| Summary: | 博士 === 國立臺灣大學 === 土木工程學研究所 === 90 === This thesis describes the combination of the dual reciprocity method (DRM) and the method of fundamental solution (MFS) as a meshless numerical method (DRM-MFS) to solve various partial differential equations. These equations include the Poisson’s equation, the Helmholtz equation, the modified Helmholtz equation, the diffusion equation, the Stokes equation, and the Navier-Stokes equations. The DRM is adopted to solve the particular solution and the MFS is used to solve the homogeneous solution.
In Chapter 1, the basic concepts of the meshless numerical method are addressed, and the historical background of the meshless numerical method is introduced concisely. Also, comparisons with other meshless numerical methods are stated.
A general formulation about the DRM-MFS is introduced and some numerical experiments about the RBF interpolation are performed in Chapter 2.
In chapter 3, numerical experiments about the DRM-MFS for time independent problems are carried out. These problems include the Poisson’s equation, the modified Helmholtz equation, and the non-assembling iterative methods.
In Chapter 4, an innovative DRM-MFS for diffusion equation based on the diffusion fundamental solution is developed mathematically. Numerical experiments about the method are also carried out. Some comparisons with the DRM-MFS based on the modified Helmholtz fundamental solution are performed.
In Chapter 5, eigenvalue problems governed by the Helmholtz equation are introduced. Techniques of singular value decomposition (SVD) are adopted to solve the resulting linear eigenvalue system. The SVD test can find the eigenvalues and the corresponding eigenfunctions with multiplicity at the same time.
In Chapter 6 and Chapter 7, some applications of the computational fluid dynamics are introduced. These problems include a cubic cavity interior flow and a sphere exterior outflow. The governing equations of these problems are the Stokes equations and the Navier-Stokes equations.
In the last chapter of this thesis, the achievements of the present works and some suggestions for further researches will be addressed briefly.
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