Absorbing Boundary Conditions for Simulating Water Waves Near Solid Bodies

The objective of this paper is to present a new method for designing absorbing or non-reflective boundary conditions (ABC) or (NRBC), illustrated by the case study of the modelling of a solid body in water, specifically the capillary gravity waves generated by its motion at the surface. The study analyses the flow of an inviscid, barotropic, and compressible fluid around the stationary solid body. The dynamic behaviour of the fluid is analysed using a two-dimensional coupled Neumann-Kelvin model extended with capillarity and inertia terms. For computational purposes, it is necessary to truncate the unbounded spatial domain with artificial boundaries and then introduce appropriate absorbing boundary conditions. The propagation of short wavelength waves in a convective fluid medium with significant differences in properties between the interior and the surface of the fluid presents a number of difficulties in the design of these conditions. The results are illustrated numerically and commented upon.

Perfectly matched layer-absorbing boundary condition for finite-element time-domain modeling of elastic wave equations

The perfectly matched layer （PML） is a highly efficient absorbing boundary condition used for the numerical modeling of seismic wave equation. The article focuses on the application of this technique to finite-element time-domain numerical modeling of elastic wave equation. However, the finite-element time-domain scheme is based on the second- order wave equation in displacement formulation. Thus, the first-order PML in velocity-stress formulation cannot be directly applied to this scheme. In this article, we derive the finite- element matrix equations of second-order PML in displacement formulation, and accomplish the implementation of PML in finite-element time-domain modeling of elastic wave equation. The PML has an approximate zero reflection coefficients for bulk and surface waves in the finite-element modeling of P-SV and SH wave propagation in the 2D homogeneous elastic media. The numerical experiments using a two-layer model with irregular topography validate the efficiency of PML in the modeling of seismic wave propagation in geological models with complex structures and heterogeneous media.

Hybrid absorbing boundary condition for threedimensional elastic wave modeling

Edge reflections are inevitable in numerical modeling of seismic wavefields, and they are usually attenuated by absorbing boundary conditions. However, the commonly used perfectly matched layer （PML） boundary condition requires special treatment for the absorbing zone, and in three-dimensional （3D） modeling, it has to split each variable into three corresponding variables, which increases the computing time and memory storage. In contrast, the hybrid absorbing boundary condition （HABC） has the advantages such as ease of implementation, less computation time, and near-perfect absorption; it is thus able to enhance the computational efficiency of 3D elastic wave modeling. In this study, a HABC is developed from two-dimensional （2D） modeling into 3D modeling based on the I st Higdon one way wave equations, and a HABC is proposed that is suitable for a 3D elastic wave numerical simulation. Numerical simulation results for a homogenous model and a complex model indicate that the proposed HABC method is more effective and has better absorption than the traditional PML method.

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.

High-Accurate Transparent Boundary Conditions for Time-Dependent Quantum Wave Packet Method

Complex absorbing potential is usually required in a time-dependent wave packet method to accomplish the calculation in a truncated region.Usually it works effectively but becomes inefficient when the wave function involves translational energy of broad range,particularly involving ultra-low energy.In this work,a new transparent boundary condition(TBC)is proposed for the time-dependent wave packet method.It in principle is of spectral accuracy when typical discrete variable representations are applied.The prominent merit of the new TBC is that its accuracy is insensitive to the translational energy distribution of the wave function,in contrast with the complex absorbing potential.Application of the new TBC is given to one-dimensional particle wave packet scatterings from a barrier with a potential well,which supports resonances states.

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.

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.

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.

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.

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.

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.

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.

An improved convolution perfectly matched layer for elastic second-order wave equation

A convolution perfectly matched layer(CPML)can efficiently absorb boundary reflection in numerical simulation.However,the CPML is suitable for the first-order elastic wave equation and is difficult to apply directly to the second-order elastic wave equation.In view of this,based on the first-order CPML absorbing boundary condition,we propose a new CPML(NCPML)boundary which can be directly applied to the second-order wave equation.We first systematically extend the first-order CPML technique into second-order wave equations,neglecting the space-varying characteristics of the partial damping coefficient in the complex-frequency domain,avoiding the generation of convolution in the time domain.We then transform the technique back to the time domain through the inverse Fourier transform.Numerical simulation indicates that the space-varying characteristics of the attenuation factor have little influence on the absorption effect and increase the memory at the same time.A number of numerical examples show that the NCPML proposed in this study is effective in simulating elastic wave propagation,and this algorithm is more efficient and requires less memory allocation than the conventional PML absorbing boundary.

On Local Nonreflecting Boundary Conditions for Time Dependent Wave Propagation

The simulation of wave phenomena in unbounded domains generally requires an artificial boundary to truncate the unbounded exterior and limit the computation to a finite region.At the artificial boundary a boundary condition is then needed,which allows the propagating waves to exit the computational domain without spurious reflection.In 1977,Engquist and Majda proposed the first hierarchy of absorbing boundary conditions,which allows a systematic reduction of spurious reflection without moving the artificial boundary farther away from the scatterer.Their pioneering work,which initiated an entire research area,is reviewed here from a modern perspective.Recent developments such as high-order local conditions and their extension to multiple scattering are also presented.Finally,the accuracy of high-order local conditions is demonstrated through numerical experiments.

Micro-Differential Boundary Conditions Modelling the Absorption of Acoustic Waves by 2D Arbitrarily-Shaped Convex Surfaces

We propose a new Absorbing Boundary Condition(ABC)for the acoustic wave equation which is derived from a micro-local diagonalization process formerly defined by M.E.Taylor and which does not depend on the geometry of the surface bearing the ABC.By considering the principal symbol of the wave equation both in the hyperbolic and the elliptic regions,we show that a second-order ABC can be constructed as the combination of an existing first-order ABC and a Fourier-Robin condition.We compare the new ABC with other ABCs and we show that it performs well in simple configurations and that it improves the accuracy of the numerical solution without increasing the computational burden.

Frequency-domain Elastic wave Simulation Based on the Nonoverlapping Domain Decomposition Method

A new wave simulation technique for the elastic wave equation in the frequency domain based on a no overlapping domain decomposition algorithm is investigated. The boundary conditions and the finite difference discrimination of the elastic wave equation are derived. The algorithm of no overlapping domain decomposition method is given. The method solves the elastic wave equation by iteratively solving sub problems defined on smaller sub domains. Numerical computations both for homogeneous and inhomogeneous media show the effectiveness of the proposed method. This method can be used in the full-waveform inversion.

一阶弹性波方程交错网格高阶差分解法

提高计算精度和运算效率是所有波场正演方法所追求的目标，本文通过将速度 （应力）对时间的奇数阶高阶寻数转化为应力（速度）对空间的导数，运用时间和空间差分精度 均可达任意阶的高阶差分法，通过交错网格技术，对一阶速度－应力弹性波方程进行了数值求 解．波场快照以及实际模型的正演结果表明，这种求解一阶弹性波方程的高阶差分解法，和 常规的差分法相比网格频散显著减小，精度明显提高，而且可以取较大的空间步长，提高计算 效率。

三维弹性波数值模拟中的吸收边界条件

在地震波传播数值模拟的过程中 ,需要使用吸收边界条件从而达到衰减人为边界反射的目的 .本文基于傍轴近似法提出了计算三维弹性波方程的吸收边界条件公式 ,表示了各边界面、边棱和角点处波场所满足的单程波方程 ,并在三维弹性波数值模拟中进行了应用 .理论模型及三维盐丘地质模型波场切片快照试算结果表明 ,该吸收边界条件可以有效地吸收人为边界反射 ,适用于较大入射角情况 ,从而消除了边界有效波信息的干扰 .由于采用四阶近似方程 ,在保证计算精度的前提下 。

各向异性弹性波场的有限元数值模拟

研究了各向异性弹性波有限元正演系统的精度和效率问题 ,提出了一种透射加衰减的组合人工边界方案 (吸收边界条件 ) ,它对各向异性波具有较好的吸收效果 ,并且有较好的稳定性能 .均匀TI介质中的模拟获得了非常清晰的波场快照 ,其波场特征与理论分析能够准确吻合 .各向异性介质模型的地表地震记录表明 ,各向异性波炮集记录在波的类型、同相轴形态、能量分布和相位等方面与各向同性波都有很大差别 .