The implementation of an improved NPML absorbing boundary condition in elastic wave modeling

在向前当模特儿的有弹性的波浪，吸收边界条件(ABC ) 被用来从模型截断边界减轻不希望得到的思考。完美地匹配的层(PML ) 证明了是最好的可得到的 ABC。然而，传统的切开 PML (SPML ) ABC 有一些严肃的劣势：例如，全球 SPML ABC 要求更多计算记忆，尽管实现是容易的。本地 SPML ABC 的实现也有一些困难，自从边和角落必须被考虑。因为卷绕旋转，传统的非切开的完美地匹配的层(NPML ) ABC 有复杂计算。在这篇论文，基于 ABC 与复杂转移频率的拉长函数(CFS ) 相结合的非切开的完美地匹配的层(NPML ) ，我们为与简单计算方程，小存储器要求，和容易的编程吸收边界条件的 PML 介绍一个新奇数字实现方法。

A note for the mechanism of high-frequency oscillation instability resulted from absorbing boundary conditions

In this paper the explanation of the mechanism of high-frequency oscillation instability resulted from absorbing boundary conditions is further improved. And we analytically prove the proposition that for one dimensional discrete model of elastic wave motion, the module of reflection factor will be greater than 1 in high frequency band when artificial wave velocity is greater than 1.5 times the ratio of discrete space step to discrete time step. Based on the proof, the frequency band in which instability occurs is discussed in detail, showing such high-frequency waves are meaningless for the numerical simulation of wave motion.

APPLICATION OF THE PML ABSORBING BOUNDARY CONDITION TO FD-TD SIMULATION

The key problem of finite-difference time-domain (FD-TD) method is the skillful application of special conditions on the boundaries of the computational domain. A new technique named Perfectly Matched Layer(PML) yields a robust Absorbing Boundary Condition(ABC) independent of the angle of incidence and the frequency of outgoing waves. In this paper, the principle of the PML technique is briefly presented. Then some problems in the application and their settlements are discussed emphatically. Finally three numerical tests and a measured result are devoted to examine the accuracy and effectiveness of this approach.

A Novel Absorbing Boundary Condition for the Frequency-Dependent Finite-Difference Time-Domain Method

A new absorbing boundary condition (ABC) for frequency dependent finite difference time domain algorithm for the arbitrary dispersive media is presented. The concepts of the digital systems are introduced to the (FD) 2TD method. On the basis of digital filter designing and vector algebra, the absorbing boundary condition under arbitrary angle of incidence are derived. The transient electromagnetic problems in two dimensions and three dimensions are calculated and the validity of the ABC is verified.

Absorbing Boundary Conditions for Hyperbolic Systems

This paper deals with absorbing boundary conditions for hyperbolic systems in one and two space dimensions.We prove the strict well-posedness of the resulting initial boundary value problem in 1D.Afterwards we establish the GKS-stability of the corresponding Lax-Wendroff-type finite difference scheme.Hereby,we have to extend the classical proofs,since the(discretized) absorbing boundary conditions do not fit the standard form of boundary conditions for hyperbolic systems.

Time Dependent Wave Propagation Modeling Using Finite Difference Scheme of 2D Wave Equation Based on Absorbing and Reflecting Boundaries

Boundary procedure is an important phenomenon in numerical simulation. To reduce or eliminate the spurious reflections significantly which is occurred in boundary is a challenging and vital approach. The appropriate artificial numerical boundaries can be applied to eliminate the effect of unnecessary spurious reflections in case of the numerical simulations of wave propagation phenomena problems. Typically, to reduce the artificial reflections, the absorbing boundary conditions are necessary. In this paper, we overview and investigate the appropriate typical absorbing boundary conditions and analyzed the boundary effect of two dimensional wave equation numerically. Reflections over the wide-ranging incident angles are complicated to eliminate, but the absorbing boundary conditions that we have applied are computationally cost efficient, easy to apply and able to reduce reflections significantly. For numerical solution, finite difference method is applied to develop numerical scheme using 2D wave equation. Using the developed numerical scheme, we obtain the numerical solution of the governing equation as an initial boundary value problem and realize the qualitative behavior of the solution in infinite space. The finite difference numerical scheme has been investigated by developing MATLAB programming language code. Numerical results have been discussed and analyzed with presenting different qualitative behavior of the numerical scheme. The accuracy and efficiency of the numerical scheme has been illustrated. The stability analysis was discussed and verified stability condition. Using the numerical scheme and absorbing boundary conditions, the boundary effects and absorption of spurious reflection of boundary have been demonstrated.

STABILITY FOR IMPOSING ABSORBING BOUNDARY CONDITIONS IN THE FINITE ELEMENT SIMULATION OF ACOUSTIC WAVE PROPAGATION＊

It is well-known that artificial boundary conditions are crucial for the efficient and accurate computations of wavefields on unbounded domains. In this paper, we investigate stability analysis for the wave equation coupled with the first and the second order absorbing boundary conditions. The computational scheme is also developed. The approach allows the absorbing boundary conditions to be naturally imposed, which makes it easier for us to construct high order schemes for the absorbing boundary conditions. A thirdorder Lagrange finite element method with mass lumping is applied to obtain the spatial discretization of the wave equation. The resulting scheme is stable and is very efficient since no matrix inversion is needed at each time step. Moreover, we have shown both abstract and explicit conditional stability results for the fully-discrete schemes. The results are helpful for designing computational parameters in computations. Numerical computations are illustrated to show the efficiency and accuracy of our method. In particular, essentially no boundary reflection is seen at the artificial boundaries.

PML Absorbing Boundary Condition for Seismic Numerical Modeling by Convolutional Differentiator in Fluid-Saturated Porous Media

The perfectly matched layer（PML） was first introduced by Berenger as an absorbing boundary condition for electromagnetic wave propagation.In this article,a method is developed to ex-tend the PML to simulating seismic wave propagation in fluid-saturated porous medium.This non-physical boundary is used at the computational edge of a Forsyte polynomial convolutional differenti-ator（FPCD） algorithm as an absorbing boundary condition to truncate unbounded media.The incor-poration of PML in Biot＇s equations is given.Numerical results show that the PML absorbing bound-ary condition attenuates the outgoing waves effectively and eliminates the reflections adequately.

HIGH-ORDER LOCAL ABSORBING BOUNDARY CONDITIONS FOR HEAT EQUATION IN UNBOUNDED DOMAINS

With the development of numerical methods the numerical computations require higher and higher accuracy. This paper is devoted to the high-order local absorbing boundary conditions （ABCs） for heat equation. We proved that the coupled system yields a stable problem between the obtained high-order local ABCs and the partial differential equation in the computational domain. This method has been used widely in wave propagation models only recently. We extend the spirit of the methodology to parabolic ones, which will become a basis to design the local ABCs for a class of nonlinear PDEs. Some numerical tests show that the new treatment is very efficient and tractable.

STABILITY ANALYSIS OF FINITE ELEMENT METHODS FORTHE ACOUSTIC WAVE EQUATION WITH ABSORBINGBOUNDARY CONDITIONS (PART I)

In Part I and Part II of this paper initial-boundary value problems of the acoustic wave equation with absorbing boundary conditions are considered. Their finite element-finite difference computational schemes are proposed. The stability of the schemes is discussed and the corresponding stability conditions are given. Part I and Part II concern the first- and the second-order absorbing boundary conditions, respectively. Finally, numerical results are presented in Part II to show the correctness of theoretical analysis. (Author abstract) 7 Refs.

Solution of the time-dependent Schrdinger equation with absorbing boundary conditions

The performances of absorbing boundary conditions （ABCs） in four widely used finite difference time domain （FDTD） methods, i.e. explicit, implicit, explicit staggered-time, and Chebyshev methods, for solving the time-dependent Schrodinger equation are assessed and compared. The computation efficiency for each approach is also evaluated. A typical evolution problem of a single Gaussian wave packet is chosen to demonstrate the performances of the four methods combined with ABCs. It is found that ABCs perfectly eliminate reflection in implicit and explicit staggered-time methods. However, small reflection still exists in explicit and Chebyshev methods even though ABCs are applied.

THE LONG WAVE OSCILLATIONS IN HARBORS WITH ABSORBING BOUNDARY CONDITIONS

This paper employs finite element method to solve shallow water equations with absorbing boundary conditions(the third kind,mixed boundary conditions).It is of practical importance in the cases that the land boundaries of the coastal area are made of porous medium allowing sea water flow in or out.The absorbing boundary conditions are treated as natural boundary conditions in wave equation finite element model.The numerical results for rectangu- lar and quarterly annular harbors indicate that the numerical solutions agree very well with ana- lytic solutions,which are also given in this paper.It is found that the land boundary absorbabili- ty may be significant to long wave oscillations,such as tidal waves in coastal harbors.

Well-Posedness of Initial Boundary Value ProblemsAssociated with Absorbing Boundary Conditions

In this paper we get one-way wave equations by using pseudo-differential operator theory,and present a set of absorbing boundary conditions based on the higher order aPProximations of oneway wave equations. An integral identity is the key point of the approximation. Also, we have provedthe well-posedness of the initial boundary value Problems related to our absorbing boundary conditionsconstructed in this artical.

Implementation of absorbing boundary conditions in dynamic simulation of the material point method

Outgoing waves arising from high-velocity impacts between soil and structure can be reflected by the conventional truncated boundaries.Absorbing boundary conditions(ABCs),to attenuate the energy of the outward waves,are necessary to ensure the proper representation of the kinematic field and the accurate quantification of impact forces.In this paper,damping layer and dashpot ABCs are implemented in the material point method(MPM)with slight adjustments.Benchmark scenarios of different dynamic problems are modelled with the ABCs configured.Feasibility of the ABCs is assessed through the velocity fluctuations at specific observation points and the impact force fluctuations on the structures.The impact forces predicted by the MPM with ABCs are verified by comparison with those estimated using a computational fluid dynamics approach.

Practical Absorbing Boundary Conditions forWave Propagation on Arbitrary Domain

This paper presents an absorbing boundary conditions(ABCs)for wave propagations on arbitrary computational domains.The purpose of ABCs is to eliminate the unwanted spurious reflection at the artificial boundaries and minimize the finite size effect.Traditional methods are usually complicate in theoretical derivation and implementation and work only for very limited types of boundary geometry.In contrast to other existing methods,our emphasis is placed on the ease of implementation.In particular,we propose a method for which the implementation can be done by fitting or learning from the simulation data in a larger domain,and it is insensitive to the geometry and space dimension of the computational domain.Furthermore,a stability criterion is imposed to ensure the stability of the proposed ABC.Numerical results are presented to demonstrate the effectiveness of our method.

Analysis of High-Order Absorbing Boundary Conditions for the Schrodinger Equation

The paper is concerned with the numerical solution of Schr¨odinger equations on an unbounded spatial domain.High-order absorbing boundary conditions for one-dimensional domain are derived,and the stability of the reduced initial boundary value problem in the computational interval is proved by energy estimate.Then a second order finite difference scheme is proposed,and the convergence of the scheme is established as well.Finally,numerical examples are reported to confirm our error estimates of the numerical methods.

An Exact Absorbing Boundary Condition for the Schr¨odinger Equation With Sinusoidal Potentials at Infinity

In this paper we study numerical issues related to the Schr ¨odinger equationwith sinusoidal potentials at infinity. An exact absorbing boundary condition in a formof Dirichlet-to-Neumann mapping is derived. This boundary condition is based on ananalytical expression of the logarithmic derivative of the Floquet solution toMathieu’sequation, which is completely new to the author’s knowledge. The implementationof this exact boundary condition is discussed, and a fast evaluation method is used toreduce the computation burden arising from the involved half-order derivative operator.Some numerical tests are given to showthe performance of the proposed absorbingboundary conditions.

Absorbing Boundary Conditions for Solving N-Dimensional Stationary Schr¨odinger Equations with Unbounded Potentials and Nonlinearities

We propose a hierarchy of novel absorbing boundary conditions for the onedimensional stationary Schr¨odinger equation with general(linear and nonlinear)potential.The accuracy of the new absorbing boundary conditions is investigated numerically for the computation of energies and ground-states for linear and nonlinear Schr¨odinger equations.It turns out that these absorbing boundary conditions and their variants lead to a higher accuracy than the usual Dirichlet boundary condition.Finally,we give the extension of these ABCs to N-dimensional stationary Schr¨odinger equations.

带矢量ABC底面的共形完全匹配层

共形完全匹配层是一种有耗各向异性媒质组成的凸且光滑的壳体,其底面一般是PEC面或PMC面,但是PEC面或PMC面会对原散射场产生反射;为了减少底面反射,将CPML原有的PEC(或PMC)底面改为矢量ABC吸收边界,并给出了带矢量ABC底面的CPML泛函公式。通过数值算例证明,这种带矢量ABC底面的CPML边界不仅减少了底面反射,而且吸收效果好,计算精度高。

基于改进的EAL边界条件的地震波场数值模拟

在地震波场数值模拟过程中,人工截断边界会产生数值反射而影响地震波场模拟的精度.应用PML(Perfectly Matched Layer)吸收边界条件是消除人工边界反射的有效手段,但是该方法计算量较大,实现较为复杂,且在某些各向异性介质中不稳定.EAL(Effective Absorbing Layer)边界条件是通过简化PML吸收边界条件的阻尼项推导出来的一种吸收边界条件.EAL的吸收项作用于波动方程的时间导数项,故EAL边界条件具有计算量小且利于并行实现的特点.应用EAL边界条件能够在保证全波形反演和逆时偏移精度的前提下,提升计算效率并降低内存消耗.本文对EAL边界条件进行两点改进:(1)引入减速阻尼项;(2)采用高斯型空间加权函数构建阻尼项.正演模拟结果表明,改进的EAL边界条件在不增加计算量和内存消耗的同时,有效提升了EAL边界条件的吸收效果且适用于复杂介质和各向异性介质.特别地,相比于PML吸收边界条件,EAL边界条件在各向异性模型中表现出良好的稳定性.