Spectral element methods on simplicial meshes,say TSEM,show both the advantages of spectral and finite element methods,i.e.,spectral accuracy and geometrical flexibility.We present a TSEM solver of the two-dimensional...Spectral element methods on simplicial meshes,say TSEM,show both the advantages of spectral and finite element methods,i.e.,spectral accuracy and geometrical flexibility.We present a TSEM solver of the two-dimensional(2D)incompressible Navier-Stokes equations,with possible extension to the 3D case.It uses a projection method in time and piecewise polynomial basis functions of arbitrary degree in space.The so-called Fekete-Gauss TSEM is employed,i.e.,Fekete(resp.Gauss)points of the triangle are used as interpolation(resp.quadrature)points.For the sake of consistency,isoparametric elements are used to approximate curved geometries.The resolution algorithm is based on an efficient Schur complement method,so that one only solves for the element boundary nodes.Moreover,the algebraic system is never assembled,therefore the number of degrees of freedom is not limiting.An accuracy study is carried out and results are provided for classical benchmarks:the driven cavity flow,the flow between eccentric cylinders and the flow past a cylinder.展开更多
The additive Schwarz preconditioner with minimal overlap is extended to triangular spectral elements(TSEM).Themethod is a generalization of the corresponding method in tensorial quadrilateral spectral elements(QSEM).T...The additive Schwarz preconditioner with minimal overlap is extended to triangular spectral elements(TSEM).Themethod is a generalization of the corresponding method in tensorial quadrilateral spectral elements(QSEM).The proposed preconditioners are based on partitioning the domain into overlapping subdomains,solving local problems on these subdomains and solving an additional coarse problem associated with the subdomain mesh.The results of numerical experiments show that the proposed preconditioner are robust with respect to the number of elements and are more efficient than the preconditioners with generous overlaps.展开更多
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 dom...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.展开更多
In this paper,we propose a systematic approach for accelerating finite element-type methods by machine learning for the numerical solution of partial differential equations(PDEs).The main idea is to use a neural netwo...In this paper,we propose a systematic approach for accelerating finite element-type methods by machine learning for the numerical solution of partial differential equations(PDEs).The main idea is to use a neural network to learn the solution map of the PDEs and to do so in an element-wise fashion.This map takes input of the element geometry and the PDE’s parameters on that element,and gives output of two operators:(1)the in2out operator for inter-element communication,and(2)the in2sol operator(Green’s function)for element-wise solution recovery.A significant advantage of this approach is that,once trained,this network can be used for the numerical solution of the PDE for any domain geometry and any parameter distribution without retraining.Also,the training is significantly simpler since it is done on the element level instead on the entire domain.We call this approach element learning.This method is closely related to hybridizable discontinuous Galerkin(HDG)methods in the sense that the local solvers of HDG are replaced by machine learning approaches.Numerical tests are presented for an example PDE,the radiative transfer or radiation transport equation,in a variety of scenarios with idealized or realistic cloud fields,with smooth or sharp gradient in the cloud boundary transition.Under a fixed accuracy level of 10^(−3) in the relative L^(2) error,and polynomial degree p=6 in each element,we observe an approximately 5 to 10 times speed-up by element learning compared to a classical finite element-type method.展开更多
This study presents an effective hybrid simulation approach for simulating broadband ground motion in complex near-fault locations.The approach utilizes a deterministic approach based on the spectral element method(SE...This study presents an effective hybrid simulation approach for simulating broadband ground motion in complex near-fault locations.The approach utilizes a deterministic approach based on the spectral element method(SEM),which is used to simulate low-frequency ground motion(f<1 Hz)by incorporating an innovative efficient discontinuous Galerkin(DG)method for grid division to accurately model basin sedimentary layers at reduced costs.It also introduces a comprehensive hybrid source model for high-frequency random scattering and a nonlinear analysis module for basin sedimentary layers.Deterministic outcomes are combined with modified three-dimensional stochastic finite fault method(3D-EXSIM)simulations of high-frequency ground motion(f>1 Hz).A fourth-order Butterworth filter with zero phase shift is employed for time-domain filtering of low-and high-frequency time series at a crossover frequency of 1 Hz,merging the low and high-frequency ground motions into a broadband time series.Taking an Ms 6.8 Luding earthquake,as an example,this hybrid method was used for a rapid and efficient simulation analysis of broadband ground motion in the region.The accuracy and efficiency of this hybrid method were verified through comparisons with actually observed station data and empirical attenuation curves.Deterministic method simulation results revealed the effects of mountainous topography,basin effects,nonlinear effects within the basin’s sedimentary layers,and a coupling interaction between the basin and the mountains.The findings are consistent with similar studies,showing that near-fault sedimentary basins significantly focus and amplify strong ground motion,and the soil’s nonlinear behavior in the basin influences ground motion to varying extents at different distances from the fault.The mountainous topography impacts the basin’s response to ground motion,leading to barrier effects.This research provides a scientific foundation for seismic zoning,urban planning,and seismic design in nearfault mountain basin regions.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
Stiffened panels have been widely utilized in fuselages and wings as critical load-bearing components. These structures are prone to be damaged under long-term and extreme loads, and their health monitoring has been a...Stiffened panels have been widely utilized in fuselages and wings as critical load-bearing components. These structures are prone to be damaged under long-term and extreme loads, and their health monitoring has been a common concern. The guided wave-based monitoring method is regarded as an efficient approach to detect the damage in stiffened plates because of its wide monitoring range and high sensitivity to micro-damage. Efficient simulation of wave propagation can theoretically demonstrate the detection mechanism of the method. In this study, a Time-Domain Spectral Finite Element Method(TD-SFEM) is adopted to study the wavefield in stiffened plates,where continuous Absorbing Layers with Increasing Damping(ALID) strategy is proposed to circumvent the disturbance of reflected waves on boundaries. After the convergence analysis, the developed TD-SFEM with ALID is validated by the finite element method first. Then, wave scattering and the influence of the stiffener are investigated in detail by comparing the results with the non-stiffened structure. Finally, the effects of the parameters of the stiffener, such as the height and width, on wave propagation are studied, respectively. The results illustrate that the proposed TDSFEM with ALID is an efficient approach to study the wave propagation in the stiffened plate and can reveal the mechanism of influence of the stiffener. It is found that the height of the stiffener changes the interference of wavefield in the plate, while the effects of the width are mainly in wave scattering and mode conversion.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
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
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
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
This paper developed a method,called compilation of Q-data and simulation of Q-spectra,combining experimental data with the aid of a microcomputer to predict and correct spectral interferences in rare-earth elements a...This paper developed a method,called compilation of Q-data and simulation of Q-spectra,combining experimental data with the aid of a microcomputer to predict and correct spectral interferences in rare-earth elements analysis by using ICP-AES. Raw spectral data of each element were obtained through experiments, followed by removing noise with Kalman smoothing and spectral averaging, and correcting the shifts of wavelength. These processed data were eventually transformed into Q-data and used in spectral simulation and intenference correction. Some fundamental problems in simulation and correction were investigated and the results indicates that Q-data are accurate enough for the correction of spectral interferences when the interferences are not too strong, and that spectral simulation is practicable in routine analysis. It is a convenient, rapid and accurate way to deal with spectral interferences in REE analvsis.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
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
文摘Spectral element methods on simplicial meshes,say TSEM,show both the advantages of spectral and finite element methods,i.e.,spectral accuracy and geometrical flexibility.We present a TSEM solver of the two-dimensional(2D)incompressible Navier-Stokes equations,with possible extension to the 3D case.It uses a projection method in time and piecewise polynomial basis functions of arbitrary degree in space.The so-called Fekete-Gauss TSEM is employed,i.e.,Fekete(resp.Gauss)points of the triangle are used as interpolation(resp.quadrature)points.For the sake of consistency,isoparametric elements are used to approximate curved geometries.The resolution algorithm is based on an efficient Schur complement method,so that one only solves for the element boundary nodes.Moreover,the algebraic system is never assembled,therefore the number of degrees of freedom is not limiting.An accuracy study is carried out and results are provided for classical benchmarks:the driven cavity flow,the flow between eccentric cylinders and the flow past a cylinder.
文摘The additive Schwarz preconditioner with minimal overlap is extended to triangular spectral elements(TSEM).Themethod is a generalization of the corresponding method in tensorial quadrilateral spectral elements(QSEM).The proposed preconditioners are based on partitioning the domain into overlapping subdomains,solving local problems on these subdomains and solving an additional coarse problem associated with the subdomain mesh.The results of numerical experiments show that the proposed preconditioner are robust with respect to the number of elements and are more efficient than the preconditioners with generous overlaps.
基金supported by the National Natural Science Foundation of China(Grant No.40774099,10874202)the National High Technology Research and Development Program of China(Grant No.2008AA06Z205)
文摘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.
基金partially supported by the NSF(Grant No.DMS 2324368)by the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin-Madison with funding from the Wisconsin Alumni Research Foundation.
文摘In this paper,we propose a systematic approach for accelerating finite element-type methods by machine learning for the numerical solution of partial differential equations(PDEs).The main idea is to use a neural network to learn the solution map of the PDEs and to do so in an element-wise fashion.This map takes input of the element geometry and the PDE’s parameters on that element,and gives output of two operators:(1)the in2out operator for inter-element communication,and(2)the in2sol operator(Green’s function)for element-wise solution recovery.A significant advantage of this approach is that,once trained,this network can be used for the numerical solution of the PDE for any domain geometry and any parameter distribution without retraining.Also,the training is significantly simpler since it is done on the element level instead on the entire domain.We call this approach element learning.This method is closely related to hybridizable discontinuous Galerkin(HDG)methods in the sense that the local solvers of HDG are replaced by machine learning approaches.Numerical tests are presented for an example PDE,the radiative transfer or radiation transport equation,in a variety of scenarios with idealized or realistic cloud fields,with smooth or sharp gradient in the cloud boundary transition.Under a fixed accuracy level of 10^(−3) in the relative L^(2) error,and polynomial degree p=6 in each element,we observe an approximately 5 to 10 times speed-up by element learning compared to a classical finite element-type method.
基金National Natural Science Foundation of China under Grant Nos.U2139208 and 52278516Key Laboratory of Earthquake Engineering and Engineering Vibration,China Earthquake Administration under Grant No.2024D15Key Laboratory of Soft Soil Characteristic and Engineering Environment,Tianjin Chengjian University under Grant No.2022SCEEKL003。
文摘This study presents an effective hybrid simulation approach for simulating broadband ground motion in complex near-fault locations.The approach utilizes a deterministic approach based on the spectral element method(SEM),which is used to simulate low-frequency ground motion(f<1 Hz)by incorporating an innovative efficient discontinuous Galerkin(DG)method for grid division to accurately model basin sedimentary layers at reduced costs.It also introduces a comprehensive hybrid source model for high-frequency random scattering and a nonlinear analysis module for basin sedimentary layers.Deterministic outcomes are combined with modified three-dimensional stochastic finite fault method(3D-EXSIM)simulations of high-frequency ground motion(f>1 Hz).A fourth-order Butterworth filter with zero phase shift is employed for time-domain filtering of low-and high-frequency time series at a crossover frequency of 1 Hz,merging the low and high-frequency ground motions into a broadband time series.Taking an Ms 6.8 Luding earthquake,as an example,this hybrid method was used for a rapid and efficient simulation analysis of broadband ground motion in the region.The accuracy and efficiency of this hybrid method were verified through comparisons with actually observed station data and empirical attenuation curves.Deterministic method simulation results revealed the effects of mountainous topography,basin effects,nonlinear effects within the basin’s sedimentary layers,and a coupling interaction between the basin and the mountains.The findings are consistent with similar studies,showing that near-fault sedimentary basins significantly focus and amplify strong ground motion,and the soil’s nonlinear behavior in the basin influences ground motion to varying extents at different distances from the fault.The mountainous topography impacts the basin’s response to ground motion,leading to barrier effects.This research provides a scientific foundation for seismic zoning,urban planning,and seismic design in nearfault mountain basin regions.
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.
基金National Natural Science Foundation of China(Nos.12072268 and 51705422)。
文摘Stiffened panels have been widely utilized in fuselages and wings as critical load-bearing components. These structures are prone to be damaged under long-term and extreme loads, and their health monitoring has been a common concern. The guided wave-based monitoring method is regarded as an efficient approach to detect the damage in stiffened plates because of its wide monitoring range and high sensitivity to micro-damage. Efficient simulation of wave propagation can theoretically demonstrate the detection mechanism of the method. In this study, a Time-Domain Spectral Finite Element Method(TD-SFEM) is adopted to study the wavefield in stiffened plates,where continuous Absorbing Layers with Increasing Damping(ALID) strategy is proposed to circumvent the disturbance of reflected waves on boundaries. After the convergence analysis, the developed TD-SFEM with ALID is validated by the finite element method first. Then, wave scattering and the influence of the stiffener are investigated in detail by comparing the results with the non-stiffened structure. Finally, the effects of the parameters of the stiffener, such as the height and width, on wave propagation are studied, respectively. The results illustrate that the proposed TDSFEM with ALID is an efficient approach to study the wave propagation in the stiffened plate and can reveal the mechanism of influence of the stiffener. It is found that the height of the stiffener changes the interference of wavefield in the plate, while the effects of the width are mainly in wave scattering and mode conversion.
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.
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.
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This paper developed a method,called compilation of Q-data and simulation of Q-spectra,combining experimental data with the aid of a microcomputer to predict and correct spectral interferences in rare-earth elements analysis by using ICP-AES. Raw spectral data of each element were obtained through experiments, followed by removing noise with Kalman smoothing and spectral averaging, and correcting the shifts of wavelength. These processed data were eventually transformed into Q-data and used in spectral simulation and intenference correction. Some fundamental problems in simulation and correction were investigated and the results indicates that Q-data are accurate enough for the correction of spectral interferences when the interferences are not too strong, and that spectral simulation is practicable in routine analysis. It is a convenient, rapid and accurate way to deal with spectral interferences in REE analvsis.
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.
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.
