Orthogonal Polynomial Series Research Articles

The validity range of CPT and Mindlin plate theory in comparison with 3-D vibrational analysis of circular plates on the elastic foundation

The validity and the range of applicability of the classical plate theory (CPT) and the first-order shear deformation plate theory, also called Mindlin plate theory (MPT), in comparison with three-dimensional (3-D) p-Ritz solution are presented for freely vibrating circular plates on the elastic foundation with different boundary conditions. In order to achieve this purpose, a study of the 3-D elasticity solution is carried out to determine the free vibration frequencies of clamped, simply supported and free circular plates resting on an elastic foundation. The Pasternak model with adding a shear layer to the Winkler model is used for describing the elastic foundation. In addition to being employed the p-Ritz algorithm, the analysis is based on the linear, small strain and 3-D elasticity theory. In this analysis method, a set of orthogonal polynomial series in a cylindrical polar coordinate system is used to arrive eigenvalue equation yielding the natural frequencies for the circular plates. The accuracy of these results is verified by appropriate convergence studies and checked with the available literature and the MPT. Furthermore, the effect of the foundation stiffness parameters, thickness-radius ratio, and different boundary conditions on the ill-conditioning of the mass matrix as well as on the vibration behavior of the circular plates is investigated. Afterwards, the validity and the range of applicability of the results obtained on the basis of the CPT and MPT for a thin and moderately thick circular plate with different values of the foundation stiffness parameters are graphically presented through comparing them with those obtained by the present 3-D p-Ritz solution. Finally, the phenomenon of mode shape switching is investigated in graphical forms for a wide range of the Winkler foundation stiffness parameters.

Circumferential wave in magneto-electro-elastic functionally graded cylindrical curved plates

Piezoelectric-piezomagnetic functionally graded materials (FGM), with a gradual change of the mechanical and electromagnetic properties, have greatly applying promises. Based on Legendre orthogonal polynomial series expansion approach, a dynamic solution is presented for the propagation of circumferential harmonic waves in piezoelectric-piezomagnetic FGM cylindrical curved plates. The materials properties are assumed to vary in the direction of the thickness according to a known variation law. The dispersion curves of the piezoelectric-piezomagnetic FGM cylindrical curved plate and the corresponding non-piezoelectric and non-piezomagnetic cylindrical curved plates are calculated to show the influences of the piezoelectricity and piezomagnetism. Electric potential and magnetic potential distributions are also obtained to illustrate the different influences of the piezoelectricity and piezomagnetism. Finally, a cylindrical curved plate at a different ratio of radius to thickness is calculated to show the influence of the ratio on the piezoelectric effect and piezomagnetic effect.

A novel method to construct stationary solutions of the Vlasov-Maxwell system: The relativistic case

A method to derive stationary solutions of the relativistic Vlasov-Maxwell system is explored. In the nonrelativistic case, a method using the Hermite polynomial series to describe the deviation from the Maxwell-Boltzmann distribution is found to be successful in deriving a few stationary solutions, including a two-dimensional one. Instead of the Hermite polynomial series, two special orthogonal polynomial series, which are appropriate to expand the deviation from the Maxwell-Jüttner distribution, are introduced in this paper. By applying this method, a new two-dimensional equilibrium is derived, which may provide an initial setup for investigations of three-dimensional relativistic collisionless reconnection of magnetic fields.

Fourier–Padé Approximants for Nikishin Systems

We study type I Fourier–Padé approximation for certain systems of functions formed by the Cauchy transform of finite Borel measures supported on bounded intervals of the real line. This construction is similar to type I Hermite–Padé approximation. Instead of power series expansions of the functions in the system, we take their development in a series of orthogonal polynomials. We give the exact rate of convergence of the corresponding approximants. The answer is expressed in terms of the extremal solution of an associated vector-valued equilibrium problem for the logarithmic potential.

Transform Methods of Computerized Tomography in Studying Electronic Structure and Fermi Surfaces of Solids

We present mathematical methods of computerized tomography, based on an analytical inversion of the Radon transform either in terms of the Fourier transforms or series of orthogonal polynomials. These techniques, so-called transform methods, are discussed for reconstructing electronic densities from both line and plane projections measured either in the two-dimensional angular correlations of annihilation radiation or in onedimensional angular correlations of annihilation radiation and the Compton scattering experiments. This paper is devoted to review all of the papers where such techniques were applied for studying electronic momentum densities and the Fermi surface of solids by angular correlations of annihilation radiation and the Compton scattering experiments.

Wave propagation in non-homogeneous magneto-electro-elastic plates

A dynamic solution is presented for the propagation of harmonic waves in imhomogeneous (functionally graded) magneto-electro-elastic plates composed of piezoelectric BaTiO 3 and magnetostrictive CoFe 2O 4. The materials properties are assumed to vary in the direction of the thickness according to a known variation law. The Legendre orthogonal polynomial series expansion approach is employed to determine the wave propagating characteristics in the plates. The dispersion curves of the imhomogeneous piezoelectric–piezomagnetic plate and the corresponding non-piezoelectric, non-piezomagnetic plates are calculated to show the influences of the piezoelectricity and piezomagnetism on the dispersion curves. They are compared with the dispersion curves of the plates with the different magnetic constants to illustrate the influential factors of the piezoelectric and piezomagnetic effect for the wave propagation in a magneto-electro-elastic plate. Electric potential and magnetic potential distributions at different wavenumbers are also obtained to illustrate the different influences of the piezoelectricity and piezomagnetism.

Wave propagation in non-homogeneous magneto-electro-elastic hollow cylinders

A dynamic solution is presented for the propagation of harmonic waves in imhomogeneous (functionally graded) magneto-electro-elastic hollow cylinders composed of piezoelectric BaTiO 3 and magnetostrictive CoFe 2O 4. The materials properties are assumed to vary in the direction of the thickness according to a known variation law. The Legendre orthogonal polynomial series expansion approach is employed to determine the wave propagating characteristics in the hollow cylinders. The dispersion curves of the imhomogeneous piezoelectric–piezomagnetic hollow cylinder and the corresponding non-piezoelectric and non-piezomagnetic hollow cylinders are calculated to show the influence of the piezoelectricity and piezomagnetism. Electric potential and magnetic potential distributions are obtained to illustrate the different influences of the piezoelectricity and piezomagnetism and the different influences of the piezoelectric effect and piezomagnetic effect on longitudinal modes and torsional modes. For the radial polarizing piezoelectric–piezomagnetic hollow cylinder, the piezoelectric effect and piezomagnetic effect take mostly on the longitudinal mode. Finally, a hollow cylinder at different ratio of radius to thickness is calculated to show the influence of the ratio on the piezoelectric effect and piezomagnetic effect.

Circumferential wave in functionally graded piezoelectric cylindrical curved plates

Functionally graded piezoelectric material (FGPM), with a gradual change of the mechanical and electric properties, has large fields of application. Based on linear three-dimensional piezoelasticity, a Legendre orthogonal polynomial series expansion approach is used for determining the characteristics of circumferential waves in FGPM cylindrical curved plates. Our results from an FGPM cylindrical curved plate with a large ratio of radius to thickness are compared with an FGPM flat plate published earlier to confirm the accuracy and range of applicability of the computational program. Circumferential wave dispersion curves for FGPM and the corresponding FGM cylindrical curved plates are calculated and the effect of piezoelectricity is shown. The influence of the ratio of radius to thickness on the piezoelectric effect and electric potential distribution is discussed. The ratio has a significant influence on the electric potential distribution. Finally, the influence of the polarizing direction on the piezoelectric effect is illustrated. The phenomenon that piezoelectricity weakens the guided wave dispersion in the FGPM cylindrical plate is observed.

3-D free vibration analysis of annular plates on Pasternak elastic foundation via p-Ritz method

In this study, a three-dimensional (3-D) free vibration analysis of thick annular plates resting on elastic foundation with different combinations of free, soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate is presented on the basis of the polynomials-Ritz method. The elastic foundation is considered as a Pasternak model with adding a shear layer to the Winkler model. The analysis procedure is based on the linear, small strain, and 3-D elasticity theory. In this analysis method, a set of orthogonal polynomial series in cylindrical polar coordinate is used to extract eigenvalue equation yielding the natural frequencies and mode shapes for the annular plates. The accuracy of these results is verified by appropriate convergence studies and checked with the available literature, finite element method (FEM) analysis and the Mindlin theory. Furthermore, the effect of the foundation stiffness parameters, thickness–radius ratio, inner–outer radius ratio and different combinations of boundary conditions on the ill-conditioning of the mass matrix as well as on the vibration behavior of the annular plates is investigated. Finally, the validity and the range of applicability of the results obtained on the basis of the Mindlin and classical plate theories for a thin and moderately thick annular plate with different values of the Winkler foundation stiffness are graphically presented through comparing them with those obtained by the present 3-D p-Ritz solution.

Wave propagation in functionally graded piezoelectric spherically curved plates

AbstractBased on linear three‐dimensional piezoelasticity, a Legendre orthogonal polynomial series expansion approach is used for determining the characteristics of guided waves in continuous functionally graded piezoelectric materials (FGPM) as spherically curved plates. The displacement components and electric potential, expanded in a series of Legendre polynomials, are introduced into the governing equations along with position‐dependent material constants so that the solution of the wave equation is reduced to an eigenvalue problem. Our results from a homogeneous anisotropic spherically curved plate are compared with those published earlier to confirm the accuracy and range of applicability of this polynomial approach. Guided‐wave dispersion curves for FGPM and the corresponding FGM spherically curved plates are calculated and the effect of piezoelectricity is shown. The mechanical displacement distribution and electric potential distribution are also illustrated. Finally, the influence of the ratio of radius to thickness on the dispersion curves is discussed. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Envelope Probability Density Functions for Fading Model in Wireless Communications

An analytical model that describes small-scale local area fading experienced by narrowband radio receivers consisting of N dominant specular waves and an arbitrary number of additional weaker waves lumped into a Gaussian distributed diffuse component was recently proposed. The exact envelope pdf of this model is in a definite integral form involving a product of N zeroth-order Bessel functions. A closed form formula for this envelope pdf is known only for N = 2. For analytic studies, a closed-form expression for this pdf for an arbitrary N = 3 is desirable. In this paper, we derived a new expression for this pdf in terms of a series of orthogonal Laguerre polynomials that can be efficiently computed for any arbitrary value of TV. In the course of our derivation, we gained much physical insight and were able to show, from our computations and analysis, that for the sum of N = 6 specular waves, the envelope pdf can be approximated closely by a Rayleigh distribution.

Computing the zeros of a Fourier series or a Chebyshev series or general orthogonal polynomial series with parity symmetries

In recent years, good algorithms have been developed for finding the zeros of trigonometric polynomials and of ordinary polynomials when written in the form of a truncated Chebyshev polynomial or Legendre polynomial series. In each case, the roots can be found from the eigenvalues of a generalized Frobenius companion matrix whose elements are trivial functions of the Fourier coefficients or Chebyshev coefficients. However, the QR method for computing the companion matrix eigenvalues has a cost that grows proportionally to N 3 where N is the polynomial degree. (By exploiting the special structure of the companion matrices, the cost can be reduced to O ( N 2 ) , but only for large N .) Here, we show that if the polynomial has definite parity, such as a trigonometric polynomial composed only of cosines or a polynomial that is a sum only of Chebyshev polynomials of odd degree, one can exploit these symmetries to halve the size of the problem. This reduces costs in the companion matrix method by a factor ranging between four and eight. For trigonometric polynomials, we give transformations that dramatically reduce costs even if the roots are found by an algorithm other than the companion matrix procedure. We further give reductions for trigonometric polynomials with double parity symmetries which save a factor of sixteen to a factor of sixty-four in the companion matrix algorithm. Special functions such as spherical harmonics, Mathieu functions, prolate spheroidal wavefunctions and Hough functions, all represented by truncated Fourier series with double parity, are a rich source of applications.

Characteristics of guided waves in anisotropic spherical curved plates

For ultrasonic non-destructive inspection of spherical curved plates, which is used in the fields of pressure vessels, spherical domes of power plants, and so on, characteristics of guided waves in spherical curved plates must be understood. Based on linear three-dimensional elasticity, an orthogonal polynomial series expansion approach is used for determining the guided wave dispersion curves and the displacements distributions in homogeneous anisotropic spherical curved plates. When the material is orthotropic or has fewer independent constants in the wave propagation direction, the coupled propagating waves can be decomposed of Lamb-like waves and SH wave. The independent SH wave is solved analytically by Bessel function. In order to test the accuracy and range of applicability of this polynomial approach, the two results of the SH wave is compared, the exact results by Bessel function and the results by the polynomial method. Guided wave dispersion curves for unidirectional composite spherical curved plates with different fiber intensity are calculated to show the effect of the fiber intensity on dispersion curves. Finally, the guided wave dispersion curves for orthotropic spherical curved plates at different radius of curvature are calculated and the inherited influences of the radius of curvature on the wave characteristics are discussed.

Characteristics of guided waves in graded spherical curved plates

An elastodynamic solution for the stress wave propagation in spherical curved plates composed of functionally graded materials (FGM) is presented. Properties of materials are assumed to vary in the direction of the thickness according to a known radial variation law (gradient field). The formulation is based on the linear three-dimensional elasticity. The Legendre orthogonal polynomial series expansion approach is used for determining the guided waves dispersion curves in functionally graded spherical curved plates. Our results from a homogeneous anisotropic spherical curved plate are compared with those published earlier to confirm the accuracy and range of applicability of this polynomial approach. Guided wave dispersion curves for graded spherical curved plates with different gradient fields are calculated, and the effects of the gradient field on the characteristics of guided waves are illustrated. Finally, dispersion curves for graded spherical curved plates at different ratios of inner radius to thickness are calculated to discover the influences of that ratio on the wave characteristics.

Guided circumferential waves in orthotropic cylindrical curved plate and the mode conversion by the end-reflection

In ultrasonic nondestructive inspection of large-diameter pipes and curved plate, longitudinal cracks are detected more efficiently by using guided circumferential waves. In the present, the study of guided circumferential waves and their application in detecting longitudinal defect were relative adequate when pipe material is isotropic. Based on linear three-dimensional elasticity, an orthogonal polynomial series expansions approach is used for determining the guided circumferential waves dispersion curves in homogeneous infinitely long orthotropic hollow cylinders. Results are compared with those published earlier and with the finite element simulation to check up the accuracy and range of applicability of this polynomial approach. Through the analysis of the displacements distributions and finite element simulation, the mode conversion of guided circumferential waves by end-reflection in cylindrical curved plate is discovered.

On orthogonal polynomial approximation with the dimensional expanding technique for precise time integration in transient analysis

We use four orthogonal polynomial series, Legendre, Chebyshev, Hermite and Laguerre series, to approximate the non-homogeneous term for the precise time integration and incorporate them with the dimensional expanding technique. They are applied to various structures subjected to transient dynamic loading together with Fourier and Taylor approximation proposed in previous works. Numerical examples show that all six methods are efficient and have reasonable precision. In particular, Legendre approximation has much higher precision and better convergence; Chebyshev approximation is also good, but only slightly inferior to Legendre approximation. The other four approximation methods usually produce results with errors hundreds of thousands of times larger. Hermite and Laguerre approximation may be useful for some special non-homogeneous terms, but do not work sufficiently well in our numerical examples. Other contributions of this paper include, a Dynamic Programming scheme for computing series coefficients, a general formula to find the assistant matrix for any polynomial series.

Recurrences and explicit formulae for the expansion and connection coefficients in series of classical discrete orthogonal polynomials

Two formulae expressing explicitly the difference derivatives and the moments of a discrete orthogonal polynomials {P n (x): Meixner, Kravchuk and Charlier} of any degree and for any order in terms of P n (x) themselves are proved. Two other formulae for the expansion coefficients of a general-order difference derivatives ∇ q f(x), and for the moments x ℓ∇ q f(x), of an arbitrary function f(x) of a discrete variable in terms of its original expansion coefficients are also obtained. Application of these formulae for solving ordinary difference equations with varying coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, is explained. An algebraic symbolic approach (using Mathematica), in order to build and solve recursively for the connection coefficients between two families of Meixner, Kravchuk and Charlier, is described. Three analytical formulae for the connection coefficients between Hahn–Charlier, Hahn–Meixner and Hahn–Kravchuk are also developed.

Generation and evaluation of orthogonal polynomials in discrete Sobolev spaces II: numerical stability

In this paper, we concern ourselves with the determination and evaluation of polynomials that are orthogonal with respect to a general discrete Sobolev inner product, that is, an ordinary inner product on the real line plus a finite sum of atomic inner products involving a finite number of derivatives. In a previous paper we provided a complete set of formulas to compute the coefficients of this recurrence. Here, we study the numerical stability of these algorithms for the generation and evaluation of a finite series of Sobolev orthogonal polynomials. Besides, we propose several techniques for reducing and controlling the rounding errors via theoretical running error bounds and a carefully chosen recurrence.

Frobenius–Padé approximants for d-orthogonal series: Theory and computational aspects

This work is about Frobenius–Padé approximants for series of orthogonal polynomials of dimension d (d∈ N) . Concerning the series, we give the projection property of partial sums, show how to compute their coefficients, and how to get the coefficients of the product of a series by a polynomial. Concerning the approximants, we essentially deal with their recursive computation. Also, we give several examples and numerical results.

Stability of parallel algorithms for polynomial evaluation

In this paper, we analyse the stability of parallel algorithms for the evaluation of polynomials written as a finite series of orthogonal polynomials. The basic part of the computation is the solution of a triangular tridiagonal linear system. This fact allows us to present a more detailed analysis. The theoretical results show that the parallel algorithms are almost as stable as their sequential counterparts for practical applications. Extensive numerical experiments confirm the theoretical conclusions.