Spectral radial basis functions for full sphere computations

The singularity of cylindrical or spherical coordinate systems at the origin imposes certain regularity conditions on the spectral expansion of any infinitely differentiable function. There are two efficient choices of a set of radial basis functions suitable for discretising the solution of a partial differential equation posed in either such geometry. One choice is methods based on standard Chebyshev polynomials; although these may be efficiently computed using fast transforms, differentiability to all orders of the obtained solution at the origin is not guaranteed. The second is the so-called one-sided Jacobi polynomials that explicitly satisfy the required behavioural conditions. In this paper, we compare these two approaches in their accuracy, differentiability and computational speed. We find that the most accurate and concise representation is in terms of one-sided Jacobi polynomials. However, due to the lack of a competitive fast transform, Chebyshev methods may be a better choice for some computationally intensive timestepping problems and indeed will yield sufficiently (although not infinitely) differentiable solutions provided they are adequately converged.     

An optimal Galerkin scheme to solve the kinematic dynamo eigenvalue problem in a full sphere

The kinematic dynamo approximation describes the generation of magnetic field in a prescribed flow of electrically-conducting liquid. One of its main uses is as a proof-of-concept tool to test hypotheses about self-exciting dynamo action. Indeed, it provided the very first quantitative evidence for the possibility of the geodynamo. Despite its utility, due to the requirement of resolving fine structures, historically, numerical work has proven difficult and reported solutions were often plagued by poor convergence. In this paper, we demonstrate the numerical superiority of a Galerkin scheme in solving the kinematic dynamo eigenvalue problem in a full sphere. After adopting a poloidal–toroidal decomposition and expanding in spherical harmonics, we express the radial dependence in terms of a basis of exponentially convergent orthogonal polynomials. Each basis function is constructed from a terse sum of one-sided Jacobi polynomials that not only satisfies the boundary conditions of matching to an electrically insulating exterior, but is everywhere infinitely differentiable, including at the origin. This Galerkin method exhibits more rapid convergence, for a given problem size, than any other scheme hitherto reported, as demonstrated by a benchmark of the magnetic diffusion problem and by comparison to numerous kinematic dynamos from the literature. In the axisymmetric flows we consider in this paper, at a magnetic Reynolds number of O(100), a convergence of 9 significant figures in the most unstable eigenvalue requires only 40 radial basis functions; alternatively, 4 significant figures requires 20 radial functions. The terse radial discretization becomes particularly advantageous when considering flows whose associated numerical solution requires a large number of coupled spherical harmonics. We exploit this new method to confirm the tentatively proposed positive growth rate of the planar flow of Bachtiar et al. [4], thereby verifying a counter-example to the Zel’dovich anti-dynamo theorem in a spherical geometry.     

Two Chebyshev Spectral Methods for Solving Normal Modes in Atmospheric Acoustics

The normal mode model is important in computational atmospheric acoustics. It is often used to compute the atmospheric acoustic field under a time-independent single-frequency sound source. Its solution consists of a set of discrete modes radiating into the upper atmosphere, usually related to the continuous spectrum. In this article, we present two spectral methods, the Chebyshev-Tau and Chebyshev-Collocation methods, to solve for the atmospheric acoustic normal modes, and corresponding programs are developed. The two spectral methods successfully transform the problem of searching for the modal wavenumbers in the complex plane into a simple dense matrix eigenvalue problem by projecting the governing equation onto a set of orthogonal bases, which can be easily solved through linear algebra methods. After the eigenvalues and eigenvectors are obtained, the horizontal wavenumbers and their corresponding modes can be obtained with simple processing. Numerical experiments were examined for both downwind and upwind conditions to verify the effectiveness of the methods. The running time data indicated that both spectral methods proposed in this article are faster than the Legendre-Galerkin spectral method proposed previously.     

配置点谱方法求解一维圆柱梯度折射率介质中的辐射传热

发展一种配置点谱方法，计算包含半透明各向异性散射介质的圆柱系统中的辐射传热.介质具有梯度折射率，并且散射反照率随空间位置变化.通过与精确解或其他方法的结果对比，验证配置点谱方法的准确性.结果表明：配置点谱方法仅采用少量的节点数就可以得到准确的辐射热流量.     

势问题的径向基函数——局部边界积分方程方法

将基于径向基函数构造的具有插值特性的近似函数和局部边界积分方程方法相结合，建立了求解势问题的径向基函数——局部边界积分方程方法，推导了相应离散方程.与其他边界积分方程的无网格方法相比，本文方法具有数值实现过程简单、计算量小、精度高的优点，并可直接施加边界条件.最后通过算例说明了该方法的有效性. 关键词： 径向基函数 无网格方法 局部边界积分方程 势问题     

光谱反射比的实际删节方法研究

为了将测量间隔较小的光谱反射比数据删节或者压缩成间隔较大的数据,给出了三种方法,即：直接选取法、线性删节法和三阶删节法,并对其特性进行了仿真验证.从仿真的结果可以看出：若将测量间隔为5 nm的数据压缩成间隔为10 nm、20 nm或30 nm的数据,则采用三阶删节方法误差最小;若将测量间隔为10 nm的数据压缩成间隔为20 nm或30 nm的数据,则直接选取法似乎结果更好一些.另外,还给出了三阶删节方法与"向量子空间法"在光谱图像压缩方面的特性比较,比较的结果发现,三阶删节方法要比"向量子空间法"更好一些.可以看出,删节方法无论是对光谱反射比的删节还是对光谱图像的压缩,其结果都能很好的满足工业需求.     

各向异性介质波导不连续问题的半解析谱单元法分析

将谱单元法与精细积分法相结合求解各向异性介质的波导不连续问题.从矢量波动方程的单变量变分形式出发,采用基于Gauss-Lobatto-Legendre多项式零点作为插值结点的谱单元,对含有各向异性介质波导的横截面进行离散,然后将问题导入哈密顿体系利用精细积分法进行求解.由于采用了谱单元法,在单元网格数较少时,可获得高准确度的计算结果;又由于利用了精细积分法,结构的纵向长度可以任意设定,克服了当人工边界设置在离介质块较远处时,计算量不断增加的缺点.研究表明半解析谱单元法可有效地应用于各向异性介质的波导不连续问题,在提高准确度的同时可大量节省计算时间.     

Spectral element method for vector radiative transfer equation

A spectral element method (SEM) is developed to solve polarized radiative transfer in multidimensional participating medium. The angular discretization is based on the discrete-ordinates approach, and the spatial discretization is conducted by spectral element approach. Chebyshev polynomial is used to build basis function on each element. Four various test problems are taken as examples to verify the performance of the SEM. The effectiveness of the SEM is demonstrated. The h and the p convergence characteristics of the SEM are studied. The convergence rate of p-refinement follows the exponential decay trend and is superior to that of h-refinement. The accuracy and efficiency of the higher order approximation in the SEM is well demonstrated for the solution of the VRTE. The predicted angular distribution of brightness temperature and Stokes vector by the SEM agree very well with the benchmark solutions in references. Numerical results show that the SEM is accurate, flexible and effective to solve multidimensional polarized radiative transfer problems.     

Molecular dynamics results for the radial distribution functions of highly asymmetric hard sphere mixtures

Molecular dynamics (MD) results for the radial distribution functions of mixtures of large and small hard spheres are reported for size ratios whose (large/small) values are 1, 2.5, 5, 7.5, and 10 in the region where the concentration of the large spheres is very small. The MD contact values of these functions are compared with formulae due to Boublik, Mansoori, Carnahan, Starling, Leland, Grundke, and Henderson, Viduna and Smith, Henderson, Trokhymchuk, Woodcock, and Chan, as well as new formulae that are considered here. The new formulae give good agreement for the large–small contact values and reasonably good agreement for the large–large contact values. The Viduna–Smith formula is satisfactory for the small–small contact value and quite reasonable for the small–large contact value. Undoubtedly, further improvements are possible. These results give insight into what may be called the colloidal limit, where the size ratio is exceedingly large while the concentration of the large spheres is exceedingly small, and into the passage to this limit.     

PGP成像光谱仪的全视场自动化光谱定标方法

为了对PGP成像光谱仪所获得的光谱数据进行定量化分析,需对PGP成像光谱仪进行光谱定标,以获得各光谱通道的中心波长、光谱分辨率及成像光谱仪的光谱弯曲等光谱特性信息。采用单色准直光法设计了一套全视场自动化的光谱定标系统,系统中引入球面镜为待测的成像光谱仪光谱定标提供准直光,通过可自动控制的折转镜改变定标入射光线的入射角,以此实现待测成像光谱仪空间维不同视场的自动化光谱定标。运用该定标系统对PGP成像光谱仪进行光谱定标实验,得到该成像光谱仪的光谱性能参数,并结合定标系统的结构特点,对实验的结果进行了精度分析。实验分析结果表明该系统对PGP成像光谱仪的中心波长定标精度达到0.1 nm,光谱分辨率定标精度达到1.3%。该研究设计的全视场自动化光谱定标系统具有结构新颖紧凑、通用性较强、光谱定标精度较高等特点,且由于自动化的控制,避免了由于人为参与定标过程所带来的额外误差。该系统可用于实现PGP成像光谱仪及其他同类型成像光谱仪的光谱定标。     

A Spectral Time-Domain Method for Computational Electrodynamics

Ever since its introduction by Kane Yee over forty years ago, the finite-difference time-domain (FDTD) method has been a widely-used technique for solving the time-dependent Maxwell's equations that has also inspired many other methods. This paper presents an alternative approach to these equations in the case of spatially-varying electric permittivity and/or magnetic permeability, based on Krylov subspace spectral (KSS) methods. These methods have previously been applied to the variable-coefficient heat equation and wave equation, and have demonstrated high-order accuracy, as well as stability characteristic of implicit time-stepping schemes, even though KSS methods are explicit. KSS methods for scalar equations compute each Fourier coefficient of the solution using techniques developed by Golub and Meurant for approximating elements of functions of matrices by Gaussian quadrature in the spectral, rather than physical, domain. We show how they can be generalized to coupled systems of equations, such as Maxwell's equations, by choosing appropriate basis functions that, while induced by this coupling, still allow efficient and robust computation of the Fourier coefficients of each spatial component of the electric and magnetic fields. We also discuss the application of block KSS methods to problems involving non-self-adjoint spatial differential operators, which requires a generalization of the block Lanczos algorithm of Golub and Underwood to unsymmetric matrices.     

ADI spectral collocation methods for parabolic problems

We discuss the Crank–Nicolson and Laplace modified alternating direction implicit Legendre and Chebyshev spectral collocation methods for a linear, variable coefficient, parabolic initial-boundary value problem on a rectangular domain with the solution subject to non-zero Dirichlet boundary conditions. The discretization of the problems by the above methods yields matrices which possess banded structures. This along with the use of fast Fourier transforms makes the cost of one step of each of the Chebyshev spectral collocation methods proportional, except for a logarithmic term, to the number of the unknowns. We present the convergence analysis for the Legendre spectral collocation methods in the special case of the heat equation. Using numerical tests, we demonstrate the second order accuracy in time of the Chebyshev spectral collocation methods for general linear variable coefficient parabolic problems.     

两个同心旋转球之间流动的谱方法

利用Stokes算子的特征函数作为基函数,用谱Galerkin方法对两个同心球之间的粘性不可压缩流动进行了研究,并作了数值模拟,并且利用谱方法得到了一个Lorenz型方程,分析了它的稳定性,证明了其吸引子的存在性。     

A non-standard auxiliary integrating sphere and correction method for the realization of diffuse reflectance

To realize spectral diffuse reflectance scale in 0/d geometric condition at National Institute of Metrology in China (NIM), a facility based on a non-standard auxiliary integrating sphere method with special structure has been built up and a correction method for the influence on the thickness of PTFE (polytetrafluoroethylene) coating at the port of auxiliary integrating sphere is mainly discussed. The uncertainty of spectral diffuse reflectance in NIM is approximately 0.25% (k = 2) in 380–800 nm. This facility has been used to establish China diffuse reflectance scales in the VIS–NIR part of the spectrum. This system and correction method presented in this paper can be easily implemented on most commercial spectrophotometers with a lower cost.     

A High-Efficiency Spectral Method for Two-Dimensional Ocean Acoustic Propagation Calculations

The accuracy and efficiency of sound field calculations highly concern issues of hydroacoustics. Recently, one-dimensional spectral methods have shown high-precision characteristics when solving the sound field but can solve only simplified models of underwater acoustic propagation, thus their application range is small. Therefore, it is necessary to directly calculate the two-dimensional Helmholtz equation of ocean acoustic propagation. Here, we use the Chebyshev–Galerkin and Chebyshev collocation methods to solve the two-dimensional Helmholtz model equation. Then, the Chebyshev collocation method is used to model ocean acoustic propagation because, unlike the Galerkin method, the collocation method does not need stringent boundary conditions. Compared with the mature Kraken program, the Chebyshev collocation method exhibits a higher numerical accuracy. However, the shortcoming of the collocation method is that the computational efficiency cannot satisfy the requirements of real-time applications due to the large number of calculations. Then, we implemented the parallel code of the collocation method, which could effectively improve calculation effectiveness.     

精确测量石英晶体旋光率的光谱分析法

提出一种精确测量石英晶体旋光率的光谱分析法。利用光学矩阵方法对测量原理进行了分析,指出通过测量由两个平行放置的偏光镜和石英晶体所组成系统的透射曲线就可以精确计算出石英晶体的旋光率;并利用分光光度计设计实验,验证了该方法的正确性。对实验数据进行了处理,拟合出了旋光色散方程,对比Lowry的公式,所得出的公式在可见光范围内的更为精准。对实验数据进行了误差分析, 结果表明：选取厚的石英晶体,长的测量波段、低的扫描速度、小的狭缝宽度都有利于提高测量精度。     

Spectral collocation method with a flexible angular discretization scheme for radiative transfer in multi-layer graded index medium

The spectral collocation method (SCM) is employed to solve the radiative transfer in multi-layer semitransparent medium with graded index. A new flexible angular discretization scheme is employed to discretize the solid angle domain freely to overcome the limit of the number of discrete radiative direction when adopting traditional S_N discrete ordinate scheme. Three radial basis function interpolation approaches, named as multi-quadric (MQ), inverse multi-quadric (IMQ) and inverse quadratic (IQ) interpolation, are employed to couple the radiative intensity at the interface between two adjacent layers and numerical experiments show that MQ interpolation has the highest accuracy and best stability. Variable radiative transfer problems in double-layer semitransparent media with different thermophysical properties are investigated and the influence of these thermophysical properties on the radiative transfer procedure in double-layer semitransparent media is also analyzed. All the simulated results show that the present SCM with the new angular discretization scheme can predict the radiative transfer in multi-layer semitransparent medium with graded index efficiently and accurately.     

A block Chebyshev–Davidson method with inner–outer restart for large eigenvalue problems

We propose a block Davidson-type subspace iteration using Chebyshev polynomial filters for large symmetric/hermitian eigenvalue problem. The method consists of three essential components. The first is an adaptive procedure for constructing efficient block Chebyshev polynomial filters; the second is an inner–outer restart technique inside a Chebyshev–Davidson iteration that reduces the computational costs related to using a large dimension subspace; and the third is a progressive filtering technique, which can fully employ a large number of good initial vectors if they are available, without using a large block size. Numerical experiments on several Hamiltonian matrices from density functional theory calculations show the efficiency and robustness of the proposed method.     

基于激光旋转加热的非导电材料高温光谱发射率测试方法与装置

光谱发射率是表征材料热物理性能的重要参数。对于非导电材料的高温光谱发射率测试,一般采用高温加热炉加热或辐射加热的方式来进行发射率测试,存在的问题是采用高温石墨炉加热时,样品可能会与高温石墨发生化学反应,从而破坏材料原有物性;采用辐射加热,一般是单向静止加热,会存在样品温场梯度非均匀分布的问题。基于激光旋转加热和样品/黑体整体一体化设计,提出了一种“样品动中测”的非导电材料高温光谱发射率测试新方法,建立了相应的测量模型,突破了传统的 “样品静中测”的局限,样品与参考黑体共形一体化设计,采用微区域光谱辐射成像方法,同时测量参考黑体和样品的光谱辐射能量与温度。建立了激光旋转加热状态下的热传导方程,对典型样品材料的温度分布进行了仿真计算,结果表明旋转样品温场分布较为均匀,分析了温场分布与红外光谱发射率测量误差间的关系,给出了适用于本测试方法的材料的热导率下限值。基于该方法,搭建了相应的测量装置,对典型材料碳化硅在1 000 K时的光谱发射率进行了测试,在4 μm处对各个典型高温温度点的光谱发射率进行了测试,得到了碳化硅材料在红外波段的光谱发射率波长变化和温度变化规律特性。与国外的测量结果进行了比对,结果较为一致,验证了激光旋转加热光谱发射率测试方法的可行性。采用此方法,不破坏样品本身的理化特性,样品加热升温速度快,测量温度范围上限高,有效减小了激光静止单向加热带来的温度不均匀性,可同时测量出样品和参考黑体的光谱辐射亮度及温度,无需另外再设计标准高温黑体,解决了现有非导电材料高温光谱发射率测试中非均匀加热和辐射能量同步比对测量的问题,可应用于多种非导电材料高温光谱发射率的测试。