Vector Arithmetic in the Triangular Grid

Vector arithmetic is a base of (coordinate) geometry, physics and various other disciplines. The usual method is based on Cartesian coordinate-system which fits both to continuous plane/space and digital rectangular-grids. The triangular grid is also regular, but it is not a point lattice: it is not closed under vector-addition, which gives a challenge. The points of the triangular grid are represented by zero-sum and one-sum coordinate-triplets keeping the symmetry of the grid and reflecting the orientations of the triangles. This system is expanded to the plane using restrictions like, at least one of the coordinates is an integer and the sum of the three coordinates is in the interval [−1,1]. However, the vector arithmetic is still not straightforward; by purely adding two such vectors the result may not fulfill the above conditions. On the other hand, for various applications of digital grids, e.g., in image processing, cartography and physical simulations, one needs to do vector arithmetic. In this paper, we provide formulae that give the sum, difference and scalar product of vectors of the continuous coordinate system. Our work is essential for applications, e.g., to compute discrete rotations or interpolations of images on the triangular grid.     

Differential Quadrature Analysis of Moving Load Problems

The differential quadrature method (DQM) has been successfully used in a variety of fields. Similar to the conventional point discrete methods such as the collocation method and finite difference method, however, the DQM has some difficulty in dealing with singular functions like the Dirac-delta function. In this paper, two modifications are introduced to overcome the difficulty encountered in solving differential equations with Dirac-delta functions by using the DQM. The moving point load is work-equivalent to loads applied at all grid points and the governing equation is numerically integrated before it is discretized in terms of the differential quadrature. With these modifications, static behavior and forced vibration of beams under a stationary or a moving point load are successfully analyzed by directly using the DQM. It is demonstrated that the modified DQM can yield very accurate solutions. The compactness and computational efficiency of the DQM are retained in solving the partial differential equations with a time dependent Dirac-delta function.     

Thermo-electro-mechanical vibration of coupled piezoelectric-nanoplate systems under non-uniform voltage distribution embedded in Pasternak elastic medium

In the present paper, the thermo-electro-mechanical vibration characteristics of a piezoelectric-nanoplate system (PNPS) embedded in a polymer matrix are investigated. The system is subjected to a non-uniform voltage distribution. The voltage distribution and in-plane preloads are very important in the resonance mode of smart composite nanostructures using PNPS. Small scale effects are taken into consideration using the nonlocal continuum mechanics. Hamilton's principle is employed to derive the nonlocal equations of motion. The governing equations are solved for various boundary conditions by using differential quadrature method (DQM). To verify the accuracy of the present results, a closed-form solution is also derived for the natural frequencies of simply supported PNPSs. The results of DQM are compared with those of exact solution and an excellent agreement is found. Finally, the effects of initial preload, temperature change, boundary conditions, aspect ratio, length-to-thickness ratio, nonlocal and non-uniform parameters on the vibration characteristics of PNPSs are studied. It is shown that the natural frequencies are quite sensitive to the non-uniform and nonlocal parameters.     

The shallow water equations in Lagrangian coordinates

Recent advances in the collection of Lagrangian data from the ocean and results about the well-posedness of the primitive equations have led to a renewed interest in solving flow equations in Lagrangian coordinates. We do not take the view that solving in Lagrangian coordinates equates to solving on a moving grid that can become twisted or distorted. Rather, the grid in Lagrangian coordinates represents the initial position of particles, and it does not change with time. We apply numerical methods traditionally used to solve differential equations in Eulerian coordinates, to solve the shallow water equations in Lagrangian coordinates. The difficulty with solving in Lagrangian coordinates is that the transformation from Eulerian coordinates results in solving a highly nonlinear partial differential equation. The non-linearity is mainly due to the Jacobian of the coordinate transformation, which is a precise record of how the particles are rotated and stretched. The inverse Jacobian must be calculated, thus Lagrangian coordinates cannot be used in instances where the Jacobian vanishes. For linear (spatial) flows we give an explicit formula for the Jacobian and describe the two situations where the Lagrangian shallow water equations cannot be used because either the Jacobian vanishes or the shallow water assumption is violated. We also prove that linear (in space) steady state solutions of the Lagrangian shallow water equations have Jacobian equal to one. In the situations where the shallow water equations can be solved in Lagrangian coordinates, accurate numerical solutions are found with finite differences, the Chebyshev pseudospectral method, and the fourth order Runge–Kutta method. The numerical results shown here emphasize the need for high order temporal approximations for long time integrations.     

Stability Analysis and Order Improvement for Time Domain Differential Quadrature Method

The differential quadrature method has been widely used in scientific and engineering computation. However, for the basic characteristics of time domain differential quadrature method, such as numerical stability and calculation accuracy or order, it is still lack of systematic analysis conclusions. In this paper, according to the principle of differential quadrature method, it has been derived and proved that the weighting coefficients matrix of differential quadrature method meets the importantV-transformation feature. Through the equivalence of the differential quadrature method and the implicit Runge-Kutta method, it has been proved that the differential quadrature method is A-stable and $s$-stage $s$-order method. On this basis, in order to further improve the accuracy of the time domain differential quadrature method, a class of improved differential quadrature method of $s$-stage $2s$-order has been proposed by using undetermined coefficients method and Padéapproximations. The numerical results show that the proposed differential quadrature method is more precise than the traditional differential quadrature method.     

A mass-conservative average flow model based on finite element method for complex textured surfaces

A mass-conservative average flow model based on the finite element method(FEM) is introduced to predict the performances of textured surfaces applied in mechanical seals or thrust bearings.In this model,the Jakobsson-Floberg-Olsson(JFO) boundary conditions are applied to the average flow model for ensuring the mass-conservative law.Moreover,the non-uniform triangular grid is utilized,which can deal with the problem of complex geometric shapes.By adopting the modeling techniques,the model proposed here is capable of dealing with complex textured surfaces.The algorithm is proved correct by the numerical experiment.In addition,the model is employed to gain further insight into the influences of the dimples with different shapes and orientations on smooth and rough surfaces on the load-carrying capacity.     

Variational calculations of vibrational energy levels for XY4 molecules: 2. Bending states of methane

A variational method for calculating excited bending states of symmetric tetrahedral pentaatomic molecules is presented based on the use of Radau coordinates and Jacobi polynomials as the basis functions. Symmetry is used both to reduce the size of secular matrix to be diagonalized and to calculate potential energy matrix elements over a reduced grid of quadrature points. Methods of treating the redundant coordinate are investigated and fitting is found to be more effective than the use of Taylor expansions. Test results are presented for methane, for which stretch-bend coupling and the contribution due to the redundant coordinate are found to be significant. Converged results are obtained for bending states significantly higher than considered in previous calculations. These states will be used as a basis for bending motions in a fully coupled stretch-bend calculation.     

求解声辐射问题的微分求积法与微分求积单元法

基于无限域中的Helmholtz波动方程,将微分求积法与微分求积单元法应用于二维及三维声辐射问题的求解,对最外层节点施加不同阶数的人工边界条件,区域内使用均匀及非均匀的节点分布方式,分析了节点分布方式及人工边界条件对计算结果的影响,比较了两种数值方法的计算精度。研究结果表明:微分求积法与微分求积单元法,前者精度更高,而后者耗时更少,在频率较低时,具备较高的效费比。人工边界条件对计算结果的影响主要体现在低频段,而节点分布方式的影响主要体现在高频段。非均匀的节点分布方式在不同频段都具备更好的计算精度。      

Application of the Generalized Differential Quadrature Method in Solving Burgers'Equations

The aim of this paper is to obtain numerical solutions of the one-dimensional, two-dimensional and coupled Burgers' equations through the generalized differential quadrature method (GDQM). The polynomial-based differential quadrature (PDQ) method is employed and the obtained system of ordinary differential equations is solved via the total variation diminishing Runge-Kutta (TVD-RK) method. The numerical solutions are satisfactorily coincident with
the exact solutions. The method can compete against the methods applied in the literature.     

Variational calculations of vibrational energy levels for XY4 molecules 1. Stretching states

A variational method for calculating excited stretching states of symmetric tetrahedral pentaatomic molecules is presented based on the use of Radau coordinates and Morse oscillator-like basis functions. Symmetry is used both to reduce the size of the secular matrix to be diag-onalized and to calculate the potential energy matrix elements over a reduced grid of quadrature points. Test results are presented for methane, silane and germane. For CH₄, stretch-bend coupling is found to be significant, whereas it is less important for the more strongly local mode SiH₄ and GeH₄ molecules. Converged results are obtained for stretching states significantly higher than considered in previous calculations. These states will be used to represent stretching motions in a fully coupled stretch-bend calculation.     

非均匀采样的功率谱反演大气湍流相位屏的快速模拟

对大气湍流功率谱非均匀采样可以有效改善传统功率谱反演法低频采样严重不足的缺陷, 实现高精度的大气湍流相位屏的模拟. 但采用的直接求和运算计算复杂度高, 相位屏的模拟速度极慢. 将非均匀快速傅里叶变换(NUFFT)引入到大气湍流相位屏的模拟, 可以实现相位屏的快速模拟. 从随机过程的谱分解出发, 将大气湍流相位随机过程表示为有限谐波分量叠加和的均方极限. 通过一个高斯核函数的卷积, 将非均匀分布的谐波复振幅映射到均匀网格空间, 进而利用快速傅里叶变换, 降低计算复杂度, 加快大气湍流相位屏的模拟速度. 以大气湍流的Kolmogorov 谱为例, 利用NUFFT仿真得到大气湍流相位屏, 并对相位屏的模拟精度、模拟速度和误差进行统计分析. 结果表明, NUFFT的引入可以实现快速、高精度的大气湍流相位屏的模拟.     

一种非结构/结构多层混合网格方法及其应用

对于粘性绕流的数值模拟,在自适应直角网格基础上,结合三角形非结构网格和结构化网格,利用其各自的优势和特点,提出一种生成混合杂交网格的思路和方法.在物面附近生成适合粘性流计算的大长宽比结构化网格,在远场分布自适应直角网格,快速离散计算空间.对于复杂的多体问题,采用三角形网格来连接各体网格,并运用网格合并的方法,保证各网格之间的光滑过渡与连接,提高网格质量.针对一些二维、三维外形的绕流问题,在上述网格基础上,采用B-L代数湍流模型和中心有限体积法,完成Navier-Stokes和Euler方程数值模拟的对比计算,结果表明网格生成和流场计算是正确的.     

2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures

This paper focuses on the dynamic behavior of functionally graded conical, cylindrical shells and annular plates. The last two structures are obtained as special cases of the conical shell formulation. The first-order shear deformation theory (FSDT) is used to analyze the above moderately thick structural elements. The treatment is developed within the theory of linear elasticity, when materials are assumed to be isotropic and inhomogeneous through the thickness direction. The two-constituent functionally graded shell consists of ceramic and metal that are graded through the thickness, from one surface of the shell to the other. Two different power-law distributions are considered for the ceramic volume fraction. The homogeneous isotropic material is inferred as a special case of functionally graded materials (FGM). The governing equations of motion, expressed as functions of five kinematic parameters, are discretized by means of the generalized differential quadrature (GDQ) method. The discretization of the system leads to a standard linear eigenvalue problem, where two independent variables are involved without using the Fourier modal expansion methodology. For the homogeneous isotropic special case, numerical solutions are compared with the ones obtained using commercial programs such as Abaqus, Ansys, Nastran, Straus, Pro/Mechanica. Very good agreement is observed. Furthermore, the convergence rate of natural frequencies is shown to be very fast and the stability of the numerical methodology is very good. Different typologies of non-uniform grid point distributions are considered. Finally, for the functionally graded material case numerical results illustrate the influence of the power-law exponent and of the power-law distribution choice on the mechanical behavior of shell structures.     

Heterodyne interferometer for displacement measurement with amplitude quadrature and noise suppression

The high precision displacement measurement in nanoscale is crucial to many applications. We present a heterodyne interferometry with differential phase to amplitude conversion scheme for displacement measurement in nanoscale. In this approach, the differential phase introduced by the displacement is converted into the amplitudes of heterodyne signals in quadrature. Meanwhile, the heterodyne signals in phase quadrature are also achieved so that the displacement can be determined from the amplitude ratio of the quadrature signals, and the direction of displacement can be determined from the phase quadrature. Since the differential phase to quadrature amplitude conversion is achieved through the optical addition and subtraction by polarization tuning, which are based on differential detection concept. Thus the proposed method benefits from the features of differential detection with common phase noise and correlated amplitude noise rejection and that of quadrature detection with real time and wide dynamic range of phase measurement. To demonstrate the capability of proposed method in differential phase measurement, we measure the displacement drove by a commercially available PZT pusher and found close agreement between the experiment and the theory. The experimental evidence of noise suppression is also found with spectral measurements, which demonstrates the resolution of displacement measurement at 60 pm and minimum detectable differential phase of 5.6 × 10⁻⁶ rad/ over 50 kHz.     

A global shallow water model using high order multi-moment constrained finite volume method and icosahedral grid

A novel accurate numerical model for shallow water equations on sphere have been developed by implementing the high order multi-moment constrained finite volume (MCV) method on the icosahedral geodesic grid. High order reconstructions are conducted cell-wisely by making use of the point values as the unknowns distributed within each triangular cell element. The time evolution equations to update the unknowns are derived from a set of constrained conditions for two types of moments, i.e. the point values on the cell boundary edges and the cell-integrated average. The numerical conservation is rigorously guaranteed. In the present model, all unknowns or computational variables are point values and no numerical quadrature is involved, which particularly benefits the computational accuracy and efficiency in handling the spherical geometry, such as coordinate transformation and curved surface.Numerical formulations of third and fourth order accuracy are presented in detail. The proposed numerical model has been validated by widely used benchmark tests and competitive results are obtained. The present numerical framework provides a promising and practical base for further development of atmospheric and oceanic general circulation models.     

Vibration analysis of a circular arch with variable cross-section using differential transformation and generalized differential quadrature

Vibration analysis of circular arches is an important subject in mechanics due to its various applications. In particular, circular arches with variable cross-section have been widely used to satisfy modern architectural and structural requirements. Recently, the generalized differential quadrature method (GDQM) and differential transformation method (DTM) were proposed by Shu and Zhou, respectively. In this study, GDQM and DTM are applied to vibration analysis of circular arches with variable cross-section. The governing equation of motion is derived and the non-dimensional natural frequencies are obtained for various boundary conditions. The concepts of differential transformation and generalized differential quadrature are briefly introduced. The results obtained by these methods are compared with previously published works. GDQM and DTM showed fast convergence, accuracy and validity in solving the vibration problem for circular arches with variable cross-sections.     

112 Gbit/s信号混传多种速率信号的非线性效应影响

主要研究在波长间隔50GHz的密集波分复用系统中,112Gbit/s偏分复用差分四相移键控信号在不同速率的相邻信道影响下经过长距离传输后的非线性容忍度。实验结果表明,112Gbit/s信号结合43Gbit/s信号或者10Gbit/s信号传输时系统得到较好的非线性容忍度,验证了在严重的非线性效应存在的情况下同步传输系统的可行性。     

Evaluation of quadrature schemes for the discrete ordinates method

After reviewing the presently available quadrature schemes for the discrete ordinates method, the accuracy of different schemes is analyzed and evaluated. It is shown from a comprehensive error analysis that the moment conditions have to satisfied not only for the principal coordinates directions, as it is mostly carried out, but for any arbitrary test direction. Among the schemes with approximately 50 discrete ordinates the DCT-020-2468 quadrature was found to give the best accuracy. The highest accuracy among all schemes is achieved by the LC-11 quadrature which requires 96 discrete ordinates. This scheme is rarely used up to date and deserves more attention for high accuracy predictions.     

多点择优推进阵面法生成曲面三角形网格

在现有曲面非结构网格生成法的基础上,提出了一种新的曲面网格生成法——多点择优推进阵面法.它可在曲面上直接进行三角形网格划分,克服了映射法的网格变形问题,并且可以在网格生成结束后,对曲面网格直接进行Laplace格点松弛光顺.该方法使用简单,不受曲面块类型的限制,且网格质量高,可以为三维非结构网格生成提供高质量的初始阵面,并给出了若干个算例.