Chebyshev Spectral Element Method Research Articles

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.

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.

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.

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.

MODIFIED VIRTUAL CRACK-CLOSURE TECHNIQUE USING SPECTRAL ELEMENT METHOD

This paper presents a general Chebyshev Spectral Element Method (CSEM) for obtaining strain-energy release rates for crack growth in two-dimensional isotropic materials. The procedure uses virtual crack-closure method. This method is developed for different orders of CSEM and suitable expressions for energy release rates are obtained. These expressions are evaluated by applying them to two Mode-I and two Mixed-mode problems. Two different classes of spectral elements (SEs) are formulated using Chebyshev interpolating functions, the inconsistent conventional SE formulation and the field consistent SE formulation. The convergence is investigated using both the formulations. A relative study on the efficiency of the CSEM with increasing order of polynomials is clearly brought out. The effect of crack-tip element size is also studied. It is observed that field consistent formulation always gives better results compared to inconsistent formulation. Comparisons with results from the literature for these problems show the efficiency of the CSEM.

Compressible Subsonic Particle-Laden Flow over a Square Cylinder

Results are presented from simulations of transient vortex shedding in a particle-laden compressible flow over a square cylinder. The carrier phase is computed by solving the Navier-Stokes equations directly with a Chebyshev spectral element method. The dispersed phase is solved by a Lagrangian method, tracking the particles by way of the Basset-Boussinesq-Oseen equation. The coupling between the two phases is one way, that is, the volume fraction of the particles is assumed to be small so the effect of particles on the carrier phase is neglected. The results show that the main effect of the carrier-phase compressibility is the compression of the fluid in front of the cylinder and its expansion behind the cylinder. The particle phase shows the well-known focusing effect for high particle to fluid density ratios. For a lower density ratio, forces such as the pressure drag, virtual mass, Basset history, and Saffman lift tend to increase the dispersion. Compressibility effects are noticeable at low density ratios. In such situations, a lower deposition of the particles at the front of the square and a lower capture of the particles behind the square is observed.

Earthquake Ground Motion Simulation Through the 2-D Spectral Element Method

The application of the 2-D Chebyshev spectral element method (SPEM) to engineering seismology problems is reviewed in this paper. The SPEM is a high-order finite element technique which solves the variational formulation of the seismic wave propagation equations. The computational domain is discretised into an unstructured grid composed by irregular quadrilateral elements. This property makes the SPEM particularly suitable to compute numerically accurate solutions of the full wave equations in complex media. The earthquake is simulated following an approach that can be considered global, that is all the factors influencing the wave propagation — source, crustal heterogeneity, fine details of the near-surface structure, and topography — are taken into account and solved simultaneously. The basic earthquake source is represented by a 2-D point double couple model. Ruptures propagating along fault segments placed on the model plane are simulated as a finite summation of elementary point sources. After a general introduction, the paper first gives an overview of the method; then it concentrates on some methodological topics of interest for practical applications, such as quadrangular mesh generation, source definition and scaling, numerical accuracy and computational efficiency. Limitations and advantages of using a 2-D approach, although sophisticated such as the SPEM, are addressed, as well. The effectiveness of the method is illustrated through two case histories, i.e. the ground shaking prediction in Catania (Sicily, Italy) for a catastrophic earthquake, and the analysis of the ground motion in the presence of a massive structure.

EARTHQUAKE GROUND MOTION SIMULATION THROUGH THE 2-D SPECTRAL ELEMENT METHOD

The application of the 2-D Chebyshev spectral element method (SPEM) to engineering seismology problems is reviewed in this paper. The SPEM is a high-order finite element technique which solves the variational formulation of the seismic wave propagation equations. The computational domain is discretised into an unstructured grid composed by irregular quadrilateral elements. This property makes the SPEM particularly suitable to compute numerically accurate solutions of the full wave equations in complex media. The earthquake is simulated following an approach that can be considered "global", that is all the factors influencing the wave propagation — source, crustal heterogeneity, fine details of the near-surface structure, and topography — are taken into account and solved simultaneously. The basic earthquake source is represented by a 2-D point double couple model. Ruptures propagating along fault segments placed on the model plane are simulated as a finite summation of elementary point sources. After a general introduction, the paper first gives an overview of the method; then it concentrates on some methodological topics of interest for practical applications, such as quadrangular mesh generation, source definition and scaling, numerical accuracy and computational efficiency. Limitations and advantages of using a 2-D approach, although sophisticated such as the SPEM, are addressed, as well. The effectiveness of the method is illustrated through two case histories, i.e. the ground shaking prediction in Catania (Sicily, Italy) for a catastrophic earthquake, and the analysis of the ground motion in the presence of a massive structure.

2-D spectral element simulations of destructive ground shaking in Catania (Italy)

This study wants to estimate the strong ground motion in the municipal area of Catania (Italy) for a catastrophic earthquake scenario. It is part of a larger research program funded by the National Research Council – National Group for the Defence Against Earthquakes (CNR-GNDT), The Catania Project, devoted to evaluating the seismic risk of a highly urbanised area, such as that of Catania, located in a seismically active region. The reference earthquake simulates the catastrophic event (M ≈ 7.2) of 1693. The ground shaking is computed solving the 2-D full-wave equation by the Chebyshev spectral element method (SPEM). Particular emphasis is given to the construction of realistic structural models, also including the finest local detail, obtained from the geophysical, geological and geotechnical data available. Simulations are performed for several sources, to account for both a change in source position and orientation, and the finite extension of the fault along its dip. Synthetic seismograms and peak ground acceleration (PGA) envelopes, calculated at the surface for four transects across the Catania area, constitute the main result of this study which can be used for practical purposes. Simulations show that ground motion is strongly influenced by both source characteristics and crustal structure. We have found that PGA values range between 0.1 g and 0.5 g, although particular site conditions strongly affect these values locally. For example, the frequencies of maximum interest in civil engineering (1.5–4 Hz) are enhanced selectively by a thick portion of surface sediments (i.e., 30–100 m for an average shear wave velocity of 500–600 m/s). An unexpected feature is the appreciable increase of PGA at large epicentral distances, which contradicts classical attenuation relations. All the results are examined through an analysis of the propagating wavefield.