A Strain-reconstructed Smoothed Finite Element Method And Its Application To The Inverse Problems

The finite element method(FEM)is one of the important numerical methods.It has been widely used in engineering and science.However,the standard FEM has some inherent disadvantages.For example,the numerical solutions of FEM are very sensitive to the mesh distortion in the analysis of large deformation.Another critical issue of FEM is the overly-stiff property,leading to relatively poor solutions in stress analysis and locking issues.In order to solve these problems,many types of meshfree methods were proposed.Since most of the shape functions in the meshfree method do not have the ? function properties,the implementation of essential boundary conditions requires some special treatments.In the framework of FEM and meshfree method,Liu and his groups proposed smoothed finite element method(S-FEM)with smoothed strain.In this thesis,the FEM,the meshfree method and the S-FEM are firstly introduced.Then we introduce the node based smoothed finite element method(NS-FEM)and the edge based smoothed finite element method(ES-FEM)respectively.Next,the construction of the smooth strain based on NS-FEM and ES-FEM is described.In addition,the strategy to overcome the volumetric locking issue for nearly incompressible materials is also analyzed in this thesis.As NS-FEM is considered as often softened model,it provides the upper bound solutions of strain energy.Although NS-FEM is very effective to solve the volumetric locking issue,the solutions of displacement in the NS-FEM are not improved.On the other hand,the ES-FEM with a close to exact stiffness is able to improve the accuracy of the displacement significantly.Thus,a strain-reconstructed smoothed finite element method proposed in this thesis combines the NS-FEM and ES-FEM,by introducing an adjustable parameter ?.The convergence of the strain-reconstructed smoothed finite element method is proved by the theory,and the relationship between the stiffness matrix and the stiffness matrix of NS-FEM and ES-FEM is analyzed.In this thesis,we have conducted the detailed analysis to determine the optimal parameter ? to obtain ultra-accurate solutions.The numerical results have demonstrated that the proposed strain-reconstructed smoothed finite element method is very effective to analyze the elasticity problems with nearly incompressible Poisson's ratio.In addition,the strain-reconstructed smoothed finite element method is applied to the inverse problem,which is very accurate to predict the status of abnormal tissue.