Chebyshev Spectral Element Research Articles

Parametric electro‐mechanical dispersive behaviors of guided waves in the functionally graded piezoelectric annular plate

AbstractThis paper investigates the propagation characteristics of guided waves in functionally graded annular plates considering the electro‐mechanical coupling. In the orthogonal circular curvilinear coordinate system, the elastodynamic wave equation of guided waves is derived for the functional graded piezoelectric annular plate. The Chebyshev spectral elements are utilized to model the annular plate, and the governing wave equation is discretized. The convergence and accuracy of the proposed method are evaluated by comparing with the results in previous publications. The parametric study on dispersive characteristics of the functionally graded piezoelectric annular plate is conducted systematically for the influences of various volume fraction index, curvature, and geometry size. Some exciting results are discussed including mode separation, mode conversion, and cutoff frequency. This research can benefit the design and development of advanced smart materials and structures.

Free vibration analysis of rectangular thin plates with corner and inner cutouts using C1 Chebyshev spectral element method

This paper presents a novel higher-order C1 continuous Chebyshev spectral element method for the free vibration analysis of rectangular thin plates with corner and inner cutouts. To achieve C1 continuity at the interfaces of adjacent elements, the Hermite interpolation functions through nonuniformly distributed Chebyshev nodes are employed to construct the element displacement field. A Gauss–Lobatto type quadrature is used to efficiently evaluate the element matrices. Comparison with other well-established methods show that the proposed element has good convergence and accuracy. The influence of cutout size, location and eccentric ratio on the natural vibration characteristics of plates are investigated.

A lumped mass Chebyshev spectral element method and its application to structural dynamic problems

A lumped mass Chebyshev spectral element method and its application to structural dynamic problems

Guided waves propagation in sandwich cylindrical structures with functionally graded graphene-epoxy core and piezoelectric surface layers

As an ideal reinforcing nanofiller, graphene platelets (GPLs) can significantly improve physical properties of nanocomposites, which have attracted considerable attention for design and development of advanced lightweight nanocomposite structures in engineering. In this paper, a sandwich cylindrical structure is considered for dispersion properties of wave propagation by axisymmetric isogeometric analysis (IGA). The sandwich structure is composed of a functionally graded (FG) nanocomposite core and piezoelectric surface layers. Graphene platelets are dispersed in the interlayer by three typical distribution patterns through the thickness. In virtue of the symmetry and the advantages of isogeometric analysis, the sandwich cylindrical structure can be described by one-dimensional representation along the radial direction. Based on Hamilton’s principle, parameterized governing equation for wave propagation is obtained with the non-uniform rational B-splines (NURBS), which leads to an second-order eigenvalue problem. The modified wave finite element (WFE) method and the Chebyshev spectral element (SE) method are utilized to verify the reliability and accuracy of this approach. Then, the effects of several significant parameters of GPLs and geometric sizes of the sandwich structure on wave propagation characteristics are discussed in details. The results of this study are beneficial to deeply understanding and control of wave propagation in advanced piezoelectric composites for the applications in structural health monitoring and nondestructive evaluation.

Chebyshev谱元法模拟波动的两种集中质量矩阵

Chebyshev谱元法模拟波动的两种集中质量矩阵

Guided wave propagation in a multilayered magneto-electro-elastic curved panel by Chebyshev spectral elements method

This paper investigates the dispersion characteristics of guided waves in a multilayered magneto-electro-elastic curved panel. Based on the Hamilton principle, the Chebyshev spectral element method is applied along the radial and circumferential directions to obtain the dispersion equation of the waves. The advantage of the spectral element method over a high degree piecewise polynomial basis function is the high order of accuracy. The influences on the dispersion characteristics is analyzed in terms of four factors: structural radian, magneto-electro effect, thickness-diameter ratio, and stacking sequence. In addition, the dispersion curves in the curved panel display a phenomenon which different from plate and hollow cylinders: mode conversion. We explain this from the perspective of physics and mathematics, separately. The structural radian and thickness-diameter ratio show the most significant effect on the mode conversion.

Error Analysis of Chebyshev Spectral Element Methods for the Acoustic Wave Equation in Heterogeneous Media

This work concerns the error analysis of the spectral element method with Gauss–Lobatto–Chebyshev collocation points with the implicit Newmark average acceleration scheme for the two-dimensional acoustic wave equation. The analysis is restricted to homogeneous Dirichlet boundary conditions, constant compressibility and variable density. The proposed error estimates are optimal with respect to the mesh parameter although suboptimal on the polynomial degree. Numerical examples illustrate the theoretical results.

Chebyshev spectral element method for natural convection in a porous cavity under local thermal non-equilibrium model

An accurate Chebyshev spectral element method is developed to simulate the natural convection in a porous cavity using the local thermal non-equilibrium model. Two horizontal walls of the cavity are adiabatic, while the left vertical wall is heated and the right side wall is cooled. The non-dimensional governing equations are derived based on the Darcy model and Boussinesq assumption. The validation test with exact solution verifies the high accuracy of the algorithm, and the numerical results for natural convection in porous square cavity reach an excellent agreement with reported solutions. Two physical problems are investigated: (1) Natural convection in porous cavity at different aspect ratios with fixed temperature on two vertical walls; (2) Natural convection in porous square cavity with temporal sinusoidal temperature on the hot wall and fixed temperature on the cold wall. The effects of aspect ratio and non-dimensional heat transfer characteristic parameters are analyzed.

Chebyshev Spectral Element Analysis for Pore-Pressure of ERDs during Construction Period

Chebyshev spectral elements are applied to dissipation analysis of pore-pressure of roller compaction earth-rockfilled dams (ERD) during their construction. Nevertheless, the conventional finite element, for its excellent adaptability to complex geometrical configuration, is the most common way of spatial discretization for the pore-pressure solution of ERDs now [1]. The spectral element method, by means of the spectral isoparametric transformation, surmounts the disadvantages of disposing with complex geometry. According to the illustration of numerical examples, one can conclude that the spectral element methods have the following obvious advantages: 1) large spectral elements can be used in spectral element methods for the domains of homogeneous material; 2) in the application of large spectral elements to spatial discretization, only a few leading terms of Chebyshev interpolation polynomial are taken to arrive at the solutions of better accuracy; 3) spectral element methods have excellent convergence as well-known. Spectral method also is used to integrate the evolution equation in time to avoid the limitation of conditional stability of time-history integration

Dynamic analysis of beam-cable coupled systems using Chebyshev spectral element method

The dynamic characteristics of a beam–cable coupled system are investigated using an improved Chebyshev spectral element method in order to observe the effects of adding cables on the beam. The system is modeled as a double Timoshenko beam system interconnected by discrete springs. Utilizing Chebyshev series expansion and meshing the system according to the locations of its connections, numerical results of the natural frequencies and mode shapes are obtained using only a few elements, and the results are validated by comparing them with the results of a finite-element method. Then the effects of the cable parameters and layout of connections on the natural frequencies and mode shapes of a fixed-pinned beam are studied. The results show that the modes of a beam–cable coupled system can be classified into two types, beam mode and cable mode, according to the dominant deformation. To avoid undesirable vibrations of the cable, its parameters should be controlled in a reasonable range, or the layout of the connections should be optimized.

Guided wave propagation in an infinite functionally graded magneto-electro-elastic plate by the Chebyshev spectral element method

This paper is to investigate the characteristics of elastic guided waves in an infinite functionally graded magneto-electro-elastic plate. Based on the Hamilton’s principle, Chebyshev spectral element method is applied along the thickness direction to obtain the dispersion equation. The convergence and accuracy of the proposed method are compared with the results of commercial software Disperse. For the infinite functionally graded plate composed of piezoelectric BaTiO3 and magnetostrictive CoFe2O4, some numerical examples about dispersion curves and wave structures are given, and the influence of the change of the volume fraction index N on dispersive behavior are analyzed. In addition, the velocity deletion, which has not been mentioned in previous studies of the infinite functionally graded magneto-electro-elastic plate, is found with the dispersion curves.

An Advective-Spectral-Mixed Method for Time-Dependent Many-Body Wigner Simulations

As a phase space language for quantum mechanics, the Wigner function approach bears a close analogy to classical mechanics and has been drawing growing attention, especially in simulating quantum many-body systems. However, deterministic numerical solutions have been almost exclusively confined to one-dimensional one-body systems and few results are reported even for one-dimensional two-body problems. This paper serves as the first attempt to solve the time-dependent many-body Wigner equation through a grid-based advective-spectral-mixed method. The main feature of the method is to resolve the linear advection in $(\boldsymbol{x},t)$-space by an explicit three-step characteristic scheme coupled with the piecewise cubic spline interpolation, while the Chebyshev spectral element method in $\boldsymbol{k}$-space is adopted for accurate calculation of the nonlocal pseudo-differential term. Not only is the time step of the resulting method not restricted by the usual CFL condition and thus a large time step is allowed, but also the mass conservation can be maintained. In particular, for the system consisting of identical particles, the advective-spectral-mixed method can also rigorously preserve physical symmetry relations. The performance is validated through several typical numerical experiments, like the Gaussian barrier scattering, electron-electron interaction and a Helium-like system, where the third-order accuracy against both grid spacing and time stepping is observed.

Effects of Cable on the Dynamics of a Cantilever Beam with Tip Mass

The dynamic effects of cable attachment on a cantilever beam with tip mass are investigated by an improved Chebyshev spectral element method. The cabled beam is modeled as a double-beam system connected by springs at several discrete locations. By utilizing high order Chebyshev polynomials as basis functions and meshing the system at the locations of connections, precise numerical results of the natural frequencies and mode shapes can be obtained using only a few elements. The accuracy of this method is validated through comparing the results of finite element method and those of spectral element method in literature. The validated method is implemented to investigate the effects of parameters, including spring stiffness, number of connections, density, and Young’s modulus of cable. The results show that the mode shapes of the cabled beam system can be classified into two types: beam mode shapes and cable mode shapes, according to their main deformation. Their corresponding natural frequencies change in very different ways with the variation of system parameters. This work can be applied to optimize the dynamic characteristics of precise spacecraft structures with cable attachments.

The Chebyshev spectral element approximation with exact quadratures

A new Chebyshev spectral element method has been developed in this paper, in which exact quadratures are used to overcome a shortfall of the Gauss–Chebyshev quadrature in variational spectral element projections. The method is validated with the Stokes and the Cauchy–Riemann problems. It is shown that an enhancement of the approximation convergence rate is attained, and numerical accuracy is much better than that from the Gauss–Lobatto–Legendre spectral element method.

WAVE PROPAGATION MODELING IN HIGHLY HETEROGENEOUS MEDIA BY A POLY-GRID CHEBYSHEV SPECTRAL ELEMENT METHOD

A wide range of applications requires the modeling of wave propagation phenomena in media with variable physical properties in the domain of interest, while highly accurate algorithms are needed to avoid unphysical effects. Spectral element methods (SEM), based on either a Chebyshev or a Legendre polynomial basis, have excellent properties of accuracy and flexibility in describing complex models, outperforming other techniques. In the standard SEM approach the computational domain is discretized by using very coarse meshes and constant-property elements, but in some cases the accuracy and the computational efficiency may be seriously reduced. For instance, a finely heterogeneous medium requires grid resolution down to the finest scales, leading to an extremely large problem dimension. In such problems the wavelength scale of interest is much larger but cannot be exploited in order to reduce the problem size. A poly-grid Chebyshev spectral element method (PG-CSEM) can overcome this limitation. In order to accurately deal with continuous variation in the properties, or even with small scale fluctuations, temporary auxiliary grids are introduced which avoid the need of using any finer global grid, and at the macroscopic level the wave field propagation is solved maintaining the SEM accuracy and computational efficiency.

A Chebyshev Spectral Elements Method for Earth Dam Seepage Analysis

Based on the theory of spectral elements method,this paper theoretically analyzed the issue of earth dam seepage under both one-dimensional and two-dimensional conditions.First,an integral form about the seepage problem was set up;then,the discrete matrix equations were formed by using Chebyshev polynomials as the basis function and the global coefficient matrix was formed in the way used in the finite-element method.Finally,according to different types of boundary conditions,water head linear array {h} was calculated and the whole dam's water head was educed from this linear array.

Implicit Chebyshev spectral element method for acoustics wave equations

In the present study, the Chebyshev spectral element method coupled with the implicit Newmark time integral method is investigated to simulate acoustic field. The approach is first applied to a numerical example. In this case, the numerical result is consistent with the reference solution, and the influence of time step and Newmark factor on numerical accuracy is studied. The resulting figures indicate that the smaller the time step and Newmark factor are, the higher the numerical accuracy will be. The division pattern of computational domain can affect the numerical accuracy, which indicates that the numerical accuracy can be improved when the grid nodes tend to be distributed uniformly in the computational domain. Secondly, a numerical example is used to investigate the absorbing boundary condition, the result of which indicates that the absorbing boundary condition absorbs the numerical reflection whereas the Dirichlet boundary condition causes the numerical reflection. Moreover, the approach is also employed for wave equation with source terms such as monopole, dipole and quadrapole, whose numerical results keep their radiation traits intactly.

The Chebyshev spectral element method using staggered predictor and corrector for elastic wave simulations

Based on strong and weak forms of elastic wave equations, a Chebyshev spectral element method (SEM) using the Galerkin variational principle is developed by discretizing the wave equation in the spatial and time domains and introducing the preconditioned conjugate gradient (PCG)-element by element (EBE) method in the spatial domain and the staggered predictor/corrector method in the time domain. The accuracy of our proposed method is verified by comparing it with a finite-difference method (FDM) for a homogeneous solid medium and a double layered solid medium with an inclined interface. The modeling results using the two methods are in good agreement with each other. Meanwhile, to show the algorithm capability, the suggested method is used to simulate the wave propagation in a layered medium with a topographic traction free surface. By introducing the EBE algorithm with an optimized tensor product technique, the proposed SEM is especially suitable for numerical simulation of wave propagations in complex models with irregularly free surfaces at a fast convergence rate, while keeping the advantage of the finite element method.

Effect of Vibrating Buildings on Free-Field Motion and on Adjacent Structures: The Bonefro (Italy) Case History

Abstract During the 2002 Molise sequence, a shock was recorded in a building that suffered heavy damage (Mucciarelli et al. , 2004). The availability of this data, along with other similar data, led to the organization of an international workshop for structural engineers aimed at the reproduction of the staged evolution of damage observed (A. Goretti and M. Mucciarelli, unpublished manuscript, 2007). The aim of this present work is to model the damaged building; the adjacent, less damaged building; and a large section of the subsoil. The model should provide some insight into two possible phenomena: (1) the soil–structure interaction (SSI), that is, the modification of the free-field seismic response induced by the presence of the buildings located on soft soil; and (2) the possible interaction between the two buildings. The study was performed using the 2D Chebyshev spectral element method (SPEM) for a 2D section. The model, with topography, includes the bedrock (limestone) with the overlaying clay layer, circa 30 m. The building is modeled by an equivalent shear-wave velocity, which is calculated from its identified mechanical properties. First, the model’s ability to reproduce the motions recorded inside and outside the building was verified. Then, one or both of the buildings were removed to study their effect. The influence of the presence of buildings on the free-field motion is about 40% on spectral values, and it is felt up to a distance of about 250 m from the buildings. The results also suggest the presence of resonance between buildings.

SPECTRAL ELEMENT METHOD FOR ACOUSTIC PROPAGATION PROBLEMS BASED ON LINEARIZED EULER EQUATIONS

A Chebyshev spectral element approximation of acoustic propagation problems based on linearized Euler equations is introduced, and the numerical approach is based on spectral elements in space with first-order Clayton–Engquist–Majda absorbing boundary conditions and implicit Newmark method in time. An initial perturbation problem has been solved to test the accuracy and stability of the numerical method. Then the sound propagation by source terms is also studied, including the radiation of a monopole and dipolar source in both stationary medium and uniform mean flow. The numerical simulation leads to good results in both accuracy and stability. Compared with the analytical solutions, the numerical results show the advantages in spectral accuracy even with relatively fewer grid points. Moreover, the implicit Newmark method in time marching also presents its superiority in stability. Finally, a problem of sound propagation in pipes is simulated as well.