meshfree methods,such as EFG,have natural advantages in thin plates and shells analysis due to the smoothness of the approximation.However,similar to the meshfree analysis of solids mentioned above,meshfree analysis of thin plates and shells is also lack of efficient and accurate numerical integration methods.This is also one of the major issues this thesis devotes to study.The government equation of thin plate bending problems is fourth-order partial differential equation,thus second order derivatives of nodal shape functions are involved in stiffness matrix.Obviously,accurate evaluation of the domain integration in thin plate bending problems is more difficult than that in solids.To this end,this thesis rationally derived the correction equations for nodal second order derivatives based on Hu-Washizu three-field variational principle.The derived correction equations apply to arbitrary order approximation and describe the relation among the corrected nodal second order derivatives,nodal shape functions and its standard derivatives.Based on this,the corresponding integration schemes are established.Taking the cubic meshfree approximation as an example,the established three-point integration scheme using triangular background integration cells is able to reflect pure bending and linear bending modes.Therefore,it is named as linear curvature smoothing method.The proposed method shows much better accuracy and efficiency than standard Gauss integration method.This thesis also extends the linear curvature smoothing method to free vibration analysis.More accurate natural frequency is obtained by the proposed method than Gauss integration method and the existing constant curvature smoothing method.Besides,this thesis also develops a nodal integration scheme for thin plate bending,which can exactly reproduce linear bending modes.Numerical analysis of thin shells is more difficult than thin-plates since both membrane and bending stresses exist.This thesis employs the geometrically exact shell model in which,the mid-surface of shell can be always mapped to a plane in parametric space.Based on this model,the nodal derivative correction equations for both membrane and bending strains are established in parametric space.The former describes the relation between the corrected nodal derivatives and nodal shape functions;the latter describes the relation among the corrected second-order nodal derivatives,nodal shape functions and their standard derivatives.Based on the developed nodal derivative correction equations,this thesis proposed an integration scheme for cubic meshfree approximation,which can exactly reproduce linear strain modes.In comparison with the standard Gauss integration method,the proposed method significantly improves the computational accuracy and efficiency,and obtains more accurate membrane and bending stress fields.Efficient meshfree analysis of thin shells is accomplished by the proposed method.Fracture of plates and shells widely exists in real engineering and is always one of the focuses of fracture mechanics.However,numerical simulation of the crack propagation in thin plates and shells is still far from mature due to the C1 continuity requirement.It is well known that an important part in simulation of crack propagation is the numerical description of strong discontinuities(cracks).In this thesis,phantom nodes are used to describe the strong discontinuities.Besides,the corrected nodal derivatives are employed to improve the accuracy and computational efficiency.As a result,numerical simulation of the crack propagation in thin plates and shells is accomplished in this thesis.