Absorbing boundary conditions for elastic waves

Absorbing boundary conditions are needed for computing numerical models of wave motions in unbounded spatial domains. The boundary conditions developed here for elastic waves are generalizations of ones developed earlier for acoustic waves. These conditions are based on compositions of simple first‐order differential operators. The formulas can be applied without modification to problems in both two and three dimensions. The boundary conditions are stable for all values of the ratio of P‐wave velocity to S‐wave velocity, and they are effective near a free surface and in a horizontally stratified medium. The boundary conditions are approximated with simple finite‐difference equations that use values of the solution only along grid lines perpendicular to the boundary. This property facilitates implementation, especially near a free surface and at other corners of the computational domain.

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LONG-TERM STABILITY ANALYSIS OF ACOUSTIC ABSORBING BOUNDARY CONDITIONS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202513500280 ◽

2013 ◽

Vol 23 (11) ◽

pp. 2129-2154 ◽

Cited By ~ 1

Author(s):

HÉLÈNE BARUCQ ◽

JULIEN DIAZ ◽

VÉRONIQUE DUPRAT

Keyword(s):

Stability Analysis ◽

Boundary Conditions ◽

Absorbing Boundary Conditions ◽

Computational Domain ◽

Acoustic Wave Equation ◽

Absorbing Boundary ◽

Term Stability ◽

Long Term Stability ◽

The Stability

This work deals with the stability analysis of a one-parameter family of Absorbing Boundary Conditions (ABC) that have been derived for the acoustic wave equation. We tackle the problem of long-term stability of the wave field both at the continuous and the numerical levels. We first define a function of energy and show that it is decreasing in time. Its discrete form is also decreasing under a Courant–Friedrichs–Lewy (CFL) condition that does not depend on the ABC. Moreover, the decay rate of the continuous energy can be determined: it is exponential if the computational domain is star-shaped and this property can be illustrated numerically.

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Stability of wide‐angle absorbing boundary conditions for the wave equation

Geophysics ◽

10.1190/1.1442750 ◽

1989 ◽

Vol 54 (9) ◽

pp. 1153-1163 ◽

Cited By ~ 19

Author(s):

R. A. Renaut ◽

J. Petersen

Keyword(s):

Wave Equation ◽

Boundary Conditions ◽

Least Squares ◽

Absorbing Boundary Conditions ◽

Computational Domain ◽

Wide Angle ◽

Courant Number ◽

Absorbing Boundary ◽

Wide Range ◽

Artificial Boundaries

Numerical solution of the two‐dimensional wave equation requires mapping from a physical domain without boundaries to a computational domain with artificial boundaries. For realistic solutions, the artificial boundaries should cause waves to pass directly through and thus mimic total absorption of energy. An artificial boundary which propagates waves in one direction only is derived from approximations to the one‐way wave equation and is commonly called an absorbing boundary. Here we investigate order 2 absorbing boundary conditions which include the standard paraxial approximation. Absorption properties are compared analytically and numerically. Our numerical results confirm that the [Formula: see text] or Chebychev‐Padé approximations are best for wide‐angle absorption and that the Chebychev or least‐squares approximations are best for uniform absorption over a wide range of incident angles. Our results also demonstrate, however, that the boundary conditions are stable for varying ranges of Courant number (ratio of time step to grid size). We prove that there is a stability barrier on the Courant number specified by the coefficients of the boundary conditions. Thus, proving stability of the interior scheme is not sufficient. Furthermore, waves may radiate spontaneously from the boundary, causing instability, even if the stability bound on the Courant number is satisfied. Consequently, the Chebychev and least‐squares conditions may be preferred for wide‐angle absorption also.

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Sweeping preconditioners for stratified media in the presence of reflections

SN Partial Differential Equations and Applications ◽

10.1007/s42985-020-00019-x ◽

2020 ◽

Vol 1 (4) ◽

Author(s):

Janosch Preuß ◽

Thorsten Hohage ◽

Christoph Lehrenfeld

Keyword(s):

Boundary Conditions ◽

Harmonic Wave ◽

Absorbing Boundary Conditions ◽

Stratified Medium ◽

Stratified Media ◽

Absorbing Boundary ◽

Small Perturbations ◽

Numerical Tests ◽

Time Harmonic ◽

Dirichlet To Neumann Operator

Abstract In this paper we consider sweeping preconditioners for time harmonic wave propagation in stratified media, especially in the presence of reflections. In the most famous class of sweeping preconditioners Dirichlet-to-Neumann operators for half-space problems are approximated through absorbing boundary conditions. In the presence of reflections absorbing boundary conditions are not accurate resulting in an unsatisfactory performance of these sweeping preconditioners. We explore the potential of using more accurate Dirichlet-to-Neumann operators within the sweep. To this end, we make use of the separability of the equation for the background model. While this improves the accuracy of the Dirichlet-to-Neumann operator, we find both from numerical tests and analytical arguments that it is very sensitive to perturbations in the presence of reflections. This implies that even if accurate approximations to Dirichlet-to-Neumann operators can be devised for a stratified medium, sweeping preconditioners are limited to very small perturbations.

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Generating and Absorbing Boundary Conditions for Free-Surface Flow Simulations in Offshore Applications

Volume 2: CFD and VIV ◽

10.1115/omae2015-41577 ◽

2015 ◽

Cited By ~ 1

Author(s):

Bülent Düz ◽

René H. M. Huijsmans ◽

Peter R. Wellens ◽

Mart J. A. Borsboom ◽

Arthur E. P. Veldman ◽

...

Keyword(s):

Boundary Conditions ◽

Cfd Simulation ◽

Free Surface Flow ◽

Surface Flow ◽

Absorbing Boundary Conditions ◽

Computational Domain ◽

Wave Impact ◽

Absorbing Boundary ◽

Floating Structures ◽

Hydrodynamic Wave

Numerical simulations of wave phenomena necessarily have to be carried out in a limited computational domain. This implies that incoming waves should be prescribed properly, and the outgoing waves should leave the domain without causing reflections. In this paper we will present an enhanced type of such generating and absorbing boundary conditions (GABC). The new approach is applied in studies of extreme hydrodynamic wave impact on rigid and floating structures in offshore and coastal engineering, for which the VOF-based CFD simulation tool ComFLOW has been developed.

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A transparent boundary technique for numerical modeling of elastic waves

Geophysics ◽

10.1190/1.1444604 ◽

1999 ◽

Vol 64 (3) ◽

pp. 963-966 ◽

Cited By ~ 4

Author(s):

Jianlin Zhu

Keyword(s):

Numerical Modeling ◽

Boundary Conditions ◽

Elastic Waves ◽

Wave Equations ◽

Normal Incidence ◽

Absorbing Boundary Conditions ◽

S Wave ◽

Absorbing Boundary ◽

S Waves ◽

Elastic Wave Equations

In numerical modeling of wave motions, strong reflections from artificial model boundaries may contaminate or mask true reflections from the interior model interfaces. Hence, developing a kind of exterior model boundary transparent to the outgoing waves is of critical importance. Among proposed solutions, e.g., Smith (1974), Kausel and Tassoulas (1981), and Higdon (1991), the most widely used may be the Clayton and Engquist (1977) method of absorbing boundary conditions, based on paraxial approximations for acoustic and elastic‐wave equations. However, absorbing boundary conditions make the reflection coefficients zero only for normal incidence, and suppression of reflected S-waves (Clayton and Engquist, 1977) becomes poorer as the ratio of P- to S-wave velocity ([Formula: see text]) becomes larger.

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Performance advantages of CPML over UPML absorbing boundary conditions in FDTD algorithm

Journal of Electrical Engineering ◽

10.1515/jee-2017-0006 ◽

2017 ◽

Vol 68 (1) ◽

pp. 47-53 ◽

Cited By ~ 1

Author(s):

Branko D. Gvozdic ◽

Dusan Z. Djurdjevic

Keyword(s):

Boundary Conditions ◽

Electromagnetic Waves ◽

Fdtd Method ◽

Absorbing Boundary Conditions ◽

Numerical Code ◽

Computational Domain ◽

Absorbing Boundary ◽

Simulation Performance ◽

Efficient Type ◽

Computer Resources

Abstract Implementation of absorbing boundary condition (ABC) has a very important role in simulation performance and accuracy in finite difference time domain (FDTD) method. The perfectly matched layer (PML) is the most efficient type of ABC. The aim of this paper is to give detailed insight in and discussion of boundary conditions and hence to simplify the choice of PML used for termination of computational domain in FDTD method. In particular, we demonstrate that using the convolutional PML (CPML) has significant advantages in terms of implementation in FDTD method and reducing computer resources than using uniaxial PML (UPML). An extensive number of numerical experiments has been performed and results have shown that CPML is more efficient in electromagnetic waves absorption. Numerical code is prepared, several problems are analyzed and relative error is calculated and presented.

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Equivalent Boundary Conditions for Heterogeneous Acoustic Media

TEMA (São Carlos) ◽

10.5540/tema.2014.015.03.0301 ◽

2014 ◽

Vol 15 (3) ◽

pp. 301

Author(s):

Manuela Longoni De Castro ◽

Julien Diaz ◽

Victor Perón

Keyword(s):

Boundary Conditions ◽

Seismic Waves ◽

Heterogeneous Media ◽

Absorbing Boundary Conditions ◽

Computational Domain ◽

Absorbing Boundary ◽

Equivalent Boundary ◽

Artificial Boundaries ◽

Acoustic Media ◽

Propagation Of Waves

In this work, we have worked on possibilities to model artificial boundaries needed in the simulation of wave propagation in acoustic heterogeneous media. Our motivation is to restrict the computational domain in the simulation of seismic waves that are propagated from the earth and transmitted to the stratified heterogeneous media composed by ocean and atmosphere. Two possibilities were studied and compared in computational tests: the use of absorbing boundary conditions on an artificial boundary in the atmosphere layer and the elimination of the atmosphere layer using an equivalent boundary condition that mimics the propagation of waves through the atmosphere. <br />

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Frequency-domain elastic wavefield simulation with hybrid absorbing boundary conditions

Journal of Geophysics and Engineering ◽

10.1093/jge/gxz038 ◽

2019 ◽

Vol 16 (4) ◽

pp. 690-706

Author(s):

Zhencong Zhao ◽

Jingyi Chen ◽

Xiaobo Liu ◽

Baorui Chen

Keyword(s):

Boundary Condition ◽

Free Surface ◽

Boundary Conditions ◽

Frequency Domain ◽

Time Domain ◽

Absorbing Boundary Conditions ◽

Elastic Media ◽

Absorbing Boundary ◽

Wavefield Simulation ◽

The Time Domain

Abstract The frequency-domain seismic modeling has advantages over the time-domain modeling, including the efficient implementation of multiple sources and straightforward extension for adding attenuation factors. One of the most persistent challenges in the frequency domain as well as in the time domain is how to effectively suppress the unwanted seismic reflections from the truncated boundaries of the model. Here, we propose a 2D frequency-domain finite-difference wavefield simulation in elastic media with hybrid absorbing boundary conditions, which combine the perfectly matched layer (PML) boundary condition with the Clayton absorbing boundary conditions (first and second orders). The PML boundary condition is implemented in the damping zones of the model, while the Clayton absorbing boundary conditions are applied to the outer boundaries of the damping zones. To improve the absorbing performance of the hybrid absorbing boundary conditions in the frequency domain, we apply the complex coordinate stretching method to the spatial partial derivatives in the Clayton absorbing boundary conditions. To testify the validity of our proposed algorithm, we compare the calculated seismograms with an analytical solution. Numerical tests show the hybrid absorbing boundary condition (PML plus the stretched second-order Clayton absorbing condition) has the best absorbing performance over the other absorbing boundary conditions. In the model tests, we also successfully apply the complex coordinate stretching method to the free surface boundary condition when simulating seismic wave propagation in elastic media with a free surface.

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