球几何中子输运保正线性间断有限元格式

针对球几何中子输运方程线性间断有限元方法计算的负中子通量问题,构造了保正线性间断有限元格式,该格式保持中子角通量0阶矩和1阶矩。现有方法计算中子角通量非负时,采用传统的线性间断有限元方法,求解线性方程组;原方法计算出现负通量,则采用构造的保正格式,求解非线性方程组。编制了球几何中子输运问题保正格式程序模块,并集成到应用程序。数值算例表明构造的保正格式计算的中子通量非负,有效降低数值误差,提高数值计算的精度。     

球几何中子输运保正线性间断有限元格式

针对球几何中子输运方程线性间断有限元方法计算的负中子通量问题,构造了保正线性间断有限元格式,该格式保持中子角通量0阶矩和1阶矩。现有方法计算中子角通量非负时,采用传统的线性间断有限元方法,求解线性方程组;原方法计算出现负通量,则采用构造的保正格式,求解非线性方程组。编制了球几何中子输运问题保正格式程序模块,并集成到应用程序。数值算例表明构造的保正格式计算的中子通量非负,有效降低数值误差,提高数值计算的精度。     

A finite difference method of solving anisotropic scattering problems

A new method of solving radiative transfer problems is described including a comparison of its speed with that of the doubling method, and a discussion of its accuracy and suitability for computations involving variable optical properties. The method uses a discretization in angle to produce a coupled set of first-order differential equations which are integrated between discrete depth points to produce a set of recursion relations for symmetric and anti-symmetric angular sums of the radiation field at alternate depth points. The formulation given here includes depth-dependent anisotropic scattering, absorption, and internal sources, and allows arbitrary combinations of specular and non-Lambertian diffuse reflection at either or both boundaries. The method is shown to be faster than the doubling method when the number of depth points and angular quadrature points is identical. Numerical tests of the method show that it can return accurate emergent intensities even for large optical depths. The method is also shown to conserve flux to machine accuracy in conservative atmospheres. Finally, several checks are made that demonstrate that the new method can compute accurate radiation fields in atmospheres with variable optical properties.     

Efficient method for solving the problems of wave diffraction by scatterers with broken boundaries

The pattern equations method is extended to solving the problems of wave scattering by bodies with piecewise smooth boundaries. The method is based on the reduction of the initial boundary-value problem to an integro-operator equation of the second kind in the scattering pattern of a body. With the use of the series expansion of the scattering pattern in angular spherical harmonics, the problem is ultimately reduced to solving an infinite algebraic system of equations in the expansion coefficients of the scattering pattern. The conditions at which this system can be solved by the method of reduction are formulated. Examples of solving the problems of wave scattering by bodies with impedance boundaries are considered. Essential advantages of the proposed method over other known methods are demonstrated.     

A simple computational method for internal polarized radiation fields of finite slab atmospheres

An extended doubling method is formulated, which provides together with the emergent radiation also the internal polarized radiation field without additional iterations. Two sets of linear regular integral relations are derived, which have to be fulfilled by the surface Green's function matrix or, equivalently, by the Stokes vector of the slab albedo problem radiation field. The integral relations refer to the half range angular variable of the direction of incidence and to the full range angular variable of the direction of light propagation, respectively.     

Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements

We propose the P_N approximation based on a finite element framework for solving the radiative transport equation with optical tomography as the primary application area. The key idea is to employ a variable order spherical harmonic expansion for angular discretization based on the proximity to the source and the local scattering coefficient. The proposed scheme is shown to be computationally efficient compared to employing homogeneously high orders of expansion everywhere in the domain. In addition the numerical method is shown to accurately describe the void regions encountered in the forward modeling of real-life specimens such as infant brains. The accuracy of the method is demonstrated over three model problems where the P_N approximation is compared against Monte Carlo simulations and other state-of-the-art methods.     

Central potential reconstruction from scattering data by the variable boundary method with the use of Bargmann potentials

An effective numerical method for solving the inverse quantum scattering problem with a fixed angular momentum is suggested. The method is based on combining the variable boundary method suggested earlier and the method of Bargmann potentials. Test calculations are evidence of high method effectiveness in various particular cases, for instance, when there are bound states and resonances.     

可变不确定集约束的鲁棒自适应波束形成方法

自适应波束形成方法在存在协方差矩阵误差和导向向量误差的失配条件中性能严重下降。最差情况性能最优。(Worst-Case Performance Optimization,WCPO)方法可以显著增强失配条件下自适应波束形成的鲁棒性,但该方法存在系统性的信号功率过估计问题,并且限定协方差矩阵不确定集与导向向量不确定集的两个关键参数需要人为指定,缺乏具备明确物理意义的求解方法和数据自适应性.本文以WCPO方法为基础,给出了信号功率过估计的理论分析以及相应的改进方法,提出了基于矩阵重构的协方差矩阵不确定集参数的自适应估计方法,最终得到一种可变不确定集约束的鲁棒自适应波束形成方法(WCPO-PCVC).相比WCPO方法,该方法消除了信号功率估计的系统性偏差,且关键参数无需人为指定。数值仿真和海试数据处理结果表明,该方法在失配条件下具有良好的干扰与旁瓣抑制能力,与常规波束形成方法相比具有更好的信号到达角分辨能力,与自适应波束形成方法相比估计的信号功率更为精准。     

Vibrations and buckling of a beam on a variable winkler elastic foundation

Two methods for solving the eigenvalue problems of vibrations and stability of a beam on a variable Winkler elastic foundation are presented and compared. The first is based on using the exact stiffness, consistent mass, and geometric stiffness matrices for a beam on a variable Winkler elastic foundation. The second method is based on adding an element foundation stiffness matrix to the regular beam stiffness matrix, for vibrations and stability analysis. With these matrices, it is possible to find the natural frequencies and mode shapes of vibrations, and buckling load and mode shape, by using a small number of segments. It is concluded that the use of the element foundation stiffness approach yields better convergence at lower computation costs.     

Symmetry of the nonlinear collision operator matrix and new prospects in the moment method for solving the Boltzmann equation

Traditionally, the moment method has been used in kinetic theory to calculate transport coefficients. Its application to the solution of more complicated problems runs into enormous difficulties associated with calculating the matrix elements of the collision operator. The corresponding formulas for large values of the indices are either lacking or are very cumbersome. In this paper relations between matrix elements are derived from very general principles, and these can be employed as simple recurrence relations for calculating all the nonlinear and linear anisotropic matrix elements from assigned linear isotropic matrix elements. Efficient programs which implement this algorithm are developed. The possibility of calculating the distribution function out to 8–10 thermal velocities is demonstrated. The results obtained open up prospects for solving many topical problems in kinetic theory. Zh. Tekh. Fiz. 69, 6–9 (September 1999)     

Seismic response of underwater members of variable cross section

This paper is concerned with a method for solving seismic response problems of a pile of variable cross section with a tip inertia subjected to a sea bottom seismic displacement. The analysis developed here is based on an elastodynamic theory in which the effects of the continuously distributed mass and rigidity of the pile are included. The method includes use of Fourier series expansion, the Laplace transform, the transfer matrix method and the residue theorem in order to deal with the complex seismic displacement and arbitrarily shaped piles; considerable simplification of the calculations is thus achieved. The theoretical results given are applicable to seismic response problems for a pile of arbitrary shape with a tip inertia, excited by arbitrary displacements. As an application of the present theoretical results, the dynamic response has been calculated for hollow piles of curved conical shape with tip inertias and double taper beams subjected to seismic displacements.     

Multi-layer evolution schemes for the finite-dimensional quantum systems in external fields

The operator-difference multi-layer (ODML) schemes for solving the time-dependent Schrödinger equation (TDSE) till six order accuracy by a time step are presented. The reduced schemes for solving a set of the coupled TDSE’s are devised by using a set of appropriate basis angular functions and a finite element method with respect to a hyper-radial variable. Convergence by a number of the basis functions and efficiency of the numerical schemes are demonstrated in the case of an exactly solvable model of the two-dimensional oscillator in time-depended electric fields.     

An approach to the anomalous commutation relations of rotational angular momenta in molecules

In molecular quantum mechanics, angular momentum operators occur which obey anomalous commutation relations. A direct method of evaluating the matrix elements of such operators is presented. The method is particularly well suited to calculations which use spherical tensor techniques; it also avoids some confusing aspects of Van Vleck's reversed angular momentum method [2]. The relationship between results derived in space- and molecule-fixed axis systems is made explicit and a consistent phase convention is established. The transformation matrix relating coupled and decoupled basis sets for problems with molecular quantization is also derived.     

In-plane free vibration analysis of cable-arch structure

Cable-stayed arch bridge is a new type of composite bridge, which utilizes the mechanical characters of cable and arch. Based on the supporting members of cable-stayed arch bridge and of erection of arch bridge using of the cantilever construction method with tiebacks, we propose a novel mechanical model of cable-arch structure. In this model, the equations governing vibrations of the cable-arch are derived according to Hamilton's principle for dynamic problems in elastic body under equilibrium state. Then, the program of solving the dynamic governing equations is ultimately established by the transfer matrix method for free vibration of uniform and variable cross-section, and the internal characteristics of the cable-arch are investigated. After analyzing step by step, the research results approve that the program is accurate; meanwhile, the mechanical model and method are both valuable and significant not only in theoretical research and calculation but also in design of engineering.     

Preventing numerical errors generated by interface-capturing schemes in compressible multi-material flows

In the present work, errors generated in computations of compressible multi-material flows using shock-capturing schemes are examined, specifically pressure oscillations (when the specific heats ratio is variable), but also temperature spikes and species conservation errors. These numerical errors are generated at material discontinuities due to an inconsistent treatment of the convective terms. Though temperature errors are irrelevant to solutions to the Euler equations, it is shown that they have the potential to lead to problems when physical diffusion is included, i.e., for the Navier–Stokes equations. These errors are studied analytically and numerically by considering the one-dimensional advection of isolated material discontinuities. A methodology preventing such errors for weighted essentially non-oscillatory (WENO) schemes is presented, in which modified WENO weights are used to solve the transport equation for mass fraction in conservative form to prevent temperature and species conservation errors. Pressure errors are prevented by solving an additional transport equation for a given function of the ratio of specific heats. Several multi-dimensional problems with various discontinuities (shocks, material interfaces and contact discontinuities), including the single-mode Richtmyer–Meshkov instability, and turbulence are considered to test the method.     

Matrix Riccati equation formulation for radiative transfer in a plane-parallel geometry

In this paper, we formulate the radiative transfer problem as an initial value problem via a pair of nonlinear matrix differential equations (matrix Riccati equations or MREs) which describe the reflection (R) and transmission (T) matrices of the specific intensities in a plane-parallel geometry. One first computes R and T matrices of some small but finite layer thickness (equivalent optical thickness τ∼0.01 and then repetitively applies the doubling method to the reflection and transmission matrices R(τ)and T(τ) until reaching the desired layer thickness. The initial matrices R(τ₀)and T(τ₀) can be computed quite accurately by either of the following methods: multiple-order, multiple-scattering approximation, iterative method or fourth-order Runge-Kutta techniques. In addition, the reflection coefficient matrix of a semi-infinite medium satisfies an algebraic matrix equation which can be solved repetitively by a matrix method. MREs offer an alternative way of solving plane-parallel radiative transport problems. This method requires only elementary matrix operations (addition, multiplication and inversion). For vector and/or beam-wave radiative transfer problems, large matrices are required to describe the physics adequately, and the MRE method provides a significant reduction in computer memory and computation time.     

低速湍流模拟的预处理技术研究

本文开展低速湍流的预处理技术研究. 该预处理技术采用守恒型变量及主控方程与湍流方程相耦合的隐式求解方法, 并为确保迭代求解稳定性, 发展了合理的参考马赫数定义、双时间步无矩阵方法迭代求解形式以及湍流源项隐式处理方法等, 从而真正实现全速湍流软件平台统一形式. 在喷管、翼型和方柱等低速湍流数值模拟中, 本文方法正确刻画了流场结构特征, 计算与理论、实验等相关结果符合较好, 具有很强的迭代收敛性和结果精度.     

An analytic and numerical solution with spectral Green's function method for transport equation in spherical geometry

Since in many cases curvilinear geometry is more appropriate than cartesian geometry for precise modeling of the complex systems for reactor calculation, we have developed the spectral Green's function (SGF) method which is employed to obtain angular and scalar flux distributions in heterogeneous sphere geometry with isotropic scattering. In this study, we showed that the neutron transport problems of homogeneous spheres could be reduced to the solution of plane geometry equation.Finally, some results are discussed and compared with those already obtained by diamond difference scheme to test the accuracy of the results. The agreement is satisfactory. SGF method is very suitable for the numerical solution of the neutron transport equation with isotropic scattering.     

变系数瞬态热传导问题边界元格式的特征正交分解降阶方法

提出了一种基于边界元法求解变系数瞬态热传导问题的特征正交分解(POD)降阶方法,重组并推导出变系数瞬态热传导问题适合降阶的边界元离散积分方程,建立了变系数瞬态热传导问题边界元格式的POD降阶模型,并用常数边界条件下建立的瞬态热传导问题的POD降阶模态,对光滑时变边界条件瞬态热传导问题进行降阶分析.首先,对一个变系数瞬态热传导问题,建立其边界域积分方程,并将域积分转换成边界积分;其次,离散并重组积分方程,获得可用于降阶分析的矩阵形式的时间微分方程组;最后,用POD模态矩阵对该时间微分方程组进行降阶处理,建立降阶模型并对其求解.数值算例验证了本文方法的正确性和有效性.研究表明:1)常数边界条件下建立的低阶POD模态矩阵,能够用来准确预测复杂光滑时变边界条件下的温度场结果;2)低阶模型的建立,解决了边界元法中采用时间差分推进技术求解大型时间微分方程组时求解速度慢、算法稳定性差的问题.     

解粒子输运问题的半随机模拟数值方法

分析了解粒子输运问题的随机模拟方法和确定性方法的优缺点,提出了零方差迭代格式和半随机模拟方法——把MC方法和确定性方法有机结合起来,达到扬长避短的目的。扼要地介绍了近年来在这方面的工作。