Orthogonal Series Expansion Research Articles

Circumferential SH Wave in the Inhomogeneous Magneto-Electro-Elastic Cylindrical Curved Plates

Based on linear magneto-electro-elasticity, the circumferential SH wave equation of transversely isotropy piezoelectricity-piezomagnetic functionally graded cylindrical curved plate is obtained. The Legendre orthogonal polynomial series expansion approach is employed to derive its asymptotic solution. The dispersion curves, electric potential and magnetic potential distributions are calculated. The influence of the piezoelectricity and piezomagnetism on the circumferential SH wave is investigated and the influential factors are analyzed. Finally, the influence of the gradient field and the ratio of the radius to thickness on the piezoelectric effect and piezomagnetic effect are illustrated.

Guided thermoelastic wave propagation in layered plates without energy dissipation

In this paper, the propagation of guided thermoelastic waves in laminated orthotropic plates subjected to stress-free, isothermal boundary conditions is investigated in the context of the Green-Naghdi (GN) generalized thermoelastic theory (without energy dissipation). The coupled wave equations and heat conduction equation are solved by the Legendre orthogonal polynomial series expansion approach. The validity of the method is confirmed through a comparison. The dispersion curves of thermal modes and elastic modes are illustrated simultaneously. Dispersion curves of the corresponding pure elastic plate are also shown to analyze the influence of the thermoelasticity on elastic modes. The displacement and temperature distributions are shown to discuss the differences between the elastic modes and thermal modes.

Error analysis of Jacobi derivative estimators for noisy signals

Recent algebraic parametric estimation techniques (see Fliess and Sira-Ramírez, ESAIM Control Optim Calc Variat 9:151–168, 2003, 2008) led to point-wise derivative estimates by using only the iterated integral of a noisy observation signal (see Mboup et al. 2007, Numer Algorithms 50(4):439–467, 2009). In this paper, we extend such differentiation methods by providing a larger choice of parameters in these integrals: they can be reals. For this, the extension is done via a truncated Jacobi orthogonal series expansion. Then, the noise error contribution of these derivative estimations is investigated: after proving the existence of such integral with a stochastic process noise, their statistical properties (mean value, variance and covariance) are analyzed. In particular, the following important results are obtained: (a) the bias error term, due to the truncation, can be reduced by tuning the parameters, (b) such estimators can cope with a large class of noises for which the mean and covariance are polynomials in time (with degree smaller than the order of derivative to be estimated), (c) the variance of the noise error is shown to be smaller in the case of negative real parameters than it was in Mboup et al. (2007, Numer Algorithms 50(4):439–467, 2009) for integer values. Consequently, these derivative estimations can be improved by tuning the parameters according to the here obtained knowledge of the parameters’ influence on the error bounds.

Generalized thermoelastic waves in functionally graded plates without energy dissipation

Abstract In this paper, a dynamic solution of the propagating thermoelastic waves in functionally graded material (FGM) plate subjected to stress-free, isothermal boundary conditions is presented in the context of the Green–Naghdi (GN) generalized thermoelastic theory. The FGM plate is composed of two orthotropic materials. The materials properties are assumed to vary in the direction of the thickness according to a known variation law. The coupled wave equation and heat conduction equation are solved by the Legendre orthogonal polynomial series expansion approach. The convergency of the method is discussed through a numerical example. The dispersion curves of the inhomogeneous thermoelastic plate and the corresponding pure elastic plate are compared to show the characteristics of thermal modes and the influence of the thermoelasticity on elastic modes. The displacement, temperature and stress distributions of elastic modes and thermal modes are shown to discuss their differences. A plate with a different gradient variation is calculated to illustrate the influence of the gradient field on the wave characteristics.

Wave Characteristics in Magneto-electro-elastic Functionally Graded Spherical Curved Plates

Piezoelectric-piezomagnetic functionally graded materials (FGM), with a gradual change of the mechanical, electric and magnetic properties, have great promise for different applications. Based on the Legendre orthogonal polynomial series expansion approach, a dynamic solution is presented for the propagation of harmonic waves in piezoelectric-piezomagnetic FGM spherical curved plates. The material 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 spherical curved plate and the corresponding non-piezoelectric, non-piezomagnetic 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 spherical curved plate at different ratio of radius to thickness is calculated to show the influence f the ratio on the piezoelectric effect and piezomagnetic effect.

Autofocusing with the help of orthogonal series transforms

Autofocusing with the help of orthogonal series transformsAn autofocus algorithm employing orthogonal series expansions is proposed. Several instances of the generic algorithm, based on discrete trigonometric, polynomial and wavelet series, are reviewed. The algorithms are easy to implement in the transform coders used in digital cameras. Formal analysis of the algorithm properties is illustrated in experiments. Some practical issues are also discussed.

Generalized thermoelastic waves in spherical curved plates without energy dissipation

In this article, the propagation of thermoelastic waves in orthotropic spherical curved plates subjected to stress-free, isothermal boundary conditions is investigated in the context of the Green–Naghdi (GN) generalized thermoelastic theory (without energy dissipation). The theoretical formulation is based on the linear GN thermoelastic theory. The coupled wave equation and heat conduction equation expressed by the displacement and temperature are obtained. By the Legendre orthogonal polynomial series expansion approach, the coupled controlling equations are solved. The convergence of the method is demonstrated through a numerical example. The dispersion curves of thermal modes and elastic modes are illustrated simultaneously. Dispersion curves of the corresponding purely elastic spherical plate are also shown to analyze the influence of thermoelasticity on elastic modes. The displacement, temperature and stress distributions of both elastic modes and thermal modes are calculated to show their differences. A thermoelastic spherical plate with a different ratio of radius to thickness is considered to show the influence of the ratio on the characteristics of thermoelastic waves.

Circumferential thermoelastic waves in orthotropic cylindrical curved plates without energy dissipation

In this article, the propagation of guided thermoelastic waves in the circumferential direction of orthotropic cylindrical curved plates subjected to stress-free, isothermal boundary conditions is investigated in the context of the Green–Naghdi (GN) generalized thermoelastic theory (without energy dissipation). The coupled wave equations and heat conduction equation are solved by the Legendre orthogonal polynomial series expansion approach. The convergency of the method is discussed through a numerical example. The dispersion curves of thermal modes and elastic modes are illustrated simultaneously. Dispersion curves of the corresponding pure elastic cylindrical plate are also shown to analyze the influence of the thermoelasticity on elastic modes. The displacement, temperature and stress distributions are shown to discuss the differences between the elastic modes and thermal modes. A thermoelastic cylindrical plate with a different ratio of radius to thickness is considered to discuss the influence of the ratio on the characteristics of circumferential thermoelastic waves.

Nonlinearity Recovering in Hammerstein System from Short Measurement Sequence

The problem of data pre-filtering for nonparametric identification of Hammerstein system from short (finite) data set is considered. The two-stage method is proposed. First, the linear dynamic block is identified using instrumental variables technique, and the inverse of the obtained model is used for output filtering. Next, the standard procedure of nonparametric regression function estimation (kernel-based, or using orthogonal series expansion) is applied, involving the filtered output sequence instead of the original one. It is shown, that for small and moderate number of data, the estimation error can be significantly reduced in comparison with standard nonparametric methods. The asymptotic properties of the method (consistency and rate of convergence) remain the same as in the classical versions of nonparametric algorithms.

Method of summation of some slowly convergent series

A new method of summation of slowly convergent series is proposed. It may be successfully applied to the summation of generalized and basic hypergeometric series, as well as some classical orthogonal polynomial series expansions. In some special cases, our algorithm is equivalent to Wynn’s epsilon algorithm, Weniger transformation [E.J. Weniger, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Computer Physics Reports 10 (1989) 189–371] or the technique recently introduced by Čížek et al. [J. Čížek, J. Zamastil, L. Skála, New summation technique for rapidly divergent perturbation series. Hydrogen atom in magnetic field, Journal of Mathematical Physics 44 (3) (2003) 962–968]. In the case of trigonometric series, our method is very similar to the Homeier’s H transformation, while in the case of orthogonal series — to the K transformation. Two iterated methods related to the proposed method are considered. Some theoretical results and several illustrative numerical examples are given.

Free vibration analysis of an eccentric hollow cylinder using exact 3D elasticity theory

The translational addition theorem for cylindrical wave functions in conjunction with the appropriate orthogonal series expansions and the pertinent boundary conditions are employed to develop an exact 3D elasticity solution for free vibrations of a simply (shear diaphragm) supported elastic circular cylinder of finite length with an eccentrically located inner circular cavity. The frequency spectrum plots of the first several eigenfrequencies are presented in a wide range of dimensionless eccentricities for selected length-to-radius and inner–outer radius ratios. Also, a detailed study on the 2D free vibration characteristics of an infinite eccentric cylinder is included. The numerical results describe the imperative influence of cavity eccentricity, mode type, and radii and length ratios on the vibrational characteristics of the hollow cylinder. It is observed that the introduction of bore eccentricity causes not only an increase in the number of resonant frequencies through the splitting of degenerate modes of the unperturbed problem, but also changes the appearance order of natural vibration modes. The accuracy of solutions is checked through appropriate convergence studies, and the validity of results is established with the aid of a commercial finite element package as well as by comparison with the data in the existing literature.

On the level crossing duration distributions of nakagami processes

The problem related to the fade and non-fade duration distributions (FDD and NFDD) of randomly distributed signals has been studied for more than forty years. However, for non-gaussian processes, closed form general solutions for these distributions are still unknown. In this paper we use an orthogonal series expansion to analyze the FDD and NFDD (also known as downcrossing and upcrossing duration distributions respectively) for the rather general case of Nakagami distributed random processes. Our results show that for asymptotic and non-asymptotic threshold levels the crossing duration distributions can be approximated, in a simple closed form way, using the gamma distribution. We back up this approximation with some goodness of fit metrics and show how the parameters of such distribution can be computed a priori analytically.

Circumferential Waves in Piezoelectric Cylindrical Curved Plate

Based on linear three-dimensional piezoelasticity, an orthogonal polynomial series expansion approach is used to determine the characteristics of guided circumferential waves in homogeneous anisotropic piezoelectric cylindrical curved plates. The displacement components and electrical 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. The dispersion curves for PZT-4 and Ba 2 NaNb 5 O 15 cylindrical curved plates are calculated. The effect of piezoelectricity is also shown. Mechanical displacement distribution and electric potential distribution are illustrated and the influence of the radius to thickness ratio on the circumferential wave characteristics is discussed. Finally, the influence of the polarizing direction on the piezoelectric effect is shown.

Wave characteristics in functionally graded piezoelectric hollow cylinders

Based on linear three-dimensional piezoelasticity, the Legendre orthogonal polynomial series expansion approach is used for determining the wave characteristics in hollow cylinders composed of the functionally graded piezoelectric materials (FGPM) with open circuit. The displacement and electric potential components, 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. Dispersion curves for FGPM and the corresponding non-piezoelectric hollow cylinders are calculated to show the piezoelectric effect. The influence of the ratio of radius to thickness is discussed. Electric potential and displacement distributions are used to show the piezoelectric effect on the flexural torsional mode. The influence of the polarizing direction on the piezoelectric effect is illustrated. For the radial and axial polarization, the piezoelectric effect reacts mostly on the longitudinal mode. For circumferential polarization, the piezoelectric effect reacts mostly on the torsional mode. In the FGPM hollow cylinder, piezoelectricity can weaken the guided wave dispersion.

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.

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.

The method of fundamental solutions with eigenfunctions expansion method for 3D nonhomogeneous diffusion equations

AbstractAfter the successful applications of the combination of the method of fundamental solutions (MFS), the method of particular solutions (MPS), and the eigenfunctions expansion method (EEM) to solve 2D homogeneous and nonhomogeneous diffusion equations by Young et al. (Young et al., Numer Meth Part Differ Equat 22 (2006), 1173), this article intends to extend the same fundamental concepts to calculate more challenging 3D nonhomogeneous diffusion equations. The nonhomogeneous diffusion equations with time‐independent source terms and boundary conditions are analyzed by the proposed meshless MFS‐MPS‐EEM model. Nonhomogeneous diffusion equation in any complex domains can be decomposed into a Poisson equation and a homogeneous diffusion equation by the principle of linear superposition. This approach is proved to be far better off than solutions by using classic method of separation of variables with inefficient multisummation of very sophisticated series expansion from special functions, which can only limit to treat very simple 3D geometries such as cube, cylinder, or sphere. Poisson equation is solved by using the MPS‐MFS model, in which the source term in the Poisson equation is first handled by the MPS based on the compactly‐supported radial basis functions and the Laplace equation is solved by the MFS. On the other hand, by utilizing the EEM, the homogeneous diffusion equation is first transformed into a Helmholtz equation, which is then solved by the MFS together with the technique of singular value decomposition (SVD) to acquire the eigenvalues and eigenfunctions. After the eigenfunctions are obtained, we can synthesize the diffusion solutions like the orthogonal Fourier series expansions but with only one summation for the series even for multidimensional problems. Numerical results for four case studies of 3D homogeneous and nonhomogeneous diffusion problems show good agreement with the analytical and other numerical solutions, such as finite element method (FEM). Thus, the present numerical scheme has provided a promising meshfree numerical approach to solve 3D nonhomogeneous diffusion equations with time‐independent source terms and boundary conditions for very irregular domains. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

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.

A computational procedure for response statistics-based optimization of stochastic non-linear FE-models

An approach for an efficient solution of response statistics-based optimization problems of non-linear FE systems under stochastic loading is presented. A sequential approximate optimization approach, where approximate stochastic analyses are used during portions of the optimization process, is implemented in the proposed formulation. In this approach, analytical approximations of the performance functions in terms of the design variables are considered during the optimization process. The analytical approximations are constructed by combining a mixed linearization approach with a stochastic response sensitivity analysis. The state of the system is defined in terms of the statistical second-moment characteristics of the structural response. The stochastic loading and the response of the system are represented by an orthogonal series expansion of the corresponding covariance matrices. In particular, a truncated Karhunen–Loève (K–L) expansion is applied. The system of non-linear equations is replaced by a statistical equivalent linear system. The evaluation of the K–L vectors is carried out by an efficient procedure that combines local linearization, modal analysis and static response of higher structural modes. An illustrative example is presented that shows the efficiency of the proposed methodology: it considers a building finite element model enforced with non-linear hysteretic devices and subject to a stochastic ground acceleration. Two types of problems are considered: a minimum structural weight design problem and an optimal non-linear device design problem.

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.