基于Abel逆变换的激光诱导等离子体辐射特性研究

激光诱导击穿光谱(LIBS)作为一种新型的物质成分测量方法已经在越来越多的领域得到广泛应用，但是与传统的分析方法相比，LIBS技术的分析性能还需进一步提高。LIBS技术的理论基础是激光诱导等离子体，从物理机理上研究等离子体特性，对LIBS系统实验参数的优化具有指导作用，也为提高LIBS技术的检测能力奠定理论基础。激光诱导等离子体是一个与空间相关的非稳态辐射源，空间分辨光谱测量是探究等离子体物理特性的重要手段之一。为研究激光诱导等离子体的辐射特性，采用1 064 nm的Nd∶YAG调Q固体激光器烧蚀合金钢样品产生等离子体，利用空间分辨装置测量二维空间的等离子体辐射光谱信号，通过分析可知实验采集的光谱信号是信号探测器测量路径上的积分光谱强度，由此计算得到的等离子体参数也是观测路径上的平均值。为了深入研究等离子体由内层到外层的辐射规律，首先测量得到等离子体路径积分光谱强度的横向空间分布，然后以等离子体为光学薄和圆柱对称的前提条件，采用三次样条函数算法对路径积分光谱强度进行Abel逆变换，反演得到等离子体由内层到外层谱线辐射率的径向空间分布。选取等离子体辐射光谱中的原子谱线Fe Ⅰ: 374.55 nm和Mn Ⅰ: 403.08 nm为研究对象，分析等离子体辐射光谱的空间分布特征，研究结果表明，等离子体辐射路径积分光谱强度的横向分布呈现出中心位置强度大边缘位置强度小的特征，这是由于等离子体膨胀扩张的结果引起的；通过Abel逆变换得到等离子体光谱辐射率的径向分布，结果表明等离子体从内层到外层谱线的辐射率经过了先增加后减小的变化规律，等离子体中心处出现辐射率的极小值，造成这种现象的主要原因是由于等离子体辐射源中心区域具有较低的电子密度；选取等离子体辐射光谱中Fe元素的11条原子谱线，采用Boltzman法分别由谱线相应的积分光谱强度和辐射率计算等离子体温度，得到等离子体温度的横向空间和径向空间的二维分布，两者具有类似的变化规律；由等离子体温度的横向空间分布可以看出，随着离样品表面距离的增加，等离子体温度呈现单调减小的趋势，等离子体中心到边缘区域等离子体温度逐渐降低，这是由等离子体膨胀扩张以及与环境气体相互作用共同的结果；由等离子体温度的径向空间分布可以看出等离子体由内层到外层等离子体温度逐渐降低，这是由于等离子体膨胀扩张冷却引起的。由此可见，采用Abel逆变换能够实现等离子体由内层到外层的辐射特性分析，为深入理解等离子体产生和演变的物理机理提供实验依据，从而为提高激光诱导击穿光谱技术的分析性能奠定理论基础。     

A stable numerical inversion of Abel's integral equation using almost Bernstein operational matrix

Many problems in physics like reconstruction of the radially distributed emissivity from the line-of-sight projected intensity, the 3-D image reconstruction from cone-beam projections in computerized tomography, etc. lead naturally, in the case of radial symmetry, to the study of Abel's type integral equation. The aim of this communication is to modify the stable algorithm proposed in [Singh VK, Pandey RK, Singh OP. New stable numerical solution of singular integral equations of Abel type by using normalized Bernstein polynomials. Applied Mathematical Sciences 2009;3(5):241–255] which is based on normalized Bernstein polynomial approximation of the projected intensity profile. So, first we construct an orthonormal family of polynomials of degree 5 from the 5th degree Bernstein polynomials B_i5 and use them as a basis to approximate the projected intensity profile. Then, a 6×6 matrix P, named as almost Bernstein operational matrix of integration is constructed and used to reduce the integral equation to a system of algebraic equation which can be solved easily. The method is quite accurate and stable even though the approximations are performed by polynomials of degree 5, as illustrated by applying the method to intensity data with and without random noise to invert and compare it with those obtained by the other methods or with the known analytical inverse. Thus it is good method for applying to experimental intensities distorted by noise.     

On the spherical harmonics and discrete ordinates methods for azimuth-dependent intensity calculations

It is known that the azimuth-averaged component of the intensity computed from the spherical harmonics method for solving the equation of radiative transfer is “exact” at the Gauss quadrature points. We show that a similar relation holds for higher terms in the Fourier expansion of the intensity but that the results are “exact” at the zeros of the associated Legendre polynomials. The relationship between the discrete ordinates and spherical harmonics methods follows from this discussion. A discrete ordinates quadrature scheme, based on the zeros of the associated Legendre polynomials, is shown to maintain the correspondence of the methods for these problems, as well as providing a better set of points than other methods in use.     

Abel inversion using Legendre wavelets expansion

A new method is presented for reconstruction of the radially distributed emissivity from the line-of-sight projected intensity. The method is based on approximating the projected intensity profile by Legendre wavelets. The coefficients of the approximation are computed using the inner product of Legendre wavelets and the intensity profile. The emissivity profile is then obtained by the combination of the functions related to Legendre wavelets. The method is more accurate and noise resistant than other methods when applied to experimental data, and there is no need for a complete noise filtering of the intensity data before applying the inversion.     

A simple Abel inversion method of interferometric data for temperature measurement in axisymmetric medium

In this work, we describe a simple method of Abel inversion for temperature measurement in a natural convection axisymmetric flow. The essence of the method is that the measured lateral fringe shift profile is fitted with a polynomial with only even powers and then Abel inverse integral is evaluated analytically. This technique is compared with recent existing methods to test the accuracy and error propagation using a simulated interferogram of natural convection flow below a downward-facing heated horizontal disk in air. For this comparison, lateral fringe profiles are simulated using temperature fields computed by solving Navier Stokes and energy equations. Through random-number generation, noise profile is artificially added to the simulated noise-free lateral fringe shift profile. The results showed that the proposed technique for Abel inversion leads to accurate temperature profiles when the lateral fringe shift profile is fitted with even-power polynomials having degrees ranging from 20 to 30.     

加性信号调制下指数形式关联噪声驱动的单模激光的光强关联时间

采用线性化近似,计算了加性信号调制下的由具有指数关联的两白噪声驱动的单模激光增益模型的光强关联时间.发现两噪声间关联程度对光强关联时间随噪声强度的变化曲线有很大的影响,两噪声间关联程度取不同值时,光强关联时间随噪声强度的变化曲线中将出现极大值(即出现共振) 或极小值(即出现抑制) .     

Application of jacobi polynomial methods to the one-speed transport equation

The transport equations associated with radiation damage studies are often solved using expansions in Legendre polynomials. The radiation damage distribution functions which satisfy these equations may be sharply peaked in the forward direction, while the Legendre polynomials, as a set, are isotropic. This situation requires the use of many terms in the Legendre expansion in order to adequately represent the distribution functions. The Jacobi polynomials, on the other hand, can have strong peaking built into their associated weight function. To test the usefulness of the Jacobi polynomials we use them to solve the simple, one-speed, neutron transport equation. The results are then compared to the exact theory and to the results of applying Legendre methods to the same problem. This sample calculation demonstrates the advantage of the Jacobi polynomials in strongly non-isotropic situations.     

基于基函数展开方法处理具有陡峭界面的Abel反演问题

在激光聚变研究中,存在大量具有陡峭不连续界面特性的靶丸等离子体诊断问题,需要借助Abel反演方法对结果进行分析。 提出了一种基于基函数展开处理不连续径向分布的Abel反演方法,采用带约束的Landweber迭代法求得优化结果。利用该算法可以对不连续阶跃分布进行比较准确的反演,获得阶跃结构特征。在理想情况下,针对不连续阶跃分布的反演方差好于10-4,即使在存在一定噪声的情况下也仍然可以获得较好的结果。该方法可用于对激光聚变研究中压缩过程中球壳的背光照相、自发光成像、光谱测量等诊断结果的数据分析。     

Spatial Moments for the Nonstationary One-Velocity Problem of Transport Theory with Isotropic Scattering. 2. Plane Instantaneous Source

The spatial distribution of the density of particles emitted by a plane infinite isotropic source with a unit surface particle density is reconstructed for the nonstationary one-velocity problem of transport theory by the method of polynomial expansions with the use of Legendre and Hermite polynomials. The diffusion approximation is examined and the boundaries of the spatiotemporal region in which this approximation is valid are estimated.     

BECOOL: Ballooning eigensolver with COOL finite elements

An incompressible variational ideal ballooning mode equation is discretized with the COOL finite element discretization scheme using basis functions composed of variable order Legendre polynomials. This reduces the second order ordinary differential equation to a special block pentadiagonal matrix equation that is solved using an inverse vector iteration method. A benchmark test of BECOOL (Ballooning Eigensolver using COOL finite elements) with second order Legendre polynomials recovers precisely the eigenvalues computed by the VVBAL shooting code. Timing runs reveal the need to determine an optimal lower order case. Eigenvalue convergence runs show that cubic Legendre polynomials construct the optimal ballooning mode equation for intensive computations.     

Quantitative visualization of temperature field in non-luminous flame by flame reaction technique

Abstract  

An experimental technique for measuring a temperature field in an axisymmetric non-luminous flame is developed using the flame reaction technique combined with the inverse Abel transformation. Flame visualization is carried out using alkali metal solution of Potassium (K), which is supplied to a premixed methane/air flame in a form of spray mist. The basic principle of this technique is based on the measurement of local emission intensity distribution visualized by the flame reaction, which is a function of temperature according to the Maxwell–Boltzmann statistics of thermodynamics. The relationship between the local intensity and the temperature is obtained from the calibration study, in which the local intensity is evaluated from the line of sight intensity by the inverse Abel transformation, and the temperature is measured by thermocouple. This technique is successfully applied to the measurement of local temperature distribution in steady and flickering premixed methane/air flame. The temperature field in the flickering flame indicates that the local temperature oscillates periodically with the flickering frequency, and the highest temperature is found along the flame front and in the merging region.     

阿贝尔逆变换数据处理算法在电弧诊断中的应用

提出了一种由等离子体辐射投影值重建发射系数的阿贝尔逆变换数据处理算法。采用计算简单、变换精度高的Bockasten插值方法实现阿贝尔逆变换积分方程的离散化。在阿贝尔逆变换前,运用傅里叶变换频域低通滤波去除实验数据中的噪声,通过多项式插值弥补数据采样率过低、提高空间分辨力,并对投影数据进行对称化处理以消除数据偏离对称对结果的影响。运用此算法对实验数据进行处理,得到电流200 A、弧长5 mm电弧温度在阴极下方0.5 mm处最高,超过22000 K,与文献中结果一致。该算法能够有效地克服噪声对变换结果的影响,运算速度快、计算精度高、稳定性好,处理大量数据时具有明显的优势。     

Reconstruction of time-averaged temperature of non-axisymmetric turbulent unconfined sooting flame by inverse radiation analysis

A multi-wavelength inversion method is extended to reconstruct the time-averaged temperature distribution in non-axisymmetric turbulent unconfined sooting flame by the multi-wavelength measured data of low time-resolution outgoing emission and transmission radiation intensities. Gaussian, β and uniform distribution probability density functions (PDF) are used to simulate the turbulent fluctuation of temperature, respectively. The reconstruction of time-averaged temperature consists of three steps. First, the time-averaged spectral absorption coefficient is retrieved from the time-averaged transmissivity data by an algebraic reconstruction technique. Then, the time-averaged blackbody spectral radiation intensity is estimated from the outgoing spectral emission radiation intensities. Finally, the time-averaged temperature is approximately reconstructed from the multi-wavelength time-averaged spectral emission radiation data by the least-squares method. Noisy input data have been used to test the performance of the proposed inversion method. The results show that the time-averaged temperature distribution can be estimated with good accuracy, even with noisy input data. The accuracy of the estimation decreases with the increase of turbulent fluctuation intensity of temperature and the effects of assumed PDF on the reconstruction of temperature are small.     

Effect of light-collection geometry on reconstruction errors in Abel inversions

The Abel inversion, used to reconstruct axisymmetric radial profiles from line-of-sight intensity measurements, is increasingly used to make spatially resolved combustion measurements. An Abel deconvolution is valid only when incoming rays are parallel, whereas most practical optical setups used for emission imaging consist of single-lens and multilens systems that collect light in a cone, over a nonzero solid angle. A ray-tracing simulation was performed to aid in understanding how optical collection geometry affects measured intensity signals and the resultant reconstructed emissivity profiles. Simulation results are compared with emission tomography measurements performed on an axisymmetric laminar diffusion flame.     

Reconstruction of emission coefficients for a non-axisymmetric coupling arc by algebraic reconstruction technique

A preliminary investigation of tomographic reconstruction of an asymmetric arc plasma has been carried out. The objective of this work aims at reconstructing emission coefficients of a non-axisymmetric coupling arc from measured intensities by means of an algebraic reconstruction technique (ART). In order to define the optimal experimental scheme for good quality with limited views, the dependence of the reconstruction quality on three configurations (four, eight, ten projection angles) are presented and discussed via a displaced Gaussian model. Then, the emission coefficients of a free burning arc are reconstructed by the ART with the ten-view configuration and an Abel inversion, respectively, and good agreement is obtained. Finally, the emission coefficient profiles of the coupling arc are successfully achieved with the ten-view configuration. The results show that the distribution of emission coefficient for the coupling arc is different from centrosymmetric shape. The ART is perfectly suitable for reconstructing emission coefficients of the coupling arc with the ten-view configuration, proving the feasibility and utility of the ART to characterize an asymmetric arc.     

An iterative method for coil sensitivity estimation in multi-coil MRI systems

This paper presents an iterative coil sensitivity estimation method for multi-coil MRI systems. The proposed method works with coil images in the magnitude image domain. It determines a region of support (RoS), a region being composed of the same type of tissues, by a region growing algorithm, which makes use of both intensities and intensity gradients of pixels. By repeating this procedure, it can determine multiple regions of support, which together cover most of the concerned image area. The union of these regions of support provides a rough estimate of the sensitivity of each coil through dividing the intensities of pixels by the average intensity inside every region of support. The obtained rough coil sensitivity estimate is further approached with the product of multiple low-order polynomials, rather than a single one. The product of these polynomials provides a smooth estimate of the sensitivity of each coil. With the obtained sensitivities of coils, it can produce a better reconstructed image, which determines more correct regions of support and yields preciser estimates of the sensitivities of coils. In other words, the method can be iteratively implemented to improve the estimation performance. The proposed method was verified through both simulated data and clinical data from different body parts. The experimental results confirm the superiority of our method to some conventional methods.     

正交各向异性板液/固界面的声反射与声透射

采用Legendre正交多项式法,对液/固界面声波反射和透射系数进行求解。利用Legendre正交多项式对正交各向异性板中的位移解进行展开,推导出板中的应力和波动控制方程。联立液/固界面的边界条件和波动控制方程,建立线性无关方程组,用以计算液/固界面的反射和透射系数及Legendre多项式的展开系数,计算所得铝板液/固界面的反射和透射系数与传递矩阵法的计算结果吻合良好。以单向纤维增强复合材料板为例,在不同的方位角下,分析了反射和透射系数随斜入射角度、入射波频率的变化关系,以及板中声场的位移分布。所取Legendre正交多项式截止阶数越大,可用来计算的频厚积范围越大。研究拓展了Legendre正交多项式法的适用范围,为材料力学性能的声学测量提供了理论基础。      

Dynamics of Gompertzian tumour growth under environmental fluctuations

We investigate the effect of correlated additive and multiplicative Gaussian white noise on the Gompertzian growth of tumours. Our results are obtained by solving numerically the time-dependent Fokker-Planck equation (FPE) associated with the stochastic dynamics. In our numerical approach we have adopted B-spline functions as a truncated basis to expand the approximated eigenfunctions. The eigenfunctions and eigenvalues obtained using this method are used to derive approximate solutions of the dynamics under study. We perform simulations to analyze various aspects, of the probability distribution, of the tumour cell populations in the transient- and steady-state regimes. More precisely, we are concerned mainly with the behaviour of the relaxation time (τ) to the steady-state distribution as a function of (i) of the correlation strength (λ) between the additive noise and the multiplicative noise and (ii) as a function of the multiplicative noise intensity (D) and additive noise intensity (α). It is observed that both the correlation strength and the intensities of additive and multiplicative noise, affect the relaxation time.     

等离子体发射和吸收系数的轴对称重建

在激光聚变靶丸等离子体诊断中,为了同时重建出等离子体发射系数和吸收系数的空间分布,建立了分层结构成像模型,利用LM（Levenberg-Marquardt）非线性最小二乘优化算法,提出了基于该模型的重建算法。数值模拟结果表明,该算法成功地重建出发射系数和吸收系数,而且重建精度高,明显优于不考虑吸收衰减影响的Abel逆变换重建结果。对于连续分布,当噪声标准差为0.01时,发射系数与吸收系数重建误差分别为0.17,0.22。     

A numerical study of fractional rheological models and fractional Newell-Whitehead-Segel equation with non-local and non-singular kernel

In the recent years, few type of fractional derivatives which have non-local and non-singular kernel are introduced. In this work, we present fractional rheological models and Newell-Whitehead-Segel equations with non-local and non-singular kernel. For solving these equations, we present a spectral collocation method based on the shifted Legendre polynomials. To do this, we extend the unknown functions and its derivatives using the shifted Legendre basis. These expansions and the properties of the shifted Legendre polynomials along with the spectral collocation method will help us to reduce the main problem to a set of nonlinear algebraic equations. Finally, The accuracy and efficiency of the proposed method are reported by some illustrative examples.