SOME PROBLEMS WITH THE METHOD OF FUNDAMENTAL SOLUTION USING RADIAL BASIS FUNCTIONS

The present work describes the application of the method of fundamental solutions （MFS） along with the analog equation method （AEM） and radial basis function （RBF） approximation for solving the 2D isotropic and anisotropic Helmholtz problems with different wave numbers. The AEM is used to convert the original governing equation into the classical Poisson＇s equation, and the MFS and RBF approximations are used to derive the homogeneous and particular solutions, respectively. Finally, the satisfaction of the solution consisting of the homogeneous and particular parts to the related governing equation and boundary conditions can produce a system of linear equations, which can be solved with the singular value decomposition （SVD） technique. In the computation, such crucial factors related to the MFS-RBF as the location of the virtual boundary, the differential and integrating strategies, and the variation of shape parameters in multi-quadric （MQ） are fully analyzed to provide useful reference.

Fundamental solution method for inverse source problem of plate equation

The elastic plate vibration model is studied under the external force. The size of the source term by the given mode of the source and some observations from the body of the plate is determined over a time interval, which is referred to be an inverse source problem of a plate equation. The uniqueness theorem for this problem is stated, and the fundamental solution to the plate equation is derived. In the case that the plate is driven by the harmonic load, the fundamental solution method （FSM） and the Tikhonov regularization technique axe used to calculate the source term. Numerical experiments of the Euler-Bernoulli beam and the Kirchhoff-Love plate show that the FSM can work well for practical use, no matter the source term is smooth or piecewise.

The Localized Method of Fundamental Solution for Two Dimensional Signorini Problems

In this work,the localized method of fundamental solution(LMFS)is extended to Signorini problem.Unlike the traditional fundamental solution(MFS),the LMFS approximates the field quantity at each node by using the field quantities at the adjacent nodes.The idea of the LMFS is similar to the localized domain type method.The fictitious boundary nodes are proposed to impose the boundary condition and governing equations at each node to formulate a sparse matrix.The inequality boundary condition of Signorini problem is solved indirectly by introducing nonlinear complementarity problem function(NCP-function).Numerical examples are carried out to validate the reliability and effectiveness of the LMFS in solving Signorini problems.

A Periodicity Analysis of the Light Curve of 3C 454.3

We analyzed the radio light curves of 3C 454.3 at frequencies 22 and 37 GHz taken from the database of Metsaeovi Radio Observatory, and found evidence of quasi-periodic activity. The light curves show great activity with very complicated non-sinusoidal variations. Two possible periods, a very weak one of 1.57 ± 0.12 yr and a very strong one of 6.15 ±0.50 yr were consistently identified by two methods, the Jurkevich method and power specmun estimation. The period of 6.15 ± 0.50 yr is consistent with results previously reported by Ciaramella et al. and Webb et al. Applying the binary black hole model to the central structure we found black hole masses of 1.53 × 10^9M⊙ and 1.86 × 10^8M⊙, and predicted that the next radio outburst is to take place in 2006 March and April.

Stellar spectra association rule mining method based on the weighted frequent pattern tree

Effective extraction of data association rules can provide a reliable basis for classification of stellar spectra. The concept of stellar spectrum weighted itemsets and stellar spectrum weighted association rules are introduced, and the weight of a single property in the stellar spectrum is determined by information entropy. On that basis, a method is presented to mine the association rules of a stellar spectrum based on the weighted frequent pattern tree. Important properties of the spectral line are highlighted using this method. At the same time, the waveform of the whole spectrum is taken into account. The experimental results show that the data association rules of a stellar spectrum mined with this method are consistent with the main features of stellar spectral types.

Electroelastic Analysis of Two-Dimensional Piezoelectric Structures by the Localized Method of Fundamental Solutions

Accurate and efficient analysis of the coupled electroelastic behavior of piezoelectric structures is a challenging task in the community of computational mechanics.During the past few decades,the method of fundamental solutions(MFS)has emerged as a popular and well-established meshless boundary collocation method for the numerical solution of many engineering applications.The classical MFS formulation,however,leads to dense and non-symmetric coefficient matrices which will be computationally expensive for large-scale engineering simulations.In this paper,a localized version of the MFS(LMFS)is devised for electroelastic analysis of twodimensional(2D)piezoelectric structures.In the LMFS,the entire computational domain is divided into a set of overlapping small sub-domains where the MFS-based approximation and the moving least square(MLS)technique are employed.Different to the classical MFS,the LMFS will produce banded and sparse coefficient matrices which makes the method very attractive for large-scale simulations.Preliminary numerical experiments illustrate that the present LMFM is very promising for coupled electroelastic analysis of piezoelectric materials.

Estimating stellar atmospheric parameters based on Lasso features

With the rapid development of large scale sky surveys like the Sloan Digital Sky Survey （SDSS）, GAIA and LAMOST （Guoshoujing telescope）, stellar spectra can be obtained on an ever-increasing scale. Therefore, it is necessary to estimate stel- lar atmospheric parameters such as Teff, log g and [Fe/H] automatically to achieve the scientific goals and make full use of the potential value of these observations. Feature selection plays a key role in the automatic measurement of atmospheric parameters. We propose to use the least absolute shrinkage selection operator （Lasso） algorithm to select features from stellar spectra. Feature selection can reduce redundancy in spectra, alleviate the influence of noise, improve calculation speed and enhance the robustness of the estimation system. Based on the extracted features, stellar atmospheric param- eters are estimated by the support vector regression model. Three typical schemes are evaluated on spectral data from both the ELODIE library and SDSS. Experimental results show the potential performance to a certain degree. In addition, results show that our method is stable when applied to different spectra.

A Novel Technique for Estimating the Numerical Error in Solving the Helmholtz Equation

In this study,we applied a defined auxiliary problem in a novel error estimation technique to estimate the numerical error in the method of fundamental solutions(MFS)for solving the Helmholtz equation.The defined auxiliary problem is substituted for the real problem,and its analytical solution is generated using the complementary solution set of the governing equation.By solving the auxiliary problem and comparing the solution with the quasianalytical solution,an error curve of the MFS versus the source location parameters can be obtained.Thus,the optimal location parameter can be identified.The convergent numerical solution can be obtained and applied to the case of an unavailable analytical solution condition in the real problem.Consequently,we developed a systematic error estimation scheme to identify an optimal parameter.Through numerical experiments,the optimal location parameter of the source points and the optimal number of source points in the MFS were studied and obtained using the error estimation technique.

On the LSP3 estimates of surface gravity for LAMOST-Kepler stars with asteroseismic measurements

Asteroseismology allows for deriving precise values of the surface gravity of stars. The accurate asteroseismic determinations now available for the large number of stars in the Kepler fields can be used to check and calibrate surface gravities that are currently being obtained spectroscopically for a huge number of stars targeted by large-scale spectroscopic surveys, such as the on-going Large Sky Area Multi-Object Fiber Spectroscopic Telescope （LAMOST） Galactic survey. The LAMOST spectral surveys have obtained a large number of stellar spectra in the Kepler fields. Stellar atmospheric parameters of those stars have been determined with the LAMOST Stellar Parameter Pipeline at Peking University （LSP3）, by template matching with the MILES empirical spectral library. In the current work, we compare surface gravities yielded by LSP3 with those of two asteroseismic samples-- the largest Kepler asteroseismic sample and the most accurate Kepler asteroseismic sample. We find that LSP3 surface gravities are in good agreement with asteroseismic values of Hekker et al., with a dispersion of -0.2 dex. Except for a few cases, asteroseismic surface gravities ofHuber et al. and LSP3 spectroscopic values agree for a wide range of surface gravities. However, some patterns in the differences can be identified upon close inspection. Potential ways to further improve the LSP3 spectroscopic estimation of stellar atmospheric parameters in the near future are briefly discussed. The effects of effective temperature and metallicity on asteroseismic determinations of surface gravities for giant stars are also discussed.

Mathematical modeling and numerical computation of the effective interfacial conditions for Stokes flow on an arbitrarily rough solid surface

The present work is concerned with a two-dimensional(2D)Stokes flow through a channel bounded by two parallel solid walls.The distance between the walls may be arbitrary,and the surface of one of the walls can be arbitrarily rough.The main objective of this work consists in homogenizing the heterogeneous interface between the rough wall and fluid so as to obtain an equivalent smooth slippery fluid/solid interface characterized by an effective slip length.To solve the corresponding problem,two efficient numerical approaches are elaborated on the basis of the method of fundamental solution(MFS)and the boundary element methods(BEMs).They are applied to different cases where the fluid/solid interface is periodically or randomly rough.The results obtained by the proposed two methods are compared with those given by the finite element method and some relevant ones reported in the literature.This comparison shows that the two proposed methods are particularly efficient and accurate.

Boundary element analysis for elastic and elastoplastic problems of 2D orthotropic media with stress concentration

Both the orthotropy and the stress concentration are common issues in modem structural engineering. This paper introduces the boundary element method （BEM） into the elastic and elastoplastic analyses for 2D orthotropic media with stress concentration. The discretized boundary element formulations are established, and the stress formulae as well as the fundamental solutions are derived in matrix notations. The numerical procedures are proposed to analyze both elastic and elastoplastic problems of 2D orthotropic me- dia with stress concentration. To obtain more precise stress values with fewer elements, the quadratic isoparametric element formulation is adopted in the boundary discretization and numerical procedures. Numerical examples show that there are significant stress concentrations and different elastoplastic behaviors in some orthotropic media, and some of the computational results are compared with other solutions. Good agreements are also observed, which demonstrates the efficiency and reliability of the present BEM in the stress concentration analysis for orthotropic media.

Localized Method of Fundamental Solutions for Acoustic Analysis Inside a Car Cavity with Sound-Absorbing Material

This paper documents the first attempt to apply a localized method of fundamental solutions(LMFS)to the acoustic analysis of car cavity containing soundabsorbing materials.The LMFS is a recently developed meshless approach with the merits of being mathematically simple,numerically accurate,and requiring less computer time and storage.Compared with the traditional method of fundamental solutions(MFS)with a full interpolation matrix,the LMFS can obtain a sparse banded linear algebraic system,and can circumvent the perplexing issue of fictitious boundary encountered in the MFS for complex solution domains.In the LMFS,only circular or spherical fictitious boundary is involved.Based on these advantages,the method can be regarded as a competitive alternative to the standard method,especially for high-dimensional and large-scale problems.Three benchmark numerical examples are provided to verify the effectiveness and performance of the present method for the solution of car cavity acoustic problems with impedance conditions.

Light curve solutions of six eclipsing binaries at the lower limit of periods for W UMa stars

Photometric observations are presented in V and I bands of six eclipsing binaries at the lower limit of the orbital periods for W UMa stars. Three of them are newly discovered eclipsing systems. The light curve solutions reveal that all shortperiod targets are contact or overcontact binaries and six new binaries are added to the family of short-period systems with estimated parameters. Four binaries have com- ponents that are equal in size and a mass ratio near 1. The phase variability shown by the V-I colors of all targets may be explained by lower temperatures on their back surfaces than those on their side surfaces. Five systems exhibit the O＇Connell effect that can be modeled by cool spots on the side surfaces of their primary components. The light curves of V1067 Her in 2011 and 2012 are fitted by diametrically opposite spots. Applying the criteria for subdivision of W UMa stars to our targets leads to ambiguous results.

Localized Method of Fundamental Solutions for Three-Dimensional Elasticity Problems: Theory

A localized version of the method of fundamental solution(LMFS)is devised in this paper for the numerical solutions of three-dimensional(3D)elasticity problems.The present method combines the advantages of high computational efficiency of localized discretization schemes and the pseudo-spectral convergence rate of the classical MFS formulation.Such a combination will be an important improvement to the classical MFS for complicated and large-scale engineering simulations.Numerical examples with up to 100,000 unknowns can be solved without any difficulty on a personal computer using the developed methodologies.The advantages,disadvantages and potential applications of the proposed method,as compared with the classical MFS and boundary element method(BEM),are discussed.

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,easyto-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 effi-ciency of the present schemes.

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.

Localized space-tim e method of fundamental solutions for three-dimensional transient diffusion problem

A localized space-time method of fundamental solutions(LSTMFS)is extended for solving three-dimensional transient diffusion problems in this paper.The interval segmentation in temporal direction is developed for the accurate simulation of long-time-dependent diffusion problems.In the LSTMFS,the whole space-time domain with nodes arranged i divided into a series of overlapping subdomains with a simple geometry.In each subdomain,the conventional method of fundamental solutions is utilized and the coefficients associated with the considered domain can be explicitly determined.By calculating a combined sparse matrix system,the value at any node inside the space-time domain can be obtained.Numerical experi-ments demonstrate that high accuracy and efficiency can be simultaneously achieved via the LSTMFS,even for the problems defined on a long-time and quite complex computational domain.

Application of Method of Fundamental Solutions in Solving Potential Flow Problems for Ship Motion Prediction

A novel panel-free approach based on the method of fundamental solutions (MFS) is proposed to solve the potential flow for predicting ship motion responses in the frequency domain according to strip theory. Compared with the conventional boundary element method (BEM), MFS is a desingularized, panel-free and integration-free approach. As a result, it is mathematically simple and easy for programming. The velocity potential is described by radial basis function (RBF) approximations and any degree of continuity of the velocity potential gradient can be obtained. Desingularization is achieved through collating singularities on a pseudo boundary outside the real fluid domain. Practical implementation and numerical characteristics of the MFS for solving the potential flow problem concerning ship hydrodynamics are elaborated through the computation of a 2D rectangular section. Then, the current method is further integrated with frequency domain strip theory to predict the heave and pitch responses of a containership and a very large crude carrier (VLCC) in regular head waves. The results of both ships agree well with the 3D frequency domain panel method and experimental data. Thus, the correctness and usefulness of the proposed approach are proved. We hope that this paper will serve as a motivation for other researchers to apply the MFS to various challenging problems in the field of ship hydrodynamics.

Domain-Decomposition Localized Method of Fundamental Solutions for Large-Scale Heat Conduction in Anisotropic Layered Materials

The localized method of fundamental solutions(LMFS)is a relatively new meshless boundary collocation method.In the LMFS,the global MFS approxima-tion which is expensive to evaluate is replaced by local MFS formulation defined in a set of overlapping subdomains.The LMFS algorithm therefore converts differential equations into sparse rather than dense matrices which are much cheaper to calcu-late.This paper makes thefirst attempt to apply the LMFS,in conjunction with a domain-decomposition technique,for the numerical solution of steady-state heat con-duction problems in two-dimensional(2D)anisotropic layered materials.Here,the layered material is decomposed into several subdomains along the layer-layer inter-faces,and in each of the subdomains,the solution is approximated by using the LMFS expansion.On the subdomain interface,compatibility of temperatures and heatfluxes are imposed.Preliminary numerical experiments illustrate that the proposed domain-decomposition LMFS algorithm is accurate,stable and computationally efficient for the numerical solution of large-scale multi-layered materials.

The Method of Fundamental Solution for a Radially Symmetric Heat Conduction Problem with Variable Coefficient

We consider an inverse heat conduction problem with variable coefficient on an annulus domain.In many practice applications,we cannot know the initial temperature during heat process,therefore we consider a non-characteristic Cauchy problem for the heat equation.The method of fundamental solutions is applied to solve this problem.Due to ill-posedness of this problem,we first discretize the problem and then regularize it in the form of discrete equation.Numerical tests are conducted for showing the effectiveness of the proposed method.