Phase measuring algorithm for extraction of isochromatics of photoelastic fringe patterns

In recent years phase-measuring techniques have been applied to the problem of extracting information of photoelastic data. We present a new phase-measuring algorithm for extraction of the isochromatics of photoelastic fringe patterns. The algorithm permits the extraction of the isochromatic phase with almost no influence from the isoclinics, thus avoiding the usual problems of low-modulation areas associated with isoclinics. The isochromatic phase map obtained with this algorithm is well suited for a full separation of the stress components in a sample. The algorithm can be used with any commercial diffuse-light circular polariscope.     

A vectorized version of a sparse matrix-vector multiply

A fast vectorized algorithm is presented for a sparse matrix-vector multiply. It can be used when the matrix, A , can be represented as a multiplitting, A = ∑ A _e. In particular, it can be applied to a matrix-vector multiply arising in finite element techniques where the matrices A _e are associated with the individual element contributions to the global matrix A . The algorithm presented here uses a data structure which is based on the individual matrices A _e and can be applied both to symmetric and to non-symmetric matrices. This algorithm would be attractive for vector architecture similar to either the CYBER 205 or the CRAY and has been implemented for both regular and irregular finite element grids on the CYBER 205. Execution times and storage requirements are compared to standard sparse and band matrix-vector multiply algorithms.     

基于遗传算法的复杂平面曲线形状误差计算

提出了归一化实数值编码的遗传算法，建立了平面复杂曲线形状误差的数学模型，并运用归一化实数值编码的遗传算法进行求解。仿真实验证明。其计算精确度非常高，可以达到任意给定的精度，非常适用于三坐标测量机。     

静力荷载作用下结构参数识别及状态评估的统计分析

基于静态结构识别方法,提出了服役结构损伤探测及状态评估的概率分析方法。首先通过有限元法把结构离散成用基本参数及核心矩阵表示的分析模型,根据两种分析模型和实际结构间的误差定义,用 Gauss-Newton法推导了在不完全测量情况下两种参数识别方法。在设计的测量情况下,进一步采用 Monte Carlo 法模拟测量数据,详细分析和比较这两种算法在有测量误差时的有效性和稳定性问题。第一种算法中的敏感矩阵不受测量误差的影响而具有比较好的性能,在确定了其识别结果的概率特性后,引入假设试验,探测和评估损伤的位置及程度。数字模拟显示了这一方法的有效性和准确性。     

琼斯矩阵在分布式光纤传感器偏振态分析中的应用

针对基于Sagnac原理的分布式光纤传感器中光波偏振态在双折射影响下所带来的干涉信号"偏振诱导衰落"问题,运用琼斯矩阵分析法,建立了传输光偏振态影响系统功率传输系数的数学模型;根据仿真分析的结果,发现使用反射镜作为反射元件,只能消除光纤圆双折射的影响,而不能消除线性双折射的影响.因此,提出了使用法拉第旋转镜提高系统抗偏振衰落能力的改进方法,仿真结果表明可以很好地消除传感光纤的线性双折射和圆双折射的影响.     

Boundary element analysis of dielectric waveguides

We apply the boundary element method to the analysis of optical waveguides. After summarizing constant and linear element algorithms for both two- and three-dimensional simulations, we introduce a new recursive series procedure for constructing the diagonal matrix elements. We then demonstrate that our method can be employed to minimize the reflectivity of optical waveguide antireflection coatings with both straight and angled facets.     

Fixed-point iteration to nonlinear finite element analysis. Part II: Formulation and implementation

The finite element formulation and implementation of the Fixed-Point Iteration (FPI) to linear/nonlinear structural static or dynamic analysis are developed. The direct and tangent formulations are presented and compared with the Newton–Raphson method (NRM). ‘Modified’ FPI algorithms have also been derived. A graphical interpretation of the method is introduced and suggested to call the FPI ‘the Saw method’. Mixing both the FPI and NRM is shown to be possible and may be useful in some applications. The overall strategies, iterative algorithms, and the appropriate norm convergence criteria necessary to implant the FPI into existing finite element programs are also included in the development. The superiority of the FPI over the NRM as seen from the development and the formulation lies in three major factors. First, the existing assembly process of element matrices is eliminated completely from the nonlinear finite element analysis. Secondly, the Gauss elimination or Crout's method is also eliminated. In the finite element terminology, this means that nonlinear structural static or dynamic responses can he obtained without recourse to the inverse of the structural stiffness matrix. Thirdly, the FPI can also be applied equally to linear structural analysis. Hence, the assembly process and the programming and storage associated with it can be removed from the existing finite element programs. While the FPI can solve problems that the NRM can, it will also be able to handle some engineering problems where the latter cannot. Buckling problems and problems where the force–displacement curve changes curvature are examples where the FPI is expected to be efficient.     

Design of asynchronous phase detection algorithms optimized for wide frequency response

In many fringe pattern processing applications the local phase has to be obtained from a sinusoidal irradiance signal with unknown local frequency. This process is called asynchronous phase demodulation. Existing algorithms for asynchronous phase detection, or asynchronous algorithms, have been designed to yield no algebraic error in the recovered value of the phase for any signal frequency. However, each asynchronous algorithm has a characteristic frequency response curve. Existing asynchronous algorithms present a range of frequencies with low response, reaching zero for particular values of the signal frequency. For real noisy signals, low response implies a low signal-to-noise ratio in the recovered phase and therefore unreliable results. We present a new Fourier-based methodology for designing asynchronous algorithms with any user-defined frequency response curve and known limit of algebraic error. We show how asynchronous algorithms designed with this method can have better properties for real conditions of noise and signal frequency variation.     

Investigation of Mueller matrices of anisotropic nonhomogeneous layers in application to an optical model of the cornea

We consider a system of anisotropic layers as an optical model of the eye cornea. Effective refraction indices for normally incident light are calculated with the assumption that each layer consists of closely packed uniform cylinders (fibrils). Jones matrix formalism is used to describe light propagation through the cornea. We calculate the Jones matrices from the experimentally measured Mueller matrices. Two algorithms are used for this purpose. The experiments have shown that ~20% of cornea area studied had the structure well described by the helical model proposed.     

基于Woodbury非线性方法的迭代算法对比分析

在结构局部非线性求解过程中,刚度矩阵仅部分元素发生改变,此时切线刚度矩阵可写成初始刚度矩阵与其低秩修正矩阵和的形式,每个增量步的位移响应可用数学中快速求矩阵逆的Woodbury公式高效求解,但通常情况下迭代计算在结构非线性分析中是不可避免的,因此迭代算法的计算性能也对分析效率有重要影响。该文以基于Woodbury非线性方法为基础,分别采用Newton-Raphson （N-R）法、修正牛顿法、3阶两点法、4阶两点法及三点法求解其非线性平衡方程,并对比分析5种迭代算法的计算性能。利用算法时间复杂度理论,得到了5种迭代算法求解基于Woodbury非线性方法平衡方程的时间复杂度分析模型,定量对比了5种迭代算法的计算效率。通过2个数值算例,从收敛速度、时间复杂度和误差等方面对比了各迭代算法的计算性能,分析了各算法适用的非线性问题。最后,计算了5种算法求解基于Woodbury非线性方法平衡方程的综合性能指标。     

Numerical analysis of Lamb waves using the finite and spectral cell methods

An accurate and efficient simulation of wave propagation phenomena plays an important role in different engineering disciplines. In structural health monitoring, for example, ultrasonic guided waves are used to detect and localize damage and to assess the structural integrity of the component part under consideration. Because of the complexity of real structures, the numerical simulation of structural health monitoring systems is a computationally demanding task. Therefore, to facilitate the analysis of wave propagation phenomena, the authors propose to combine the finite cell method with the spectral element method. The ensuing novel method is referred to as the spectral cell method. Because it does not rely on body‐fitted meshes, the resulting approach eliminates all discretization difficulties encountered in conventional finite element methods. Moreover, with the aid of mass lumping, it paves the way for the use of explicit time‐integration algorithms. In the first part of the paper, we show that using a lumped mass matrix instead of the consistent one has no detrimental effect on the accuracy of the spectral element method. We introduce the spectral cell method in the second part, showing that, when applied to wave propagation analysis, the spectral cell method yields results comparable with other standard higher order finite element approaches.Copyright © 2014 John Wiley & Sons, Ltd.     

On the algebraic characterization of a Mueller matrix in polarization optics. II. Necessary and sufficient conditions for Jones-derived Mueller matrices

Abstract

We show that every Mueller matrix, that is a real 4 × 4 matrix M which transforms Stokes vectors into Stokes vectors, may be factored as M = L ₂ KL ₁ where L ₁ and L ₂ are orthochronous proper Lorentz matrices and K is a canonical Mueller matrix having only two different forms, namely a diagonal form for type-I Mueller matrices and a non-diagonal form (with only one non-zero off-diagonal element) for type-II Mueller matrices. Using the general forms of Mueller matrices so derived, we then obtain the necessary and sufficient conditions for a Mueller matrix M to be Jones derived. These conditions for Jones derivability, unlike the Cloude conditions which are expressed in terms of the eigenvalues of the Hermitian coherency matrix T associated with M, characterize a Jones-derived matrix M through the G eigenvalues and G eigenvectors of the real symmetric N matrix N = [Mtilde]GM associated with M. Appending the passivity conditions for a Mueller matrix onto these Jones-derivability conditions, we then arrive at an algebraic identification of the physically important class of passive Jones-derived Mueller matrices.     

Efficient parallel algorithms for elastic–plastic finite element analysis

This paper presents our new development of parallel finite element algorithms for elastic–plastic problems. The proposed method is based on dividing the original structure under consideration into a number of substructures which are treated as isolated finite element models via the interface conditions. Throughout the analysis, each processor stores only the information relevant to its substructure and generates the local stiffness matrix. A parallel substructure oriented preconditioned conjugate gradient method, which is combined with MR smoothing and diagonal storage scheme are employed to solve linear systems of equations. After having obtained the displacements of the problem under consideration, a substepping scheme is used to integrate elastic–plastic stress–strain relations. The procedure outlined controls the error of the computed stress by choosing each substep size automatically according to a prescribed tolerance. The combination of these algorithms shows a good speedup when increasing the number of processors and the effective solution of 3D elastic–plastic problems whose size is much too large for a single workstation becomes possible.     

Obstacle identification using the TRAC algorithm with a second-order ABC

We consider obstacle identification using wave propagation. In such problems, one wants to find the location, shape, and size of an unknown obstacle from given measurements. We propose an algorithm for the identification task based on a time-reversed absorbing condition (TRAC) technique. Here, we apply the TRAC method to time-dependent linear acoustics, although our methodology can be applied to other wave-related problems as well, such as elastodynamics. There are two main contributions of our identification algorithm. The first contribution is the development of a robust and effective method for obstacle identification. While the original paper presented criteria for accepting or rejecting regions that enclose the obstacle, we use these criteria to develop an algorithm that automatically identifies the location of the obstacle. The second contribution is the utilization of an improved absorbing boundary condition (ABC) for the identification. We use the second-order Engquist-Majda ABC, and we implement it with a finite element scheme. To our knowledge, this is the first time that the second-order Engquist-Majda ABC is employed with the finite element method, as this boundary condition does not naturally fit in finite element schemes in its original form. Numerical experiments for the algorithms are presented.     

Analytical phase-measuring technique for retardation measurements

An analytical method for phase detection in retardation measurements is proposed. The experimental setup is based on a simple linear polariscope with a λ plate. The intensity modulation at the output of the polariscope is measured when the wavelength is changed and a grid of phase-shifted intensity values is recorded. The phase difference between the components of the light propagating along the principal axes of the birefringent sample is determined with Greivenkamp's algorithm employed in phase-stepping interferometry. Error analysis for the new method is performed. Simplified algorithms for faster data analysis are proposed. The accuracy attained is comparable with the accuracy of known phase-detection methods used in retardation measurements.     

Algorithm based on rigorous coupled-wave analysis for diffractive optical element design

Diffractive optical element design is an important problem for many applications and is usually achieved by the Gerchberg-Saxton or the Yang-Gu algorithm. These algorithms are formulated on the basis of monochromatic wave propagation and the far-field assumption, because the Fourier transform is used to model the wave propagation. We propose an iterative algorithm (based on rigorous coupled-wave analysis) for the design of a diffractive optical element. Since rigorous coupled-wave analysis (instead of Fourier transformation) is used to calculate the light-field distribution behind the optical element, the diffractive optical element can thus be better designed. Simulation results are provided to verify the proposed algorithm for designing a converging lens. Compared with the well-known Gerchberg-Saxton and Yang-Gu algorithms, our method provides 7.8% and 10.8%, respectively, improvement in converging the light amplitude when a microlens is desired. In addition, the proposed algorithm provides a solution that is very close to the solution obtained by the simulated annealing method (within 1.89% error).     

A finite element solution of a tidal current problem in the seto inland sea by using the ICCG method

When the finite element method is applied to the analysis of tidal currents in an inland sea with many islands, a system of linear equations with large band and sparse coefficient matrix is solved at each time step, and therefore the finite element methods usually suffer a severe economic disadvantage for practical calculations. The method used in this paper for solving a system of linear equations with large band and sparse coefficient matrix is the incomplete Cholesky conjugate gradient (ICCG) method: The ICCG method was compared with other methods such as the Gaussian elimination method, the Gauss–Seidel method and the conjugate gradient method. This method showed significant improvement in computation time and it can overcome the disadvantage that the efficiency to solve the matrix equations which appear in the finite element analysis of tidal currents usually diminishes as the bandwidth grows. The simulation results of tidal currents in the Seto Inland Sea of Japan were compared with field data and good agreements were obtained.     

Study of validity of the single-diode model for solar cells by I–V curves parameters extraction using a simple numerical method

The I–V measurement of solar cells is one of the most employed electrical characterization techniques in the photovoltaics research field due to the valuable information one can obtain from such a curve. Parameters like Voc, Isc and the maximum power can be easily observed at a glance. Furthermore, additional information can be extracted by a more exhaustive analysis involving the equivalent electrical circuit of a solar cell which is based on ideal electrical components with a well-defined behavior. This equivalent circuit is typically assumed to be formed by a current source in parallel with a single diode and a shunt resistor, connected to a series resistor. However, this model does not take into account the separate contributions of the different carrier transport mechanisms in solar cells; for example, carrier diffusion and recombination-generation in the depletion region. Therefore, the single diode model may not be suitable in many practical cases. In this work, a simple numerical method was developed to extract the parameters for both single diode and double diode models from experimental I–V curves of solar cells. The developed numerical algorithm was applied to extract the parameters for a published benchmark solar cell which has been used for testing this kind of algorithms. The extracted parameters using our simple method are comparable with other more sophisticated and computer power demanding algorithms. Thereafter, we applied the developed algorithm to extract the single-diode parameters from simulated “experimental” I–V curves, where two carrier transport mechanisms are present, trying to understand under what conditions the single diode approximation would provide meaningful parameters for such experimental curves. It is shown that the extracted parameters can vary strongly, particularly for the dark saturation current and ideality factor, without much variation of the root mean square error between the experimental data and the model, causing these values to be unreliable and its physical interpretation misleading. We show that the same algorithm can be applied to a double diode (two exponentials) model providing physically meaningful parameters without much computing power requirements. In summary, there is no further justification for using a single diode model to interpret the experimental I–V curves of real solar cells.     

Full-field phase error compensation method based on relationship between unwrapped phase and phase error

The three-dimensional reconstruction in phase-measuring profilometry (PMP) usually involves the phase error caused by the gamma effect of the projector. In this study, the relationship between the unwrapped phase and the phase error of every pixel is analysed, and an effective full-field phase error compensation method based on this relationship is proposed for the reduction of every pixel error. In our optimized PMP system, the full-field phase error can be detected by directly fitting the unwrapped phase of the reference plane. In addition, the relationship between the unwrapped phase and the phase error can be established by creating a phase-error lookup table for the phase error compensation of every pixel. The experimental results demonstrate the effectiveness of the proposed method in practical PMP, and the measurement errors can be reduced by a factor of least 10.     

From formal solutions to computational methods avoiding passages to the limit

Before the advent of digital computers, the so-called formal solutions were the only available solutions to differential equations. Formal solutions can be closed solutions, or solutions involving infinite algorithms. The latter involve an infinite number of algebraic operations. Truncation becomes thus necessary, and the concepts of truncation error and convergence become vital. Once digital computers became available, other kinds of computational methods could be used and it became convenient to distinguish between computational methods like finite difference and finite element methods, in which numerical analysis starts before integration, and those like classical integral methods and boundary element methods, in which numerical analysis starts after integration. The classical finite difference method, in which a mesh is required, is a particular case of the generalised difference methods, characterised by a local interpolation around each node together with the collocation technique. The generalised difference method may be regarded as a modality of the meshless techniques. The finite element method differs of the finite difference method in that the approximate solution is generated respectively by variational and by collocation techniques. Hybrid and block elements are dual generalisations of the finite element method in which compatibility and equilibrium are respectively allowed within each element. Also in what concerns the methods in which numerical analysis starts after integration, bold steps have been given toward their generalisation, like those avoiding passages to the limit.