Spectral Element Viscosity Methods for Nonlinear Conservation Laws on the Semi-Infinite Interval

In this paper we propose a spectral element: vanishing viscosity (SEW) method for the conservation laws on the semi-infinite interval. By using a suitable mapping, the problem is first transformed into a modified conservation law in a bounded interval, then the well-known spectral vanishing viscosity technique is generalized to the multi-domain case in order to approximate this trarsformed equation more efficiently. The construction details and convergence analysis are presented. Under a usual assumption of boundedness of the approximation solutions, it is proven that the solution of the SEW approximation converges to the uniciue entropy solution of the conservation laws. A number of numerical tests is carried out to confirm the theoretical results.

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.

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.

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.

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.

An efficient source wavefield reconstruction scheme using single boundary layer values for the spectral element method

In the adjoint-state method, the forward-propagated source wavefield and the backward-propagated receiver wavefield must be available simultaneously either for seismic imaging in migration or for gradient calculation in inversion. A feasible way to avoid the excessive storage demand is to reconstruct the source wavefield backward in time by storing the entire history of the wavefield in perfectly matched layers. In this paper, we make full use of the elementwise global property of the Laplace operator of the spectral element method (SEM) and propose an efficient source wavefield reconstruction method at the cost of storing the wavefield history only at single boundary layer nodes. Numerical experiments indicate that the accuracy of the proposed method is identical to that of the conventional method and is independent of the order of the Lagrange polynomials, the element type, and the temporal discretization method. In contrast, the memory-saving ratios of the conventional method versus our method is at least N when using either quadrilateral or hexahedron elements, respectively, where N is the order of the Lagrange polynomials used in the SEM. A higher memorysaving ratio is achieved with triangular elements versus quadrilaterals. The new method is applied to reverse time migration by considering the Marmousi model as a benchmark. Numerical results demonstrate that the method is able to provide the same result as the conventional method but with about 1/25 times lower storage demand. With the proposed wavefield reconstruction method, the storage demand is dramatically reduced;therefore, in-core memory storage is feasible even for large-scale three-dimensional adjoint inversion problems.

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.

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.

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.

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.

Finite Difference Preconditioners for Legendre Based Spectral Element Methods on Elliptic Boundary Value Problems

Finite difference type preconditioners for spectral element discretizations based on Legendre-Gauss-Lobatto points are analyzed. The latter is employed for the approximation of uniformly elliptic partial differential problems. In this work, it is shown that the condition number of the resulting preconditioned system is bounded independently of both of the polynomial degrees used in the spectral element method and the element sizes. Several numerical tests verify the h-p independence of the proposed preconditioning.

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.

C1-Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations

This paper is devoted to Professor Benyu Guo's open question on the C1-conforming quadrilateral spectral element method for fourth-order equations which has been endeavored for years. Starting with generalized Jacobi polynomials on the reference square, we construct the C1-conforming basis functions using the bilinear mapping from the reference square onto each quadrilateral element which fall into three categories-interior modes, edge modes, and vertex modes. In contrast to the triangular element, compulsively compensatory requirements on the global C1-continuity should be imposed for edge and vertex mode basis functions such that their normal derivatives on each common edge are reduced from rational functions to polynomials, which depend on only parameters of the common edge. It is amazing that the C1-conforming basis functions on each quadrilateral element contain polynomials in primitive variables, the completeness is then guaranteed and further confirmed by the numerical results on the Petrov-Galerkin spectral method for the non-homogeneous boundary value problem of fourth-order equations on an arbitrary quadrilateral. Finally, a C1-conforming quadrilateral spectral element method is proposed for the biharmonic eigenvalue problem, and numerical experiments demonstrate the effectiveness and efficiency of our spectral element method.

A MPI PARALLEL PRECONDITIONED SPECTRAL ELEMENT METHOD FOR THE HELMHOLTZ EQUATION

Spectral element method is well known as high-order method, and has potential better parallel feature as compared with low order methods. In this paper, a parallel preconditioned conjugate gradient iterative method is proposed to solving the spectral element approximation of the Helmholtz equation. The parallel algorithm is shown to have good performance as compared to non parallel cases, especially when the stiffness matrix is not memorized. A series of numerical experiments in one dimensional case is carried out to demonstrate the efficiency of the proposed method.

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.

Spectral-element simulations of elastic wave propagation in exploration and geotechnical applications

We apply the spectral-element method（SEM）,a high-order finite-element method（FEM） to simulate seismic wave propagation in complex media for exploration and geotechnical problems. The SEM accurately treats geometrical complexities through its flexible FEM mesh and accurately interpolates wavefields through high-order Lagrange polynomials. It has been a numerical solver used extensively in earthquake seismology. We demonstrate the applicability of SEM for selected 2D exploration and geotechnical velocity models with an open-source SEM software package SPECFEM2D. The first scenario involves a marine survey for a salt dome with the presence of major internal discontinuities,and the second example simulates seismic wave propagation for an open-pit mine with complex surface topography. Wavefield snapshots,synthetic seismograms,and peak particle velocity maps are presented to illustrate the promising use of SEM for industrial problems.

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.

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

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

High-Order Spectral Stochastic Finite Element Analysis of Stochastic Elliptical Partial Differential Equations

This study presents an experiment of improving the performance of spectral stochastic finite element method using high-order elements. This experiment is implemented through a two-dimensional spectral stochastic finite element formulation of an elliptic partial differential equation having stochastic coefficients. Deriving this spectral stochastic finite element formulation couples a two-dimensional deterministic finite element formulation of an elliptic partial differential equation with generalized polynomial chaos expansions of stochastic coefficients. Further inspection of the performance of resulting spectral stochastic finite element formulation with adopting linear and quadratic (9-node or 8-node) quadrilateral elements finds that more accurate standard deviations of unknowns are surprisingly predicted using quadratic quadrilateral elements, especially under high autocorrelation function values of stochastic coefficients. In addition, creating spectral stochastic finite element results using quadratic quadrilateral elements is not unacceptably time-consuming. Therefore, this study concludes that adopting high-order elements can be a lower-cost method to improve the performance of spectral stochastic finite element method.

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.