Research On First Order FEM And VIV Simulation Of Circular Cylinder

In this thesis, the variational method based on the partition Galerkin’s equation is proposed, which is based on the summary of scholars on the the finite element method of elastic mechanics and fluid mechanics. A first-order finite element algorithm is proposed to improve stress accuracy in conventional finite element method for elastic mechanics. A system of first-order fluid dynamics equations without pressure terms is presented for the difficulty of solving the incompressible Navier-Stokes equation. And the VIV simulation of circular cylinder at high mass ration and high Reynolds number is simulated and analyzed. The main research work is as follows:(1) The partition galerkin equations are proposed, the continuous conditions of displacement and stress on the interface are relaxed. The lower of continuity of the weak forms is analyzed. Based on the partition Galerkin(partition weighted residuals), the system is established, which provides a theoretical basis for the construction of various elements, especially the quasi-conforming elements and the hybrid elements. For the problems of elasticity, the partition weak form, the partition generalized virtual work equation and the partition variational principle are derived based on the partition Galerkin equation. The composition mode of integral form solution is analyzed. The conditions for choosing the right test function are presented. The variational method based on the partition Galerkin equation provides the theoretical basis for the first-order finite element method.(2) The first-order finite element method is applied to the two-dimensional and three-dimensional elasticity problems. First, the first-order weak form of the elastic problem is derived, and then finite element programs are coded by using Free Fem++ software. Typical examples are used to compare the accuracy of the first-order algorithm and the conventional finite element method. The results of numerical calculation show that the accuracy of the stress and displacement are on the same order and the improvement of stress accuracy can be achieved by using the first-order algorithm, and the time of the finite element method is less than that of the conventional finite element method for the same accuracy. The first-order finite element method provides a new method for improving the accuracy of finite element method.(3) The incompressible Navier-Stokes equation and its finite element method are introduced, and the characteristic based split scheme is introduced, and the simulation of a flow around 2D circular cylinder is presented. The first-order finite element method is applied to the incompressible viscous fluid. For incompressible viscous flow, a first-order fluid dynamic equation system without pressure term is presented. Based on the finite element method, the same order interpolation is used for stress and velocity. Numerical examples for steady flow between two parallel plates and nonsteady viscous fluid flow around 2D circular cylinder are calculated respectively, and the results are compared with the exact solution and the standard test examples respectively. Using finite element method to solve the proposed first-order equations of fluid dynamics, stress and velocity are interpolated with the same order, which can avoid the decrese of accuracy induced by repeatition of velocity derivatives in the process of solving Navier Stokes equations.(4) The problem of vortex induced vibration of high mass ratio cylinders at high Reynolds number is studied. A detailed introduction to the theory of SST turbulence model and some relevant parameters is presented. A fluid structure interaction simulation is carried by using mass-spring-damping system and SST turbulence with the CFD software using the Galerkin least square finite element method. The results of fluid-structure-interaction analysis are consistent with the experimental results, and the correctness of the method is proved and it can be used on the condition of the same type of problems. Numerical simulation results show that the SST turbulence model is suitable for the flow problems of boundary layer separation caused by strong adverse pressure gradient, and the upper branch appears at high Reynolds number for high mass ratio circular cylinder, which provides the reference for the study on this kind of engineering problem.