arXiv subjects

W. Chen

Publications and source records attributed to W. Chen.

At least 289 records · Page 16Linked to original sources

arXiv2002

Some addenda on distance function wavelets

This report will add some supplements to the recently finished report series on the distance function wavelets (DFW). First, we define the general distance in terms of the Riesz potential, and then, the distance function Abel wavelets are derived via the fractional integral and Laplacian. Second, the DFW Weyl transform is found to be a shifted Laplace potential DFW. The DFW Radon transform is also presented. Third, we present a conjecture on truncation error formula of the multiple reciprocity Laplace DFW series and discuss its error distributions in terms of node density distributions. Forth, we point out that the Hermite distance function interpolation can be used to replace overlapping in the domain decomposition in order to produce sparse matrix. Fifth, the shape parameter is explained as a virtual extra axis contribution in terms of the MQ-type Possion kernel. The report is concluded with some remarks on a range of other issues.

math.NA↗

arXiv2002

A meshless, integration-free, and boundary-only RBF technique

Based on the radial basis function (RBF), non-singular general solution and dual reciprocity method (DRM), this paper presents an inherently meshless, integration-free, boundary-only RBF collocation techniques for numerical solution of various partial differential equation systems. The basic ideas behind this methodology are very mathematically simple. In this study, the RBFs are employed to approximate the inhomogeneous terms via the DRM, while non-singular general solution leads to a boundary-only RBF formulation for homogenous solution. The present scheme is named as the boundary knot method (BKM) to differentiate it from the other numerical techniques. In particular, due to the use of nonsingular general solutions rather than singular fundamental solutions, the BKM is different from the method of fundamental solution in that the former does no require the artificial boundary and results in the symmetric system equations under certain conditions. The efficiency and utility of this new technique are validated through a number of typical numerical examples. Completeness concern of the BKM due to the only use of non-singular part of complete fundamental solution is also discussed.

cs.CE↗

arXiv2002

RBF-based meshless boundary knot method and boundary particle method

This paper is concerned with the two new boundary-type radial basis function collocation schemes, boundary knot method (BKM) and boundary particle method (BPM). The BKM is developed based on the dual reciprocity theorem, while the BPM employs the multiple reciprocity technique. Unlike the method of fundamental solution, the wto methods use the nonsingular general solutions instead of singular fundamental solution to circumvent the controversial artificial boundary outside physical domain. Compared with the boundary element method, both the BKM and BPM are meshfree, superconvergent, meshfree, integration free, symmetric, and mathematically simple collocation techniques for general PDEs. In particular, the BPM does not require any inner nodes for inhomogeneous problems. In this study, the accuracy and efficiency of the two methods are numerically demonstrated to some 2D, 3D Helmholtz and convection-diffusion problems under complicated geometries.

cs.CE↗

arXiv2002

A Lyapunov Formulation for Efficient Solution of the Poisson and Convection-Diffusion Equations by the Differential Quadrature Method

Civan and Sliepcevich [1, 2] suggested that special matrix solver should be developed to further reduce the computing effort in applying the differential quadrature (DQ) method for the Poisson and convection-diffusion equations. Therefore, the purpose of the present communication is to introduce and apply the Lyapunov formulation which can be solved much more efficiently than the Gaussian elimination method. Civan and Sliepcevich [2] first presented DQ approximate formulas in polynomial form for partial derivatives in tow-dimensional variable domain. For simplifying formulation effort, Chen et al. [3] proposed the compact matrix form of these DQ approximate formulas. In this study, by using these matrix approximate formulas, the DQ formulations for the Poisson and convection-diffusion equations can be expressed as the Lyapunov algebraic matrix equation. The formulation effort is simplified, and a simple and explicit matrix formulation is obtained. A variety of fast algorithms in the solution of the Lyapunov equation [4-6] can be successfully applied in the DQ analysis of these two-dimensional problems, and, thus, the computing effort can be greatly reduced. Finally, we also point out that the present reduction technique can be easily extended to the three-dimensional cases.

cs.CE↗

arXiv2002

New advances in dual reciprocity and boundary-only RBF methods

This paper made some significant advances in the dual reciprocity and boundary-only RBF techniques. The proposed boundary knot method (BKM) is different from the standard boundary element method in a number of important aspects. Namely, it is truly meshless, exponential convergence, integration-free (of course, no singular integration), boundary-only for general problems, and leads to symmetric matrix under certain conditions (able to be extended to general cases after further modified). The BKM also avoids the artificial boundary in the method of fundamental solution. An amazing finding is that the BKM can formulate linear modeling equations for nonlinear partial differential systems with linear boundary conditions. This merit makes it circumvent all perplexing issues in the iteration solution of nonlinear equations. On the other hand, by analogy with Green's second identity, this paper also presents a general solution RBF (GSR) methodology to construct efficient RBFs in the dual reciprocity and domain-type RBF collocation methods. The GSR approach first establishes an explicit relationship between the BEM and RBF itself on the ground of the weighted residual principle. This paper also discusses the RBF convergence and stability problems within the framework of integral equation theory.

cs.CE↗

arXiv2002

Relationship between boundary integral equation and radial basis function

This paper aims to survey our recent work relating to the radial basis function (RBF) from some new views of points. In the first part, we established the RBF on numerical integration analysis based on an intrinsic relationship between the Green's boundary integral representation and RBF. It is found that the kernel function of integral equation is important to create efficient RBF. The fundamental solution RBF (FS-RBF) was presented as a novel strategy constructing operator-dependent RBF. We proposed a conjecture formula featuring the dimension affect on error bound to show the independent-dimension merit of the RBF techniques. We also discussed wavelet RBF, localized RBF schemes, and the influence of node placement on the RBF solution accuracy. The centrosymmetric matrix structure of the RBF interpolation matrix under symmetric node placing is proved. The second part of this paper is concerned with the boundary knot method (BKM), a new boundary-only, meshless, spectral convergent, integration-free RBF collocation technique. The BKM was tested to the Helmholtz, Laplace, linear and nonlinear convection-diffusion problems. In particular, we introduced the response knot-dependent nonsingular general solution to calculate varying-parameter and nonlinear steady convection-diffusion problems very efficiently. By comparing with the multiple dual reciprocity method, we discussed the completeness issue of the BKM. Finally, the nonsingular solutions for some known differential operators were given in appendix. Also we expanded the RBF concepts by introducing time-space RBF for transient problems.

cs.CE↗

arXiv2002

New Insights in Boundary-only and Domain-type RBF Methods

This paper has made some significant advances in the boundary-only and domain-type RBF techniques. The proposed boundary knot method (BKM) is different from the standard boundary element method in a number of important aspects. Namely, it is truly meshless, exponential convergence, integration-free (of course, no singular integration), boundary-only for general problems, and leads to symmetric matrix under certain conditions (able to be extended to general cases after further modified). The BKM also avoids the artificial boundary in the method of fundamental solution. An amazing finding is that the BKM can formulate linear modeling equations for nonlinear partial differential systems with linear boundary conditions. This merit makes it circumvent all perplexing issues in the iteration solution of nonlinear equations. On the other hand, by analogy with Green's second identity, we also presents a general solution RBF (GSR) methodology to construct efficient RBFs in the domain-type RBF collocation method and dual reciprocity method. The GSR approach first establishes an explicit relationship between the BEM and RBF itself on the ground of the potential theory. This paper also discusses some essential issues relating to the RBF computing, which include time-space RBFs, direct and indirect RBF schemes, finite RBF method, and the application of multipole and wavelet to the RBF solution of the PDEs.

cs.CE↗

arXiv2002

Definitions of distance function in radial basis function approach

Very few studies involve how to construct the efficient RBFs by means of problem features. Recently the present author presented general solution RBF (GS-RBF) methodology to create operator-dependent RBFs successfully [1]. On the other hand, the normal radial basis function (RBF) is defined via Euclidean space distance function or the geodesic distance [2]. This purpose of this note is to redefine distance function in conjunction with problem features, which include problem-dependent and time-space distance function.

cs.CE↗

arXiv2002

Orthonormal RBF wavelet and ridgelet-like series and transforms for high-dimensional problems

This paper developed a systematic strategy establishing RBF on the wavelet analysis, which includes continuous and discrete RBF orthonormal wavelet transforms respectively in terms of singular fundamental solutions and nonsingular general solutions of differential operators. In particular, the harmonic Bessel RBF transforms were presented for high-dimensional data processing. It was also found that the kernel functions of convection-diffusion operator are feasible to construct some stable ridgelet-like RBF transforms. We presented time-space RBF transforms based on non-singular solution and fundamental solution of time-dependent differential operators. The present methodology was further extended to analysis of some known RBFs such as the MQ, Gaussian and pre-wavelet kernel RBFs.

cs.SC↗

arXiv2002

Symmetric boundary knot method

The boundary knot method (BKM) is a recent boundary-type radial basis function (RBF) collocation scheme for general PDEs. Like the method of fundamental solution (MFS), the RBF is employed to approximate the inhomogeneous terms via the dual reciprocity principle. Unlike the MFS, the method uses a nonsingular general solution instead of a singular fundamental solution to evaluate the homogeneous solution so as to circumvent the controversial artificial boundary outside the physical domain. The BKM is meshfree, superconvergent, integration free, very easy to learn and program. The original BKM, however, loses symmetricity in the presense of mixed boundary. In this study, by analogy with Hermite RBF interpolation, we developed a symmetric BKM scheme. The accuracy and efficiency of the symmetric BKM are also numerically validated in some 2D and 3D Helmholtz and diffusion reaction problems under complicated geometries.

cs.CE↗

arXiv2002

Distance function wavelets - Part III: "Exotic" transforms and series

Part III of the reports consists of various unconventional distance function wavelets (DFW). The dimension and the order of partial differential equation (PDE) are first used as a substitute of the scale parameter in the DFW transforms and series, especially with the space and time-space potential problems. It is noted that the recursive multiple reciprocity formulation is the DFW series. The Green second identity is used to avoid the singularity of the zero-order fundamental solution in creating the DFW series. The fundamental solutions of various composite PDEs are found very flexible and efficient to handle a borad range of problems. We also discuss the underlying connections between the crucial concepts of dimension, scale and the order of PDE through the analysis of dissipative acoustic wave propagation. The shape parameter of the potential problems is also employed as the "scale parameter" to create the non-orthogonal DFW. This paper also briefly discusses and conjectures the DFW correspondences of a variety of coordinate variable transforms and series. Practically important, the anisotropic and inhomogeneous DFW's are developed by using the geodesic distance variable. The DFW and the related basis functions are also used in making the kernel distance sigmoidal functions, which are potentially useful in the artificial neural network and machine learning. As or even worse than the preceding two reports, this study scarifies mathematical rigor and in turn unfetter imagination. Most results are intuitively obtained without rigorous analysis. Follow-up research is still under way. The paper is intended to inspire more research into this promising area.

cs.CE↗

arXiv2002

High-order fundamental and general solutions of convection-diffusion equation and their applications with boundary particle method

In this study, we presented the high-order fundamental solutions and general solutions of convection-diffusion equation. To demonstrate their efficacy, we applied the high-order general solutions to the boundary particle method (BPM) for the solution of some inhomogeneous convection-diffusion problems, where the BPM is a new truly boundary-only meshfree collocation method based on multiple reciprocity principle. For the sake of completeness, the BPM is also briefly described here.

cs.CE↗

arXiv2002

Distance function wavelets - Part II: Extended results and conjectures

Report II is concerned with the extended results of distance function wavelets (DFW). The fractional DFW transforms are first addressed relating to the fractal geometry and fractional derivative, and then, the discrete Helmholtz-Fourier transform is briefly presented. The Green second identity may be an alternative devise in developing the theoretical framework of the DFW transform and series. The kernel solutions of the Winkler plate equation and the Burger's equation are used to create the DFW transforms and series. Most interestingly, it is found that the translation invariant monomial solutions of the high-order Laplace equations can be used to make very simple harmonic polynomial DFW series. In most cases of this study, solid mathematical analysis is missing and results are obtained intuitively in the conjecture status.

cs.CE↗

arXiv2002

Distance function wavelets - Part I: Helmholtz and convection-diffusion transforms and series

This report aims to present my research updates on distance function wavelets (DFW) based on the fundamental solutions and the general solutions of the Helmholtz, modified Helmholtz, and convection-diffusion equations, which include the isotropic Helmholtz-Fourier (HF) transform and series, the Helmholtz-Laplace (HL) transform, and the anisotropic convection-diffusion wavelets and ridgelets. The latter is set to handle discontinuous and track data problems. The edge effect of the HF series is addressed. Alternative existence conditions for the DFW transforms are proposed and discussed. To simplify and streamline the expression of the HF and HL transforms, a new dimension-dependent function notation is introduced. The HF series is also used to evaluate the analytical solutions of linear diffusion problems of arbitrary dimensionality and geometry. The weakness of this report is lacking of rigorous mathematical analysis due to the author's limited mathematical knowledge.

cs.CE↗

arXiv2002

A quasi-RBF technique for numerical discretization of PDE's

Atkinson developed a strategy which splits solution of a PDE system into homogeneous and particular solutions, where the former have to satisfy the boundary and governing equation, while the latter only need to satisfy the governing equation without concerning geometry. Since the particular solution can be solved irrespective of boundary shape, we can use a readily available fast Fourier or orthogonal polynomial technique O(NlogN) to evaluate it in a regular box or sphere surrounding physical domain. The distinction of this study is that we approximate homogeneous solution with nonsingular general solution RBF as in the boundary knot method. The collocation method using general solution RBF has very high accuracy and spectral convergent speed and is a simple, truly meshfree approach for any complicated geometry. More importantly, the use of nonsingular general solution avoids the controversial artificial boundary in the method of fundamental solution due to the singularity of fundamental solution.

cs.CE↗

arXiv2002

Diffuse Neutron Scattering Study of Relaxor Ferroelectric (1-x)Pb(Zn1/3Nb2/3)O3-xPbTiO3(PZN-xPT)

Diffuse neutron scattering is a valuable tool to obtain information about the size and orientation of the polar nanoregions that are a characteristic feature of relaxor ferroelectrics. In this paper, we present new diffuse scattering results obtained on Pb(Zn1/3Nb2/3)O3 (PZN for short) and (1-x)Pb(Zn1/3Nb2/3)O3-xPbTiO3(PZN-xPT)single crystals (with x=4.5 and 9%), around various Bragg reflections and along three symmetry directions in the [100]-[011] zone. Diffuse scattering is observed around reflections with mixed indices, (100), (011) and (300), and along transverse and diagonal directions only. No diffuse scattering is found in longitudinal scans. The diffuse scattering peaks can be fitted well with a Lorentzian function, from which a correlation length is extracted. The correlation length increases with decreasing temperatures down to the transition at Tc, first following a Curie-Weiss law, then departing from it and becoming flat at very low temperatures. These results are interpreted in terms of three temperature regions: 1) dynamic polarization fluctuations (i.e. with a finite lifetime) at high temperatures, 2) static polarization reorientations (condensation of polar nanoregions) that can still reorient as a unit (relaxor behavior) at intermediate temperatures and 3) orientational freezing of the polar nanoregions with random strain fields in pure PZN or a structural phase transition in PZN-xPT at low temperatures. The addition of PT leads to a broadening of the diffuse scattering along the diagonal ([111]) relative to the transverse ([100]) direction, indicating a change in the orientation of the polar regions. Also, with the addition of PT, the polar nanoregions condense at a higher temperature above Tc.

cond-mat.mtrl-sci↗

arXiv2002

Echoes in X-ray Binaries

We present a method of analysing the correlated X-ray and optical/UV variability in X-ray binaries, using the observed time delays between the X-ray driving lightcurves and their reprocessed optical echoes. This allows us to determine the distribution of reprocessing sites within the binary. We model the time-delay transfer functions by simulating the distribution of reprocessing regions, using geometrical and binary parameters. We construct best-fit time-delay transfer functions, showing the regions in the binary responsible for the reprocessing of X-rays. We have applied this model to observations of the Soft X-ray Transient, GRO j1655-40. We find the optical variability lags the X-ray variability with a mean time delay of 19.3$pm{2.2}$ seconds. This means that the outer regions of the accretion disc are the dominant reprocessing site in this system. On fitting the data to a simple geometric model, we derive a best-fit disk half-opening angle of 13.5$^{+2.1}_{-2.8}$ degrees, which is similar to that observed after the previous outburst by cite{orosz97}. This disk thickening has the effect of almost entirely shielding the companion star from irradiation at this stage of the outburst.

astro-ph↗

arXiv2002

Ferroelectric Ordering in the Relaxor Pb(Mg1/3Nb2/3)O3 as Evidenced by Low-Temperature Phonon Anomalies

Neutron scattering measurements of the lowest-energy TO phonons in the relaxor Pb(Mg1/3Nb2/3)O3 (PMN) are reported for 10<=T<=750 K. The soft mode, which is overdamped by the polar nanoregions below the Burns temperature T_d = 620 K, surprisingly recovers below 220 K. The square of the soft mode energy hw0^2 increases linearly with decreasing temperature, and is consistent with the behavior of a ferroelectric soft mode. At 10 K, hw0 reaches 11 meV, the same value observed in ferroelectric Pb(Zn1/3Nb2/3)O3 at low-T. An unusual broadening of the TA phonon starts at T_d and disappears at 220 K, coincident with the recovery of the TO mode. These dynamics suggest that a well-developed ferroelectric state is established below 220 K.

cond-mat.mtrl-sci↗