极坐标系下非分裂PML及时域有限元实现

在弹性波传播的数值模拟中,吸收边界被广泛应用于截取有限空间进行无限空间问题的分析.完全匹配层（perfect matched layer, PML）吸收边界较其他吸收边界条件具有更优越的吸收性能,已被成功应用于直角坐标系下的弹性波方程正演模拟.考虑极坐标系下二阶弹性波动方程,通过采用辅助函数的方法,提出了一种非分裂格式的完全匹配层吸收边界条件.并且基于Galerkin近似技术,给出了非对称以及轴对称条件下的时域有限元计算格式.通过数值算例分析了该极坐标系下分裂格式的完全匹配层吸收边界的有效性.     

An energy-momentum conserving scheme for Hamiltonian wave equation based on multiquadric trigonometric quasi-interpolation

Based on multiquadric trigonometric quasi-interpolation, the paper proposes a meshless symplectic scheme for Hamiltonian wave equation with periodic boundary conditions. The scheme first discretizes the equation in space using an iterated derivative approximation method based on multiquadric trigonometric quasi-interpolation and then in time with an appropriate symplectic scheme. This in turn yields a finite-dimensional semi-discrete Hamiltonian system whose energy and momentum (approximations of the continuous ones) are invariant with respect to time. The key feature of the scheme is that it conserves both the energy and momentum of the Hamiltonian system for both uniform and scattered centers, while classical energy-momentum conserving schemes are only for uniform centers. Numerical examples provided at the end of the paper show that the scheme is efficient and easy to implement.     

Efficient finite-difference multi-scheme approach to the simulation of seismic waves in anisotropic media

This paper presents an original multi-scheme approach to the numerical simulation of seismic wave propagation in models with anisotropic formations. To simulate wave propagation in the anisotropic parts of the model, the Lebedev scheme is used. This scheme is rather universal, but highly expensive in terms of computational efficiency. In the main part of the model, a highly efficient standard staggered grid scheme is proposed. The two schemes are coupled to ensure convergence of the reflection/propagation coefficients with a prescribed order. The algorithm combines the universality of the Lebedev scheme and the efficiency of the standard staggered grid scheme.     

一类带有空间时间白噪音随机弹性方程的全离散差分格式

随机弹性方程在结构工程中有许多应用.本文研究一类由空间时间白噪音扰动的随机弹性方程的全离散有限差分格式.通过引入新的函数,将随机弹性方程表示成一阶方程组的形式,然后对噪音项进行分片常数逼近,构造了带有空间时间白噪音随机弹性方程的全离散差分格式.基于对Gronwall不等式和Burkholder不等式的应用,证明了格式的L~p收敛性并得到了收敛阶.在数值实验中结合Monte-Carlo方法,所得实验结果与理论分析是一致的.     

Modified Legendre Rational Spectral Method for the Whole Line

A mutually orthogonal system of rationalSome approximation results are establishedfunctions on the whole line is introduced.As an example of applications, a modified Legendre rational spectral scheme is given for the Dirac equation. Its numerical solution keeps the same conservation as the genuine solution. This feature not only leads to reasonable numerical simulation of nonlinear waves, but also simplifies the analysis. The convergence of the proposed scheme is proved. Numerical results demonstrate the efficiency of this new approach and coincide with the analysis well.     

Computation, Dimensionality, and Zero-Dissipation limit of the Ginzburg-Landau Wave Equation

In this paper, convergence is proved and optimal error estimatesare obtained for the Galerkin approximation scheme for computingspatially period solutions of the Ginzburg-Landau wave equation.Using this convergent scheme, we carry out a series of numericalexperiments towards an understanding of the finite dimensionalityof the wave motion evolving from a small neighbourhd of thezero solution. In the zerodissipation Limit, we show that thesolutions approach those of the cubic SchrÖdinger equation.     

Dispersion analysis of triangle-based spectral element methods for elastic wave propagation

We study the numerical dispersion/dissipation of Triangle-based Spectral Element Methods (TSEM) of order N????1 when coupled with the Leap-Frog (LF) finite difference scheme to simulate the elastic wave propagation over a structured triangulation of the 2D physical domain. The analysis relies on the discrete eigenvalue problem resulting from the approximation of the dispersion relation. First, we present semi-discrete dispersion graphs by varying the approximation polynomial degree and the number of discrete points per wavelength. The fully-discrete ones are then obtained by varying also the time step. Numerical results for the TSEM, resp. TSEM-LF, are compared with those of the classical Quadrangle-based Spectral Element Method (QSEM), resp. QSEM-LF.     

Composite generalized Laguerre–Legendre pseudospectral method for Fokker–Planck equation in an infinite channel

In this paper, we propose a composite generalized Laguerre–Legendre pseudospectral method for the Fokker–Planck equation in an infinite channel, which behaves like a parabolic equation in one direction, and behaves like a hyperbolic equation in other direction. We establish some approximation results on the composite generalized Laguerre–Legendre–Gauss–Radau interpolation, with which the convergence of proposed composite scheme follows. An efficient implementation is provided. Numerical results show the spectral accuracy in space of this approach and coincide well with theoretical analysis. The approximation results and techniques developed in this paper are also very appropriate for many other problems on multiple-dimensional unbounded domains, which are not of standard types.     

Solitary waves in a tapered prestressed fluid-filled elastic tube

In the present work, treating the arteries as a tapered, thin walled, long and circularly conical prestressed elastic tube and using the longwave approximation, we have studied the propagation of weakly nonlinear waves in such a fluid-filled elastic tube by employing the reductive perturbation method. By considering the blood as an incompressible inviscid fluid the evolution equation is obtained as the Korteweg-de Vries equation with a variable coefficient. It is shown that this type of equations admit a solitary wave type of solution with variable wave speed. It is observed that, the wave speed increases with distance for positive tapering while it decreases for negative tapering.     

Solitary waves in a tapered prestressed fluid-filled elastic tube

In the present work, treating the arteries as a tapered, thin walled, long and circularly conical prestressed elastic tube and using the longwave approximation, we have studied the propagation of weakly nonlinear waves in such a fluid-filled elastic tube by employing the reductive perturbation method. By considering the blood as an incompressible inviscid fluid the evolution equation is obtained as the Korteweg-de Vries equation with a variable coefficient. It is shown that this type of equations admit a solitary wave type of solution with variable wave speed. It is observed that, the wave speed increases with distance for positive tapering while it decreases for negative tapering.     

Solitary waves in a fluid-filled thin elastic tube with variable cross-section

The present work treats the arteries as a thin walled prestressed elastic tube with variable cross-section and uses the longwave approximation to study the propagation of weakly nonlinear waves in such a fluid-filled elastic tube by employing the reductive perturbation method. By considering the blood as an incompressible inviscid fluid, the evolution equation is obtained as the Korteweg–de Vries equation with a variable coefficient. It is shown that this type of equations admits a solitary wave type of solution with variable wave speed. It is observed that, for soft biological tissues with an exponential strain energy function the wave speed increases with distance for narrowing tubes while it decreases for expanding tubes.     

Summatory and integral equations in periodic diffraction problems for elastic waves at defects in stratified media

We consider the diffraction problem for an elastic wave on a periodic set of defects located at the interface of stratified media. We reduce the mentioned problem to a pair summatory functional equation with respect to coefficients of the expansion of the desired wave by quasiperiodic waves (the Floquet waves). Using the method of integral identities, we reduce the pair equation to a regular infinite system of linear equations. One can solve this system by the truncation method. We prove that the integral identity is the necessary and sufficient condition for the solvability of the auxiliary overspecified problem for a system of equations in a half-plane in the elasticity theory. We obtain integral equations of the second kind which are equivalent to the initial diffraction problem.     

The three-dimensional contact problem for a wedge-shaped valve

The three-dimensional contact problem for an elastic wedge-shaped valve, situated in a wedge-shaped cavity in an elastic space, is investigated. A regular asymptotic method is used to solve the integral equation of this problem. The method is effective for a contact area relatively far from the edge of the wedge-shaped cavity. Calculations are carried out. The solutions of the three-dimensional auxiliary problems on the equilibrium of an elastic wedge-shaped cavity and an elastic wedge are based on well-known Green's functions, constructed using Fourier and Kontorovich–Lebedev integral transformations.     

Popular ansatz methods and solitary wave solutions of the Kuramoto-Sivashinsky equation

Some methods to look for exact solutions of nonlinear differential equations are discussed. It is shown that many popular methods are equivalent to each other. Several recent publications with “new” solitary wave solutions for the Kuramoto-Sivashinsky equation are analyzed. We demonstrate that all these solutions coincide with the known ones.      

Weakly nonlinear waves in a linearly tapered elastic tube filled with a fluid

In the present work, treating the arteries as a tapered, thin-walled, long and circularly conical prestressed elastic tube and using the long-wave approximation, we have studied the propagation of weakly nonlinear waves in such a fluid-filled elastic tube by employing the reductive perturbation method. By considering the blood as an incompressible inviscid fluid the evolution equation is obtained as the Korteweg-de Vries equation with a variable coefficient. It is shown that this type of equations admits a solitary wave type of solution with variable wave speed. It is observed that the wave speed increases with the scaled time parameter τ for positive tapering while it decreases for negative tapering, as expected.     

Spectrum comparison for a conserved reaction-diffusion system with a variational property

We are dealing with a two-component system of reaction-diffusion equations with conservation of a mass in a bounded domain subject to the Neumann or the periodic boundary conditions. We consider the case that the conserved system is transformed into a phase-field type system. Then the stationary problem is reduced to that of a scalar reaction-diffusion equation with a nonlocal term. We study the linearized eigenvalue problem of an equilibrium solution to the system, and compare the eigenvalues with ones of the linearized problem arising from the scalar nonlocal equation in terms of the Rayleigh quotient. The main theorem tells that the number of negative eigenvalues of those problems coincide. Hence, a stability result for nonconstant solutions of the scalar nonlocal reaction-diffusion equation is applicable to the system.     

A Novel Time‐Domain BIEM for Wave Propagation Analysis in Anisotropic Solids With Cracks

Analysis of elastic wave propagation in anisotropic solids with cracks is of particular interest to quantitative non‐destructive testing and fracture mechanics. For this purpose, a novel time‐domain boundary integral equation method (BIEM) is presented in this paper. A finite crack in an unbounded elastic solid of general anisotropy subjected to transient elastic wave loading is considered. Two‐dimensional plane strain or plane stress condition is assumed. The initial‐boundary value problem is formulated as a set of hypersingular time‐domain traction boundary integral equations (BIEs) with the crack‐opening‐displacements (CODs) as unknown quantities. A time‐stepping scheme is developed for solving the hypersingular time‐domain BIEs. The scheme uses the convolution quadrature formula of Lubich [1] for temporal convolution and a Galerkin method for spatial discretization of the BIEs. An important feature of the present time‐domain BIEM is that it uses the Laplace‐domain instead of the more complicated time‐domain Green's functions. Fourier integral representations of Laplace‐domain Green's functions are applied. No special technique is needed in the present time‐domain BIEM for evaluating hypersingular integrals.     

DFT modal analysis of spectral element methods for the 2D elastic wave equation

The DFT modal analysis is a dispersion analysis technique that transforms the equations of a numerical scheme to the discrete Fourier transform domain sampled in the mesh nodes. This technique provides a natural matching of exact and approximate modes of propagation. We extend this technique to spectral element methods for the 2D isotropic elastic wave equation, by using a Rayleigh quotient approximation of the eigenvalue problem that characterizes the dispersion relation, taking full advantage of the tensor product representation of the spectral element matrices. Numerical experiments illustrate the dependence of dispersion errors on the grid resolution, polynomial degree, and discretization in time. We consider spectral element methods with Chebyshev and Legendre collocation points.     

A high‐order finite difference method for 1D nonhomogeneous heat equations

In this article a sixth‐order approximation method (in both temporal and spatial variables) for solving nonhomogeneous heat equations is proposed. We first develop a sixth‐order finite difference approximation scheme for a two‐point boundary value problem, and then heat equation is approximated by a system of ODEs defined on spatial grid points. The ODE system is discretized to a Sylvester matrix equation via boundary value method. The obtained algebraic system is solved by a modified Bartels‐Stewart method. The proposed approach is unconditionally stable. Numerical results are provided to illustrate the accuracy and efficiency of our approximation method along with comparisons with those generated by the standard second‐order Crank‐Nicolson scheme as well as Sun‐Zhang's recent fourth‐order method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009