Further Development of Local MQ-DQ Method and Its application in CFD

Abstract

In the past two decades, a group of mesh-free methods were developed based on radial basis functions (RBFs). Local multiquadric-differential quadrature (MQ-DQ) method is a newly developed method which falls into this group. Compared with other RBF methods, the local MQ-DQ method mainly has two advantages. First, it is a local method, which makes it feasible to solve large scale problems. Second, it is based on derivative approximation instead of function approximation. Thus it can be well applied to both linear and nonlinear problems. The effectiveness of this method has been proven by its applications to various kinds of fluid flow problems. However, the research on the local MQ-DQ method is still in the preliminary stage. More work is required to further reveal its basic properties and improve its performance in solving fluid flow problems.

In this thesis, we firstly derived the formulas for the finite difference (FD) schemes based on the MQ function approximation instead of the low order polynomial approximation and named them as MQ-FD methods, which can be considered as special cases of the local MQ-DQ method. The effect of the shape parameter c in MQ on the formulas of the MQ-FD methods is analyzed. One interesting observation is that when c goes to infinity, the MQ-FD formulas of derivative approximation are the same as those given by the conventional FD schemes. Another observation is that as compared with the conventional FD schemes, the MQ-FD methods may solve periodic boundary value problems more accurately. However, for general boundary value problems, the accuracy may not be as high as that using the conventional FD schemes.

Secondly, this thesis focused on improving the flexibility and efficiency of the local MQ-DQ method. An efficient local stencil adaptive algorithm was developed and combined with the local MQ-DQ method. The combined method bears the properties of both local MQ-DQ method for mesh-free numerical discretization and local stencil adaptive algorithm for high computational efficiency. Moreover, a hybrid technique which combines this mesh-free method with conventional FD scheme was adopted to further improve its efficiency. In this technique, the local MQ-DQ method is applied for the spatial discretization in the region around the curved boundary while conventional FD scheme is applied in the rest of the flow domain taking advantage of its high computational efficiency.

Finally, the local MQ-DQ method was extended to simulate fluid flow problems with curved boundary in three-dimensional (3D) space. An error estimate was provided for the 3D local MQ-DQ method to study the influence of shape parameter and the number of supporting points on its numerical accuracy. It was observed that the convergence rate can be improved by increasing the number of supporting points. The problem of flow past a sphere was simulated by the 3D local MQ-DQ method to demonstrate its capability and flexibility in solving 3D fluid flow problems with curved boundary. The obtained numerical results showed that it is a promising scheme for solving 3D fluid flow problems with curved boundary.