Improved Meshless Computational Methods for Solving Multi-Dimensional Problems

Abstract

In this dissertation, we investigate and improve the meshless numerical methods for various kinds of engineering problems governed by partial differential equations (PDEs). In particular, we extend the applicability of the meshless numerical methods to solve higher dimensional problems under complicated geometries. The stability and efficiency of the improved meshless numerical methods are studied and verified via numerical comparisons with existing numerical methods. The first chapter of this dissertation gives an overview of meshless methods and the subsequent chapters introduce three newly improved meshless methods respectively.

The meshless Finite Block Method (FBM) which solves PDEs in a direct way is
proposed in which the technique of Lagrange interpolation with equally spaced nodes is applied to construct first order differential matrices from which the higher order differential matrices are obtained. By employing the mapping technique, the physical domain is mapped into a normalized domain for two-dimensional or three-dimensional problems with 8 seeds or 20 seeds respectively. Besides, the roots of Chebyshev polynomial of first kind are considered in FBM for the first time. Several numerical examples and comparisons with other numerical methods are presented to validate the stability and high accuracy of FBM.

The Finite Integration Method (FIM) has been recently developed in which the trapezoidal rule for numerical quadrature is used for meshless computation. The resultant integration matrix arised from FIM is lower triangular and hence gives an unconditional stable and accurate numerical approximation. This advantage is also enjoyed for high dimensional problems. By using piecewise polynomials, we extend the FIM to Generalized Finite Integration Method without and with Volterra operator (GFIM and GFIM-V). In GFIM-V, the computational cost and accuracy of the approximation can be adjusted by the degree of piecewise polynomial and the selection of nodal points. Furthermore, by taking advantage of the differential matrix in FBM, the boundary conditions in FIM, GFIM and GFIM-V can be easily formulated and handled. These advantages make the improved meshless numerical methods (FIM, GFIM, GFIM-V) applicable to solve higher-order higher-dimensional problems. The application of these meshless methods and the combination of FBM, FIM and GFIM-V will be illustrated via several numerical examples respectively.

Finally, we give a new mathematical model for the modified bi-Helmholtz equation which can be used in the reconstruction of 3D implicit surfaces with the Method of Fundamental Solutions (MFS). In the algorithm, we illustrate how to properly determine a parameter of the mathematical model so that spurious surfaces can be avoided. Five surface reconstruction examples are presented to validate the proposed numerical model.

Date of Award	24 Aug 2020
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Yiu Chung Benny HON (Supervisor)