Study On Fully And High Order Smoothing Technology With The Framework Of Finite Element Method And Its Application

Advanced numerical methods are widely used in automotive industry,aeronautics and astronautics,oceaneering,civil engineering,bioengineering and other fields,finite element method is an effective method to solve all kinds of complicated problems.With the further development of science and technology,both the accuracy and stability of the solution are required in the process of product design and analysis of engineering structure.The traditional finite element method shows more and more limitations,such as the accuracy of the solutions are depended on the quality of grids,locking phenomenon,grids should be matched with the boundary of the elements when solving the discontinuous problems,and so on.The smoothed finite element method(S-FEM)based on the framework of traditional finite element method(FEM)was proposed by coupling with the strain smoothing technique,which inherits the advantages of the traditional FEM,meanwhile,it has its idiosyncrasies,such as high accuracy,high efficiency and upper bound property.However,only the stiffness matrix was smoothed in the traditional S-FEM,and the mass matrix is still computed by the traditional Gauss integration.Therefore,the coordinate mapping and the computation of the Jacobean matrix cannot be completely avoided.On the other hand,the traditional S-FEM is mostly based on linear interpolation,which can not provide the expected result in the polygonal(2D)/polyhedral elements(3D)or the higher order elements,and even provide less accurate than the traditional polygon finite element method.In this paper,the works are based on the smoothing technique,not only the strain is smoothed and but also the smoothing technique is extended to the calculation of the consistent mass matrix,and called it as fully smoothed technique.Besides,the fully smoothed technique was coupled with the extended finite element method(XFEM)and used for the analysis of weak discontinuity problem.Moreover,in order to further improve the theory of S-FEM and SmXFEM(Smooted eXtended Finite Element Method),the linear smoothing technique was introduced into the higher ord er elements,and employed for the analysis of 2D plane problem,laminated composite plates.The works completed in this paper are as follows:1)The fully smoothed integration for the 2D plane problem is studied.Except for using the smoothing technique for the strain,we introduce a special integral scheme to calculate the consistent mass matrix based on the combination of Gauss divergence theorem with the evaluation of an indefinite integral.This unifies the integral form of the stiffness matrix and the consistent mass matrix,and we called it as the fully smoothing integration.The accuracy of the special integral scheme and the sensitivity to the quality of the elements are verified by the examples of free vibration and forced vibration.2)The S-FEM is extended into the axisymmetric problem.The stiffness matrix and the consistent mass matrix are constructed by combining Gauss divergence theorem with the evaluation of an indefinite integral.Applying the special integral and smoothing technique,all the domain integrals in stiffness matrix and mass matrix can be smoothed and rewritten as boundary integrals of smoothing cells.The static and structure dynamic analysis of axisymmetric problems are respectively solved based on the CS-FEM,NS-FEM,ES-FEM,which verifiy the validity of the proposed approach.3)A fully smoothed axisymmetric XFEM is developed and employed to solve the axisymmetric problem with weak discontinuity.The method is based on the framework of CS-FEM and coupling with the XFEM.The smoothed strain and the consistent mass matrix are computed over the each smoothing cell,all the domain integrals are transformed into boundary integrals of smoothing cells,and on this basis we can obtain the smoothed stiffness matrix and consistent mass ma trix.Comparing with the traditional XFEM,the proposed technique does not require sub-triangulation for the purpose of numerical integration,no coordinate mapping is required,and the computation of the derivative of shape function can also be avoided.T he accuracy and convergence properties of the proposed technique are demonstrated with a series of numerical examples in elastostatics and elastodynamics with weak discontinuities.Finally,the proposed method is used to calculate the effective material pa rameters of carbon nanotube composites.4)The linear smoothing technique is introduced into the higher order 2D element(Q8).The modified strain matrix is computed by the divergence theorem between the nodal shape functions and their derivatives using Ta ylor‘s expansion of the weak form,the defect of low accuracy of traditional smoothing technique in higher order elements is improved.The improved accuracy and superior convergence rates are numerically demonstrated with a few benchmark problems,and it i s employed to analyze the stress concentration around cutouts for the laminated composites.5)A linear smoothed eight-node Reissner-Mindlin plate element(Q8 plate element)based on the first order shear deformation theory is developed.The proposed Q8 plate element has good performance to alleviate shear locking phenomenon,yield more reasonable results for the distorted meshes,and maintain a high rate of convergence.Meanwhile,all the computations are based on the global Cartesian coordinate system,which avoids the coordinate mapping and the calculation of the derivative of shape function is not required.Several numerical examples are presented to demonstrate the accuracy,convergence and the sensitive to the mesh distortion of the proposed method.Finally,it is applied to the static and free vibration analysis of composite plates.