Meshfree Barycentric Interpolation Collocation Method And Its Applications In Variational Inequality

We introduces the barycentric Lagrange interpolation collocation method(BLIM),barycentric rational interpolation collocation method(BRIM),high-order barycentric rational interpolation collocation method(HBRIM) in details and gives the algorithm framework of problems described by variational inequality. Furthermore, BLIM, BRIM, HBRIM are applied to the elastic-plastic torsion problem, fourth-order obstacle problem, and viscoelastic dynamic frictionless contact problem with adhesion.The main work is as follows:In chapter2, it presents three kinds of barycentric interpolation shape functions and constructs barycentric interpolation collocation method. The algorithm framework for a kind of second order elliptic partial differential equations is given. In the numerical examples, we compare the effectiveness of these methods. The effect of the number of nodes, nodal distribution and parameters are also discussed. Finally, we apply this method into the elastic-plastic torsion problem. Comparing the numerical results with exact solution, we find this method efficient.In chapter 3, in order to eliminate the bad effects of poles, five extensions of barycentric interpolation collocation method are introduced by changing the weights and nodal distribution. One and two dimensional numerical examples are realized, we compare the effectiveness of these five methods.In chapter 4, it introduces the obstacle problem described by a kind of fourth-order elliptic variational inequality. Reformulated the original problem by duality method, it yieds the algorithm of obstacle problem by the coupling of the iteration Uzawa method and the barycentric interpolation collocation method. Numerical examples are given. It also discussed the influence of the obstacle parameters to the results.In chapter 5, We apply the barycentric interpolation collocation method to the viscoelastic dynamic frictionless contact problem with adhesion. Applying the time difference and spatial barycentric interpolation collocation method, we construct the decoupling barycentric interpolation collocation method for this dynamic model. Numerical examples are presented. Comparing the numerical results with the FEM method and the DQM method, it shows the good efficiency and precision.