A new adaptive cell average spectral element method(SEM)is proposed to solve the time-dependent Wigner equation for transport in quantum devices.The proposed cell average SEM allows adaptive non-uniform meshes in phas...A new adaptive cell average spectral element method(SEM)is proposed to solve the time-dependent Wigner equation for transport in quantum devices.The proposed cell average SEM allows adaptive non-uniform meshes in phase spaces to reduce the high-dimensional computational cost of Wigner functions while preserving exactly the mass conservation for the numerical solutions.The key feature of the proposed method is an analytical relation between the cell averages of the Wigner function in the k-space(local electron density for finite range velocity)and the point values of the distribution,resulting in fast transforms between the local electron density and local fluxes of the discretized Wigner equation via the fast sine and cosine transforms.Numerical results with the proposed method are provided to demonstrate its high accuracy,conservation,convergence and a reduction of the cost using adaptive meshes.展开更多
Spectral element methods are well established in the field of wave propagation,in particular because they inherit the flexibility of finite element methods and have low numerical dispersion error.The latter is experim...Spectral element methods are well established in the field of wave propagation,in particular because they inherit the flexibility of finite element methods and have low numerical dispersion error.The latter is experimentally acknowledged,but has been theoretically shown only in limited cases,such as Cartesian meshes.It is well known that a finite element mesh can contain distorted elements that generate numerical errors for very large distortions.In the present work,we study the effect of element distortion on the numerical dispersion error and determine the distortion range in which an accurate solution is obtained for a given error tolerance.We also discuss a double-grid calculation of the spectral element matrices that preserves accuracy in deformed geometries.展开更多
In this paper,we propose a spectral vanishing viscosity method for the triangular spectral element computation of high Reynolds number incompressible flows.This can be regarded as an extension of a similar stabilizati...In this paper,we propose a spectral vanishing viscosity method for the triangular spectral element computation of high Reynolds number incompressible flows.This can be regarded as an extension of a similar stabilization technique for the standard spectral element method.The difficulty of this extension lies in the fact that a suitable definition of spectral vanishing viscosity operator in non-structured elements does not exist,and it is not clear that if a suitably defined spectral vanishing viscosity provides desirable dissipation for the artificially accumulated energy.The main contribution of the paper includes:1)a well-defined spectral vanishing viscosity operator is proposed for non-standard spectral element methods for the Navier-Stokes equations based on triangular or tetrahedron partitions;2)an evaluation technique is introduced to efficiently implement the stabilization term without extra computational cost;3)the accuracy and efficiency of the proposed method is carefully examined through several numerical examples.Our numerical results show that the proposed method not only preserves the exponential convergence,but also produces improved accuracy when applied to the unsteady Navier-Stokes equations having smooth solutions.Especially,the stabilized triangular spectral element method efficiently stabilizes the simulation of high Reynolds incompressible flows.展开更多
In this paper,we present a IP_N×IP_N spectral element method and a detailed comparison with existing methods for the unsteady incompressible Navier-Stokes equa- tions.The main purpose of this work consists of:(i)...In this paper,we present a IP_N×IP_N spectral element method and a detailed comparison with existing methods for the unsteady incompressible Navier-Stokes equa- tions.The main purpose of this work consists of:(i) detailed comparison and discussion of some recent developments of the temporal discretizations in the frame of spectral el- ement approaches in space;(ii) construction of a stable IP_N×IP_N method together with a IP_N→IP_(N-2) post-filtering.The link of different methods will be clarified.The key feature of our method lies in that only one grid is needed for both velocity and pressure variables,which differs from most well-known solvers for the Navier-Stokes equations. Although not yet proven by rigorous theoretical analysis,the stability and accuracy of this one-grid spectral method are demonstrated by a series of numerical experiments.展开更多
The strong motion of a small long and narrow basin caused by a moderate scenario earthquake is simulated by using the spectral-element method and the parallel computing technique.A total of five different geometrical ...The strong motion of a small long and narrow basin caused by a moderate scenario earthquake is simulated by using the spectral-element method and the parallel computing technique.A total of five different geometrical profiles within the basin are used to analyze the generation and propagation of surface waves and their relation to the basin structures in both the time and frequency domain.The amplification effects are analyzed by the distribution of peak ground velocity(PGV)and cumulative kinetic energy(Ek) in the basin.The results show that in the 3D basin,the excitation of the fundamental and higher surface wave modes are similar to that of the 2D model.Small bowls in the basin have great influence on the amplification and distribution of strong ground motion,due to their lateral resonances when the wavelengths of the lateral surface waves are comparable to the size of the bowls.Obvious basin edge effects can be seen at the basin edge closer to the source for constructive interference between direct body waves and the basin-induced surface waves.The Ek distribution maps show very large values in small bowls and some corners in the basin due to the interference of waves propagating in different directions.A high impedance contrast model can excite more surface wave modes,resulting in longer shaking durations as well as more complex seismograms and PGV and Ek distributions.展开更多
A diagonal or lumped mass matrix is of great value for time-domain analysis of structural dynamic and wave propagation problems,as the computational efforts can be greatly reduced in the process of mass matrix inversi...A diagonal or lumped mass matrix is of great value for time-domain analysis of structural dynamic and wave propagation problems,as the computational efforts can be greatly reduced in the process of mass matrix inversion.In this study,the nodal quadrature method is employed to construct a lumped mass matrix for the Chebyshev spectral element method(CSEM).A Gauss-Lobatto type quadrature,based on Gauss-Lobatto-Chebyshev points with a weighting function of unity,is thus derived.With the aid of this quadrature,the CSEM can take advantage of explicit time-marching schemes and provide an efficient new tool for solving structural dynamic problems.Several types of lumped mass Chebyshev spectral elements are designed,including rod,beam and plate elements.The performance of the developed method is examined via some numerical examples of natural vibration and elastic wave propagation,accompanied by their comparison to that of traditional consistent-mass CSEM or the classical finite element method(FEM).Numerical results indicate that the proposed method displays comparable accuracy as its consistent-mass counterpart,and is more accurate than classical FEM.For the simulation of elastic wave propagation in structures induced by high-frequency loading,this method achieves satisfactory performance in accuracy and efficiency.展开更多
In this paper, the spectral element method(SEM)is improved to solve the moving load problem. In this method, a structure with uniform geometry and material properties is considered as a spectral element, which means t...In this paper, the spectral element method(SEM)is improved to solve the moving load problem. In this method, a structure with uniform geometry and material properties is considered as a spectral element, which means that the element number and the degree of freedom can be reduced significantly. Based on the variational method and the Laplace transform theory, the spectral stiffness matrix and the equivalent nodal force of the beam-column element are established. The static Green function is employed to deduce the improved function. The proposed method is applied to two typical engineering practices—the one-span bridge and the horizontal jib of the tower crane. The results have revealed the following. First, the new method can yield extremely high-precision results of the dynamic deflection, the bending moment and the shear force in the moving load problem.In most cases, the relative errors are smaller than 1%. Second, by comparing with the finite element method, one can obtain the highly accurate results using the improved SEM with smaller element numbers. Moreover, the method can be widely used for statically determinate as well as statically indeterminate structures. Third, the dynamic deflection of the twin-lift jib decreases with the increase in the moving load speed, whereas the curvature of the deflection increases.Finally, the dynamic deflection, the bending moment and the shear force of the jib will all increase as the magnitude of the moving load increases.展开更多
Guided waves are generally considered as a powerful approach for crack detection in structures,which are commonly investigated using the finite element method(FEM).However,the traditional FEM has many disadvantages in...Guided waves are generally considered as a powerful approach for crack detection in structures,which are commonly investigated using the finite element method(FEM).However,the traditional FEM has many disadvantages in solving wave propagation due to the strict requirement of mesh density.To tackle this issue,this paper proposes an efficient time-domain spectral finite element method(SFEM)to analyze wave propagation in cracked structures,in which the breathing crack is modeled by definiiig the spectral gap element.Moreover,novel orthogonal polynomials and Gauss-Lobatto-Legendre quadrature rules are adopted to construct the spectral element.Meanwhile,a separable hard contact is utilized to simulate the breathing behavior.Finally,a comparison of the numerical results between the FEM and the SFEM is conducted to demonstrate the high efficiency and accuracy of the proposed method.Based on the developed SFEM,the nonlinear features of waves and influence of the incident mode are also studied in detail,which provides a helpful guide for a physical understanding of the wave propagation behavior in structures with breathing cracks.展开更多
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 ...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.展开更多
In this paper,a new strategy for a sub-element-based shock capturing for discontinuous Galerkin(DG)approximations is presented.The idea is to interpret a DG element as a col-lection of data and construct a hierarchy o...In this paper,a new strategy for a sub-element-based shock capturing for discontinuous Galerkin(DG)approximations is presented.The idea is to interpret a DG element as a col-lection of data and construct a hierarchy of low-to-high-order discretizations on this set of data,including a first-order finite volume scheme up to the full-order DG scheme.The dif-ferent DG discretizations are then blended according to sub-element troubled cell indicators,resulting in a final discretization that adaptively blends from low to high order within a single DG element.The goal is to retain as much high-order accuracy as possible,even in simula-tions with very strong shocks,as,e.g.,presented in the Sedov test.The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement and parallel computing.The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.展开更多
Spectral element method（SEM） for elastic media is well known for its great flexibility and high accuracy in solving problems with complex geometries.It is an advanced choice for wave simulations.Due to anelasticity ...Spectral element method（SEM） for elastic media is well known for its great flexibility and high accuracy in solving problems with complex geometries.It is an advanced choice for wave simulations.Due to anelasticity of earth media,SEM for elastic media is no longer appropriate.On fundamental of the second-order elastic SEM,this work takes the viscoelastic wave equations and the vertical transversely isotropic（VTI） media into consideration,and establishes the second-order SEM for wave modeling in viscoelastic VTI media.The second-order perfectly matched layer for viscoelastic VTI media is also introduced.The problem of handling the overlapped absorbed corners is solved.A comparison with the analytical solution in a twodimensional viscoelastic homogeneous medium shows that the method is accurate in the wave-field modeling.Furtherly,numerical validation also presents its great flexibility in solving wave propagation problems in complex heterogeneous media.This second-order SEM with perfectly matched layer for viscoelastic VTI media can be easily applied in wave modeling in a limited region.展开更多
A global spherical Fourier-Legendre spectral element method is proposed to solve Poisson equations and advective flow over a sphere. In the meridional direction, Legendre polynomials are used and the region is divided...A global spherical Fourier-Legendre spectral element method is proposed to solve Poisson equations and advective flow over a sphere. In the meridional direction, Legendre polynomials are used and the region is divided into several elements. In order to avoid coordinate singularities at the north and south poles in the meridional direction, Legendre-Gauss-Radau points are chosen at the elements involving the two poles. Fourier polynomials are applied in the zonal direction for its periodicity, with only one element. Then, the partial differential equations are solved on the longitude-latitude meshes without coordinate transformation between spherical and Cartesian coordinates. For verification of the proposed method, a few Poisson equations and advective flows are tested. Firstly, the method is found to be valid for test cases with smooth solution. The results of the Poisson equations demonstrate that the present method exhibits high accuracy and exponential convergence. High- precision solutions are also obtained with near negligible numerical diffusion during the time evolution for advective flow with smooth shape. Secondly, the results of advective flow with non-smooth shape and deformational flow are also shown to be reasonable and effective. As a result, the present method is proved to be capable of solving flow through different types of elements, and thereby a desirable method with reliability and high accuracy for solving partial differential equations over a sphere.展开更多
The tidal Love numbers of the Moon are a set of nondimensional parameters that describe the deformation responses of the Moon to the tidal forces of external celestial bodies.They play an important role in the theoret...The tidal Love numbers of the Moon are a set of nondimensional parameters that describe the deformation responses of the Moon to the tidal forces of external celestial bodies.They play an important role in the theoretical calculation of the Moon’s tidal deformation and the inversion of its internal structure.In this study,we introduce the basic theory for the theoretical calculation of the tidal Love numbers and propose a new method of solving the tidal Love numbers:the spectral element method.Moreover,we explain the mathematical theory and advantages of this method.On the basis of this new method,using 10 published lunar internal structure reference models,the lunar surface and lunar internal tidal Love numbers were calculated,and the influence of different lunar models on the calculated Love numbers was analyzed.Results of the calculation showed that the difference in the second-degree lunar surface Love numbers among different lunar models was within 8.5%,the influence on the maximum vertical displacement on the lunar surface could reach±8.5 mm,and the influence on the maximum gravity change could reach±6μGal.Regarding the influence on the Love numbers inside the Moon,different lunar models had a greater impact on the Love numbers h_(2) and l_(2) than on k_(2) in the lower lunar mantle and core.展开更多
The hierarchical reconstruction （HR） [Liu, Shu, Tadmor and Zhang, SINUM ＇07] has been successfully applied to prevent oscillations in solutions computed by finite volume, Runge-Kutta discontinuous Galerkin, spectra...The hierarchical reconstruction （HR） [Liu, Shu, Tadmor and Zhang, SINUM ＇07] has been successfully applied to prevent oscillations in solutions computed by finite volume, Runge-Kutta discontinuous Galerkin, spectral volume schemes for solving hyperbolic conservation laws. In this paper, we demonstrate that HR can also be combined with spectral/hp element method for solving hyperbolic conservation laws. An orthogonal spectral basis written in terms of Jacobi polynomials is applied. High computational efficiency is obtained due to such matrix-free algorithm. The formulation is conservative, and essential nomoscillation is enforced by the HR limiter. We show that HR preserves the order of accuracy of the spectral/hp element method for smooth solution problems and generate essentially non-oscillatory solutions profiles for capturing discontinuous solutions without local characteristic decomposition. In addition, we introduce a postprocessing technique to improve HR for limiting high degree numerical solutions.展开更多
Spectral element methods (SEM) are superior to general finite element methods (FEM) in achieving high order accuracy through p-type refinement. Owing to orthogonal polynomials in both expansion and test functions, the...Spectral element methods (SEM) are superior to general finite element methods (FEM) in achieving high order accuracy through p-type refinement. Owing to orthogonal polynomials in both expansion and test functions, the discretization errors in SEM could be reduced exponentially to machine zero so that the spectral convergence rate can be achieved. Inherited the advantage of FEM, SEM can enhance resolution via both h-type and p-type mesh-refinement. A penalty method was utilized to compute force fields in particulate flows involving freely moving rigid particles. Results were analyzed and comparisons were made;therefore, this penalty-implemented SEM was proven to be a viable method for two-phase flow problems.展开更多
Site effects study has always been a key research topic in earthquake engineering.This study proposes a hybrid method to analyze large-scale three-dimensional sedimentary basin under Rayleigh(R)wave incidence.The prop...Site effects study has always been a key research topic in earthquake engineering.This study proposes a hybrid method to analyze large-scale three-dimensional sedimentary basin under Rayleigh(R)wave incidence.The proposed hybrid method includes two steps:1)calculate the free field responses of layered sites subjected to R-wave using the frequency-wavenumber method;2)Simulate the local site region using spectral element method with the equivalent forces input computed from the free field responses.A comprehensive verification study is conducted demonstrating the accuracy of this method.To investigate the effect of sedimentary basin on R-wave propagation,a parametric study is performed on the medium impedance contrast ratio of sedimentary basins and the incident seismic wave predominant frequency,revealing the scattering patterns of sedimentary basins under R-wave incidence.Finally,a practical case of the Wudu Basin in the Tibetan Plateau region of China is simulated.Results indicate significant amplification of R-wave by sedimentary basin,and the proposed hybrid method could serve as a reliable and efficient approach for large-scale R-wave propagation simulation.展开更多
This paper studies the bandgap characteristics of a locally resonant metamaterial beam with time delays.The dispersion relations are addressed based on transfer matrix method.The governing equations of motion of the b...This paper studies the bandgap characteristics of a locally resonant metamaterial beam with time delays.The dispersion relations are addressed based on transfer matrix method.The governing equations of motion of the beam in the frequency domain are given according to spectral element method.The amplitude-frequency responses of the forced beam are determined by solving linear algebraic equations.The obtained results show that the time-delayed feedback control has great relationships with the location,width and number of the bandgaps.It is interesting that the time delay can change the direction of the movement of the bandgap and give rise to the generation of multiple bandgaps.The influences of different combinations of control parameters on the bandgap properties are shown,such as broadening effects.展开更多
The accurate simulation of wave propagation in real media requires properly taking the attenuation into account,which leads to wave dissipation together with its causal companion,wave dispersion.In this study,to obtai...The accurate simulation of wave propagation in real media requires properly taking the attenuation into account,which leads to wave dissipation together with its causal companion,wave dispersion.In this study,to obtain a weak formulation of heterogenous viscoacoustic wave propagation in an infinite domain,the viscoacoustic medium is first characterized by its frequency-dependent complex bulk compliance instead of the classically used complex bulk modulus.Then,a mechanical model using serially connected standard linear solids(SSLS)is built to obtain the rational approximation of the complex bulk compliance whose parameters are calculated via an adapted nonlinear optimization method.Utilizing the obtained bulk compliance-based constitutive relation,a novel second-order viscoacoustic wave equation in the frequency domain is derived,of which the weak formulation can be physically explained as the virtual work equation and can thus be discretized using a continuous spectral element method in space.Additionally,a new method is introduced to address the convolution terms involved in the inverse Fourier transform,whose accurate time integration can then be achieved using an explicit time scheme,which avoids the transient growth that exists in the classical method.The resulting full time-space decoupling scheme can handle wave propagation in arbitrary heterogeneous media.Moreover,to treat the wave propagation in an infinite domain,a perfectly matched layer in weak formulation is derived for the truncation of the infinite domain via complex coordinate stretching of the virtual work equation.With only minor modification,the resulting perfectly matched layer can be implemented using the same time scheme as for the wave equation inside the truncated domain.The accuracy,numerical stability,and versatility of the new proposed scheme are demonstrated with numerical examples.展开更多
Basin effect was first described following the analysis of seismic ground motion associated with the 1985 MW8.1 earthquake in Mexico.Basins affect the propagation of seismic waves through various mechanisms,and severa...Basin effect was first described following the analysis of seismic ground motion associated with the 1985 MW8.1 earthquake in Mexico.Basins affect the propagation of seismic waves through various mechanisms,and several unique phenomena,such as the basin edge effect,basin focusing effect,and basin-induced secondary waves,have been observed.Understanding and quantitatively predicting these phenomena are crucial for earthquake disaster reduction.Some pioneering studies in this field have proposed a quantitative relationship between the basin effect on ground motion and basin depth.Unfortunately,basin effect phenomena predicted using a model based only on basin depth exhibit large deviations from actual distributions,implying the severe shortcomings of single-parameter basin effect modeling.Quaternary sediments are thick and widely distributed in the Beijing-Tianjin-Hebei region.The seismic media inside and outside of this basin have significantly different physical properties,and the basin bottom forms an interface with strong seismic reflections.In this study,we established a three-dimensional structure model of the Quaternary sedimentary basin based on the velocity structure model of the North China Craton and used it to simulate the ground motion under a strong earthquake following the spectral element method,obtaining the spatial distribution characteristics of the ground motion amplification ratio throughout the basin.The back-propagation(BP)neural network algorithm was then introduced to establish a multi-parameter mathematical model for predicting ground motion amplification ratios,with the seismic source location,physical property ratio of the media inside and outside the basin,seismic wave frequency,and basin shape as the input parameters.We then examined the main factors influencing the amplification of seismic ground motion in basins based on the prediction results,and concluded that the main factors influencing the basin effect are basin shape and differences in the physical properties of media inside and outside the basin.展开更多
In the paper an important issue of vibrations of the transmission line in real conditions was analyzed.Such research was carried out by the authors of this paper taking into account the cross-section of the cable bein...In the paper an important issue of vibrations of the transmission line in real conditions was analyzed.Such research was carried out by the authors of this paper taking into account the cross-section of the cable being in use on the transmission line.Analysis was performed for the modern ACSR high voltage transmission line with span of 213.0 m.The purpose of the investigation was to analyze the vibrations of the power transmission line in the natural environment and compare with the results obtained in the numerical simulations.Analysis was performed for natural and wind excited vibrations.The numerical model was made using the Spectral Element Method.In the spectral model,for various parameters of stiffness,damping and tension force,the system response was checked and compared with the results of the accelerations obtained in the situ measurements.A frequency response functions(FRF)were calculated.The credibility of the model was assessed through a validation process carried out by comparing graphical plots of FRF functions and numerical values expressing differences in acceleration amplitude(MSG),phase angle differences(PSG)and differences in acceleration and phase angle total(CSG)values.Particular attention was paid to the hysteretic damping analysis.Sensitivity of the wave number was performed for changing of the tension force and section area of the cable.The next aspect constituting the purpose of this paper was to present the wide possibilities of modelling and simulation of slender conductors using the Spectral Element Method.The obtained results show very good accuracy in the range of both experimental measurements as well as simulation analysis.The paper emphasizes the ease with which the sensitivity of the conductor and its response to changes in density of spectral mesh division,cable cross-section,tensile strength or material damping can be studied.展开更多
基金S.H.Shao is partially supported by China Scholarship Council(CSC)and he also thanks Dr.Biegel for providing the reference[12].TLu is sponsored by SRF for ROCS,SEM and gratefully acknowledges the NSFC(Grant No.10701005)+2 种基金the support of NKBRP 2006 CB302705.WCai thanks the support of the United States Army Research Office(Grant No.W911NF-07-1-0492)a NSFC support(No.10828101).
文摘A new adaptive cell average spectral element method(SEM)is proposed to solve the time-dependent Wigner equation for transport in quantum devices.The proposed cell average SEM allows adaptive non-uniform meshes in phase spaces to reduce the high-dimensional computational cost of Wigner functions while preserving exactly the mass conservation for the numerical solutions.The key feature of the proposed method is an analytical relation between the cell averages of the Wigner function in the k-space(local electron density for finite range velocity)and the point values of the distribution,resulting in fast transforms between the local electron density and local fluxes of the discretized Wigner equation via the fast sine and cosine transforms.Numerical results with the proposed method are provided to demonstrate its high accuracy,conservation,convergence and a reduction of the cost using adaptive meshes.
基金funded in part by the SPICE-Marie Curie RTN project(contract MRTN-CT-2003-504267)supported by the ICTP Programme for Training and Research in Italian Laboratories,Trieste,Italy,and by CNPq,Brazil,under grant 314553/2009-6.
文摘Spectral element methods are well established in the field of wave propagation,in particular because they inherit the flexibility of finite element methods and have low numerical dispersion error.The latter is experimentally acknowledged,but has been theoretically shown only in limited cases,such as Cartesian meshes.It is well known that a finite element mesh can contain distorted elements that generate numerical errors for very large distortions.In the present work,we study the effect of element distortion on the numerical dispersion error and determine the distortion range in which an accurate solution is obtained for a given error tolerance.We also discuss a double-grid calculation of the spectral element matrices that preserves accuracy in deformed geometries.
基金NNW2018-ZT4A06 project and NSFC grant 11971408Lizhen Chen is partially supported by Grant U1930402.
文摘In this paper,we propose a spectral vanishing viscosity method for the triangular spectral element computation of high Reynolds number incompressible flows.This can be regarded as an extension of a similar stabilization technique for the standard spectral element method.The difficulty of this extension lies in the fact that a suitable definition of spectral vanishing viscosity operator in non-structured elements does not exist,and it is not clear that if a suitably defined spectral vanishing viscosity provides desirable dissipation for the artificially accumulated energy.The main contribution of the paper includes:1)a well-defined spectral vanishing viscosity operator is proposed for non-standard spectral element methods for the Navier-Stokes equations based on triangular or tetrahedron partitions;2)an evaluation technique is introduced to efficiently implement the stabilization term without extra computational cost;3)the accuracy and efficiency of the proposed method is carefully examined through several numerical examples.Our numerical results show that the proposed method not only preserves the exponential convergence,but also produces improved accuracy when applied to the unsteady Navier-Stokes equations having smooth solutions.Especially,the stabilized triangular spectral element method efficiently stabilizes the simulation of high Reynolds incompressible flows.
基金partially supported by National NSF of China under Grant 10602049The research of the second author was partially supported by National NSF of China under Grant 10531080+1 种基金the Excellent Young Teachers Program by the Ministry of Education of China973 High Performance Scientific Computation Research Program 2005CB321703.
文摘In this paper,we present a IP_N×IP_N spectral element method and a detailed comparison with existing methods for the unsteady incompressible Navier-Stokes equa- tions.The main purpose of this work consists of:(i) detailed comparison and discussion of some recent developments of the temporal discretizations in the frame of spectral el- ement approaches in space;(ii) construction of a stable IP_N×IP_N method together with a IP_N→IP_(N-2) post-filtering.The link of different methods will be clarified.The key feature of our method lies in that only one grid is needed for both velocity and pressure variables,which differs from most well-known solvers for the Navier-Stokes equations. Although not yet proven by rigorous theoretical analysis,the stability and accuracy of this one-grid spectral method are demonstrated by a series of numerical experiments.
基金National Natural Science Foundation of China under Grant No.51078337,No.51108431 and No.91315301
文摘The strong motion of a small long and narrow basin caused by a moderate scenario earthquake is simulated by using the spectral-element method and the parallel computing technique.A total of five different geometrical profiles within the basin are used to analyze the generation and propagation of surface waves and their relation to the basin structures in both the time and frequency domain.The amplification effects are analyzed by the distribution of peak ground velocity(PGV)and cumulative kinetic energy(Ek) in the basin.The results show that in the 3D basin,the excitation of the fundamental and higher surface wave modes are similar to that of the 2D model.Small bowls in the basin have great influence on the amplification and distribution of strong ground motion,due to their lateral resonances when the wavelengths of the lateral surface waves are comparable to the size of the bowls.Obvious basin edge effects can be seen at the basin edge closer to the source for constructive interference between direct body waves and the basin-induced surface waves.The Ek distribution maps show very large values in small bowls and some corners in the basin due to the interference of waves propagating in different directions.A high impedance contrast model can excite more surface wave modes,resulting in longer shaking durations as well as more complex seismograms and PGV and Ek distributions.
基金Supported by:Joint Research Fund for Earthquake Science,launched by the National Natural Science Foundation of China and the China Earthquake Administration under Grant No.U2039208。
文摘A diagonal or lumped mass matrix is of great value for time-domain analysis of structural dynamic and wave propagation problems,as the computational efforts can be greatly reduced in the process of mass matrix inversion.In this study,the nodal quadrature method is employed to construct a lumped mass matrix for the Chebyshev spectral element method(CSEM).A Gauss-Lobatto type quadrature,based on Gauss-Lobatto-Chebyshev points with a weighting function of unity,is thus derived.With the aid of this quadrature,the CSEM can take advantage of explicit time-marching schemes and provide an efficient new tool for solving structural dynamic problems.Several types of lumped mass Chebyshev spectral elements are designed,including rod,beam and plate elements.The performance of the developed method is examined via some numerical examples of natural vibration and elastic wave propagation,accompanied by their comparison to that of traditional consistent-mass CSEM or the classical finite element method(FEM).Numerical results indicate that the proposed method displays comparable accuracy as its consistent-mass counterpart,and is more accurate than classical FEM.For the simulation of elastic wave propagation in structures induced by high-frequency loading,this method achieves satisfactory performance in accuracy and efficiency.
基金supported by the National Key Technology R&D Program (Grant 2011BAJ02B01-02)the National Natural Science Foundation of China (Grant 11602065)
文摘In this paper, the spectral element method(SEM)is improved to solve the moving load problem. In this method, a structure with uniform geometry and material properties is considered as a spectral element, which means that the element number and the degree of freedom can be reduced significantly. Based on the variational method and the Laplace transform theory, the spectral stiffness matrix and the equivalent nodal force of the beam-column element are established. The static Green function is employed to deduce the improved function. The proposed method is applied to two typical engineering practices—the one-span bridge and the horizontal jib of the tower crane. The results have revealed the following. First, the new method can yield extremely high-precision results of the dynamic deflection, the bending moment and the shear force in the moving load problem.In most cases, the relative errors are smaller than 1%. Second, by comparing with the finite element method, one can obtain the highly accurate results using the improved SEM with smaller element numbers. Moreover, the method can be widely used for statically determinate as well as statically indeterminate structures. Third, the dynamic deflection of the twin-lift jib decreases with the increase in the moving load speed, whereas the curvature of the deflection increases.Finally, the dynamic deflection, the bending moment and the shear force of the jib will all increase as the magnitude of the moving load increases.
基金the National Natural Sclenee Foundation of China(Grant No.51704222)China Pastdoctoral Science Foundation(Grant No.2018M633570)Fundamental Research Funds for the Cemtal Unveritiee(Grant No.3102017090004).
文摘Guided waves are generally considered as a powerful approach for crack detection in structures,which are commonly investigated using the finite element method(FEM).However,the traditional FEM has many disadvantages in solving wave propagation due to the strict requirement of mesh density.To tackle this issue,this paper proposes an efficient time-domain spectral finite element method(SFEM)to analyze wave propagation in cracked structures,in which the breathing crack is modeled by definiiig the spectral gap element.Moreover,novel orthogonal polynomials and Gauss-Lobatto-Legendre quadrature rules are adopted to construct the spectral element.Meanwhile,a separable hard contact is utilized to simulate the breathing behavior.Finally,a comparison of the numerical results between the FEM and the SFEM is conducted to demonstrate the high efficiency and accuracy of the proposed method.Based on the developed SFEM,the nonlinear features of waves and influence of the incident mode are also studied in detail,which provides a helpful guide for a physical understanding of the wave propagation behavior in structures with breathing cracks.
基金supported by the National Basic Research Program of China (Grant 2013CB733004)
文摘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.
文摘In this paper,a new strategy for a sub-element-based shock capturing for discontinuous Galerkin(DG)approximations is presented.The idea is to interpret a DG element as a col-lection of data and construct a hierarchy of low-to-high-order discretizations on this set of data,including a first-order finite volume scheme up to the full-order DG scheme.The dif-ferent DG discretizations are then blended according to sub-element troubled cell indicators,resulting in a final discretization that adaptively blends from low to high order within a single DG element.The goal is to retain as much high-order accuracy as possible,even in simula-tions with very strong shocks,as,e.g.,presented in the Sedov test.The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement and parallel computing.The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.
基金financially supported by the National Natural Science Foundation of China (Grant No.41304077)Postdoctoral Science Foundation of China (Grant No.2013M531744,2014T70740)+1 种基金Key Laboratory of Geospace Environment and Geodesy (Grant No.12-02-03)Subsurface Multi-scale Imaging Laboratory (Grant No.SMIL-2014-01)
文摘Spectral element method（SEM） for elastic media is well known for its great flexibility and high accuracy in solving problems with complex geometries.It is an advanced choice for wave simulations.Due to anelasticity of earth media,SEM for elastic media is no longer appropriate.On fundamental of the second-order elastic SEM,this work takes the viscoelastic wave equations and the vertical transversely isotropic（VTI） media into consideration,and establishes the second-order SEM for wave modeling in viscoelastic VTI media.The second-order perfectly matched layer for viscoelastic VTI media is also introduced.The problem of handling the overlapped absorbed corners is solved.A comparison with the analytical solution in a twodimensional viscoelastic homogeneous medium shows that the method is accurate in the wave-field modeling.Furtherly,numerical validation also presents its great flexibility in solving wave propagation problems in complex heterogeneous media.This second-order SEM with perfectly matched layer for viscoelastic VTI media can be easily applied in wave modeling in a limited region.
基金supported by the Shandong Post-Doctoral Innovation Fund(Grant No.201303064)the Qingdao Post-Doctoral Application Research Project+1 种基金the National Basic Research(973) Program of China(Grant No.2012CB417402 and 2010CB950402)the National Natural Science Foundation of China(Grant No.41176017)
文摘A global spherical Fourier-Legendre spectral element method is proposed to solve Poisson equations and advective flow over a sphere. In the meridional direction, Legendre polynomials are used and the region is divided into several elements. In order to avoid coordinate singularities at the north and south poles in the meridional direction, Legendre-Gauss-Radau points are chosen at the elements involving the two poles. Fourier polynomials are applied in the zonal direction for its periodicity, with only one element. Then, the partial differential equations are solved on the longitude-latitude meshes without coordinate transformation between spherical and Cartesian coordinates. For verification of the proposed method, a few Poisson equations and advective flows are tested. Firstly, the method is found to be valid for test cases with smooth solution. The results of the Poisson equations demonstrate that the present method exhibits high accuracy and exponential convergence. High- precision solutions are also obtained with near negligible numerical diffusion during the time evolution for advective flow with smooth shape. Secondly, the results of advective flow with non-smooth shape and deformational flow are also shown to be reasonable and effective. As a result, the present method is proved to be capable of solving flow through different types of elements, and thereby a desirable method with reliability and high accuracy for solving partial differential equations over a sphere.
基金supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB4 1000000)the National Natural Science Foundation of China (Grant Nos. 42104006, 41974023, 42174101, 41874094, 41874026)the self-deployed foundation of the State Key Laboratory of Geodesy and Earth’s Dynamics (Grant No. S21L6404)
文摘The tidal Love numbers of the Moon are a set of nondimensional parameters that describe the deformation responses of the Moon to the tidal forces of external celestial bodies.They play an important role in the theoretical calculation of the Moon’s tidal deformation and the inversion of its internal structure.In this study,we introduce the basic theory for the theoretical calculation of the tidal Love numbers and propose a new method of solving the tidal Love numbers:the spectral element method.Moreover,we explain the mathematical theory and advantages of this method.On the basis of this new method,using 10 published lunar internal structure reference models,the lunar surface and lunar internal tidal Love numbers were calculated,and the influence of different lunar models on the calculated Love numbers was analyzed.Results of the calculation showed that the difference in the second-degree lunar surface Love numbers among different lunar models was within 8.5%,the influence on the maximum vertical displacement on the lunar surface could reach±8.5 mm,and the influence on the maximum gravity change could reach±6μGal.Regarding the influence on the Love numbers inside the Moon,different lunar models had a greater impact on the Love numbers h_(2) and l_(2) than on k_(2) in the lower lunar mantle and core.
基金Research was supported in part by NSF grant DMS-0800612Research was supported by Applied Mathematics program of the US DOE Office of Advanced Scientific Computing ResearchThe Pacific Northwest National Laboratory is operated by Battelle for the U.S. Department of Energy under Contract DE-AC05-76RL01830
文摘The hierarchical reconstruction （HR） [Liu, Shu, Tadmor and Zhang, SINUM ＇07] has been successfully applied to prevent oscillations in solutions computed by finite volume, Runge-Kutta discontinuous Galerkin, spectral volume schemes for solving hyperbolic conservation laws. In this paper, we demonstrate that HR can also be combined with spectral/hp element method for solving hyperbolic conservation laws. An orthogonal spectral basis written in terms of Jacobi polynomials is applied. High computational efficiency is obtained due to such matrix-free algorithm. The formulation is conservative, and essential nomoscillation is enforced by the HR limiter. We show that HR preserves the order of accuracy of the spectral/hp element method for smooth solution problems and generate essentially non-oscillatory solutions profiles for capturing discontinuous solutions without local characteristic decomposition. In addition, we introduce a postprocessing technique to improve HR for limiting high degree numerical solutions.
文摘Spectral element methods (SEM) are superior to general finite element methods (FEM) in achieving high order accuracy through p-type refinement. Owing to orthogonal polynomials in both expansion and test functions, the discretization errors in SEM could be reduced exponentially to machine zero so that the spectral convergence rate can be achieved. Inherited the advantage of FEM, SEM can enhance resolution via both h-type and p-type mesh-refinement. A penalty method was utilized to compute force fields in particulate flows involving freely moving rigid particles. Results were analyzed and comparisons were made;therefore, this penalty-implemented SEM was proven to be a viable method for two-phase flow problems.
基金supported by the National Natural Science Foundation of China(Grant Nos.U2139208 and 52178495).
文摘Site effects study has always been a key research topic in earthquake engineering.This study proposes a hybrid method to analyze large-scale three-dimensional sedimentary basin under Rayleigh(R)wave incidence.The proposed hybrid method includes two steps:1)calculate the free field responses of layered sites subjected to R-wave using the frequency-wavenumber method;2)Simulate the local site region using spectral element method with the equivalent forces input computed from the free field responses.A comprehensive verification study is conducted demonstrating the accuracy of this method.To investigate the effect of sedimentary basin on R-wave propagation,a parametric study is performed on the medium impedance contrast ratio of sedimentary basins and the incident seismic wave predominant frequency,revealing the scattering patterns of sedimentary basins under R-wave incidence.Finally,a practical case of the Wudu Basin in the Tibetan Plateau region of China is simulated.Results indicate significant amplification of R-wave by sedimentary basin,and the proposed hybrid method could serve as a reliable and efficient approach for large-scale R-wave propagation simulation.
文摘This paper studies the bandgap characteristics of a locally resonant metamaterial beam with time delays.The dispersion relations are addressed based on transfer matrix method.The governing equations of motion of the beam in the frequency domain are given according to spectral element method.The amplitude-frequency responses of the forced beam are determined by solving linear algebraic equations.The obtained results show that the time-delayed feedback control has great relationships with the location,width and number of the bandgaps.It is interesting that the time delay can change the direction of the movement of the bandgap and give rise to the generation of multiple bandgaps.The influences of different combinations of control parameters on the bandgap properties are shown,such as broadening effects.
基金National Natural Science Foundation of China under Grant No.U2039209the National Key R&D Program of China under Grant No.2022YFC3004303+1 种基金the Heilongjiang Natural Science Foundation for Distinguished Young Scholars under Grant No.JQ2022E006Heilongjiang Natural Science Foundation Joint Guidance Project under Grant No.LH2021E122。
文摘The accurate simulation of wave propagation in real media requires properly taking the attenuation into account,which leads to wave dissipation together with its causal companion,wave dispersion.In this study,to obtain a weak formulation of heterogenous viscoacoustic wave propagation in an infinite domain,the viscoacoustic medium is first characterized by its frequency-dependent complex bulk compliance instead of the classically used complex bulk modulus.Then,a mechanical model using serially connected standard linear solids(SSLS)is built to obtain the rational approximation of the complex bulk compliance whose parameters are calculated via an adapted nonlinear optimization method.Utilizing the obtained bulk compliance-based constitutive relation,a novel second-order viscoacoustic wave equation in the frequency domain is derived,of which the weak formulation can be physically explained as the virtual work equation and can thus be discretized using a continuous spectral element method in space.Additionally,a new method is introduced to address the convolution terms involved in the inverse Fourier transform,whose accurate time integration can then be achieved using an explicit time scheme,which avoids the transient growth that exists in the classical method.The resulting full time-space decoupling scheme can handle wave propagation in arbitrary heterogeneous media.Moreover,to treat the wave propagation in an infinite domain,a perfectly matched layer in weak formulation is derived for the truncation of the infinite domain via complex coordinate stretching of the virtual work equation.With only minor modification,the resulting perfectly matched layer can be implemented using the same time scheme as for the wave equation inside the truncated domain.The accuracy,numerical stability,and versatility of the new proposed scheme are demonstrated with numerical examples.
基金funded by the General Program of the National Natural Science Foundation of China(No.42174070)the General Program of the Beijing Natural Science Foundation(No.8222035).
文摘Basin effect was first described following the analysis of seismic ground motion associated with the 1985 MW8.1 earthquake in Mexico.Basins affect the propagation of seismic waves through various mechanisms,and several unique phenomena,such as the basin edge effect,basin focusing effect,and basin-induced secondary waves,have been observed.Understanding and quantitatively predicting these phenomena are crucial for earthquake disaster reduction.Some pioneering studies in this field have proposed a quantitative relationship between the basin effect on ground motion and basin depth.Unfortunately,basin effect phenomena predicted using a model based only on basin depth exhibit large deviations from actual distributions,implying the severe shortcomings of single-parameter basin effect modeling.Quaternary sediments are thick and widely distributed in the Beijing-Tianjin-Hebei region.The seismic media inside and outside of this basin have significantly different physical properties,and the basin bottom forms an interface with strong seismic reflections.In this study,we established a three-dimensional structure model of the Quaternary sedimentary basin based on the velocity structure model of the North China Craton and used it to simulate the ground motion under a strong earthquake following the spectral element method,obtaining the spatial distribution characteristics of the ground motion amplification ratio throughout the basin.The back-propagation(BP)neural network algorithm was then introduced to establish a multi-parameter mathematical model for predicting ground motion amplification ratios,with the seismic source location,physical property ratio of the media inside and outside the basin,seismic wave frequency,and basin shape as the input parameters.We then examined the main factors influencing the amplification of seismic ground motion in basins based on the prediction results,and concluded that the main factors influencing the basin effect are basin shape and differences in the physical properties of media inside and outside the basin.
文摘In the paper an important issue of vibrations of the transmission line in real conditions was analyzed.Such research was carried out by the authors of this paper taking into account the cross-section of the cable being in use on the transmission line.Analysis was performed for the modern ACSR high voltage transmission line with span of 213.0 m.The purpose of the investigation was to analyze the vibrations of the power transmission line in the natural environment and compare with the results obtained in the numerical simulations.Analysis was performed for natural and wind excited vibrations.The numerical model was made using the Spectral Element Method.In the spectral model,for various parameters of stiffness,damping and tension force,the system response was checked and compared with the results of the accelerations obtained in the situ measurements.A frequency response functions(FRF)were calculated.The credibility of the model was assessed through a validation process carried out by comparing graphical plots of FRF functions and numerical values expressing differences in acceleration amplitude(MSG),phase angle differences(PSG)and differences in acceleration and phase angle total(CSG)values.Particular attention was paid to the hysteretic damping analysis.Sensitivity of the wave number was performed for changing of the tension force and section area of the cable.The next aspect constituting the purpose of this paper was to present the wide possibilities of modelling and simulation of slender conductors using the Spectral Element Method.The obtained results show very good accuracy in the range of both experimental measurements as well as simulation analysis.The paper emphasizes the ease with which the sensitivity of the conductor and its response to changes in density of spectral mesh division,cable cross-section,tensile strength or material damping can be studied.
