Legendre Polynomial Method Research Articles

Two-Dimensional Legendre Polynomial Method for Internal Tide Signal Extraction

This study employs the two-dimensional Legendre polynomial fitting (2-D LPF) method to fit M2 tidal harmonic constants from satellite altimetry data within the region of 53°E–131°E, 34°S–6°N, extracting internal tide signals acting on the sea surface. The M2 tidal harmonic constants are derived from the sea surface height (SSH) data of the TOPEX/Poseidon (T/P), Jason-1, Jason-2, and Jason-3 satellites via t-tide analysis. We validate the 2-D LPF method against the 300 km moving average (300 km smooth) method and the one-dimensional Legendre polynomial fitting (1-D LPF) method. Through cross-validation across 42 orbits, the optimal polynomial orders are determined to be seven for 1-D LPF, and eight and seven for the longitudinal and latitudinal directions in 2-D LPF, respectively. The 2-D LPF method demonstrated superior spatial continuity and smoothness of internal tide signals. Further single-orbit correlation analysis confirmed generally higher correlation with topographic and density perturbations (correlation coefficients: 0.502, 0.620, 0.245; 0.420, 0.273, −0.101), underscoring its accuracy. Overall, the 2-D LPF method can use all regional data points, overcoming the limitations of single-orbit approaches and proving its effectiveness in extracting internal tide signals acting on the sea surface.

Modeling and analysis of acoustoelastic guided waves propagation in anisotropic fiber reinforced polymer composite plates under initial gradient stress

Process-induced initial gradient stress is inevitable during the fabrication of fiber reinforced polymer (FRP) composites. In this research, governing wave equations with variable coefficients are established for FRP plates under initial gradient stress, based on the acoustoelastic theories. The Legendre polynomial (LP) method is utilized to derive the solving algorithm. The dispersion curves are obtained without layer slicing, which provides a more realistic analysis model for gradient stress. The accuracy and convergence of the proposed method are discussed through numerical examples. The influences of initial gradient stress on dispersion slowness curves of Lamb and SH waves are analyzed, considering the parabolic initial gradient stress distribution, in which it’s common in the manufacturing process of FRP composites. The effect of initial gradient stress distribution on the phase velocity is investigated in detail. The influence of the anisotropic effect of FRP on guided wave propagation under initial gradient stress is discussed. The numerical analysis shows that the behavior of guided waves is influenced not only by the initial gradient stress but also dominated by propagation direction in anisotropic FRP composite plates. These results can provide a useful reference for the testing based on guided waves for FRP composite, especially when the initial gradient stress is encountered.

Generalized thermoelastic wave response in a hollow cylinder with temperature-dependent properties based on the memory-dependent derivative of the heat conduction model

High-temperature pipelines are susceptible to defects due to the long-term service in high-temperature environments. The guided wave technology is promising for online non-destructive testing. Before implementing it in engineering, it is essential to better understand guided wave characteristics, especially attenuation, which significantly decreases the test distance and the resolution. However, describing the large attenuation is challenging due to the strong thermoelastic coupling, resulting in limitations of available models. To address these limitations, this paper aims to introduce a memory-dependent generalized thermoelastic model for investigating guided waves propagating in a hollow cylinder with temperature-dependent material properties. The analytical solutions are obtained using the Legendre polynomial method. The influence of the memory-dependent effect, temperature variation, and boundary conditions are studied. Some interesting findings are revealed: (1) The time delay factor is more dominant when adjusting κ and n to accurately describe guided wave propagation at high temperatures. (2): The mode conversion is accompanied by the attenuation and group velocity jump when the adjacent flexural torsional and longitudinal modes intersect, which is advisable to avoid when choosing the excitation frequency. (3) The relationship between phase velocity and attenuation versus temperature for quasi-elastic modes is nonlinear, while that for thermal modes is approximately linear.

A comprehensive review on fractional-order optimal control problem and its solution

Abstract This article presents a comprehensive literature survey on fractional-order optimal control problems. Fractional-order differential equation is extensively used nowadays to model real-world systems accurately, which exhibit fractal dimensions, memory effects, as well as chaotic behaviour. These versatile features attract engineers to concentrate more on this, and it is widely used in the broad domain of science and technology. The mentioned numerical tools take the necessary optimal conditions into account, which makes it a two-point boundary value problem of non-integer order. In this review article, some numerical approaches for the approximation have been stated for obtaining the solution to fractional optimal control problems (FOCPs). Here, few numerical approaches including Grunwald-Letnikov approximation, Adams type predictor-corrector method, generalized Euler’s method, Caputo-Fabrizio method Bernoulli and Legendre polynomials method, Legendre operational method, and Ritz’s and Jacobi’s method are treated as an advanced method to obtain the solution of FOCP. Fractional delayed optimal control is selected for our investigation. It refers to a type of control problem where the control action is delayed by a fractional amount of time. In other words, the control input at a given time depends not only on the current state of the system but also on its past state at fractional times. The fractional delayed optimal control problem is formulated as an optimization problem that seeks to minimize a cost function subject to a set of constraints that represent the dynamics of the system and the fractional delay in the control input. The solution to this problem typically involves the use of fractional polynomials types, i.e. Chebyshev and Bassel polynomials.

Axisymmetric free vibration modeling of a functionally graded piezoelectric resonator by a double Legendre polynomial method

Axisymmetric free vibration modeling of a functionally graded piezoelectric resonator by a double Legendre polynomial method

Modeling of hollow cylinder piezoelectric resonator with current excitation by a double Legendre polynomial method

This article presents an extension of the double Legendre polynomial method to model and evaluate the axisymmetric free vibration problem for the axially polarized piezoelectric resonator. The aim is to show the benefit of the employed method. The polynomial series was used to define the mechanical displacements and electrical potential at the same time. The proposed approach considerably improves computing efficiency by avoiding much of the numerical integration calculations. As a result, the electromechanical coupling coefficient shows that the fundamental mode is the most piezoactive mode in the structure. Furthermore, the electrical input impedance and field profiles are easily presented.

Study on propagation characteristics of ultrasonic guided wave and detection of the defect in resin bolts

The nondestructive testing of the bolt anchoring quality based on ultrasonic guided wave technology is significant to ensure project security. In this paper, an improved Legendre polynomial method (ILPM) is presented to study the longitudinal axisymmetric guided waves in a full-length bonding resin bolt which is a typical bilayer structure. The ILPM employs analytical integral calculation instead of numerical integral in the traditional Legendre polynomial method (TLPM), leading to superior computational efficiency. The effectiveness of the ILPM is validated, and the convergence and computational efficiency are also discussed. The attenuation and group velocity curves of guided waves are solved for the resin bolt. Through the theoretical analysis of dispersion characteristics, the optimal excitation frequencies for experimental detection can be found. Meanwhile, the guided wave detection system of a resin bolt is established to verify the theoretical results. Based on the Self-Sending and Self-Receiving experiment, the resin bolts with different anchoring defects (debonding length and incision size) are studied. The influences of L-mode defect detection ability and defect size parameters on the amplitude of the guided wave signal are analyzed. The experimental results show that L-mode can be used to detect the defects in the bolt, and the defect size can be characterized by the defect echo amplitude.

A comparative study for calculating dispersion curves in viscoelastic multi-layered plates

This study examines and critically compares various commonly used models for calculating complex wavenumbers in viscoelastic orthotropic multi-layer materials (e.g. fiber reinforced polymer laminates). Both exact 3D elasticity theory based methods and approximate methods are considered for wave propagation analysis: (i) global matrix method (GMM), (ii) stiffness matrix method (SMM), (iii) hybrid compliance-stiffness matrix method (HCSMM), (iv) semi analytical finite element (SAFE) method, (v) Legendre polynomial method (LPM), and (vi) fifth order shear deformation theory (5-SDT). A detailed mathematical formulation is provided for each of these techniques.The accuracy and the computational efficiency for calculating complex wavenumbers over a wide frequency range are analyzed for both homogenized and multi-layered viscoelastic orthotropic media. This information is helpful to select the proper method to be used in inversion schemes for material characterization. A comparative analysis is also performed on the computed through thickness displacement fields within the solid medium. The different implemented methods have been compiled in a toolbox with a graphical user interface, called “The Dispersion Box”, and can be freely downloaded from Orta et al. (2022).

Propagation and attenuation of Lamb waves in functionally graded fractional viscoelastic soft plates with a pre-deformation

An improved understanding of the wave propagation in incompressible soft structures with a pre-deformation, especially the grasp of viscoelastic effect, could fundamentally catalyze technical developments in many areas including medicine, and rubber. Based on the nonlinear elasticity and linear incremental theories, a fractional-order hyper-viscoelastic model is established to derive dynamic equations of Lamb waves in functionally graded pre-deformed viscoelastic soft plates. The fractional-order Kelvin-Voigt model is applied to concern the viscoelasticity. An improved Legendre polynomial method is employed to analytically solve wave equations. The complex wave solutions are obtained without the iterative. The correctness of the present method is validated by a comparison with the finite element method. The influence of viscoelasticity and pre-formation on dispersion and attenuation curves is investigated. Some new wave phenomena are revealed: Phase velocity and attenuation of Lamb waves can be controlled by altering stretch ratios λi, and the pre-stretching restrains the viscoelastic effect; The minimum attenuation frequency can be adjusted by altering the material component distributions. Results provide the foundation for the design of biomaterials with the controllable viscoelasticity to greatly promote their applications in the regenerative medicine.

Elastic wave attenuation in a functionally graded viscoelastic couple stress plate, sandwiched between two elastic half-spaces

Taking the viscoelasticity and size-dependent effect into account, the reflection and transmission of elastic waves through functionally graded viscoelastic plate sandwiched in two elastic half-spaces are investigated. The extended analytical integration Legendre polynomial method and Kelvin-Voigt viscoelastic model in the context of the modified couple stress theory are utilized to perform the derivation. The global matrix method is used to confirm the validity of the extended analytical integration Legendre polynomial method. Compared with the global matrix method, the extended analytical integration Legendre polynomial method provides an efficient numerical calculation method without layering and iterative root seeking. The influences of the couple stress, viscoelasticity and inhomogeneity of medium on elastic wave reflection and transmission are discussed. In addition, the attenuation caused by the viscoelasticity is analyzed. It is found that the couple stress effect weakens the influence of the viscoelasticity, while the viscoelasticity strengthens the couple stress effect. The couple stress effect decreases the attenuation, while the inhomogeneity increases the attenuation.

Complete guided wave in piezoelectric nanoplates: A nonlocal stress expansion polynomial method

Taking the size-dependent and piezoelectric effect into account, the complete guided waves in nonlocal piezoelectric nanoplates are investigated. The Legendre polynomial method based on stress and electric displacement expansion is proposed to obtain the complete dispersion curves in the complex domain. The nonlocal and piezoelectric effects on propagative and evanescent waves are discussed. It is found that the nonlocal effect improves the phase velocity of complex Lamb wave, while reduces the attenuation and frequency bandwidth of complex Lamb wave. The stiffness-softening effect of the nonlocal structure reduces the phase velocity, cut-off frequency and escape frequency of propagative waves, while the stiffness-hardening effect of the piezoelectric structure increases the phase velocity, cut-off frequency and escape frequency of propagative waves. In addition, the nonlocal effect increases the piezoelectric effect, while the piezoelectric effect weakens the nonlocal effect on phase velocity.

Guided waves in a functionally graded 1-D hexagonal quasi-crystal plate with piezoelectric effect

For the purpose of design and optimization for piezoelectric quasi-crystal transducers, guided waves in a functionally graded 1-D hexagonal piezoelectric quasi-crystal plate are investigated. In this paper, a model combined with the Bak’s and elastohydrodynamic models is utilized to derive governing equations of wave motion, and real, pure imaginary, and complex roots of governing equations are calculated by using the modified Legendre polynomial method. Subsequently, dispersion curves and displacements of phonon and phason modes are illustrated. Then, guided waves in functionally graded 1-D hexagonal piezoelectric quasi-crystal plates with different quasi-periodic directions are studied. And the phonon-phason coupling effect on Lamb and SH waves are analyzed. Accordingly, some interesting results are obtained: The phonon-phason coupling just affects Lamb waves in the x- and z-direction quasi-crystal plates, and SH waves in the y-direction quasi-crystal plate. Besides, frequencies of propagative phason modes decrease as phonon-phason coupling coefficients Ri increase. Furthermore, a variation in the polarization has a more significant influence on phonon modes, and a variation in the quasi-periodic direction has a more significant influence on phason modes.

Guided waves propagation in multi-layered porous materials by the global matrix method and Biot theory

In this research, a numerical approach for analyzing the analytical solution of guided wave propagation characteristics in multilayer two-phased porous media is presented. Combing the global matrix method and Biot theory, and introducing the modulus-porosity-poisson’s ratio relation, the global dispersion equations of multi-layered porous materials are established. Also, the complex boundary conditions in multi-layered two-phase porous media are considered simultaneously. In order to confirm the feasibility and accuracy of the proposed method, the dispersion curves of the guided waves propagation in a single-layer porous graphite layer and a triple-layer graphite/copper/graphite model (porosity close to 0) were numerically calculated. The results were compared with the propagation characteristics of the guided waves in the corresponding stacked sequential laminates without porosity, which is done by the state-vector formalism and the Legendre polynomials method based on our previous work. Then, this approach is further applied to lithium ion battery. The change of porosity in graphite is used to simulate the state of charge of lithium-ion battery. The influence of porosity change on mode coupling effect of guided waves is analyzed, and the mapping relationship between porosity and corresponding dispersion curves is explored. Meanwhile, the research shows that, with the decrease of porosity (which means the state of charge increases), the phase velocity of the fundamental modes gradually increases, making the corresponding time of flight decrease gradually. Moreover, the theoretical model captures a meaningful relationship between state of charge and acoustic behavior. It gives theoretical support for nondestructive evaluation and quantitative estimation of the state characteristics of lithium-ion batteries.

Alternative Legendre Polynomials Method for Nonlinear Fractional Integro-Differential Equations with Weakly Singular Kernel

In this paper, we present a numerical scheme for finding numerical solution of a class of weakly singular nonlinear fractional integro-differential equations. This method exploits the alternative Legendre polynomials. An operational matrix, based on the alternative Legendre polynomials, is derived to be approximated the singular kernels of this class of the equations. The operational matrices of integration and product together with the derived operational matrix are utilized to transform nonlinear fractional integro-differential equations to the nonlinear system of algebraic equations. Furthermore, the proposed method has also been analyzed for convergence, particularly in context of error analysis. Moreover, results of essential numerical applications have also been documented in a graphical as well as tabular form to elaborate the effectiveness and accuracy of the proposed method.

Optimization of coupled Lamb wave parameters for defect detection in anisotropic composite three-layer with Kelvin-Voigt viscoelasticity using Legendre polynomial method

An extension of the Legendre polynomial method is proposed, which allows to model coupled Lamb wave propagation in anisotropic multilayers with Kelvin-Voigt viscoelastic properties in a carbon-epoxy-layer sequence [0/ϕ/0]. In spite of complicated nonlinear dependences of the constituent equations on the viscoelastic properties, a linear relationship is found between the attenuation curves calculated by Hysteretic model, and the attenuation curves calculated by the Kelvin-Voigt model. In view of verifying the feasibility of using wave propagation analysis for non-destructive testing of composite materials, predictions are made concerning the dependence of the experimentally accessible phase velocities and attenuation coefficients of the modes on (virtual, defect induced, changes of) the viscoelastic constants in general and the fiber orientation in particular. In order to get a deeper insight in the material property sensitivity of the waves and in how to excite and detect them efficiently, also the through-thickness of displacements and energies profiles of different wave modes and frequencies are investigated, and how these depend on the viscoelastic moduli Cij.

Analysis of Longitudinal Guided Wave Propagation in the Functionally Graded Hollow Cylinder Using State-Vector Formalism and Legendre Polynomial Hybrid Approach

A numerical approach for analyzing the guided wave propagation behavior in a functionally graded material (FGM) hollow cylinder is presented. Based on the three-dimensional linear elasticity theory, the state matrix form of displacement and stress component is introduced into the differential governing equation. Combining with boundary conditions, the dispersion characteristic equations can be constructed. Meanwhile, taking the orthogonal completeness of Legendre series into consideration, the exhausting root findings of transcendental equation can be avoided, and it transforms the redundant integral operators into analytical expressions by means of its recursive properties. Moreover, it is not only unnecessary to stratify FGMs, and introducing the univariate nonlinear regression to modelling the arbitrary gradient distribution, and transforming the redundant integral operator into an analytical expression by means of recursive property. It should be pointed out that the coefficient matrix consists only zero and constant terms, which are related to the elastic constants and normalized wave number. And it provides a fast and effectively approach to extracting the dispersion curves, displacement distribution and stress profile, simultaneously. Meanwhile, the proposed method can get rid of the exhausted root-finding algorithm, and accordingly distinguish the different modes and the dispersion curves simultaneously. Finally, the typical non-stratified computation of dispersion curves of longitudinal guided wave for FGM hollow cylinder is realized. The accuracy of the proposed method is further demonstrated through comparison with the available data from Legendre polynomial method. Several numerical cases about iron based alumina FGM cylinder are studied, and the convergence of the present polynomial approach is discussed. Then the effect of the circumferential wave number, gradient variation, diameter-thickness ratio on the dispersion characteristics are illustrated. Moreover, the distribution characteristics of displacement and stress profiles of F(1,1) mode at a given frequency are discussed in detail. It is shown that the state-vector formalism and Legendre polynomial hybrid approach is a powerful tool in solving wave motions in FGM elastic cylinder, which lays a theoretical foundation for the non-destructive evaluation and quantitative characterization of the gradient structure characteristics of functionally graded hollow cylinder.

Active contour driven by adaptively weighted signed pressure force combined with Legendre polynomial for image segmentation

This paper proposes an active contour driven by adaptively weighted signed pressure force (SPF) combined with the Legendre polynomial method for image segmentation. First, an adaptively weighted global average intensity (GAI) term is defined wherein GAI differences are the weighted factors of the interior and exterior region-driving centers. Second, an adaptively weighted Legendre polynomial intensity (LPI) term is defined which adopts the Legendre polynomial intensity average differences as the weighted factors of the interior and exterior region-driving centers. Finally, the GAI and LPI terms are introduced into a novel SPF function and a coefficient is applied to weight their effect degrees; a new edge stopping function (ESF) is defined and combined with the region-based method to robustly converge the curve to the boundary of the object. Experiments demonstrate that this method is highly accurate and computationally efficient for images with inhomogeneous intensity, blurred edge, low contrast, and noise problems. Moreover, the segmentation results are independent of the initial contour.

Guided wave propagation in functionally graded fractional viscoelastic plates: A quadrature-free Legendre polynomial method

Compared to the traditional integer-order viscoelastic model, a fractional-order derivative viscoelastic model is shown to be advantageous. A quadrature-free Legendre polynomial method combining the Weyl definition of fractional order derivative is for the first time employed to solve guided waves in functionally graded fractional viscoelastic plates. The presented method avoids a lot of numerical integration calculation in traditional polynomial method and greatly improves the computational efficiency. The effects of different graded fields and fractional orders on the dispersion and attenuation curves are illustrated. The displacement distributions, Poynting vectors and power flows are analyzed to reveal guided wave characteristics.

Guided wave propagating in a 1-D hexagonal piezoelectric quasi-crystal plate

In the context of Bak’s model, guided waves in a 1-D hexagonal piezoelectric quasi-crystal plate are investigated by applying the Legendre polynomial method. Three cases of quasi-periodic directions are discussed. The dispersion curves, phonon, and phason displacement distributions are illustrated. Some new wave phenomena are revealed: The phase velocity of Lamb wave phason modes decreases as the phonon–phason coupling parameters, $$R_{i}$$ , increase. Phason displacements and the electric potential have consistent distributions with those of phonon displacement components in the quasi-periodic direction. These obtained results lay the theoretical basis for the design and optimization of piezoelectric devices.

Application of state vector formalism and Legendre polynomial hybrid method in the longitudinal guided wave propagation analysis of composite multi-layered pipes

In this research, we applied a polynomial hybrid approach for modeling longitudinal guided waves propagating in anisotropic composites multi-layered pipes. Theoretically, dispersion characteristic equations in cylindrical coordinate system were derived by introducing the state vector form of displacement and stress components. In virtue of the orthogonality completeness and recursion properties of Legendre polynomial series, the dispersion curves of longitudinal guided waves for arbitrary multi-layered anisotropic pipes can be obtained efficiently and accurately. As an alternative serial approach, it solves the wave propagation problem of complex anisotropic cylinders, and also avoids the tedious integral operations in the traditional Legendre polynomial method. The reliability of the numerical results of longitudinal guided waves propagating in arbitrary multi-layered anisotropic pipes was investigated based on the state matrix and Legendre polynomial hybrid method. At first, we calculated the multi-layered pipes composed of isotropic materials (steel and aluminum) with different diameter-thickness ratios and circumferential orders, and the results are consistent with the ones from the global matrix method. Then, the numerical analysis for a triple-layered adhesive pipe was implemented. Finally, the composite multi-layered pipes with at most 16 layers, which are the T300/914 in different orientations, were studied. Meanwhile, the effect of the layer number, fiber angle and circumferential order on the propagation characteristics of longitudinal guided waves were analyzed. To demonstrate the generality of this approach, we compared the dispersion curves of flat plate case with the results from the cylindrical case using the same physical properties by making the geometrical diameter to thickness ratio infinity. Furthermore, the displacements and stress distributions were also illustrated at given frequencies.