Three-dimensional simulations of strong ground motion in the Shidian basin based upon the spectral-element method

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.

High-precision solution to the moving load problem using an improved spectral element method

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.

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

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.

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

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

Seismic wave modeling in viscoelastic VTI media using spectral element method

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.

A IP_N×IP_N Spectral Element Projection Method for the Unsteady Incompressible Navier-Stokes Equations

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.

A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

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.

Theoretical calculation of tidal Love numbers of the Moon with a new spectral element method

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.

SPECTRAL/HP ELEMENT METHOD WITH HIERARCHICAL RECONSTRUCTION FOR SOLVING NONLINEAR HYPERBOLIC CONSERVATION LAWS

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.

Physics-based seismic analysis of ancient wood structure:fault-to-structure simulation

Based on the domain reduction method,this study employs an SEM-FEM hybrid workflow which integrates the advantages of the spectral element method(SEM)for flexible and highly efficient simulation of seismic wave propagation in a three-dimensional(3D)regional-scale geophysics model and the finite element method(FEM)for fine simulation of structural response including soil-structure interaction,and performs a physics-based simulation from initial fault rupture on an ancient wood structure.After verification of the hybrid workflow,a large-scale model of an ancient wood structure in the Beijing area,The Tower of Buddhist Incense,is established and its responses under the 1665 Tongxian earthquake and the 1730 Yiheyuan earthquake are simulated.The results from the simulated ground motion and seismic response of the wood structure under the two earthquakes demonstrate that this hybrid workflow can be employed to efficiently provide insight into the relationships between geophysical parameters and the structural response,and is of great significance toward accurate input for seismic simulation of structures under specific site and fault conditions.

On the Monotonicity of Q^(3) Spectral Element Method for Laplacian

The monotonicity of discrete Laplacian, i.e., inverse positivity of stiffness matrix, implies discrete maximum principle, which is in general not true for high order accurate schemes on unstructured meshes. On the other hand,it is possible to construct high order accurate monotone schemes on structured meshes. All previously known high order accurate inverse positive schemes are or can be regarded as fourth order accurate finite difference schemes, which is either an M-matrix or a product of two M-matrices. For the Q3spectral element method for the two-dimensional Laplacian, we prove its stiffness matrix is a product of four M-matrices thus it is unconditionally monotone. Such a scheme can be regarded as a fifth order accurate finite difference scheme. Numerical tests suggest that the unconditional monotonicity of Q^(k) spectral element methods will be lost for k ≥ 9 in two dimensions, and for k ≥ 4 in three dimensions. In other words, for obtaining a high order monotone scheme, only Q^(2) and Q^(3) spectral element methods can be unconditionally monotone in three dimensions.

A Global Spectral Element Model for Poisson Equations and Advective Flow over a Sphere

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.

Measurements in Situ and Spectral Analysis of Wind Flow Effects on Overhead Transmission Lines

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.

Spectral Element Simulation of Rotating Particle in Viscous Flow

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.

3D numerical simulation on fluid-structure interaction of structure subjected to underwater explosion with cavitation

In the underwater-shock environment, cavitation occurs near the structural surface. The dynamic response of fluid-structure interactions is influenced seriously by the cavitation effects. It is also the difficulty in the field of underwater explosion. With the traditional boundary element method and the finite element method （FEM）, it is difficult to solve the nonlinear problem with cavitation effects subjected to the underwater explosion. To solve this problem, under the consideration of the cavitation effects and fluid compressibility, with fluid viscidity being neglected, a 3D numerical model of transient nonlinear fluid-structure interaction subjected to the underwater explosion is built. The fluid spectral element method （SEM） and the FEM are adopted to solve this model. After comparison with the FEM, it is shown that the SEM is more precise than the FEM, and the SEM results are in good coincidence with benchmark results and experiment results. Based on this, combined with ABAQUS, the transient fluid-structure interaction mechanism of the 3D submerged spherical shell and ship stiffened plates subjected to the underwater explosion is discussed, and the cavitation region and its influence on the structural dynamic responses are presented. The paper aims at providing references for relevant research on transient fluid-structure interaction of ship structures subjected to the underwater explosion.

Petrov-Galerkin Spectral Element Method for Mixed Inhomogeneous Boundary Value Problems on Polygons

The authors investigate Petrov-Galerkin spectral element method. Some results on Legendre irrational quasi-orthogonal approximations are established, which play important roles in Petrov-Galerkin spectral element method for mixed inhomogeneous boundary value problems of partial differential equations defined on polygons. As examples of applications, spectral element methods for two model problems, with the spectral accuracy in certain Jacobi weighted Sobolev spaces, are proposed. The techniques developed in this paper are also applicable to other higher order methods.

An H^m-conforming spectral element method on multi-dimensional domain and its application to transmission eigenvalues

We develop an Hm-conforming(m 1) spectral element method on multi-dimensional domain associated with the partition into multi-dimensional rectangles. We construct a set of basis functions on the interval [-1, 1] that are made up of the generalized Jacobi polynomials(GJPs) and the nodal basis functions.So the basis functions on multi-dimensional rectangles consist of the tensorial product of the basis functions on the interval [-1, 1]. Then we construct the spectral element interpolation operator and prove the associated interpolation error estimates. Finally, we apply the H2-conforming spectral element method to the Helmholtz transmission eigenvalues that is a hot problem in the field of engineering and mathematics.

Some progress in spectral methods Dedicated to Professor Shi Zhong-Ci on the Occasion of his 80th Birthday

In this paper, we review some results on the spectral methods. We first consider the Jacobi spectral method and the generalized Jacobi spectral method for various problems, including degenerated and singular differential equations. Then we present the generalized Jacobi quasi-orthogonal approximation and its applica- tions to the spectral element methods for high order problems with mixed inhomogeneous boundary conditions. We also discuss the related spectral methods for non-rectangular domains and the irrational spectral methods for unbounded domains. Next, we consider the Hermite spectral method and the generalized Hermite spec- tral method with their applications. Finally, we consider the Laguerre spectral method and the generalized Laguerre spectral method for many problems defined on unbounded domains. We also present the generalized Laguerre quasi-orthogonal approximation and its applications to certain problems of non-standard type and exterior problems.

Time-domain Spectral Finite Element Method for Wave Propagation Analysis in Structures with Breathing Cracks

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.

Adaptive Conservative Cell Average Spectral Element Methods for Transient Wigner Equation in Quantum Transport

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.