二维声场预测的快速多极基本解法

传统基本解法在二维大规模模型的声场求解过程中,系统方程形成和求解的计算量正比于自由度N的二次方O(N2)和三次方O(N3),求解效率低;为此,引入快速多极子算法并采用广义极小残差法迭代求解,提出一种用于二维声场预测的快速多极基本解法。对无限长圆柱体及二维类车体辐射模型的仿真结果表明,当N为3000时,分别采用快速多极基本解法与传统基本解法求解所需的时间比值约为百分之四,且N越大比值越小;最终实现系统方程的形成和求解的计算量降低到正比于自由度O(N),提高了对二维大规模模型声场预测计算效率。      

A multilayer method of fundamental solutions for Stokes flow problems

The method of fundamental solutions (MFS) is a meshless method for the solution of boundary value problems and has recently been proposed as a simple and efficient method for the solution of Stokes flow problems. The MFS approximates the solution by an expansion of fundamental solutions whose singularities are located outside the flow domain. Typically, the source points (i.e. the singularities of the fundamental solutions) are confined to a smooth source layer embracing the flow domain. This monolayer implementation of the MFS (monolayer MFS) depends strongly on the location of the user-defined source points: On the one hand, increasing the distance of the source points from the boundary tends to increase the convergence rate. On the other hand, this may limit the achievable accuracy. This often results in an unfavorable compromise between the convergence rate and the achievable accuracy of the MFS. The idea behind the present work is that a multilayer implementation of the MFS (multilayer MFS) can improve the robustness of the MFS by efficiently resolving different scales of the solution by source layers at different distances from the boundary. We propose a block greedy-QR algorithm (BGQRa) which exploits this property in a multilevel fashion. The proposed multilayer MFS is much more robust than the monolayer MFS and can compute Stokes flows on general two- and three-dimensional domains. It converges rapidly and yields high levels of accuracy by combining the properties of distant and close source points. The block algorithm alleviates the overhead of multiple source layers and allows the multilayer MFS to outperform the monolayer MFS.     

Mathematical and numerical studies on meshless methods for exterior unbounded domain problems

The method of fundamental solutions (MFS) is an efficient meshless method for solving boundary value problems in an exterior unbounded domain. The numerical solution obtained by the MFS is accurate, while the corresponding matrix equation is ill-conditioned. A modified MFS (MMFS) with the proper basis functions is proposed by the introduction of the modified Trefftz method (MTM). The concrete expressions of the corresponding condition numbers are given in mathematical forms and the solvability by these methods is mathematically proven. Thereby, the optimal parameter minimizing the condition number is also mathematically given. Numerical experiments show that the condition numbers of the matrices corresponding to the MTM and the MMFS are reduced and that the numerical solution by the MMFS is more accurate than the one by the conventional method.     

Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points

The purpose of this study is to propose a high-accuracy and fast numerical method for the Cauchy problem of the Laplace equation. Our problem is directly discretized by the method of fundamental solutions (MFS). The Tikhonov regularization method stabilizes a numerical solution of the problem for given Cauchy data with high noises. The accuracy of the numerical solution depends on a regularization parameter of the Tikhonov regularization technique and some parameters of the MFS. The L-curve determines a suitable regularization parameter for obtaining an accurate solution. Numerical experiments show that such a suitable regularization parameter coincides with the optimal one. Moreover, a better choice of the parameters of the MFS is numerically observed. It is noteworthy that a problem whose solution has singular points can successfully be solved. It is concluded that the numerical method proposed in this paper is effective for a problem with an irregular domain, singular points, and the Cauchy data with high noises.     

An adaptive greedy technique for inverse boundary determination problem

In this paper, the method of fundamental solutions (MFS) is employed for determining an unknown portion of the boundary from the Cauchy data specified on parts of the boundary. We propose a new numerical method with adaptive placement of source points in the MFS to solve the inverse boundary determination problem. Since the MFS source points placement here is not trivial due to the unknown boundary, we employ an adaptive technique to choose a sub-optimal arrangement of source points on various fictitious boundaries. Afterwards, the standard Tikhonov regularization method is used to solve ill-conditional matrix equation, while the regularization parameter is chosen by the L-curve criterion. The numerical studies of both open and closed fictitious boundaries are considered. It is shown that the proposed method is effective and stable even for data with relatively high noise levels.     

Galerkin Formulations of the Method of Fundamental Solutions

In this paper, we introduce two Galerkin formulations of the Method of Fundamental Solutions (MFS). In contrast to the collocation formulation of the MFS, the proposed Galerkin formulations involve the evaluation of integrals over the boundary of the domain under consideration. On the other hand, these formulations lead to some desirable properties of the stiffness matrix such as symmetry in certain cases. Several numerical examples are considered by these methods and their various features compared.     

基于压缩感知的目标频空电磁散射特性快速分析

矩量法是求解目标电磁散射问题的一种常用数值方法,因其精度较高而被广泛应用.应用矩量法求解目标频空电磁散射特性时,随着入射波的角度和频率的变化,需要间隔很小的角度和频率步长反复求解矩量法生成的矩阵方程,运算量极大.为解决此类问题,本文结合压缩感知理论和渐近波形估计形成一种新的有效计算方法.首先,基于压缩感知理论引入一种富含空间信息的新型入射源,其次,在该入射源照射下应用渐近波形估计技术求解,从而快速实现目标频空电磁散射特性分析.     

A Practical Algorithm for Determining the Optimal Pseudo-Boundary in the Method of Fundamental Solutions

One of the main difficulties in the application of the method of fundamental solutions (MFS) is the determination of the position of the pseudo-boundary on which are placed the singularities in terms of which the approximation is expressed. In this work, we propose a simple practical algorithm for determining an estimate of the pseudo-boundary which yields the most accurate MFS approximation when the method is applied to certain boundary value problems. Several numerical examples are provided.     

Chemical Imaging of Phase-Separated Polymer Blends by Fluorescence Microscopy

Blends of poly(vinylacetate) (PVAc) and poly(cyclohexylmethacrylate) (PCHMA) labeled by copolymerization with 4-methacryloylamine-4-nitrostilbene (Sb), with (1-pyrenylmethyl)methacrylate (Py), or with 3-(methacryloylamine)propyl-N-carbazole (Cbz) were prepared by casting dilute solutions in tetrahydrofurane (THF) or chloroform. Films about 10 m thick were formed. Phase separation in two types of domains is observed by transmission optical microscopy (TOM) and epifluorescence microscopy (EFM): small craters of 1 to 10 m placed at the polymer-air interface and larger domains, on the scale of 100 m. The morphology of samples depends on the composition of the polymer blend and on solvent. The green fluorescence of Sb, the violet of Py, or the blue of Cbz provides imaging of the distribution of PCHMA in the different domains and in the matrix. It is thus observed that (i) superficial craters and large domains are formed mainly by PCHMA and (ii) the matrix is composed of PVAc in films cast from THF and it is a blend of the two polymers, homogeneous at the submicrometric scale, for chloroform. The emission intensity of Py, recorded by microfluorescence spectroscopy (MFS), yields a mapping similar to imaging detection. It is remarkable that in films cast from chloroform, the smaller domains are distributed with a 2D hexatic order disrupted by dislocations and disclinations, whereas in films cast from THF, a larger heterogeneity is found, denoting different mechanisms of solvent evaporation.     

Singular meshless method using double layer potentials for exterior acoustics

Time-harmonic exterior acoustic problems are solved by using a singular meshless method in this paper. It is well known that the source points cannot be located on the real boundary, when the method of fundamental solutions (MFS) is used due to the singularity of the adopted kernel functions. Hence, if the source points are right on the boundary the diagonal terms of the influence matrices cannot be derived. Herein we present an approach to obtain the diagonal terms of the influence matrices of the MFS for the numerical treatment of exterior acoustics. By using the regularization technique to regularize the singularity and hypersingularity of the proposed kernel functions, the source points can be located on the real boundary and therefore the diagonal terms of influence matrices are determined. We also maintain the prominent features of the MFS, that it is free from mesh, singularity, and numerical integration. The normal derivative of the fundamental solution of the Helmholtz equation is composed of a two-point function, which is one of the radial basis functions. The solution of the problem is expressed in terms of a double-layer potential representation on the physical boundary based on the potential theory. The solutions of three selected examples are used to compare with the results of the exact solution, conventional MFS, boundary element method, and Dirichlet-to-Neumann finite element method. Good numerical performance is demonstrated by close agreement with other solutions.     

An Improved Formulation of Singular Boundary Method

This study proposes a new formulation of singular boundary method (SBM) to solve the 2D potential problems, while retaining its original merits being free of integration and mesh, easy-to-program, accurate and mathematically simple without the requirement of a fictitious boundary as in the method of fundamental solutions (MFS). The key idea of the SBM is to introduce the concept of the origin intensity factor to isolate the singularity of fundamental solution so that the source points can be placed directly on the physical boundary. This paper presents a new approach to derive the analytical solution of the origin intensity factor based on the proposed subtracting and adding-back techniques. And the troublesome sample nodes in the ordinary SBM are avoided and the sample solution is also not necessary for the Neumann boundary condition. Three benchmark problems are tested to demonstrate the feasibility and accuracy of the new formulation through detailed comparisons with the boundary element method (BEM), MFS, regularized meshless method (RMM) and boundary distributed source (BDS) method.     

Preconditioning based on Calderon’s formulae for periodic fast multipole methods for Helmholtz’ equation

Solution of periodic boundary value problems is of interest in various branches of science and engineering such as optics, electromagnetics and mechanics. In our previous studies we have developed a periodic fast multipole method (FMM) as a fast solver of wave problems in periodic domains. It has been found, however, that the convergence of the iterative solvers for linear equations slows down when the solutions show anomalies related to the periodicity of the problems. In this paper, we propose preconditioning schemes based on Calderon’s formulae to accelerate convergence of iterative solvers in the periodic FMM for Helmholtz’ equations. The proposed preconditioners can be implemented more easily than conventional ones. We present several numerical examples to test the performance of the proposed preconditioners. We show that the effectiveness of these preconditioners is definite even near anomalies.     

Three-dimensional elliptic solvers for interface problems and applications

Second-order accurate elliptic solvers using Cartesian grids are presented for three-dimensional interface problems in which the coefficients, the source term, the solution and its normal flux may be discontinuous across an interface. One of our methods is designed for general interface problems with variable but discontinuous coefficient. The scheme preserves the discrete maximum principle using constrained optimization techniques. An algebraic multigrid solver is applied to solve the discrete system. The second method is designed for interface problems with piecewise constant coefficient. The method is based on the fast immersed interface method and a fast 3D Poisson solver. The second method has been modified to solve Helmholtz/Poisson equations on irregular domains. An application of our method to an inverse interface problem of shape identification is also presented. In this application, the level set method is applied to find the unknown surface iteratively.     

Three Boundary Meshless Methods for Heat Conduction Analysis in Nonlinear FGMs with Kirchhoff and Laplace Transformation

This paper presents three boundary meshless methods for solving problems of steady-state and transient heat conduction in nonlinear functionally graded materials (FGMs). The three methods are, respectively, the method of fundamental solution (MFS), the boundary knot method (BKM), and the collocation Trefftz method (CTM) in conjunction with Kirchhoff transformation and various variable transformations. In the analysis, Laplace transform technique is employed to handle the time variable in transient heat conduction problem and the Stehfest numerical Laplace inversion is applied to retrieve the corresponding time-dependent solutions. The proposed MFS, BKM and CTM are mathematically simple, easy-to-programming, meshless, highly accurate and integration-free. Three numerical examples of steady state and transient heat conduction in nonlinear FGMs are considered, and the results are compared with those from meshless local boundary integral equation method (LBIEM) and analytical solutions to demonstrate the efficiency of the present schemes.     

Function projective synchronization of identical and non-identical modified finance and Shimizu–Morioka systems

In this paper, function projective synchronizations (FPS) of identical and non-identical modified finance systems (MFS) and Shimizu?CMorioka system (S-MS) are studied via active control technique. The technique is applied to construct a response system which synchronizes with a given drive system for a desired function relation between identical MFS, identical S-MS and between MFS and S-MS. The results are validated via numerical simulations.     

A Complete 2D Stability Analysis of Fast MHD Shocks in an Ideal Gas

 An algorithm of numerical testing of the uniform Lopatinski condition for linearized stability problems for 1-shocks is suggested. The algorithm is used for finding the domains of uniform stability, neutral stability, and instability of planar fast MHD shocks. A complete stability analysis of fast MHD shock waves is first carried out in two space dimensions for the case of an ideal gas. Main results are given for the adiabatic constant γ=5/3 (mono-atomic gas), that is most natural for the MHD model. The cases γ=7/5 (two-atomic gas) and γ>5/3 are briefly discussed. Not only the domains of instability and linear (in the usual sense) stability, but also the domains of uniform stability, for which a corresponding linearized stability problem satisfies the uniform Lopatinski condition, are numerically found for different given angles of inclination of the magnetic field behind the shock to the planar shock front. As is known, uniform linearized stability implies the nonlinear stability, that is local existence of discontinuous shock front solutions of a quasilinear system of hyperbolic conservation laws. Received: 12 March 2002 / Accepted: 8 November 2002 Published online: 10 February 2003 Communicated by P. Constantin     

Triple differential cross section in ionization of hydrogen by electron impact

A novel model is proposed to study the ionization of atomic hydrogen by fast election impact in coplanar asymmetric geometry making use of the post form of the transition matrix element for the energy shell and the two-potential formula. Based on the approximation of projectile plane waves and three-body problems, the transition matrix element is decomposed into two parts: the structure and scattering factor and the correlation factor. The contributions of these factors to triple differential cross sections are investigated using the method of asymptotic and convergent series.     

Improvement of metal–ferroelectric–silicon structures without buffer layers between Si and ferroelectric films

Si-based metal–ferroelectric–semiconductor (MFS) structures without buffer layers between Si and ferroelectric films have been developed by depositing SrBi₂Ta₂O₉ (SBT) directly on n-type (100)-oriented Si. Some effective processes are adopted to improve the electrical properties of these MFS structures. Contrary to the conventional MFS structures with top electrodes directly on ferroelectrics, our MFS structures have been developed with thin dense SiO₂ films deposited between ferroelectric films and top electrodes. Due to the SiO₂ films, the leakage current densities of MFS structures are reduced to 2×10^-8 A/cm² under the bias of 5 V. The C-V electrical properties of the MFS structures are greatly improved after annealing at 400 °C in N₂ ambient for 1 h. The C-V memory windows are increased to 3 V, which probably results from the decrease of the interface trap density at the Si/SBT interface. Received: 7 September 1999 / Accepted: 24 November 1999 / Published online: 2 August 2000     

The Method of Fundamental Solutions for Solving Convection-Diffusion Equations with Variable Coefficients

A meshless method based on the method of fundamental solutions (MFS) is proposed to solve the time-dependent partial differential equations with variable coefficients. The proposed method combines the time discretization and the one-stage MFS for spatial discretization. In contrast to the traditional two-stage process, the one-stage MFS approach is capable of solving a broad spectrum of partial differential equations. The numerical implementation is simple since both closed-form approximate particular solution and fundamental solution are easier to find than the traditional approach. The numerical results show that the one-stage approach is robust and stable.     

Solution of Two-Dimensional Stokes Flow Problems Using Improved Singular Boundary Method

In this paper, an improved singular boundary method (SBM), viewed as one kind of modified method of fundamental solution (MFS), is firstly applied for the numerical analysis of two-dimensional (2D) Stokes flow problems. The key issue of the SBM is the determination of the origin intensity factor used to remove the singularity of the fundamental solution and its derivatives. The new contribution of this study is that the origin intensity factors for the velocity, traction and pressure are derived, and based on that, the SBM formulations for 2D Stokes flow problems are presented. Several examples are provided to verify the correctness and robustness of the presented method. The numerical results clearly demonstrate the potentials of the present SBM for solving 2D Stokes flow problems.