Computation of Fresnel Integrals. II

This paper describes an improved method for computing Fresnel integrals with an error of less than 1 × 10⁻⁹. The method is based on a known approximate formula for a different integral which is due to Boersma and referenced by Abramowitz and Stegun.     

Stokes flow around an asymmetric channel divider; a computational approach using MATLAB

While computational methods for solving Stokes-flow problems have existed for some time, these have depended on specialized codes developed specifically, for this type of problem. This work shows how to combine traditional applied mathematics and a modern over-the-counter software package Matlab to solve and study Stokes flow in a channel with a splitter plate. Specifically exact unidirectional flow solutions are used as a basis for choosing boundary conditions for Matlab to anticipate the boundary conditions of a Stokes flow. A method for selecting zeroth and first-order approximate boundary conditions is presented, along with a suggestion for finding a second-order approximation. It is also shown that small errors made in choosing the approximate boundary conditions do not grow as one moves away from the boundary into the interior of the flow. Finally several computational examples using this approach are presented.     

Three Improvements in Reduction and Computation of Elliptic Integrals

Three improvements in reduction and computation of elliptic integrals are made. 1. Reduction formulas, used to express many elliptic integrals in terms of a few standard integrals, are simplified by modifying the definition of intermediate “basic integrals.” 2. A faster than quadratically convergent series is given for numerical computation of the complete symmetric elliptic integral of the third kind. 3. A series expansion of an elliptic or hyperelliptic integral in elementary symmetric functions is given, illustrated with numerical coefficients for terms through degree seven for the symmetric elliptic integral of the first kind. Its usefulness for elliptic integrals, in particular, is important.     

Fresnel arrays for the study of Lamb waves in laser ultrasonics

An experimental technique is proposed to analyze and control laser-generated Lamb waves propagating in aluminum plates. The technique consists in forming on the surface of the specimen an array of concentric arc sources by passing the laser beam through a Fresnel lens. The spacing between the illuminated arcs produces a forcing wavelength for which only a few specific frequencies (those which satisfy the dispersion relation) can propagate in the sample. Dispersion curves can be obtained by measuring the frequency content of the received signals for a range of wavenumbers. The technique offers a relatively high signal-to-noise ratio because of the narrow-band nature of the method and because of the confocal geometry of the source distribution. Reasonably good agreement is obtained between theoretical and experimental dispersion curves especially for the lower modes, thus showing that the proposed technique may have some potential for some specific applications in laser ultrasonic nondestructive evaluation.     

菲涅耳微透镜列阵衍射效率的测试

介绍了测量菲涅耳微透镜列阵的衍射效率的两种不同方法及其系统，方法简单易行，并实际测试了具有微小单元尺寸的菲涅耳微透镜列阵的衍射效率。     

Characteristics of lateral-shearing nulling interferometry by use of double Fresnel rhombs

Abstract

Nulling interferometer for directly imaging exoplanets with a segmented-mirror telescope is investigated. Lateral shearing interferometry is applied to a segmented-mirror telescope such as the Thirty Meter Telescope. Use of a pair of double Fresnel rhombs in Mach–Zehnder interferometer achieves achromatic nulling and lateral shearing simultaneously. In this paper, computer simulations of the lateral interferometry with the Fresnel rhombs in the near infrared region are carried out to analyze the effects of segmentation errors, tip-tilt errors, amplitude and optical-path difference errors on its nulling performance.     

上海高性能计算公共服务平台

高性能计算已被公认为是继理论和实验之后,人类认识世界的第三大方法,它是衡量一个国家或地区经济技术综合实力的重要标志,同时又是当今世界发达国家竞相争夺的一个战略制高点。本文介绍了高性能计算的国内外现状和发展趋势,以及上海建立高性能计算公共服务平台的战略目标和其发挥作用的应用领域。     

空间用平板形菲涅耳透镜的设计和光学效率研究

研究平板形菲涅耳太阳聚光透镜的光学问题,给出了线聚焦和点聚焦透镜的设计方法,讨论了设计参数对透镜光学效率的影响,为这类透镜的最优设计提供理论依据。     

Observation of kinoform-style multilayer Fresnel zone plate by scanning ion microscopy

We investigated a kinoform-style Fresnel zone plate (FZP) that is a high-efficient optical element for hard X-ray focusing. The Cu/Al kinoform-style FZPs were fabricated with a sputtered-sliced method. The FZP was composed of 450 layers (30 layers, 15 pairs) of Al, Cu/Al composite, and Cu. The microstructure of the Cu/Al composite layer was observed by scanning ion microscopy based on a focused ion beam (FIB). In the Cu/Al composite layer, the SIM images obtained by grain orientation contrast observation indicated that the Cu grains were grown gradually with increasing Cu concentration.     

Efficient direct computation of the pseudo-inverse and its gradient

The pseudo-inverse (also called the Moore–Penrose inverse or the generalized inverse) has many uses in engineering in fields such as control design, structural dynamics and identification. Efficient computation of the pseudo-inverse can greatly ease the computational burden associated with these techniques. In addition, the gradient of the pseudo-inverse may be needed for sensitivity analysis or optimization. Typical methods for computing the pseudo-inverse require the singular value or eigenvalue decomposition of the appropriate matrices. Moreover, if the gradient is required, it is either computed with finite differences, or by taking the gradient of the Singular Value Decomposition (SVD) and eigen decomposition of the appropriate matrices. However, this is a very difficult task, if possible at all. This paper develops a direct method of computing the gradient of the pseudo-inverse of well-conditioned systems with respect to a scalar. The paper begins by revisiting a direct method for computing the pseudo-inverse developed by Greville for matrices with independent columns. When applied to a square, fully populated, non-symmetric case, with independent columns, it was found that the approach can be up to 8 times faster than the conventional approach of using the SVD. Rectangular cases are shown to yield similar levels of speed increase. A method is then presented which is a direct approach for computing the gradient of the pseudo-inverse that previously did not exist. To help illustrate the algorithms, simple MATLAB code is provided. © 1997 John Wiley & Sons, Ltd.     

含有三次非线性的粘性阻尼双线性迟滞振动系统的响应计算

研究了一类无记忆恢复力中含有位移三次非线性的粘性阻尼双线性迟滞振动系统的响应计算问题，提出了一种Fourier级数展开法与系统处于特殊点时的运动方程相结合的近似解法，该方法可靠、编程简单，具有较好的工程实用性。     

Numerical Evaluation of Diffraction Integrals

This paper describes a simple numerical integration method for diffraction integrals which is based on elementary geometrical considerations of the manner in which different portions of the incident wavefront contribute to the diffracted field. The method is applicable in a wide range of cases as the assumptions regarding the type of integral are minimal, and the results are accurate even when the wavefront is divided into only a relatively small number of summation elements. Higher accuracies can be achieved by increasing the number of summation elements and/or incorporating Simpson’s rule into the basic integration formula. The use of the method is illustrated by numerical examples based on Fresnel’s diffraction integrals for circular apertures and apertures bounded by infinite straight lines (slits, half planes). In the latter cases, the numerical integration formula is reduced to a simple recursion formula, so that there is no need to perform repetitive summations for every point of the diffraction profile.     

修正型凸轮曲线的设计算法

通过修正型凸轮曲线在平压模切机输纸机构中的应用原理的论述,介绍了一种平行分度凸轮曲线设计算法的原理。     

计算模型修正的频率响应函数方法

本文提出一种有限元动力分析计算模型修正的频率响应函数方法。这个方法综合考虑了系统的质量、刚度,特别是阻尼矩阵的修正,由位置矩阵将修正量集中体现在测量自由度上,利用结构系统的频率响应函数有限元分析值和实测值,计算出模型的质量、阻尼和刚度矩阵的修正量。文中算例结果表明,本方法有较好的修正结果。     

Asymptotic expansions of the Biot-Savart law for a slender vortex with core variation

The Method of Matched Asymptotic Expansion of Singular Integrals (MAESI) is used to expand the Biot-Savart law in terms of different parameters. This method is first used to find, in terms of the small distance r to a line vortex, the first orders of the known expansion of the potential flow induced by this line vortex. This method is also used to easily compare two equations of motion of a slender vortex filament: the one obtained in an ad-hoc way by a cut-off line-integral technique and the other derived from the Navier-Stokes equations by Callegari and Ting. Finally, this method is used to give the inner expansion of the flow induced by a slender vortex in terms of its slenderness . This is the first inner expansion up to order one in terms of of the Biot-Savart law for a slender vortex. An application of this inner expansion is then given to find the induced velocity of a family of non-circular vortex rings with axisymmetric axial-core variation. In order to understand the time-evolution of these initial conditions to the Navier-Stokes equations, a short time scale is introduced. A quasi-hyperbolic system that describes the leading-order dynamics of the axisymmetric axial core variation on a curved slender vortex filament is finally extracted from the Navier-Stokes equations.     

基于进化计算的地毯图案生成方法

新疆民族地毯图案具有独特的地域性、民族性和风格,如何保留和传承其风格特征是图案创新设计的难点.提出一种基于进化计算的创新设计方法.可通过分解地毯图案,分析其组成元素,抽取具有表征风格的特征予以进化计算,再生新型纹样,设计中还可以调整图案内容及结构来丰富花型变化,并通过不同顺序关系中元素组合增强图案的表现力,仿真实验表明该方法实现了生成具有风格化的创新图案,借助于主客观评价模型得到很高地认可.     

消声瓦声学性能计算方法研究

首先建立均匀圆柱腔中弹性波的计算模型,然后利用该模型建立复合过渡型声腔结构消声瓦吸声性能计算的理论模型,并应用这一理论模型计算分析现有声腔结构消声瓦及几种变形的声腔结构声腔结构形式的消声瓦的吸声性能,讨论材料环境温度和材料参数变化对吸声性能的影响,最后总结消声瓦声腔结构和不同材料参数对吸声性能影响的基本规律.     

Remarks on the Gauss quadrature rule for a particular class of finite-part integrals

The method of finite-part (Hadamard-type or hypersingular) integrals is already a standard approach in applied numerical methods in engineering and, mainly, in acoustics, fluid dynamics, elasticity and fracture mechanics. Here we study the convergence of the non-classical (possessing a negative node, outside the integration interval, as well as a negative weight corresponding to this node) Gauss-Legendre quadrature rule for finite-part integrals on the interval [0,1] with the weight function 1/y and we prove its validity for integrands possessing a continuous first derivative. The case of integrands possessing a higher-order derivative is also considered and the rate of convergence is established. These results are based on the derivation of a bound for the negative node in the above quadrature rule and, further, on the theory of convergence of the classical Gauss quadrature rule. Numerical results, verifying the results of this paper, are also displayed. The present results can be generalized to the Gauss-Laguerre similarly non-classical quadrature rule for finite-part integrals (on the interval [0], ∞)) as well as to the two special, but important in engineering applications, non-classical Gaussian quadrature rules for Cauchy-type principal-value integrals and Mangler-type finite-part integrals have been proposed by Tsamasphyros and Dimou.     

Computation of stress intensity factors for a 2-D body from displacements at an arbitrary location

An approach to stress intensity factor computation is presented which allows accurate estimation using a simple finite element program and a coarse mesh of elements. The stress intensity factor is obtained from what we term the Crack-Displacement (C-D) Factor. This involves the rate of change of displacements with crack length at the same location remote from the crack due to two loading conditions. One loading condition is merely that applied to the cracked body. The other loading condition is a virtual line force applied at the location at which the displacement rates are computed. The accuracy of the procedure is demonstrated for uniform tensile stresses applied to a center-cracked panel and an edge-cracked strip.     

A mathematical model of the edge-defined film-fed growth process

A model is presented to simulate the steady-state growth of a fiber produced by the edge-defined film-fed growth (EFG) process. Equations describing the axisymmetric transport of heat in the melt and fiber are discussed. Heat transfer between the system and the surrounding environment is assumed to take place via convection and radiation. Given the fiber translation velocity, the external temperature profile, the die temperature, and the input flow rate or the pressure, asymptotic solutions for the temperature profiles in the melt and fiber, and the melt/gas and solidifying interfacial shapes are developed in the limit of a small melt slenderness ratio (fiber radius/approximate melt height). The effect of process parameters on the shape of the fiber system is investigated.