Study Of The Meshfree Method Based On Stabilized Conforming Nodal Integration And RPIM Shape Function

Meshfree method is a new numerical method developing from more than ten years ago. Having advantages in non-mesh, adaptation, accuracy etc., it is used in many fields that the finite element method (FEM) can not solve, such as dynamic crack propagation, large deformation, material breakage, high speed impact, and so on. As the collocation methods have poor stability, the Galerkin methods are adopted in this paper. Because the local weak form meshfree methods can not easily deal with complex boundaries and make the system matrix unsymmetrical, which results in excessive time of solution, so the main study on meshfree methods in this thesis is global weak form meshfree methods.The global weak form meshfree method deriving from FEM is a well developed meshfree method because of its advantages in stability and efficiency. But this method also has obvious disadvantages: require background cell for gauss integration; can not ensure the continuity of the displacement in cell and on the relevant boundaries; the integral results are not exact because of the complicated shape functions. The stabilized conforming nodal integration (SCNI) based on the strain smoothing theory can easily avoid these difficulties. In addition, the radial basis point interpolation method (RPIM) shape function is accepted in order to enforce the essential boundary conditions easily. Regretfully, the conditional RPIM can not satisfy the global conformability, making the energy principle equation unbalanced, and the form of conforming RPIM is also very complicated. Consequently, the conditional RPIM combined with SCNI will form a new global weak form meshfree method, which not only satisfies the global conformability but also imposes the essential boundary conditions exactly.The key technology of global weak form meshfree methods is first presented in detail. The equivalent integral weak form of elasticity control equation is formed by means of weighted residual method. Then the SCNI and RPIM shape function are detailedly introduced. The following work is the applications in static mechanics of the global weak form meshfree method based on these two factors. The relative flow chart is also formed depending on the solution procedure of common weak form methods. In numerical examples, the patch test, cantilever beam and infinite plate with a circular hole are selected to establish the convergence, accuracy, stability, and robustness of the present method.Then the applications in dynamic mechanics of the above meshfree method are investigated in this thesis for the first time, and the main energy is focused on the vibration analysis of structure. After the essential equation of weak form meshfree method is developed, solutions of two typical dynamic problems, free vibration and forced vibration of structure, are fully presented. In the end, the cantilever beam is used again in order to describe the dynamics. Combing the present meshfree method with the above two solutions about dynamics problems, we can get stable, accurate, and convergent results. These works provide new thoughts about the study on meshfree methods in dynamic problems.