Orthogonal Polynomial Approach Research Articles

Guided waves in multilayered plates: An improved orthogonal polynomial approach

Conventional orthogonal polynomial approach can solve the multilayered plate only when the material properties of two adjacent layers do not change significantly. This paper developed an improved orthogonal polynomial approach to solve wave propagation in multilayered plates with very dissimilar material properties. Through numerical comparisons among the exact solution, the results from the conventional polynomial approach and from the improved polynomial approach, the validity of the improved polynomial approach is illustrated. Finally, it is shown that the conventional polynomial approach can not yield correct continuous normal stress profiles. The improved orthogonal polynomial approach has overcome this drawback.

Wave Propagation in Piezoelectric Rods with RectangularCross Sections

The ring ultrasonic transducers are widely used in the ocean engineering and medical fields. This paper employs an extended orthogonal polynomial approach to solve the guided wave propagation in two-dimensional structures, i.e. piezoelectric rings with rectangular cross-sections. The extended polynomial approach can overcome the drawbacks of the conventional orthogonal polynomial approach which can be used to solve wave propagation in one-dimensional structures. Through numerical comparison with the available results for a rectangular aluminum bar, the validity of the present approach is illustrated. The dispersion curves and displacement and electric potential distributions of various rectangular piezoelectric rings are calculated, and the effects of different radius to thickness ratios, width to height ratios and polarizing directions on the dispersion curves are illustrated.

Orthogonal polynomial approach to estimation of poles of rational functions from data on open curves

This paper deals with the problem of finding poles of rational functions from function values on open curves in the complex plane. For this problem, Nara and Ando recently proposed an algorithm that reduces the problem to a system of linear equations through contour integration. The main aim of this paper is to analyze and improve this algorithm by giving a new interpretation to the algorithm in terms of orthogonal polynomials. It is demonstrated that the system of linear equations is not always uniquely solvable and that this difficulty can be remedied by doubling the number of the linear equations. Moreover, to cope with discretization errors caused by numerical integration, we introduce new polynomials similar, in spirit, to discrete orthogonal polynomials, which yield an algorithm free from discretization errors.

Guided waves in general anisotropic layered rectangular rods: An extended orthogonal polynomial approach

The orthogonal polynomial approach has been used to solve the guided wave propagation in structures for about 20 years, but was limited until now to one-dimensional structures, that is, structures with a finite dimension in only one direction, such as horizontally infinite flat plates and axially infinite hollow cylinders. This paper extends the orthogonal polynomial approach to two-dimensional structures, that is, structures with finite dimensions in two directions, with illustration on the multilayered rectangular rod. Through numerical comparison with the available results, the validity of the extended polynomial approach is illustrated. The dispersion curves and displacement distributions of various layered rectangular rods are obtained to show their guided wave characteristics.

Instantons and extreme value statistics of random matrices

We discuss the distribution of the largest eigenvalue of a random N x N Hermitian matrix. Utilising results from the quantum gravity and string theory literature it is seen that the orthogonal polynomials approach, first introduced by Majumdar and Nadal, can be extended to calculate both the left and right tail large deviations of the maximum eigenvalue. This framework does not only provide computational advantages when considering the left and right tail large deviations for general potentials, as is done explicitly for the first multi-critical potential, but it also offers an interesting interpretation of the results. In particular, it is seen that the left tail large deviations follow from a standard perturbative large N expansion of the free energy, while the right tail large deviations are related to the non-perturbative expansion and thus to instanton corrections. Considering the standard interpretation of instantons as tunnelling of eigenvalues, we see that the right tail rate function can be identified with the instanton action which in turn can be given as a simple expression in terms of the spectral curve. From the string theory point of view these non-perturbative corrections correspond to branes and can be identified with FZZT branes.

The propagation of coupled Lamb waves in multilayered arbitrary anisotropic composite laminates

Based on linear three-dimensional elasticity theory, the wave equations of coupled Lamb waves in multilayered arbitrary anisotropic composite laminates are derived using a Legendre orthogonal polynomial approach. The elastodynamic solution for the propagation of coupled Lamb waves in composite plates is also presented to determine the characteristics of coupled Lamb waves. To verify the applicability and validity of the method, two cases of bi-layered plates formed with isotropic components and anisotropic components, respectively, are primarily manipulated for comparison with earlier known results. Next, the dispersion curves, displacements and stress distributions of Lamb waves in multilayered anisotropic laminates are calculated. The effects of coupling and fiber orientation on the characteristics of the Lamb waves are illustrated. The potential usefulness of the fundamental modes of the coupled Lamb waves is discussed in detail.

Free-ultrasonic waves in multilayered piezoelectric plates: An improvement of the Legendre polynomial approach for multilayered structures with very dissimilar materials

In 1999, Legendre polynomial approach has been proposed to solve wave propagation in the multilayered piezoelectric plate. However, it can deal with the multilayered plate only when the material properties of two adjacent layers do not change significantly. In this paper, an improvement of the Legendre polynomial approach is promoted to solve wave propagation in multilayered piezoelectric plates whatever the dissimilarities of the layer material properties. Detailed formulations are given to highlight the differences from the conventional Legendre polynomial approach. The validity of the proposed improved approach is illustrated through a numerical comparison between the improved method’s results and the exact solution obtained from the reverberation-ray matrix method. It is shown that the conventional orthogonal polynomial approach cannot conceptually calculate accurately, neither the discontinuous distribution of the normal electric field nor the continuous distributions of the normal stress and normal electric displacement, even in multilayered plates with similar layer material properties while the proposed improved polynomial approach overcomes these major drawbacks. Finally, the influences of the stacking sequences and volume fractions on dispersion curves are illustrated. It is also found that the stress and electric displacement of high frequency waves always distribute on the layer with lower wave speed.

Wave propagation in multilayered piezoelectric spherical plates

This paper proposes an improvement of the Legendre polynomial series method to solve the harmonic wave propagation in multilayered piezoelectric spherical plates, which are used in point-focusing transducers. The conventional Legendre polynomial method can deal with the multilayered structures only when the material properties of two adjacent layers do not change significantly and cannot obtain correctly normal stress and normal electric displacement shapes unlike the proposed improved orthogonal polynomial approach which overcomes these drawbacks. Detailed formulations are given to highlight its differences from the conventional Legendre polynomial approach. Through the comparisons of numerical results given by an exact solution (obtained from the reverberation-ray matrix method), and by the conventional polynomial approach and the improved polynomial approach, the validity of the proposed approach is illustrated. The influences of the radius-to-thickness ratio on dispersion curves, stress and electric displacement distributions are discussed. It is found that three factors determine the distribution of mechanical energy and electric energy at higher frequencies: radius-to-thickness ratio, wave speed, and position of the component material.

Wave propagation in the circumferential direction of general multilayered piezoelectric cylindrical plates

The Legendre polynomial approach has been proposed to solve wave propagation in multilayered flat plates and functionally graded structures for more than ten years, but it can deal with a multilayered plate only when the material properties of two adjacent layers do not change significantly. In this paper, an improvement of the Legendre polynomial approach is proposed to solve wave propagation in what, from now on, we will call general multilayered piezoelectric cylindrical plates, to mean indifferently with or without very dissimilar materials. Detailed formulations are given to highlight the differences from the conventional Legendre polynomial approach. Through numerical comparisons among the exact solution (from the reverberation-ray matrix), the conventional polynomial approach, and the improved polynomial approach, the validity of the proposed approach is illustrated. Then, the influences of the radius-to-thickness ratio on the dispersion curves, the stress, and electric displacement distributions are discussed. It is shown that the conventional orthogonal polynomial approach cannot obtain correct continuous normal stress and normal electric displacement shapes, unlike the improved orthogonal polynomial approach, which overcomes these drawbacks. It is also found that three factors determine the distribution of mechanical energy and electric energy at higher frequencies: the radius-to-thickness ratio, the wave speed of component material, and the position of the component material.

A note on the invariant distribution of a quasi-birth-and-death process

The aim of this paper is to give an explicit formula of the invariant distribution of a quasi-birth-and-death process in terms of the block entries of the transition probability matrix using a matrix-valued orthogonal polynomials approach. We will show that the invariant distribution can be computed using the squared norms of the corresponding matrix-valued orthogonal polynomials, no matter if they are or not diagonal matrices. We will give an example where the squared norms are not diagonal matrices, but nevertheless we can compute its invariant distribution.

Elastic Waves in Piezoelectric Spherical Curved Plates

Based on linear three-dimensional piezoelasticity, an orthogonal polynomial approach is used for determining the elastic wave characteristics of piezoelectric spherical 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. Guided wave dispersion curves for PZT-4 spherical curved plates are calculated. Corresponding mechanical displacement and electric potential distributions are illustrated. The influence of the ratio of radius to thickness on the wave characteristics is discussed.

Non-singlet QCD analysis in the NNLO approximation

A non-singlet QCD analysis of the structure function up to NNLO is performed based on the orthogonal polynomials approach. The results of valence quark distributions up to NNLO are in good agreement with the available theoretical models.

Time‐domain rotational angle identification of a beam structure utilizing the dynamic transverse deflection signal

The real time signal identification of rotation angle for a beam structure is studied in this work. A laser displacement sensor traced transverse displacement signal was used to identify the rotation angle of a beam. An orthogonal polynomial approach was used to expand the transverse displacement signal into orthogonal functions and their associated coefficients. These coefficients were used to identify the orthogonally expanded coefficients associated with the rotation angles of the beam structure. The rotation angle response of the beam can finally be synthesized and checked against the finite element code predicted result. We also conducted impact hammer tests on the root of a metallic cantilever beam. Both the loading history and the transverse displacement history at a specific location on the beam specimen were recorded. Similar to the numerical synthesis process, the test determined a displacement signal can also be used to identify the rotation angle of the beam. Through numerical and experimental verification, the currently proposed analysis seems to be able to identify the dynamic rotation angle response of the beam structure closely, when the prescribed transverse displacement input signal of the beam is given.

Binding dynamics of single-stranded DNA binding proteins to fluctuating bubbles inbreathing DNA

We investigate the dynamics of a single local denaturation zone in a DNA molecule, aso-called DNA bubble, in the presence of single-stranded DNA binding proteins (SSBs). Inparticular, we develop a dynamical description of the process in terms of a two-dimensionalmaster equation for the time evolution of the probability distribution of having a bubble of sizem withn bound SSBs, forthe case when m and n arethe slowest variables in the system. We derive explicit expressions for the equilibrium statistical weights for agiven m andn, which depend onthe statistical weight u associated with breaking a base-pair interaction, the loop closure exponentc, the cooperativityparameter σ0, theSSB size λ, andbinding strength κ. These statistical weights determine, through the detailed balance condition, thetransfer coefficient in the master equation. For the case of slow and fast bindingdynamics the problem can be reduced to one-dimensional master equations. In thelatter case, we perform explicitly the adiabatic elimination of the fast variablen. Furthermore, we find that for the case that the loop closure isneglected and the binding dynamics is vanishing (but with arbitraryσ0) the eigenvalues and the eigenvectors of the master equation can be obtained analytically,using an orthogonal polynomial approach. We solve the general case numerically (i.e.,including SSB binding and the loop closure) as a function of statistical weightu, binding proteinsize λ, andbinding strength κ, and compare to the fast and slow binding limits. In particular, we find that the presenceof SSBs in general increases the relaxation time, compared to the case when no bindingproteins are present. By tuning the parameters, we can drive the system from regularbubble fluctuation in the absence of SSBs to full denaturation, reflecting experimental andin vivo situations.

Percolation of clusters with a residence time in the bond definition: Integral equation theory

We consider the clustering and percolation of continuum systems whose particles interact via the Lennard-Jones pair potential. A cluster definition is used according to which two particles are considered directly connected (bonded) at time t if they remain within a distance d, the connectivity distance, during at least a time of duration tau, the residence time. An integral equation for the corresponding pair connectedness function, recently proposed by two of the authors [Phys. Rev. E 61, R6067 (2000)], is solved using the orthogonal polynomial approach developed by another of the authors [Phys. Rev. E 55, 426 (1997)]. We compare our results with those obtained by molecular dynamics simulations.

Unquenched QCD Dirac operator spectra at nonzero baryon chemical potential

The microscopic spectral density of the QCD Dirac operator at nonzero baryon chemical potential for an arbitrary number of quark flavors was derived recently from a random matrix model with the global symmetries of QCD. In this paper we show that these results and extensions thereof can be obtained from the replica limit of a Toda lattice equation. This naturally leads to a factorized form into bosonic and fermionic QCD-like partition functions. In the microscopic limit these partition functions are given by the static limit of a chiral Lagrangian that follows from the symmetry breaking pattern. In particular, we elucidate the role of the singularity of the bosonic partition function in the orthogonal polynomials approach. A detailed discussion of the spectral density for one and two flavors is given.

Identification of nonlinear systems via piecewise general orthogonal polynomials operator

Based on a hybrid orthogonal function, a new linear, continuous and bounded operator, piecewise general orthogonal polynomials operator (PGOPO), is proposed. Its main properties and operational rules have been strictly constructed based on the theory of convergence in the mean square, and are useful to discuss the orthogonal polynomial approach in the proper mathematics frame. Then applying the PGOPO method to solve the identification problem of nonlinear systems, the convergence analysis of PGOPO approximation method is given. Finally, using the PGOPO method to solve the parameter identification of two kinds of time-varying systems, the simulation studies show that the algorithm is simple and more effective than that of general orthogonal polynomials.

One-dimensional partially asymmetric simple exclusion process with open boundaries: orthogonal polynomials approach

The stationary state of the partially asymmetric simple exclusion process with open boundaries is reconsidered. The so-called matrix product ansatz is employed. This enables us to construct the stationary state in the form of the product of the matrices D and E. Noticing the fact that the matrix C( = D+E) for the model is related to certain q-orthogonal polynomials, the model is analysed for a wide range of parameters. The current and the correlation length are evaluated in the thermodynamic limit. It turns out that the phase diagram for the correlation length is richer than that for the totally asymmetric case.

Equivalence between voltage-processing methods and discrete orthogonal Legendre polynomial (DOLP) approach

There are three methods for solving the least-squares estimation (LSE) problem. (1) the power method; (2) the voltage-processing method (square-root method); and (3) the discrete orthogonal Legendre polynomial (DOLP) method. The first involves a matrix inversion and is sensitive to computer round-off errors. The second and third do not require a matrix inversion and are not as sensitive to computer round-off errors. It is shown that the voltage-processing LSE methods (Givens, Householder, and Gram-Schmidt) become the discrete orthogonal Legendre polynomial (DOLP) LSE method when the data can be modeled by a polynomial function and the times between measurements are equal. Furthermore, when the data can be modeled by a polynomial function and the time between measurements are equal, the DOLP is the preferred method because it does not require an orthonormal transformation and it does not require the back-substitution method.

Orthogonal polynomial approach to fluids with internal degrees of freedom: The case of polar, polarizable molecules

The molecules of real liquids have internal degrees of freedom that may couple with the external coordinates of position and orientation so that they affect and are affected by the microscopic liquid structure. For cases where the internal coordinates possess a Boltzmann-like distribution, a procedure was recently proposed [Phys. Rev. E 55, 426 (1997)] whereby the internal coordinates are incorporated into the conventional integral equation formulation of classical liquid state theory with no approximations beyond some reliable closure relation familiar from simple liquids. The basis of the procedure is expansions in special orthogonal polynomials of the internal coordinates. Here we use this technique to obtain the structural, thermodynamic, and electrostatic properties of a classical liquid of polar polarizable molecules, with classical Drude oscillators modeling the internal variable of fluctuating polarization. Sample results obtained using several approximate closures are compared with simulation data.