C.C. Tsai

The method of fundamental solutions for eigenproblems in domains with and without interior holes

The main purpose of the present paper is to provide a general method of fundamental solution (MFS) formulation for two- and three-dimensional eigenproblems without spurious eigenvalues. The spurious eigenvalues are avoided by utilizing the mixed potential method. Illustrated problems in the annular and concentric domains are studied analytically and numerically to demonstrate the issue of spurious eigenvalues by the discrete and continuous versions of the MFS with and without the mixed potential method. The proposed numerical method is then verified with the exact solutions of the benchmark problems in circular and spherical domains with and without holes. Further studies are performed in a three-dimensional peanut shaped domain. In the spirit of the MFS, this scheme is free from meshes, singularities and numerical integrations.

DIRECT APPROACH TO SOLVE NONHOMOGENEOUS DIFFUSION PROBLEMS USING FUNDAMENTAL SOLUTIONS AND DUAL RECIPROCITY METHODS

Abstract This paper describes a combination of the method of fundamental solutions (MFS) and the dual reciprocity method (DRM) as a mesh‐free numerical method (MFS‐DRM model) to solve 2D and 3D nonhomogeneous diffusion problems. Using our method, the homogeneous solutions of the diffusion equations are solved by the MFS, and the DRM, based on the radial basis functions (RBF) of the thin plate splines (TPS), is employed to solve for particular solutions. The present scheme is free from the frequently used Laplace transform and the finite difference discretization method to deal with the time derivative term in the governing equation. By properly placing the source points in the time‐space domain, the solution is advanced in time until a steady state solution (if one exists) is reached. Since the present method does not need mesh discretization and nodal connectivity, the computational effort and memory storage required are minimal as compared to other domain‐oriented numerical schemes such as FDM, FEM, FVM, etc. Test results obtained for 2D and 3D diffusion problems show good comparability with analytical solutions and other numerical solutions, such as those obtained by the MFS‐DRM model based on the modified Helmholtz fundamental solutions. Thus the present numerical scheme has provided a promising mesh‐free numerical tool to solve nonhomogeneous diffusion problems with space‐time unification for diffusion fundamental solutions.

The method of fundamental solutions and domain decomposition method for degenerate seepage flownet problems

Abstract This paper proposes an innovative method of fundamental solutions (MFS) which is used along with the domain decomposition method (DDM) to solve degenerate boundary problems in ground water flows governed by the Laplace equations. The method is utilized to study flownets generated in the presence of sheet piles, by decomposing regions into sub‐domains along the degenerate sheet‐pile boundary. After validating two degenerate seepage flownet problems, the method is then applied to solve three practical flownet problems with sheet piles: (1) impermeable dam, (2) semi‐circular soil stratum and (3) sloping soil stratum. The predicted flownets show good agreement with available literature. Moreover, the proposed meshless scheme is easier to implement to measure degenerate seepage flownets as compared to other numerical schemes.

The Method of Fundamental Solutions with Dual Reciprocity for thin Plates on Winkler Foundations with Arbitrary Loadings

This paper describes the combination of the method of fundamental solutions (MFS) and the dual reciprocity method (DRM) as a meshless numerical method to solve problems of thin plates resting on Winkler foundations under arbitrary loadings, where the DRM is based on the augmented polyharmonic splines constructed by splines and monomials. In the solution procedure, the arbitrary distributed loading is first approximated by the augmented polyharmonic splines (APS) and thus the desired particular solution can be represented by the corresponding analytical particular solutions of the APS. Thereafter, the complementary solution is solved formally by the MFS. In the mathematical derivations, the real coefficient operator in the governing equation is decomposed into two complex coefficient operators. In other words, the solutions obtained by the MFS-DRM are first treated in terms of these complex coefficient operators and then converted to real numbers in suitable ways. Furthermore, the boundary conditions of lateral displacement, slope, normal moment, and effective shear force are all given explicitly for the particular solutions of APS as well as the kernels of MFS. Finally, numerical experiments are carried out to validate these analytical formulas.

The method of fundamental solutions for solving options pricing models

Abstract This paper provides a foundation of the method of fundamental solutions (MFS) for the Options Pricing models governed by the Black–Scholes equation in which both the European option and American options are considered. In the solution procedure, no artificial boundary conditions are imposed for both datum and infinite sides of the stock prices. In the cases of the European options, no time marching procedures are required and numerical results are verified with the exact solutions. Since the free boundary conditions are considered for the American options, boundary update procedure is thus applied. At the same time, numerical results are compared with the results in the literatures. These numerical results indicate the MFS is an effective and robust meshless numerical solution for solving the Options Pricing models.

METHOD OF FUNDAMENTAL SOLUTIONS FOR PLATE VIBRATIONS IN MULTIPLY CONNECTED DOMAINS

This paper develops the method of fundamental solutions (MFS) to solve eigenfrequencies of plate vi- brations of multiply connected domains. The complex-valued MFS combined with the mix potential method are utilized in order to avoid the spurious eigenvalues. The benchmarked problems of annular plates with clamped, simply supported and free boundary conditions are studied analytically as well as numerically. Wherein the results demonstrate that all true eigenvalues are contained and no spurious ei- genvalues are included. In the analytical studies, the continuous version of the MFS is utilized to obtain the eigenequation by applying the degenerate kernels and Fourier series. The proposed numerical method is free from singularities, meshes, and numerical integrations and thus can be easily utilized to solve plate vibrations free from spurious eigenvalues in multiply connected domains.