Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates

A method other than the extended-boundary-condition method (EBCM) to compute the T matrix for electromagnetic scattering is presented. The separation-of-variables method (SVM) is used to solve the electromagnetic scattering problem for a spheroidal particle and to derive its T matrix in spheroidal coordinates. A transformation is developed for transforming the T matrix in spheroidal coordinates into the corresponding T matrix in spherical coordinates. The T matrix so obtained can be used for analytical calculation of the optical properties of ensembles of randomly oriented spheroids of arbitrary shape by use of an existing method to average over orientational angles. The optical properties obtained with the SVM and the EBCM are compared for different test cases. For mildly aspherical particles the two methods yield indistinguishable results. Small differences appear for highly aspherical particles. The new approach can be used to compute optical properties for arbitrary values of the aspect ratio. To test the accuracy of the expansion coefficients of the spheroidal functions for arbitrary arguments, a new testing method based on the completeness relation of the spheroidal functions is developed.     

纤维复合材料的热膨胀系数

提出了一种利用压电光声技术测量材料热膨胀系数的实验方法,并测试了单向复合材料C/C、C/Al的横向、纵向的热膨胀系数。根据已有的理论计算方法与实验结果对该方法的测试结果进行验证,证明了该检测方法的可靠性,进而又测量了C/C、C/Al材料在任一方向上的热膨胀系数。这种方法克服了理论计算过程复杂以及常规手段无法测量任一方向上热膨胀系数的缺陷。     

A direct determination of weight functions for arbitrary three-dimensional cracks – a combined analytical and numerical approach

A combined analytical and numerical method is proposed for the determination of the weight functions of stress intensity factors of cracks in an arbitrary three-dimensional elastic body. Having defined the weight functions for a given geometry of a structure, the stress intensity factors for arbitrary loading conditions can be obtained by a simple inner product of the weight function and a traction vector. Traditionally weight functions are defined in the two ways; the one is defined by the hyper-singular term of the eigen-function expansion of the displacement field of a cracked body, and the other is defined by the variation of displacement field with respect to a virtual extension of a crack. In the present paper, the weight functions for stress intensity factors are defined by applying the Maxwell-Betti's reciprocal theorem to an original problem and the auxiliary problems subjected to three kinds of force-couples acting on the crack surfaces near the limiting periphery of an arbitrary three-dimensional crack. In the present formulation, weight functions can be calculated by using a general-purpose finite element code combined with analytical expressions near the condensation point, where hyper-singularities exist. The validity of the method is confirmed by two- and three-dimensional illustrative examples.     

Computation of the External Magnetic Field, Near-Field or Far-Field, From a Circular Cylindrical Magnetic Source Using Toroidal Functions

A method is developed for computing the magnetic field from a circular or noncircular cylindrical magnetic source. A Fourier series expansion is introduced which yields an alternative to the more familiar spherical harmonic solution, Elliptic integral solution, or Bessel function solution. This alternate formulation coupled with a method called charge simulation allows one to compute the external magnetic field from an arbitrary magnetic source in terms of a toroidal expansion which is valid on any finite hypothetical external observation cylinder. In other words, the magnetic scalar potential or the magnetic field intensity is computed on a exterior cylinder which encloses the magnetic source. Also, one can compute an equivalent multipole distribution of the real magnetic source valid for points close to the circular cylindrical boundary where the more familiar spherical multipole distribution is not valid. This method can also be used to accurately compute the far field where a finite-element formulation is known to be inaccurate     

Determination of the orientation distribution function from arbitrary pole figure regions

A full pole figure is required to calculate the series expansion coefficients of the pole figure, by integration. The orientation distribution function (ODF) is then calculated by solving a system of linear equations relating the coefficients of the pole figures and those of the ODF. Using a method proposed by Bunge which does not require integration at all, but assuming a fit between the theoretical ODF and the isolated experimental points, we calculated the ODF from various arbitrary chosen regions of the pole figures. This technique was applied to analyse the experimental data obtained from the measurement of three, two and one pole figures. Various tests described in this paper allow us to suggest an empirical rule indicating that the number of experimental data points should be three times higher than the number of required series expansion coefficients of the ODF.     

A fast algorithm for expansion over spherical harmonics

Algorithms for expansion over spherical harmonics are often used in electrostatic field calculation, calculation of the density functions in quantum chemistry and calculation of molecular surfaces. It usually includes expansion over spherical harmonics of degrees to several dozens. The usual method is to use an integration method over some grid on the unit sphere and in fact is a multiplication of the matrix of values of spherical harmonics in the grid points by a vector of values of the expanding function in the set of points. This algorithm executes O(NL²) operations whereN is the number of the grid points andL is the maximal degree of the spherical harmonics involved. We provide an algorithm of complexity O(NLlog² L) for multiplication of the matrix of values of spherical harmonics in points of an arbitrary grid on the unit sphere. The algorithm is based on interrelation between spherical harmonics and Legendre polynomials and on a fast algorithm for expansion over Legendre polynomials.     

Mixed-mode two-dimensional crack problem by fractal two level finite element method

Fractal two level finite element method (F2LFEM) has been extended to calculate the mixed mode stress intensity factor in a two-displacement cracked body. The complete eigenfunction expansion of displacement by Williams is employed for the global interpolation function, the factors K_I and K_II can be easily computed for any arbitrary loading on any boundary. Results are obtained for some slant crack problems in finite sheets and are compared with known results where available.     

Nonlinear dynamics of rectangular plates: investigation of modal interaction in free and forced vibrations

Nonlinear vibrations of thin rectangular plates are considered, using the von kármán equations in order to take into account the effect of geometric nonlinearities. Solutions are derived through an expansion over the linear eigenmodes of the system for both the transverse displacement and the Airy stress function, resulting in a series of coupled oscillators with cubic nonlinearities, where the coupling coefficients are functions of the linear eigenmodes. A general strategy for the calculation of these coefficients is outlined, and the particular case of a simply supported plate with movable edges is thoroughly investigated. To this extent, a numerical method based on a new series expansion is derived to compute the Airy stress function modes, for which an analytical solution is not available. It is shown that this strategy allows the calculation of the nonlinear coupling coefficients with arbitrary precision, and several numerical examples are provided. Symmetry properties are derived to speed up the calculation process and to allow a substantial reduction in memory requirements. Finally, analysis by continuation allows an investigation of the nonlinear dynamics of the first two modes both in the free and forced regimes. Modal interactions through internal resonances are highlighted, and their activation in the forced case is discussed, allowing to compare the nonlinear normal modes (NNMs) of the undamped system with the observable periodic orbits of the forced and damped structure.     

Approximations of polydispersed extinction

A method to evaluate the polydispersed extinction efficiency rapidly is presented. The method can be used for any shape for which a monodisperse code or expression is known and any polydispersion for which a second moment inverse can be computed. Integration can be performed over any interval of the distribution function. Additionally, and if required, arbitrary accuracy can be obtained. This approach is applied to spheres and randomly oriented spheroids with nth-order log-normal and modified gamma particle distributions.     

Convergence study of the truncated Karhunen–Loeve expansion for simulation of stochastic processes

A random process can be represented as a series expansion involving a complete set of deterministic functions with corresponding random coefficients. Karhunen–Loeve (K–L) series expansion is based on the eigen‐decomposition of the covariance function. Its applicability as a simulation tool for both stationary and non‐stationary Gaussian random processes is examined numerically in this paper. The study is based on five common covariance models. The convergence and accuracy of the K–L expansion are investigated by comparing the second‐order statistics of the simulated random process with that of the target process. It is shown that the factors affecting convergence are: (a) ratio of the length of the process over correlation parameter, (b) form of the covariance function, and (c) method of solving for the eigen‐solutions of the covariance function (namely, analytical or numerical). Comparison with the established and commonly used spectral representation method is made. K–L expansion has an edge over the spectral method for highly correlated processes. For long stationary processes, the spectral method is generally more efficient as the K–L expansion method requires substantial computational effort to solve the integral equation. The main advantage of the K–L expansion method is that it can be easily generalized to simulate non‐stationary processes with little additional effort. Copyright © 2001 John Wiley & Sons, Ltd.     

正交异性矩形薄板的稳定性分析

本文求得了正交异性矩形薄板屈曲位移函数微分方程的一般解。可用以求解任意边界矩形板的稳定问题。以两邻边简支，另两边自由的正方形板为例进行了计算。     

Fundamental solutions for half plane with an oblique edge crack

An elastic half plane with an oblique edge crack is considered in this paper. A pair of concentrated forces or point dislocations is assumed to act at an arbitrary point in the half plane. The half plane with an edge crack is first mapped into a unit circle by a rational mapping function so that the following analysis can be carried out on the mapped plane analytically. Then the complex stress functions are derived by separating the whole problem into two parts; one is the principal part corresponding to the infinite plane acted on by concentrated forces or dislocations, the other is the holomorphic part, which can be determined by making use of the property of regularity of complex stress functions. The stress intensity factors of the crack can be calculated with different inclined angles of the crack, and the displacement and stress components at an arbitrary position in the half plane can be expressed explicitly.     

A Comparison of Stability and Bifurcation Criteria for a Compressible Elastic Cube

A version of Rivlin’s cube problem is considered for compressible materials. The cube is stretched along one axis by a fixed amount and then subjected to equal tensile loads along the other two axes. A number of general results are found. Because of the homogeneous trivial and non-trivial deformations exact bifurcation results can be found and an exact stability analysis through the second variation of the energy can be performed. This problem is then used to compare results obtained using more general methods. Firstly, results are obtained for a more general bifurcation analysis. Secondly, the exact stability results are compared with stability results obtained via a new method that is applicable to inhomogeneous problems. This new stability method allows a full nonlinear stability analysis of inhomogeneous deformations of arbitrary, compressible or incompressible, hyperelastic materials. The second variation condition expressed as an integral involving two arbitrary perturbations is replaced with an equivalent nonlinear third order system of ordinary differential equations. The positive definiteness condition is thereby reduced to the simple numerical evaluation of zeros of a well behaved function.     

连续小波变换快速带通滤波实现算法的研究

针对连续小波变换(Continuous wavelet transform,CWT)计算量较大的问题,提出了一种利用带通滤波实现CWT的快速算法.根据CWT的定义,可将某一尺度下的小波函数看作一带通滤波器的传递函数,于是对信号和小波函数分别进行采样后,再利用快速傅里叶变换实现两个序列的线性卷积,便能求出相应尺度下的小波系数.推导了任意尺度对应的频率与尺度、小波中心频率、信号采样频率、小波函数采样频率之间的关系式,并分析了算法的计算量.仿真和应用结果表明,该法速度很快,精度较高,并且尺度可以任意取值,非常适合于工程应用.     

A semi-analytical integration method for the numerical simulation of nonlinear visco-elasto-plastic materials

A semi-analytic integration method is proposed, which can be used in numerical simulation of the mechanical behavior of nonlinear viscoelastic and viscoplastic materials with arbitrary stress nonlinearity. The method is based upon the formalism of Prony series expansion of the creep response function and accepts arbitrary stress protocols as input data. An iterative inversion technique is presented, which allows for application of the method in routines that provide strain and require stress as output. The advantage with respect to standard numerical integration methods such as the Runge-Kutta method is that it remains numerically stable even for integration over very long time steps during which strain may change considerably due to creep or recovery effects. The method is particularly suited for materials, whose viscoelastic and viscoplastic processes cover a very wide range of retardation times. In the case of simulation protocols with phases of slowly varying stress, computation time is significantly reduced compared to the standard integration methods of commercial finite element codes. An example is given that shows how the method can be used in three dimensional (3D) constitutive equations. Implemented into a Finite Element (FE) code, the method significantly improves convergence of the implicit time integration, allowing longer time increments and reducing drastically computing time. This is shown in the case of a single element exposed to a creep and recovery cycle. Some simulations of non-homogeneous boundary value problems are shown in order to illustrate the applicability of the method in 3D FE modeling.     

Green's functions for composite media

A method is given on construction of Green's functions for finding field excited by a point (or line) source in a layered composite medium. The concept of Green's function for a composite medium as a whole is introduced in a rigorous manner by showing that it possesses all properties of the Green's function for an uniform medium. The appropriate Green's function for the whole composite medium is constructed through two solutions of the corresponding homogeneous equation over the whole multiregion. The method enables one to write down, directly, the explicit expression for the field in an arbitrary region produced by a point (or line) source located in another arbitrary region without resort to images, Fourier-Bessel integrals, integral transforms of quasi-orthogonal functions. It is an extension of the existing method, which constructs the Green's functions for uniform media through the generalized Fourier expansions and two solutions of the corresponding homogeneous equations, to cases heretofore beyond its scope.     

Imaging via the van cittert zernike theorem using triple-correlations

Abstract

The van Cittert Zernike theorem relates the intensity distribution over an incoherent object to its spatial correlation. Thus if one can measure the spatial correlation function then it is possible to determine the intensity distribution by solving a Fredholm integral equation of the first kind. Since we are operating in the optical regime we can only measure the modulus of the correlation function. A third-order correlation technique, the Karhunen-Loevé expansion, is developed for determining the phase of the correlation function using singular value decomposition. Note that two inversion problems must be solved in order to determine the object intensity: (1) phase of the correlation function from its measured modulus via triple correlations, (2) object intensity from the complex-valued spatial correlation via constrained minimization. Representative numerical results are discussed.     

The effects of displacement discontinuity derivatives on wave propagation—I: Three-dimensional elastic solid

The fully dynamic, 3-dimensional analysis of largely arbitrary normal or tangential displacement discontinuities prescribed over arbitrary regions of a plane in an unbounded elastic solid is presented. Instead of invoking the dynamic Betti-Rayleigh theorem, the analysis employs two approaches which also rely on Green's function methods. The expressions generated for the displacement indicate the dependence of important solution characteristics on derivatives of the prescribed discontinuities, especially their Laplacian. Numerical results are presented for the problem of a non-uniformly moving edge-screw dislocation and a dislocation over a non-uniformly expanding rectangular area.     

Generalized theory of resonance excitation by sound scattering from an elastic spherical shell in a nonviscous fluid

This work presents the general theory of resonance scattering (GTRS) by an elastic spherical shell immersed in a nonviscous fluid and placed arbitrarily in an acoustic beam. The GTRS formulation is valid for a spherical shell of any size and material regardless of its location relative to the incident beam. It is shown here that the scattering coefficients derived for a spherical shell immersed in water and placed in an arbitrary beam equal those obtained for plane wave incidence. Numerical examples for an elastic shell placed in the field of acoustical Bessel beams of different types, namely, a zero-order Bessel beam and first-order Bessel vortex and trigonometric (nonvortex) beams are provided. The scattered pressure is expressed using a generalized partial-wave series expansion involving the beam-shape coefficients (BSCs), the scattering coefficients of the spherical shell, and the half-cone angle of the beam. The BSCs are evaluated using the numerical discrete spherical harmonics transform (DSHT). The far-field acoustic resonance scattering directivity diagrams are calculated for an albuminoidal shell immersed in water and filled with perfluoropropane gas, by subtracting an appropriate background from the total far-field form function. The properties related to the arbitrary scattering are analyzed and discussed. The results are of particular importance in acoustical scattering applications involving imaging and beam-forming for transducer design. Moreover, the GTRS method can be applied to investigate the scattering of any beam of arbitrary shape that satisfies the source-free Helmholtz equation, and the method can be readily adapted to viscoelastic spherical shells or spheres.