Development of a novel meshless method - random integral quadrature (RIQ) method and its engineering application for solving integral equations

• Main Author: Zou, Hua
• Other Authors: Li Hua
• Format: Theses and Dissertations
• Language: English
• Published: 2013
• Subjects: DRNTU::Engineering::Mathematics and analysis::Simulations
• Online Access: https://hdl.handle.net/10356/52045
• Institution: Nanyang Technological University

As is well known, meshless methods are often accepted as an important numerical technique, and are increasingly studied in recent years. Objectives of developing a meshless method are to overcome some drawbacks in conventional numerical techniques, such as the finite element method. However, the main challenges we often encounter here includes the construction of an appropriate interpolation function, which is able to interpolate the field variables with uniformly or randomly distributed field nodes that are scattered in regular or irregular domains, in order to achieve more accurate computation. A novel meshless method, termed the random integral quadrature (RIQ) method, is developed for solving the generalized integral equations in this work. By the RIQ method, the integral governing equations are discretized directly with randomly or uniformly distributed field nodes. This is achieved by discretizing the integral governing equations first through the generalized integral quadrature (GIQ) technique over a set of background virtual nodes, and then by interpolating the function values of the virtual nodes over a set of the field nodes through the Kriging interpolation technique. The RIQ method is first validated by the second kind of Fredholm integral equations and the second kind of Volterra integral equations defined in 1-D, 2-D and 3-D integral domains, and then applied for solving the integral equations with irregular integral domains, the Volterra nonlinear integral equations, and the peridynamic problems. The theoretical analysis together with the numerical case studies has proved the accuracy, efficiency and wide applications of the RIQ method.