The collocation and meshless methods for differential equations in R(2)

In recent years, meshless methods have become popular ones to solve differential equations. In this thesis, we aim at solving differential equations by using Radial Basis Functions, collocation methods and fundamental solutions (MFS). These methods are meshless, easy to understand, and even easier to implement

Boundary knot method: A meshless, exponential convergence, integration-free, and boundary-only RBF technique

Based on the radial basis function (RBF), non-singular general solution and dual reciprocity principle (DRM), this paper presents an inheretnly meshless, exponential convergence, integration-free, boundary-only collocation techniques for numerical solution of general partial differential equation systems. The basic ideas behind this methodology are very mathematically simple and generally effective. The RBFs are used in this study to approximate the inhomogeneous terms of system equations in terms of the DRM, while non-singular general solution leads to a boundary-only RBF formulation. The present method is named as the boundary knot method (BKM) to differentiate it from the other numerical techniques. In particular, due to the use of non-singular general solutions rather than singular fundamental solutions, the BKM is different from the method of fundamental solution in that the former does no need to introduce the artificial boundary and results in the symmetric system equations under certain conditions. It is also found that the BKM can solve nonlinear partial differential equations one-step without iteration if only boundary knots are used. The efficiency and utility of this new technique are validated through some typical numerical examples. Some promising developments of the BKM are also discussed.Comment: 36 pages, 2 figures, Welcome to contact me on this paper: Email: [email protected] or [email protected]

Transient thermal modelling of substation connectors by means of dimensionality reduction

This paper proposes a simple, fast and accurate simulation approach based on one-dimensional reduction and the application of the finite difference method (FDM) to determine the temperatures rise in substation connectors. The method discretizes the studied three-dimensional geometry in a finite number of one-dimensional elements or regions in which the energy rate balance is calculated. Although a one-dimensional reduction is applied, to ensure the accuracy of the proposed transient method, it takes into account the three-dimensional geometry of the analyzed system to determine for all analyzed elements and at each time step different parameters such as the incremental resistance of each element or the convective coefficient. The proposed approach allows fulfilling both accuracy and low computational burden criteria, providing similar accuracy than the three-dimensional finite element method but with much lower computational requirements. Experimental results conducted in a high-current laboratory validate the accuracy and effectiveness of the proposed method and its usefulness to design substation connectors and other power devices and components with an optimal thermal behavior.Postprint (published version