二维三温辐射扩散方程组两层预条件子的自适应求解

针对实际应用中若干典型三温线性系统,分析求解二维三温辐射扩散方程离散线性系统的代数两层预条件子(PCTL)的算法效率.结果表明,PCTL的算法效率与三个温度之间的耦合强度以及单温子系统对角占优性强弱程度有很大关系.为此,通过刻画三温线性系统的耦合强度和单温子系统对角占优性特征,提出一种PCTL中子系统的自适应求解算法.数值结果表明,可以显著改善PCTL的算法效率.对于实际数值模拟应用中37个典型三温线性系统,相对于经典AMG算法,算法整体加速2.5倍.数值实验表明算法具有很强的鲁棒性.     

基于层流动能湍流模型的数值模拟方法

从层流动能出发提出数值模拟原则;综合考虑自然、旁路和分离流转捩的因素构建实用的层流动能湍流模型,结合预处理和基本求解技术发展出适于转捩流动的数值模拟方法和程序.针对预处理技术,以Weiss-Smith矩阵为基础,考虑湍流粘性的影响;针对基本离散格式和边界条件,结合模型方程进行对角占优强化等特殊处理.最后通过平板边界层和典型翼型,特别是低雷诺数翼型的数值模拟,验证数值方法的有效性和鲁棒性.算例表明本文的方法能够为求解更复杂的流动提供参考.     

分立位置表象中双原子分子振动能级的计算

用离散的位置基矢作为连续位置基矢的近似,构建分立位置表象.在分立位置表象中,哈密顿算符矩阵具有对角占优、带状稀疏的特点,矩阵元不用做积分运算,具有特别简单的解析表达式.分别对双原子分子在Morse势、Murrell-Sorbie势和双阱势中运动情形,进行了数值计算.结果表明,在分立位置表象中计算双原子分子振动能级,方法简便,稳定性好,计算精度高.     

解弹性力学第二类边界积分方程的求积法与分裂外推

利用Sidi奇异求积公式,提出了解曲边多角形域上线性弹性力学第二类边界积分方程的求积法,即离散矩阵的每个元素的生成只需赋值不需计算任何奇异积分.通过估计离散矩阵的特征值和利用Anselone聚紧收敛理论,证明了近似解的收敛性;同时得到了误差的多参数渐近展开式;通过并行地解粗网格上的离散方程,利用分裂外推获得了高精度近似解和后验误差.     

准一维无序系统的电子结构

利用负本征值理论的态密度计算方法，研究了准一维双链无序系统的电子结构。并针对系统大小，对角和非对角无序程等各种参数，探讨了电子的局域化，系统能量分布范围等问题。在引进无序参量后，对角无序主要引起电子局域态的增多，而非对角无序则使系统的能量分布范围发生变化。     

环状非有心球谐振子势场赝自旋对称性的三对角化表示

提出一个包含非有心电耦极矩势的环状谐振子模型，在能够负载波动算子三对角化矩阵表示的完全平方可积L2空间讨论了这一势场的赝自旋对称性．利用三对角化矩阵方案，给出了波函数角向分量和径向分量展开系数满足的三项递推关系式．角向波函数和径向波函数分别以Jacobi多项式和Laguerre多项式表示，由径向分量展开系数递推关系式的对角化条件得到束缚态的能量谱．并以Descartes多项式的符号法则讨论了能量方程的代数结构．     

非均匀对角减载最小方差无失真响应多目标分辨

针对最小方差无失真响应(Minimum Variance Distortionless Response,MVDR)方法在起伏非相关噪声环境下多目标分辨性能严重下降的问题,提出一种非均匀对角减载MVDR (Inhomogeneous Diagonal Unloading MVDR,IDU-MVDR)方法。该方法首先对协方差矩阵进行非均匀对角减载,然后实施MVDR方法。各阵元上的对角减载量通过求解半正定优化问题获得,优化问题中最大化减载量之和,但约束减载后协方差矩阵的最小特征值是一个较小的正值。数值仿真表明,IDUMVDR方法可通过非均匀对角减载消除大部分非相关噪声,但保留小部分噪声分量.因此IDU-MVDR方法较MVDR方法分辨力更高,空间谱中背景级更低、弱目标谱峰更加明显,并且具备一定的稳健性.海上实验结果与数值仿真相一致,验证了IDU-MVDR方法的有效性.      

广义量子主方程：教程综述和最新进展

广义量子主方程(GQME)为模拟嵌入在量子环境中的开放量子体系的约化动力学提供了一种通用且严格的计算方法. 开放量子体系的动力学在能量、电荷以及量子相干转移过程和光化学反应中至关重要. 量子系统通常被定义为我们感兴趣的自由度,例如捕光分子的电子态或凝聚态体系中的特定振动模式. 系统周围的环境也被称为热浴,必须考虑它对系统的影响. 例如,广义量子主方程理论中用投影算符方法对其进行描述. 本综述总结了广义量子主方程的两种标准形式,即时间卷积形式的Nakajima-Zwanzig GQME和无卷积形式的广义量子主方程. 在更流行的NZ-GQME形式中,记忆核刻画了非马尔可夫和非微扰效应,给出了约化密度矩阵的精确量子动力学. 总结了几种通过含有分子信息但无投影算符的时间关联函数作为输入信息,进而求解含投影算符的记忆核的方法. 特别值得一提的是近期提出的NZ-GQME改进版方法,该方法是基于将哈密顿量划分为更通用的对角和非对角部分. 上述系统相关的热浴时间关联函数可以通过数值精确或近似量子动力学方法计算. 本文将有助于理解广义量子主方程的理论背景,并且展望通过GQME与量子计算技术的结合解决使用当今最先进的经典超级计算机无法解决的与量子动力学和量子信息相关的复杂问题.     

关于表象选择的讨论

量子力学中大量出现的是求解定态薛定谔方程 H　=E　(1)亦即求解H的本征问题.在微扰论中求解H的本征问题,归结为H矩阵的对角化问题.从物理上看,也就是最终化到H算符的自身表象中去.但我们总得先有个已知的工作表象,然后再把它设法对角化,这就出现了表象选择问题.下面以简　微扰     

包含非中心电耦极矩的环状非谐振子势场赝自旋对称性的三对角化表示

提出了一个包含非中心电耦极矩分量的环状非谐振子势模型,在能够负载Dirac波动算子三对角化表示的完全平方可积L~2空间讨论了这一势场的赝自旋对称性.利用三对角化矩阵方案,使得求解Dirac方程转换为寻求波函数展开系数满足的三项递推关系式.角向波函数和径向波函数分别以Jacobi多项式和Laguerre多项式表示.由径向分量展开系数递推关系式的对角化条件得到束缚态的能量谱,显示出这一势模型具有严格的赝自旋对称性     

大型稀疏线性方程组的改进ICCG方法

有限元线性方程组的系数矩阵一般具有稀疏性和对称性的特点,全稀疏存贮方法就是利用这些特点,只存贮对称部分的非零元素,采用链表式管理,既节省存贮空间,又便于动态更改.在带双门槛值ICCG方法的基础上,加上适当的对角元修正策略,得到一种新的改进的ICCG方法,能够确保方程组高效准确的分解和求解.数值算例证明,该算法在时间和存贮上都较为占优,可靠高效,能够应用于有限元线性方程组的求解.     

Synchronized parameter optimization of the double freeform lenses illumination system used for the CF-LCoS pico-projectors

A novel synchronized optimization method of multiple freeform surfaces is proposed and applied to double lenses illumination system design of CF-LCoS pico-projectors. Based on Snell's law and the energy conservation law, a series of first-order partial differential equations are derived for the multiple freeform surfaces of the initial system. By assigning the light deflection angle to each freeform surface, multiple surfaces can be obtained simultaneously by solving the corresponding equations, meanwhile the restricted angle on CF-LCoS is guaranteed. In order to improve the spatial uniformity, the multi-surfaces are synchronously optimized by using simplex algorithm for an extended LED source. Design example shows that the double lenses based illumination system, which employs a single 2 mm×2 mm LED chip and a CF-LCoS panel with a diagonal of 0.59 inches satisfies the needs of pico-projector. Moreover, analytical result indicates that the design method represents substantial improvement and practical significance over traditional CF-LCoS projection system, which could offer outstanding performance with both portability and low cost. The synchronized optimization design method could not only realize collimating and uniform illumination, but also could be introduced to other specific light conditions.     

Exact dynamic and static stiffness matrices of shear deformable thin-walled beam-columns

Total potential energy of non-symmetric thin-walled beam-columns in the general form is presented by introducing the displacement field based on semitangential rotations and deriving transformation equations between displacement and force parameters defined at the arbitrary axis and the centroid-shear center axis, respectively. Next, governing equations and force-deformation relations are derived from the total potential energy for a shear-deformable, uniform beam element and a system of linear eigenproblem with non-symmetric matrices is constructed based on 14 displacement parameters. And then explicit expressions for displacement parameters are derived and exact dynamic stiffness matrices are determined using force-deformatin relationships. In addition, the modified numerical method to eliminate multiple zero eigenvalues and to evaluate the exact static stiffness matrix is developed for spatial stability analysis. Finally, in order to demonstrate the validity and the accuracy of this study, the spatially coupled natural frequencies and buckling loads are evaluated and compared with analytical solutions or results analyzed by thin-walled beam elements and ABAQUS's shell elements.     

On the numerical approximation of high-frequency acoustic multiple scattering problems by circular cylinders

The aim of this paper is to propose a numerical strategy for computing the solution of two-dimensional time-harmonic acoustic multiple scattering problems at high-frequency. The scatterers are assumed to be circular, leading therefore to semi-analytical representation formulae of the scattered field through the solution of a large linear system of equations. Taking advantage of the special block Toeplitz structure of the matrix of the linear system, a fast iterative and preconditioned numerical method yielding large memory savings is proposed. Several numerical experiments for general configurations are presented to show the efficiency of the numerical method.     

一种精确求解三原子反应散射的传播子

The exact short time propagator, in a form similar to the Crank-Nicholson method but in the spirit of spectrally transformed Hamiltonian, was proposed to solve the triatomic reactive time-dependent schrödinger equation. This new propagator is exact and unconditionally convergent for calculating reactive scattering processes with large time step sizes. In order to improve the computational efficiency, the spectral difference method was applied. This resulted the Hamiltonian with elements confined in a narrow diagonal band. In contrast to our previous theoretical work, the discrete variable representation was applied and resulted in full Hamiltonian matrix. As examples, the collision energy-dependent probability of the triatomic H+H₂ and O+O₂ reaction are calculated. The numerical results demonstrate that this new propagator is numerically accurate and capable of propagating the wave packet with large time steps. However, the efficiency and accuracy of this new propagator strongly depend on the mathematical method for solving the involved linear equations and the choice of preconditioner.     

A general theory of harmonic wave propagation in linear periodic systems with multiple coupling

A general theory is presented of harmonic wave propagation in one-dimensional periodic systems with multiple coupling between adjacent periodic elements. The motion of each element is expressed in terms of a finite number of displacement coordinates. The nature and number of different wave propagation constants at any frequency are discussed, and the energy flow associated with waves having real, complex or imaginary propagation constants is investigated. Kinetic and potential energy functions are derived for the propagating waves and a generalized Rayleigh's Quotient and Rayleigh's Principle for the complex wave motion have been found. This is extended to yield a generalized Rayleigh-Ritz method of finding approximate, yet accurate, relationships between the frequencies and propagation constants of the propagating waves. The effect of damping is also considered, and a special class of “damped forced waves” is postulated for hysteretically damped periodic systems. An energy definition for the loss factor of these waves is found. Briefly considered is the two-dimensional multi-coupled periodic system in which a simple wave motion analogous to a plane wave propagates across the whole system.     

Algebraic Multigrid Preconditioning for Finite Element Solution of Inhomogeneous Elastic Inclusion Problems in Articular Cartilage

In studying biomechanical deformation in articular cartilage, the presence of cells (chondrocytes) necessitates the consideration of inhomogeneous elasticity problems in which cells are idealized as soft inclusions within a stiff extracellular matrix. An analytical solution of a soft inclusion problem is derived and used to evaluate iterative numerical solutions of the associated linear algebraic system based on discretization via the finite element method, and use of an iterative conjugate gradient method with algebraic multigrid preconditioning (AMG-PCG). Accuracy and efficiency of the AMG-PCG algorithm is compared to two other conjugate gradient algorithms with diagonal preconditioning (DS-PCG) or a modified incomplete LU decomposition (Euclid-PCG) based on comparison to the analytical solution. While all three algorithms are shown to be accurate, the AMG-PCG algorithm is demonstrated to provide significant savings in CPU time as the number of nodal unknowns is increased. In contrast to the other two algorithms, the AMG-PCG algorithm also exhibits little sensitivity of CPU time and number of iterations to variations in material properties that are known to significantly affect model variables. Results demonstrate the benefits of algebraic multigrid preconditioners for the iterative solution of assembled linear systems based on finite element modeling of soft elastic inclusion problems and may be particularly advantageous for large scale problems with many nodal unknowns.     

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.     

Duality of interfacial energies and correlation functions

Duality transformations are used to derive a relationship between energies associated with interfaces between regions of different phases of one system and correlation functions of the dual system. For the two-dimensional Ising model, the relation is confirmed for the row/column and diagonal interfaces.     

Preconditioned characteristic boundary conditions for solution of the preconditioned Euler equations at low Mach number flows

Preconditioned characteristic boundary conditions (BCs) are implemented at artificial boundaries for the solution of the two- and three-dimensional preconditioned Euler equations at low Mach number flows. The preconditioned compatibility equations and the corresponding characteristic variables (or the Riemann invariants) based on the characteristic forms of preconditioned Euler equations are mathematically derived for three preconditioners proposed by Eriksson, Choi and Merkle, and Turkel. A cell-centered finite volume Roe’s method is used for the discretization of the preconditioned system of equations on unstructured meshes. The accuracy and performance of the preconditioned characteristic BCs applied at artificial boundaries are evaluated in comparison with the non-preconditioned characteristic BCs and the simplified BCs in computing steady low Mach number flows. The two-dimensional flow over the NACA0012 airfoil and three-dimensional flow over the hemispherical headform are computed and the results are obtained for different conditions and compared with the available numerical and experimental data. The sensitivity of the solution to the size of computational domain and the variation of the angle of attack for each type of BCs is also examined. Indications are that the preconditioned characteristic BCs implemented in the preconditioned system of Euler equations greatly enhance the convergence rate of the solution of low Mach number flows compared to the other two types of BCs.