Studies On The Strain Reconstruction Point Interpolation Meshfree Method (SC-PIM)

Finite Element Method (FEM) is one of the important numerical methods.In many areas of the actual mechanics, physics, electromagnetic problems,it has a very wide applications. However, there are many nonlinearproblems in engineering practice, such as large deformation, crackpropagation and other problems. When using the finite element method tosolve nonlinear problems, the method may produces large deformation,resulting in distortion cells, in this situation, it will greatly affectthe accuracy of the FEM. We know these problems are caused due to the grid,so naturally, people tend to have a idea of get rid of the grid. Therefore,MeshFree Methods (MFree) was born.Most essential difference between MFree and FEM is that MFree doesnot depend on the grid, building shape function which is based on the nodesof local support domain. Point Interpolation Method (PIM) is one of theMeshFree Methods, it uses the information of nodes in local support domainfor interpolation to construct shape functions and it can overcome meshdependence. Usually, it uses polynomial basis shape functions, which isconsistent and has function in nature. PIM not only has a simplecalculation, but also can add to boundary conditions easily. Also, becausemany implementation details are similar between PIM and FEM, manytechniques in FEM can be used in PIM. A node-based smoothed pointinterpolation method (NS-PIM) is formed by using smoothed gradienttechniques and constructing a smoothed strain. A large number of practicalapplications find NS-PIM has a soften function, it can produce an upperbound strain. However, the traditional finite element method can oftenget a lower bound strain, the stiffness is too hard. Strain reconstructedpoint interpolation meshfree method (SC-PIM) proposed in this articlecombines the two ideas, by introducing an adjustable parameter, makes these strains by MFree and FEM do a linear combination. By adjusting thisparameter, we can get an upper bound solution, a lower bound solution anda superconvergent solution. This paper presents the detailed constructionof SC-PIM and gives a convergence proof. Through several numericalexperiments of static mechanics and dynamic heat conduction, we test theeffectiveness of SC-PIM.