Smoothed Finite Element Method For Solving Contact And Obstacle Scattering Problems

The smoothed finite element method?S-FEM?regarded as a weakened weak?W2?formulation method has been developed for solving various engineering mechanics problems.In this paper,finite element method?FEM?and S-FEM are used for solving solid mechanics,contact,scattering problems which have a wide range of biomedical backgrounds and applications.Firstly,a four-noded triangular?Tr4?element with one curved edge is first used to model the curved boundaries of complex problem domains,which deals with the lower accurate issue of linear triangular?Tr3?element on the curved boundaries.In this paper,finite element meshes can be created through mixing the linear Tr3 and the proposed Tr4?Tr3-4?elements,in which Tr3 elements are for interior and straight boundaries,and Tr4 elements are for the curved boundaries.Compared to the standard FEM-Tr3,FEM using mixed Tr3-4meshes?FEM-Tr3-4?model can significantly improve the accuracy of the solutions on the curved boundaries because of accurate approximation of the curved boundaries.Several solid mechanics problems are conducted,which validates the effectiveness of this method.Secondly,the theory of the S-FEM is further studied.S-FEM is a W2method based on the smoothing domain,such as cell-based,node-based and edge-based soomthing domains.It works well with distorted meshes.The G space theory offers the theoretical base for these W2 methods that use smoothing operations.We prove mathematically the stability and convergence of S-FEM,which also indicates the interpolation functions constructed using the S-FEM models belong to a G_h,0^s space.Thirdly,the cell-based S-FEM?CS-FEM?based on quadrilateral mesh,the node-based S-FEM?NS-FEM?and the edge-based S-FEM?ES-FEM?based on Tr3 mesh are used for 2D contact problems,combining with the formulation of linear complementarity problems?LCPs?.This model can efficiently avoid iterations at every step to improve the performance of the present algorithm.In this paper,the modified Coulomb friction contact model with tangential strength and normal adhesion is considered,which includes sticking-slipping,contacting-departing and bonding-debonding processes in a unified formulation.Smoothed Galerkin weak-form with contact boundary is deduced,in which the stiffness is implemented using the CS-FEM,NS-FEM and ES-FEM.Contact interface equations are discretized through contact point-pairs.These discretized equations are converted into LCP and solved efficiently using the Lemke method.Intensive numerical examples are given to investigate the effects of contact parameters and functionally gradient materials on contact behaviors and consider a two-dimensional contact model of femur and meniscus.The numerical results demonstrate that 1)based on Q4 mesh,all CS-FEM models are softer than FEM-Q4 model;the strain energy solutions,obtained using several CS-FEM models,are monotonically decreasing with the number of the SDs for each element increasing.2)Based on Tr3 mesh,the upper bound solutions in strain energy can be obtained using NS-FEM model,and the higher accurate solution can be obtained using ES-FEM model.Fourthly,an ES-FEM based on Tr3 mesh?ES-FEM-T3?is used for solving elastic wave scattering problem.The Navier equation and Helmholtz equations with coupled boundary for the elastic wave scattering are studied.Their smoothed Galerkin weak forms combining with the perfectly matched layer?PML?technique are derived to create effective S-FEM models.The PML can truncate the unbounded domain into bounded domain and eliminate wave reflections on the boundary.Some numerical experiments demonstrate that ES-FEM-T3 is more stable and accuracy than the standard FEM for the elastic wave scattering.Lastly,we consider an inverse obstacle scattering problem in an open space filled with a homogeneous and isotropic elastic medium.A coupled boundary value problem for scalar potential is obtained by using Helmholtz decomposition for the Navier equations.Further,the scattering problem is truncated into a coupled boundary value problem in a bounded domain using an exact transparent boundary condition.The domain derivatives for Helmholtz decomposition are derived for the displacement with respect to the variation of the surface.Using the domain derivatives,a continuation method with respect to the frequency is developed for reconstructing the surface of obstacle.Two numerical examples are presented to validate the effectiveness of the proposed method.