Chia-Ming Fan

The method of fundamental solutions and condition number analysis for inverse problems of Laplace equation

This paper investigates the applications of the method of fundamental solutions together with the conditional number analysis to solve various inverse 2D Laplace problems involving under-specified and/or over-specified boundary conditions. Through the method of fundamental solutions and the condition number analysis, it is numerically found that solutions of inverse Laplace problems can be obtained without iteration or regularization for small noise levels, since the method of fundamental solutions is a boundary-type meshless numerical method that can automatically satisfy the governing equation. However for larger values of noise levels regularization is still necessary to obtain promising result. The present paper mainly focuses on the two types of numerical predictions of inverse 2D Laplace problems: (1) Cauchy problem, and (2) shape identification problem. Good quantitative agreement with the analytical solutions and other numerical methods for small perturbed boundary data is observed by using present meshless numerical scheme.

THE METHOD OF APPROXIMATE PARTICULAR SOLUTIONS FOR SOLVING ELLIPTIC PROBLEMS WITH VARIABLE COEFFICIENTS

A new version of the method of approximate particular solutions (MAPSs) using radial basis functions (RBFs) has been proposed for solving a general class of elliptic partial differential equations. In the solution process, the Laplacian is kept on the left-hand side as a main differential operator. The other terms are moved to the right-hand side and treated as part of the forcing term. In this way, the close-form particular solution is easy to obtain using various RBFs. The numerical scheme of the new MAPSs is simple to implement and yet very accurate. Three numerical examples are given and the results are compared to Kansas method and the method of fundamental solutions.

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.

Modified Collocation Trefftz Method for the Geometry Boundary Identification Problem of Heat Conduction

In this article, a meshless numerical algorithm is proposed for the boundary identification problem of heat conduction, one kind of inverse problem. In the geometry boundary identification problem, the Cauchy data is given for part of the boundary. The Neumann boundary condition is given for the other portion of the boundary, whose spatial position is unknown. In order to stably solve the inverse problem, the modified collocation Trefftz method, a promising boundary-type meshless method, is adopted for discretizing this problem. Since the spatial position for part of the boundary is unknown, the numerical discretization results in a system of nonlinear algebraic equations (NAEs). Then, the exponentially convergent scalar homotopy algorithm (ECSHA) is used to efficiently obtain the convergent solution of the system of NAEs. The ECSHA is insensitive to the initial guess of the evolutionary process. In addition, the efficiency of the computation is greatly improved, since calculation of the inverse of the Jacobian matrix can be avoided. Four numerical examples are provided to validate the proposed meshless scheme. In addition, some factors that might influence the performance of the proposed scheme are examined through a series of numerical experiments. The stability of the proposed scheme can be proven by adding some noise to the boundary conditions.

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 Local Radial Basis Function Collocation Method for Solving Two-Dimensional Inverse Cauchy Problems

In this study, inverse Cauchy problems, which are governed by the Poisson equation, inhomogeneous Helmholtz equation, and inhomogeneous convection-diffusion-reaction equation, are analyzed by the local radial basis function collocation method (LRBFCM). In the inverse Cauchy problem, overspecified boundary conditions are given along part of the boundary and no boundary condition is imposed on the rest of the boundary. The inverse problems are generally very unstable and ill-posed, so the inverse Cauchy problem is very difficult to solve stably using any numerical scheme. The LRBFCM is one kind of domain-type meshless method and can get rid of mesh generation and numerical quadrature. In addition, the localization in LRBFCM can reduce the ill-conditioning problem and full matrix. Therefore, in this study the LRBFCM is adopted to analyze two-dimensional inverse Cauchy problems. Five numerical examples are provided to verify the proposed meshless scheme. In addition, the stability of the proposed scheme is validated by adding noise into boundary conditions.

Generalized finite difference method for solving two-dimensional inverse Cauchy problems

In this paper, a meshless numerical scheme is adopted for solving two-dimensional inverse Cauchy problems which are governed by second-order linear partial differential equations. In Cauchy problems, over-specified boundary conditions are imposed on portions of the boundary while on parts of boundary no boundary conditions are imposed. The application of conventional numerical methods to Cauchy problems yields highly ill-conditioned matrices. Hence, small noise added in the boundary conditions will tremendously enlarge the computational errors. The generalized finite difference method (GFDM), which is a newly developed domain-type meshless method, is adopted to solve in a stable manner the two-dimensional Cauchy problems. The GFDM can overcome time-consuming mesh generation and numerical quadrature. Besides, Cauchy problems can be solved stably and accurately by the GFDM. We present three numerical examples to validate the accuracy and the simplicity of the meshless scheme. In addition, different levels of noise are added into the boundary conditions to verify the stability of the proposed method.

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.

A Hybrid Cartesian/Immersed-Boundary Finite-Element Method for Simulating Heat and Flow Patterns in a Two-Roll Mill

This article describes numerical investigations of the flow and heat patterns in a two-roll mill using the immersed-boundary finite-element method over a fixed Cartesian grid. The second-order projection method is used to advance the solution in time, and a structured linear-triangle element is employed for the spatial discretization. An easily implemented interpolation scheme is adopted to allow accurate imposition of the boundary conditions on an arbitrary shape. Two numerical experiments are carried out, including two-roll-mill flow generated by the two inner cylinders rotating independently in fixed locations and moving around the center of cavity. The physical characteristics, streamline topologies, and temperature contours are discussed for a range of the rotating velocities and Reynolds numbers. The accuracy and robustness of the developed numerical model are validated by results obtained from the unstructured finite element method.

Application of the Generalized Finite-Difference Method to Inverse Biharmonic Boundary-Value Problems

In this article, the generalized finite-difference method (GFDM), one kind of domain-type meshless method, is adopted for analyzing inverse biharmonic boundary-value problems. In inverse problems governed by fourth-order partial differential equations, overspecified boundary conditions are imposed at part of the boundary, and, on the other hand, part of the boundary segment lacks enough boundary conditions. The ill-conditioning problems will appear when conventional numerical simulations are used for solving the inverse problems. Thus, small perturbations added in the boundary conditions will result in problems of instability and large numerical errors. In this article, we adopt the GFDM to stably and accurately analyze the inverse problems governed by fourth-order partial differential equations. The GFDM is truly free from time-consuming mesh generation and numerical quadrature. Six numerical examples are provided to validate the accuracy and the simplicity of the GFDM. Furthermore, different levels of noise are added into the boundary conditions to verify the satisfying stability of the GFDM.