Yong Duan

Meshless Radial Basis Function Method for Transient Electromagnetic Computations

We propose a novel numerical method to simulate transient electromagnetic problems. The time derivatives are still tackled with the customary explicit leapfrog time scheme. But in the space domain, the fields at the collocation points are expanded into a series of radial basis functions and are treated with a meshless method procedure. Our method solves numerically Maxwells equations with various assigned boundary conditions and current source excitation. Furthermore, the numerical stability condition of our method is obtained through a one-dimensional (1-D) wave equation and thus the relationship between control parameters is accounted for. To verify the accuracy and effectiveness of the new formulation, we compare the results of the proposed method with those of the conventional finite-difference time-domain method through a 1-D case study with different boundary conditions.

Lanczos-type variants of the COCR method for complex nonsymmetric linear systems

Motivated by the celebrated extending applications of the well-established complex Biconjugate Gradient (CBiCG) method to deal with large three-dimensional electromagnetic scattering problems by Pocock and Walker [M.D. Pocock, S.P. Walker, The complex Bi-conjugate Gradient solver applied to large electromagnetic scattering problems, computational costs, and cost scalings, IEEE Trans. Antennas Propagat. 45 (1997) 140-146], three Lanczos-type variants of the recent Conjugate A-Orthogonal Conjugate Residual (COCR) method of Sogabe and Zhang [T. Sogabe, S.-L. Zhang, A COCR method for solving complex symmetric linear systems, J. Comput. Appl. Math. 199 (2007) 297-303] are explored for the solution of complex nonsymmetric linear systems. The first two can be respectively considered as mathematically equivalent but numerically improved popularizing versions of the BiCR and CRS methods for complex systems presented in Sogabes Ph.D. Dissertation. And the last one is somewhat new and is a stabilized and more smoothly converging variant of the first two in some circumstances. The presented algorithms are with the hope of obtaining smoother and, hopefully, faster convergence behavior in comparison with the CBiCG method as well as its two corresponding variants. This motivation is demonstrated by numerical experiments performed on some selective matrices borrowed from The University of Florida Sparse Matrix Collection by Davis.

Coupling three-field formulation and meshless mixed Galerkin methods using radial basis functions

In this work, we solve the elliptic partial differential equation by coupling the meshless mixed Galerkin approximation using radial basis function with the three-field domain decomposition method. The formulation has been adopted to increase the efficiency of the numerical technique by decreasing the error and dealing with the ill conditioning of the linear system caused by the radial basis function. Convergence analysis of the coupled technique is treated and numerical results of some solved examples are given at the end of this paper.

Coupling projection domain decomposition method and Kansa's method in electrostatic problems

Abstract This paper present a novel approach for solving electrostatic problems associated with an asymmetrical shielded stripline and shielded coupled-striplines. This novel approach is based on combination of radial basis functions-based meshless unsymmetric collocation method (also, Kansas method) with projection domain decomposition method. Under this new method, we just need to solve a Steklov–Poincare interface equation and the original problem is solved by computing a series of independent subproblems. Two real problems are solved by the proposed approach to demonstrate the accuracy and efficiency.

Meshless Galerkin method based on regions partitioned into subdomains

In this paper, we will combine the domain decomposition method with so-called meshless Galerkin method for PDE using RBF. The convergence of this new method and numerical examples are given.

A comparative study of iterative solutions to linear systems arising in quantum mechanics

This study is mainly focused on iterative solutions with simple diagonal preconditioning to two complex-valued nonsymmetric systems of linear equations arising from a computational chemistry model problem proposed by Sherry Li of NERSC. Numerical experiments show the feasibility of iterative methods to some extent when applied to the problems and reveal the competitiveness of our recently proposed Lanczos biconjugate A-orthonormalization methods to other classic and popular iterative methods. By the way, experiment results also indicate that application specific preconditioners may be mandatory and required for accelerating convergence.

SOLVING HELMHOLTZ EQUATION BY MESHLESS RADIAL BASIS FUNCTIONS METHOD

In this paper, we propose a brief and general process to compute the eigenvalue of arbitrary waveguides using meshless method based on radial basis functions (MLM-RBF) interpolation. The main idea is that RBF basis functions are used in a point matching method to solve the Helmholtz equation only in Cartesian system. Two kinds of boundary conditions of waveguide problems are also analyzed. To verify the e-ciency and accuracy of the present method, three typical waveguide problems are analyzed. It is found that the results of various waveguides can be accurately determined by MLM-RBF.

Application of the RBF-based meshless method to solve 2-D time domain Maxwell’s equations

Radial basis functions (RBF), as a meshless technique, is widely applied to solve partial differential equations. In this paper, a meshless RBF method is applied to time domain Maxwells equations and calculates a two-dimensional (2-D) cavity case. The main idea is that the fields in the space domain are expanded into a series of radial basis functions and are treated with a meshless method procedure, and the time derivatives are still tackled with the customary difference scheme. The 2-D cavity numerical experiment has been used to validate the propose technique.

Coupling Projection Domain Decomposition Method and Meshless Collocation Method Using Radial Basis Functions in Electromagnetics

This paper presents an efficient meshless approach for solving electrostatic problems. This novel approach is based on combination of radial basis functions-based meshless unsymmetric collocation method with projection domain decomposition method. Under this new method, we just need to solve a Steklov-Poincare interface equation and the original problem is solved by computing a series of independent sub-problems. An electrostatic problem is used as an example to illustrate the application of the proposed approach. Numerical results that demonstrate the accuracy and efficiency of the method are stated.

On condition number of meshless collocation method using radial basis functions

In this paper, we will give an estimate for the condition number of meshless collocation method using radial basis functions.