Orthogonal Polynomial Series Research Articles

Vibration analysis of stiffened multi-plate structure based on a modified variational principle

In order to study the vibration characteristics of flow-induced open cavity structures, the dynamic model of stiffened multi-plate is established. The first-order shear deformable plate theory and the Timoshenko beam theory are used to model the displacement fields of isotropic plates and stiffeners, respectively. A modified variational principle combined with a multi-segment partitioning procedure is employed to formulate the discretized equations of motion. The stiffeners are considered as discrete elements, and the energy contributions are included into the system energy functional by using the displacement compatibility conditions. The displacement and rotation components of each plate segment are expanded by a duplicate series of Chebyshev orthogonal polynomials of first kind. The convergence and accuracy of the present results for isotropic stiffened plates with different boundary conditions have been validated using comparisons with the published data and those obtained from the finite element analyses. Free vibration and dynamic responses of stiffened multi-plates with either longitudinal or orthogonally oriented stiffeners are discussed. The mathematical model and methodology presented in this paper may be used as an appropriate numerical tool in the analysis and design of stiffened multi-plate structures.

Non-intrusive polynomial chaos for efficient uncertainty analysis in parametric roll simulations

Monte Carlo analyses are generally considered the standard for uncertainty analysis. While accurate, these analyses can be expensive computationally. Recently, polynomial chaos has been proposed as an alternative approach to the estimation of uncertainty distributions (Hosder et al. A non-intrusive polynomial chaos method for uncertainty propagation in CFD simulations. In: 44th AIAA aerospace sciences meeting and exhibit, Reno, Nevada, 2006; Wu et al. Uncertainty analysis for parametric roll using non-intrusive polynomial chaos. In: Proceedings of the 12th international ship stability workshop, Washington, DC, USA, 2011). This approach works by representing the function as a series of orthogonal polynomials; the weights for which can be calculated via several methods. Previous studies have demonstrated the usefulness of this technique for comparatively simple systems such as parametric roll modeled by the Mathieu equation with normally distributed parameter values (Wu et al. Uncertainty analysis for parametric roll using non-intrusive polynomial chaos. In: Proceedings of the 12th international ship stability workshop, Washington, DC, USA, 2011). In the present work, a polynomial chaos method is applied to a nonlinear computational ship dynamics model with normally distributed input parameters. Test cases were selected where parametric roll was expected to potentially occur. The resulting probability distributions are compared to the results of a Monte Carlo analysis. In general, these results demonstrate good agreement between Monte Carlo simulation and polynomial chaos in the absence of capsize with significant computation gains found with polynomial chaos. Overall, we conclude that polynomial chaos is an effective tool for reducing simulation time costs when studying parametric roll, and potentially other ship dynamics phenomena, particularly in the absence of capsize-like bifurcations.

Sharp estimates for the rate of convergence of double Fourier series in classical orthogonal polynomials

Sharp estimates are obtained for the convergence rate of “triangular” and “hyperbolic” partial sums of Fourier series in orthogonal (Laguerre, Hermite, Jacobi) polynomials in the classes of differentiable functions of two variables characterized by a generalized modulus of continuity. The proofs are based on the generalized shift operator and generalized modulus of continuity for functions from L2 having generalized partial derivatives in Levi’s sense.

Domain of convergence for a series of orthogonal polynomials

Let {pk}k=0∞ be the orthogonal polynomials with certain exponential weights. In this paper, we prove that under certain mild conditions on exponential weights class, a series of the form ∑bkpk converges uniformly and absolutely on compact subsets of an open strip in the complex plane, and diverges at every point outside the closure of this strip.

Wave propagation in functionally graded piezoelectric rods with rectangular cross-section

For the purpose of design and optimization of functionally graded piezoelectric material (FGPM) transducers, wave propagation in FGPM structures has received much attention in the past twenty years. But research focused essentially on semi-infinite structures and one-dimensional structures, i.e., structures with a finite dimension in only one direction, such as horizontally infinite flat plates and axially infinite hollow cylinders. This paper proposes a double orthogonal polynomial series approach to solve the wave propagation problem in a two-dimensional (2D) FGPM structure, namely, an FGPM rod with a rectangular cross-section. The dispersion curves and electric potential distributions are illustrated.

Спектрально-аналитический метод распознавания неточных повторов в символьных последовательностях

Proposed are theoretical basis and algorithmic implementation of spectral-analytical method of recognition of repeats in character sequences. The theoretical justification is based on the theorem on equivalent representation of the character sequence by the vector of continuous characteristic functions. Comparison of fragments of characteristic functions is performed in the standard metric in Euclidean space of expansion coefficients of the Fourier series of orthogonal polynomials. An essential feature of this approach is the ability to evaluate repeats at different scales. Another important feature is the possibility of efficient parallelization of data. In the development of algorithms we preferred scheme of computing with a minimal amount of references to memory, implying repetitive calculations and evaluations on demand. In this paradigm, proposed is an algorithm for calculating the coefficients of expansions in the orthogonal polynomials through the use of recurrence relations. It is shown that the algorithm for calculating the coefficients of expansions in the orthogonal polynomials can be effectively vectorized by computing with a fixed vector length. Parallelization and vectorization implemented using the OpenMP standard and extension Cilk Plus of language C/C++. The developed method effectively scales, depending on the parameters of the problem and the number of processor cores on systems with shared memory.

Computation of the Likelihood in Biallelic Diffusion Models Using Orthogonal Polynomials

In population genetics, parameters describing forces such as mutation, migration and drift are generally inferred from molecular data. Lately, approximate methods based on simulations and summary statistics have been widely applied for such inference, even though these methods waste information. In contrast, probabilistic methods of inference can be shown to be optimal, if their assumptions are met. In genomic regions where recombination rates are high relative to mutation rates, polymorphic nucleotide sites can be assumed to evolve independently from each other. The distribution of allele frequencies at a large number of such sites has been called “allele-frequency spectrum” or “site-frequency spectrum” (SFS). Conditional on the allelic proportions, the likelihoods of such data can be modeled as binomial. A simple model representing the evolution of allelic proportions is the biallelic mutation-drift or mutation-directional selection-drift diffusion model. With series of orthogonal polynomials, specifically Jacobi and Gegenbauer polynomials, or the related spheroidal wave function, the diffusion equations can be solved efficiently. In the neutral case, the product of the binomial likelihoods with the sum of such polynomials leads to finite series of polynomials, i.e., relatively simple equations, from which the exact likelihoods can be calculated. In this article, the use of orthogonal polynomials for inferring population genetic parameters is investigated.

Wave propagation in graded rings with rectangular cross-sections

In this paper, a double orthogonal polynomial series method is proposed to investigate the guided wave propagation in a two-dimensional (2-D) structure, namely, a FGM ring with a rectangular cross-section. Two kinds of graded rings are considered: material gradient directions being in the radial direction and in the axial direction respectively. Numerical comparison with available reference results for a straightly homogeneous rectangular bar illustrates the validity of the proposed method. The dispersion curves and displacement distributions of various FGM rings, which have different radius to thickness ratios, different material gradient directions and different thickness to height ratios, are calculated to reveal the guided wave characteristics.

Dispersion curves of 2D rods with complex cross-sections: double orthogonal polynomial approach

Orthogonal polynomial series approach has been used to analyze the guided wave propagation in structures for about 20 years. These structures have always one infinite dimension in the waveguiding direction and, sometimes a regular finite cross-section (axially infinite solid or hollow cylinder for instance) and mostly often an infinite cross-section (infinite flat plate or half-space for instance). This paper presents a double orthogonal polynomial approach to investigate guided wave propagation in structures with only one infinite dimension in the waveguiding direction and a finite but complicated cross-sectional geometries as, rectangular type, L-type, 工-type and 回-type cross-sections. Through a numerical comparison with results available in literature, the validity of the extended polynomial approach is illustrated for a specific geometry. The dispersion properties of guided waves in rods with complex cross-sections as mentioned above are discussed.

Guided wave characteristics in functionally graded piezoelectric rings with rectangular cross-sections

For the purpose of design and optimization of functionally graded piezoelectric material (FGPM) transducers, wave propagation in FGPM structures has received much attention in the past twenty years. But previous research efforts have been focused essentially on semi-infinite structures and one-dimensional structures, i.e., structures with a finite dimension in only one direction, such as horizontally infinite flat plates and axially infinite hollow cylinders. This paper proposes a double orthogonal polynomial series approach to solving the wave propagation problem in a two-dimensional FGPM structure, namely an FGPM ring with a rectangular cross-section. By numerical comparison with the available reference results for a purely elastic homogeneous rectangular rod, the validity of the extended polynomial approach is illustrated. The dispersion curves and the electric potential distributions of various FGPM rectangular rings with different material gradient directions, different polarization directions, different radius to thickness ratios, and different width to thickness ratios are calculated to reveal the guided wave characteristics.

Guided wave in multilayered piezoelectric–piezomagnetic bars with rectangular cross-sections

For the purpose of design and optimization of multilayered piezoelectric–piezomagnetic material (PPC) transducers, wave propagation in these structures has received much attention in past ten years. But the research objects of previous works are only for semi-infinite structures and one-dimensional structures, i.e., structures with a finite dimension in only one direction, such as horizontally infinite flat plates and axially infinite hollow cylinders. This paper proposes a double orthogonal polynomial series approach to solve the wave propagation problem in a two-dimensional (2-D) structure, namely, a layered PPC bar with a rectangular cross-section. Through numerical comparison with the available reference results for a purely elastic layered rectangular bar, the validity of the double polynomial approach is illustrated. The dispersion curves and mechanical displacement profiles of various layered PPC rectangular bars are calculated to reveal the wave propagation characteristics.

Wave propagation in layered piezoelectric rectangular bar: An extended orthogonal polynomial approach

Wave propagation in multilayered piezoelectric structures has received much attention in past forty years. But the research objects of previous research works are only for semi-infinite structures and one-dimensional structures, i.e., structures with a finite dimension in only one direction, such as horizontally infinite flat plates and axially infinite hollow cylinders. This paper proposes an extension of the orthogonal polynomial series approach to solve the wave propagation problem in a two-dimensional (2-D) piezoelectric structure, namely, a multilayered piezoelectric bar with a rectangular cross-section. Through numerical comparison with the available reference results for a purely elastic multilayered rectangular bar, the validity of the extended polynomial series approach is illustrated. The dispersion curves and electric potential distributions of various multilayered piezoelectric rectangular bars are calculated to reveal their wave propagation characteristics.

Dynamic analysis of composite laminated and sandwich hollow bodies of revolution based on three-dimensional elasticity theory

This paper presents a semi-analytical procedure for solving linear vibration problems of composite laminated and sandwich hollow bodies of revolution with arbitrary combinations of boundary constraints in the framework of the three-dimensional theory of elasticity. Multilevel partitioning hierarchy, viz., multilayered body of revolution, individual layer and layer segment, is adopted in the theoretical analysis. The appropriate continuity constraints on common interfaces are imposed by means of a modified variational principle combined with the least-squares weighted residual method. The displacement field of each layer segment is characterized by a mixed series of basis functions, i.e., Fourier series and orthogonal polynomials. Numerical examples concerning the free vibrations of composite laminated and sandwich hollow cylinders, cones, and spheres, are presented to show the performance of the method, and comparisons of the present results are made with solutions available in the literature and those obtained from finite element analyses. With regard to the forced vibration problems, steady-state vibration responses of a sandwich hollow cylinder under a uniformly distributed normal harmonic pressure are analyzed, and time-domain solutions of composite laminated and sandwich hollow spheres subjected to various impulsive loads, including a rectangular pulse, a triangular pulse, a half-sine pulse and an exponential pulse, are also examined. Numerical experiments show that the present method is accurate, efficient and reliable for predicting the full spectrum of vibration behaviors of multilayered hollow bodies of revolution.

Optimal Insurance Purchase Strategies via Optimal Multiple Stopping Times

In this paper we demonstrate how to develop analytic closed form solutions to optimal multiple stopping time problems of direct relevance to applications in insurance for Operational Risk. Within this context we study a class of insurance products where the policy holder has the option to insure k of its annual Operational Risk losses in a horizon of T years. This involves a choice of k out of T years in which to apply the insurance policy coverage by making claims against losses in the given year. The insurance product structure presented can accommodate any kind of annual mitigation, but we present two basic generic insurance policy structures that can be combined to create more complex types of coverage. Following the Loss Distributional Approach (LDA) with Poisson distributed annual loss frequencies and Inverse-Gaussian loss severities we are able to characterize in closed form analytical expressions for the multiple optimal decision strategy that minimizes the expected Operational Risk loss over the next T years. For the cases where the combination of insurance policies and LDA model does not lead to closed form expressions for the multiple optimal decision rules, we also develop a principled class of closed form approximations to the optimal decision rule. These approximations are developed based on a class of orthogonal Askey polynomial series basis expansion representations of the annual loss compound process distribution and functions of this annual loss.

Orthogonal Polynomials Approximation and Balanced Truncation for a Lowpass Filter

Abstract This paper considers the problem of orthogonal polynomial approximation based balanced truncation for a lowpass filter. The proposed method combines the system properties of balanced truncation, the computational effectiveness of proper orthogonal decomposition and the approximation capability of the orthogonal polynomials approximation. Orthogonal polynomials series expansion of the reachability and observability gramians is used in order to avoid solving large-scale Lyapunov equations and thus, significantly reducing the computational effort for obtaining the balancing transformation. The proposed method is applied for model reduction of a lowpass analog filter. Different sets of orthonormal functions are obtained from Legendre, Laguerre and Chebyshev orthogonal polynomials and the corresponding reduced order models are compared. The approximation precision is measured by the relative mean square error between the outputs of the full order model and the obtained reduced order models.

Effects of initial stresses on guided waves in unidirectional plates

The guided wave propagation in unidirectional plates under gravity, homogeneous initial stress in the thickness direction and inhomogeneous initial stress in the wave propagating direction is investigated in this paper based on the theory of mechanics of incremental deformations. The Legendre orthogonal polynomial series expansion method is used to solve the coupled wave equation. Two different wave propagating directions, the fiber orientation and the vertical fiber orientation, are discussed respectively. The effects of the initial stresses on the Lamb-like wave and shear-horizontal (SH) wave are respectively investigated. The effects of the initial stresses on the dispersion curves and on the displacement and stress distributions are discussed.

Technique for expanding lateral-shearing interferograms, based on the use of an expansion of the wave-front function in Chebyshev polynomials

This paper proposes a technique for expanding lateral-shearing interferograms, based on a representation of the desired wave-front function in the form of a finite expansion series in orthogonal Chebyshev polynomials. The information losses accompanying the recording and interpretation of the shearing interferograms are analyzed, along with a way of reducing them. An iteration procedure is considered for finding the wave-front function from known difference functions, using the genetic algorithm. Its efficiency is demonstrated, using test calculations as an example.

Toroidal wave in multilayered spherical curved plates

For the purpose of ultrasonic nondestructive inspection of layered spherical curved plates, characteristics of guided waves in multilayered spherical curved plates are studied using an improved orthogonal polynomial series method. Conventional polynomial series method has been used to solve wave motion in the multilayered flat plate for about four decades. However, it can deal with the layered structure only when the material properties of two adjacent layers do not change significantly. It also can not obtain correct and continuous normal stress profiles. Numerical comparisons verify that the improved polynomial method overcomes these drawbacks. Then, the influences of radius to thickness ratio, volume fraction and stacking sequence on dispersion curves and stress distributions are discussed. It is found that the radius to thickness ratio, the wave speed of component material and the position of the component material determine the distribution of mechanical energy at higher frequencies.

Vibration studies of rotating cylindrical shells with arbitrary edges using characteristic orthogonal polynomials in the Rayleigh–Ritz method

A general approach is presented for the vibration studies of rotating cylindrical shells having arbitrary edges. The present analysis is based on the Sanders' shell theory and the effects of centrifugal and Coriolis forces as well as initial hoop tension due to rotating are all taken into account. By taking the characteristic orthogonal polynomial series as the admissible functions, the Rayleigh–Ritz method is employed to derive the frequency equations of rotating cylinders with classical homogeneous boundary conditions. Further, utilizing artificial springs to simulate the elastic constraints imposed on the cylinders’ edges, one can derive the frequency equations of rotating cylindrical shells with more general boundary conditions by considering the strain energy of artificial springs during the Rayleigh–Ritz procedure. To validate the approach proposed in this paper, a series of comparison and convergence studies are performed and the investigations demonstrate high accuracy and low computational cost of the present approach. Finally, some further numerical results are given to illustrate the influence of the variations of spring stiffness on the frequencies of rotating cylinders.

Generalizations and specializations of generating functions for Jacobi, Gegenbauer, Chebyshev and Legendre polynomials with definite integrals

In this paper we generalize and specialize generating functions for classical orthogo- nal polynomials, namely Jacobi, Gegenbauer, Chebyshev and Legendre polynomials. We derive a generalization of the generating function for Gegenbauer polynomials through extension a two element sequence of generating functions for Jacobi polynomials. Specializations of generat- ing functions are accomplished through the re-expression of Gauss hypergeometric functions in terms of less general functions. Definite integrals which correspond to the presented orthogonal polynomial series expansions are also given.