Zernike annular polynomials and atmospheric turbulence

Imaging through atmospheric turbulence by systems with annular pupils is discussed using the Zernike annular polynomials. Fourier transforms of these polynomials are derived analytically to facilitate the calculation of variance and covariance of the aberration coefficients. Zernike annular shape functions are derived and used to calculate the Strehl ratio and the residual phase structure and mutual coherence functions when a certain number of modes are corrected using, say, a deformable mirror. Special cases of long- and short-exposure images are also considered. The results for systems with a circular pupil are obtained as a special case of the annular pupil.     

Rays and fields in general astigmatic resonators

General astigmatic (GA) resonators are discussed in detail. Eigenrays, eigenmodes and eigenvalues (Gouy-factors) of this resonator are evaluated. A stability diagram for such resonators is introduced, which clearly depicts the stable and unstable regions for rays as well as for fields. Eigenrays and their stability regions are evaluated using the ABCD-law. For the beam propagation Collins' integral and the second moment method are applied. The eigenfunctions for rectangular symmetry are the generalized Hermite polynomials multiplied by the Gaussian exponential factor. It is shown that for general astigmatic resonators these polynomials are the product of normal Hermite polynomials. The generating function of the eigenfunctions depends on the special resonator. It is a useful tool for all calculations and it is determined. Furthermore it is shown that the second moment characterization of the modes is a useful and easy to handle procedure to evaluate beam width, beam divergence, radius of curvature and twist of the generalized Gauss–Hermite functions.     

Modelling non-Gaussian random fields using transformations and Hermite polynomials

In the context of modelling residual roughness on nominally flat moderately polished metal surfaces, a method is proposed for solving problems related to sample function properties and/or special points such as maxima, minima, saddle points for random fields having non-Gaussian height distributions by recasting them in terms of the corresponding problems for the much more tractable Gaussian random fields by means of transformations. Special reference is made to the expansion of the transformations in series of Hermite polynomials. While the use of Hermite polynomials in connection with transformations of random fields and the useful results they yield with regard to covariance functions are well known, this paper derives the most general explicit formula for the expectation of any product of several Hermite polynomials in correlated Gaussian arguments thereby allowing their application to the higher moments of the transformed random field, in particular, to the third moment, which may be used to measure skewness.     

Non-symmetric extension of a small flaw into a plane crack at a constant rate under polynomial-form loadings

The superposition-related problems which arise from the non-symmetric extension at a constant rate of a plane crack from a small flaw by the removal of normal and shear tractions along the crack plane is treated. It is assumed that these tractions are, or can be approximated by, polynomials in the spatial and time variables in the crack plane. Because any more general polynomial can be obtained by proper combination, attention is focused on polynomials homogeneous of degree $\text{n ? 0}$ which have n + 1 terms with arbitrary constant coefficients. Expressions for the stresses and displacements are readily obtained as single integrals of analytic functions and observations concerning the dynamic intensity factors at the crack edges are made. Certain previously obtained results follow as special cases of the present work.     

矩形孔径上折射率均匀性数据拟合研究

泽尼克圆多项式在圆形光瞳的正交性和能够代表经典像差而被广泛应用到波前分析中,用泽尼克圆多项式作为矩形光瞳基底函数,通过推导得到在矩形光瞳上正交的多项式.这个在矩形光瞳上正交的多项式不仅是唯一的,而且也能够表示经典像差,就像泽尼克圆多项式在表示圆形光瞳时具有这样的特性一样.矩形光瞳上正交多项式像泽尼克圆多项式一样即可以用...     

Polynomial stress functions of anisotropic plane problems and their applications in hybrid finite elements

In this paper, systematic approaches to determine the polynomial stress functions for anisotropic plane problems are presented based on the Lekhnitskii’s theory of anisotropic elasticity. It is demonstrated that, for plane problems, there are at most four independent polynomials for arbitrary n-th order homogeneous polynomial stress functions: three independent polynomials for n equal to two and four for n greater than or equal to three. General expressions for such polynomial stress functions are derived in explicit forms. Unlike the isotropic case, the polynomials for anisotropic problems are functions of material constants, because the elastic constants cannot be eliminated in the governing equation for general anisotropic cases. The polynomials can be used as analytical trial functions to develop the new 8-node hybrid element (ATF-Q8) for anisotropic problems. This ATF-Q8 element demonstrated excellent performance in comparison with traditional numerical methods through several testing examples.     

Orthonormal polynomials in wavefront analysis: analytical solution

Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. In recent papers, we derived closed-form polynomials that are orthonormal over a hexagonal pupil, such as the hexagonal segments of a large mirror. We extend our work to elliptical, rectangular, and square pupils. Using the circle polynomials as the basis functions for their orthogonalization over such pupils, we derive closed-form polynomials that are orthonormal over them. These polynomials are unique in that they are not only orthogonal across such pupils, but also represent balanced classical aberrations, just as the Zernike circle polynomials are unique in these respects for circular pupils. The polynomials are given in terms of the circle polynomials as well as in polar and Cartesian coordinates. Relationships between the orthonormal coefficients and the corresponding Zernike coefficients for a given pupil are also obtained. The orthonormal polynomials for a one-dimensional slit pupil are obtained as a limiting case of a rectangular pupil.     

On the increasingly flat radial basis function and optimal shape parameter for the solution of elliptic PDEs

For the interpolation of continuous functions and the solution of partial differential equation (PDE) by radial basis function (RBF) collocation, it has been observed that solution becomes increasingly more accurate as the shape of the RBF is flattened by the adjustment of a shape parameter. In the case of interpolation of continuous functions, it has been proven that in the limit of increasingly flat RBF, the interpolant reduces to Lagrangian polynomials. Does this limiting behavior implies that RBFs can perform no better than Lagrangian polynomials in the interpolation of a function, as well as in the solution of PDE? In this paper, arbitrary precision computation is used to test these and other conjectures. It is found that RBF in fact performs better than polynomials, as the optimal shape parameter exists not in the limit, but at a finite value.     

Higher-order finite element modelling of laminated composite plates

P-version finite elements based on higher-order theory are developed for the two-dimensional modelling of general bending and cylindrical bending of thin-to-thick laminated composite plates. In the case of general laminated plate elements, three displacement fields are used. In the special case of cylindrically bent laminated plate elements, two displacement fields are needed. In each case the displacement is expressed as the product of two functions—one in terms of out-of-plane co-ordinates alone and the other in terms of in-plane co-ordinates. The shape functions used to build the displacement fields are based on integral of Legendre' polynomials. The quality and performance of the elements are evaluated in terms of convergence characteristics of displacements and stresses. The predicted response quantities are compared with those available in the published literature based on analytical as well as conventional finite element models.     

Integration of triangular finite elements containing corrective rational functions

Rational corrective functions are often supplemented to complete polynomials in order to create conforming triangular finite elements for plate bending. Although such elements containing complete polynomials of degree 3 or 4 have long been known, conforming triangular elements for plate bending, which contain complete polynomials of arbitrary degree p ? 5 and new corrective rational functions, have been given only recently. Integration of rational functions usually requires numerical quadrature. It is shown here that these new corrective rational functions fall into two classes. For those in one class, explicit closed form integration formulas are available. For those in the other class, integration can be performed very efficiently by summing series whose general term is O(k^d) for large k, where d ? 5. These formulas also apply to the older third and fourth order elements.     

Mathematical approximation techniques in biophysical models

The mathematical characterization of biophysical problems often leads to certain transcendental functions which must be evaluated to fully appreciate the significance of the mathematical model. In illustration, many mathematical models concerned with wave propagation and transmission of fluids through elastic tubes require treatment of the various kinds of Bessel functions and their ratios. In my recent work (The Special Functions and Their Approximations, Volumes 1 and 2, Academic Press, 1969) polynomial, rational and infinite series expansions in series of Chebyshev polynomials are developed for a wide class of transcendental functions which include Bessel functions as special cases. In numerous instances, tables of coefficients are presented. In the present paper, we show how these ideas can be exploited to yield economic methods for the evaluation of Bessel functions arising in the problems cited. As the approximations can be given in closed form, they can be easily applied to further simplify available closed form analyses.     

Parametric models of nonstationary Gaussian processes

Three parametric representations are developed for approximating a general nonstationary Gaussian process X(t). The representations: (1) are based on the Bernstein and other interpolation polynomials, spline functions, and an extension of a sampling theorem for stationary processes; (2) consist of finite sums of specified deterministic functions with random amplitudes depending on X(t); and (3) converge to X(t) as the number of these functions increases. However, their convergence rates differ. Numerical results for a nonstationary Ornstein-Uhlenbeck process show that the interpolation polynomials have the slowest rate of convergence. The parametric representations based on spline functions and the extended sampling theorem have similar convergence rates. The paper also presents methods for generating realizations of X(t) based on the three parametric models of this process.     

Optimum interpolation functions for boundary element method

In the Boundary Element Method (BEM) the density functions are approximated by interpolation functions which are chosen to satisfy appropriate continuity requirements. The error of approximation inside an element depends upon the location of the collocation points that are used in constructing the interpolation functions. The location of collocation points also affects the nodal values of the density function and, hence, the total error in the analysis if boundary conditions are satisfied in a collocation sense. In this paper, we minimize the error inside the element using the L₁ norm to obtain the optimum location of collocation points. Results show that irrespective of the continuity requirement at the element end, the location of collocation points computed by the algorithm presented in this paper results in an error that is less than the error corresponding to uniformly spaced collocation points. Results for optimum location of collocation points and the average error are presented for Lagrange polynomials up to order fifteen and for Hermite polynomials that ensure continuity up to the seventh order of derivative at the element end. The information of the optimum location of interpolation points for Lagrange and Hermite polynomials should be useful to other researchers in BEM who could incorporate it into their current programs without making significant changes that would be needed for incorporating the algorithm. The algorithm presented is independent of the BEM application in two-dimensions, provided that the density functions are approximated by polynomials and is applicable to direct and indirect formulations. Two numerical examples show the application of the algorithm to an elastostatic problem in which one boundary is represented by integrals of the Direct BEM while the other boundary by the Indirect BEM and a fracture mechanics problem by Direct method in which the crack is represented by displacement discontinuity density function.     

Vibrations of skewed cantilevered triangular,trapezoidal and parallelogram Mindlin plates with considering corner stress singularities

Based on the Mindlin shear deformation plate theory, a method is presented for determining natural frequencies of skewed cantilevered triangular, trapezoidal and parallelogram plates using the Ritz method, considering the effects of stress singularities at the clamped re‐entrant corner. The admissible displacement functions include polynomials and corner functions. The admissible polynomials form a mathematically complete set and guarantee the solution convergent to the exact frequencies when sufficient terms are used. The corner functions properly account for the singularities of moments and shear forces at the re‐entrant corner and accelerate the convergence of the solution. Detailed convergence studies are carried out for plates of various shapes to elucidate the positive effects of corner functions on the accuracy of the solution. The results obtained herein are compared with those obtained by other investigators to demonstrate the validity and accuracy of the solution. Copyright © 2005 John Wiley & Sons, Ltd.     

Uncertainty reduction in residual stress measurements by an optimised inverse solution using nonconsecutive polynomials

Many destructive methods for measuring residual stresses such as the slitting method require an inverse analysis to solve the problem. The accuracy of the result as well as an uncertainty component (the model uncertainty) depends on the basis functions used in the inverse solution. The use of a series expansion as the basis functions for the inverse solution was analysed in a previous work for the particular case where functions orders grew consecutively. The present work presents a new estimation of the model uncertainty and a new improved methodology to select the final basis functions for the case where the basis is composed of polynomials. Including nonconsecutive polynomial orders in the basis generates a larger space of possible solutions to be evaluated and allows the possibility to include higher-order polynomials. The paper includes a comparison with two other inverse analyses methodologies applied to synthetically generated data. With the new methodology, the final error is reduced and the uncertainty estimation improved.     

A generalization of prolate spheroidal functions with more uniform resolution to the triangle

Prolate spheroidal functions constitute a one-parameter (α) family of orthogonal functions in the interval. For α = 0, they are the Legendre polynomials. For larger α, the prolate spheroidal functions oscillate more uniformly than the Legendre polynomials, and provide more uniform resolution in the interval. The prolate spheroidal functions can be obtained by adding a zeroth-order term to the Sturm–Liouville equation for the Legendre polynomials. Here, the Sturm–Liouville equation for orthogonal polynomials in the triangle is modified in a similar fashion. The modification maintains the self-adjointness and symmetry properties of the original Sturm–Liouville equation, so that the new eigenfunctions are orthogonal and give spectrally accurate approximations of smooth functions with arbitrary boundary conditions in the triangle. The properties of the new eigenfunctions mimic those in the interval. For larger α, the new eigenfunctions provide more uniform resolution in the triangle.     

Uniform rational approximation of functions of several variables

We present a program which has given excellent results for uniform approximation of functions by polynomials, rational functions, generalized polynomials, and generalized rational functions. The algorithm is described in detail and several examples are discussed. The approximation is done over a finite point set, which is commonly a set of real numbers or points in the plane (in the latter case we are doing what is often known as surface fitting). Input to and output from the program is in tabular form. The method used is a linear programming approach known as the differential correction algorithm, which has been shown by several authors to always converge in theory (quadratically in some situations). In practice, we have obtained convergence in nearly every case, and quadratic convergence in most cases. The program can also be used for simultaneous approximation of several functions.     

Solution of the two-dimensional wave equation by using wave polynomials

The paper demonstrates a specific power-series-expansion technique to solve approximately the two-dimensional wave equation. As solving functions (Trefftz functions) so-called wave polynomials are used. The presented method is useful for a finite body of certain shape geometry. Recurrent formulas for the wave polynomials and their derivatives are obtained in the Cartesian and polar coordinate system. The accuracy of the method is discussed and some examples are shown.     

A complex potential-variational formulation for thermo-mechanical buckling analysis of flat laminates with an elliptic cutout

A semi-analytical method is developed for pre-buckling and buckling analyses of thin, symmetrically laminated composite panels with an elliptical cutout at an arbitrary location and orientation under general thermo-mechanical loading conditions. Both the pre-buckling and buckling analyses are based on the principle of stationary potential energy utilizing complex potential functions and complete polynomials. The complex potential functions capture the steep stress gradients and local deformations around the cutout, and the “complete” polynomials improve the global buckling response of the laminate. The complex potential functions in the pre-buckling state automatically satisfy the in-plane equilibrium equations, thus reducing the first variation of the total potential energy in terms of line integrals only. Because the complex potential functions for out-of-plane displacements are augmented by the “complete” polynomials, the area integral terms in the second variation of the total potential energy, referred to as the Treftz criterion, are retained in the buckling analysis. The kinematic boundary conditions are idealized by employing extensional and rotational springs (elastic restraints) with appropriate stiffness values. Based on the numerous validation problems, this analysis is proven credible for predicting the buckling load of rectangular and non-rectangular panels with a cutout.