Relationship between Liao and Clayton-Engquist absorbing boundary conditions: Acoustic case

1995 ◽  
Vol 85 (3) ◽  
pp. 954-956
Author(s):  
Ningya Cheng ◽  
Chuen Hon Cheng
Keyword(s):  
Boundary Condition ◽  
Reflection Coefficient ◽  
Boundary Conditions ◽  
Closed Form ◽  
Differential Form ◽  
Short Note ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary Condition ◽  
Absorbing Boundary ◽  
Close Relationship

Abstract In this short note, we derive the differential form of Liao's multi-transmitting formula. The reflection coefficient of the multi-transmitting formula in the acoustic case is obtained in closed form. These formulas are compared with Clayton-Engquist absorbing boundary condition and are shown to have a very close relationship.

An optimal absorbing boundary condition for elastic wave modeling

Geophysics ◽  
1995 ◽  
Vol 60 (1) ◽  
pp. 296-301 ◽  
Author(s):  
Chengbin Peng ◽  
M. Nafi Toksöz
Keyword(s):  
Boundary Condition ◽  
Wave Propagation ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Elastic Wave ◽  
Short Note ◽  
Absorbing Boundary Conditions ◽  
Parallel Computer ◽  
Absorbing Boundary Condition ◽  
Absorbing Boundary

Absorbing boundary conditions are widely used in numerical modeling of wave propagation in unbounded media to reduce reflections from artificial boundaries (Lindman, 1975; Clayton and Engquist, 1977; Reynolds, 1978; Liao et al., 1984; Cerjan et al., 1985; Randall, 1988; Higdon, 1991). We are interested in a particular absorbing boundary condition that has maximum absorbing ability with a minimum amount of computation and storage. This is practical for 3-D simulation of elastic wave propagation by a finite‐difference method. Peng and Toksöz (1994) developed a method to design a class of optimal absorbing boundary conditions for a given operator length. In this short note, we give a brief introduction to this technique, and we compare the optimal absorbing boundary conditions against those by Reynolds (1978) and Higdon (1991) using examples of 3-D elastic finite‐difference modeling on an nCUBE-2 parallel computer. In the Appendix, we also give explicit formulas for computing coefficients of the optimal absorbing boundary conditions.

An absorbing boundary condition for acoustic-wave propagation using a mesh-free method

Geophysics ◽  
2016 ◽  
Vol 81 (4) ◽  
pp. T145-T154 ◽  
Author(s):  
Junichi Takekawa ◽  
Hitoshi Mikada
Keyword(s):  
Boundary Condition ◽  
Wave Propagation ◽  
Acoustic Wave ◽  
Boundary Conditions ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary Condition ◽  
Acoustic Wave Propagation ◽  
Absorbing Boundary ◽  
Mesh Free ◽  
Mesh Free Method

We have developed an absorbing boundary condition for acoustic-wave propagation using a mesh-free method without sacrificing the flexibility of the mesh-free framework. When we simulate acoustic-wave propagation using a numerical method, artificial reflections from model edges induced by a truncated computational domain should be avoided. Although many absorbing boundary conditions have been developed, most of them have been based on a regular latticed alignment of grids or nodes, and the efficiency of such absorbing boundary conditions for irregular arrangement of grids or nodes has not been examined yet. We have studied the artificial reflections generated at the boundaries of a model for a mesh-free method, and we have proposed a novel approach for suppressing the artifacts. The method uses a hybrid approach with a transition zone, in which the wavefield is estimated by a weighted average of solutions from the one- and two-way wave equations. Numerical experiments indicate that the proposed method can provide good performance in suppression of the artificial edge reflections even for irregular distributions of calculation points in the vicinity of model edges.

Absorbing boundary condition for the elastic wave equation

Geophysics ◽  
1988 ◽  
Vol 53 (5) ◽  
pp. 611-624 ◽  
Author(s):  
C. J. Randall
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Finite Difference ◽  
Elastic Wave ◽  
Grazing Incidence ◽  
Scalar Fields ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary Condition ◽  
Elastic Wave Equation ◽  
Absorbing Boundary

Extant absorbing boundary conditions for the elastic wave equation are generally effective only for waves nearly normally incident upon the boundary. High reflectivity is exhibited for waves traveling obliquely to the boundary. In this paper, a new and efficient absorbing boundary condition for two‐dimensional and three‐dimensional finite‐difference calculations of elastic wave propagation is presented. Compressional and shear components of the incident vector displacement fields are separated by calculating intermediary scalar potentials, allowing the use of Lindman’s boundary condition for scalar fields, which is highly absorbing for waves incident at any angle. The elastic medium is assumed to be homogeneous in the region immediately adjacent to the boundary. The reflectivity matrix of the resulting absorbing boundary for elastic waves is calculated, including the effects of finite‐difference truncation error. For effectively all angles of incidence, reflectivities are much smaller than those of the commonly employed paraxial absorbing boundaries, and the boundary condition is stable for any physical Poisson’s ratio. The nearly complete absorption predicted by the reflectivity matrix calculations, even at near grazing incidence, is demonstrated in a finite‐difference application.

Absorbing boundary conditions for acoustic media

Geophysics ◽  
1985 ◽  
Vol 50 (6) ◽  
pp. 892-902 ◽  
Author(s):  
R. G. Keys
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Plane Waves ◽  
Basic Element ◽  
Absorbing Boundary Conditions ◽  
Boundary Operator ◽  
Order Differential Operator ◽  
Absorbing Boundary ◽  
Boundary Operators

By decomposing the acoustic wave equation into incoming and outgoing components, an absorbing boundary condition can be derived to eliminate reflections from plane waves according to their direction of propagation. This boundary condition is characterized by a first‐order differential operator. The differential operator, or absorbing boundary operator, is the basic element from which more complicated boundary conditions can be constructed. The absorbing boundary operator can be designed to absorb perfectly plane waves traveling in any two directions. By combining two or more absorption operators, boundary conditions can be created which absorb plane waves propagating in any number of directions. Absorbing boundary operators simplify the task of designing boundary conditions to reduce the detrimental effects of outgoing waves in many wave propagation problems.

scholarly journals Assessment of a PML Boundary Condition for Simulating an MRI Radio Frequency Coil

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
Yunsuo Duan ◽  
Tamer S. Ibrahim ◽  
Bradley S. Peterson ◽  
Feng Liu ◽  
Alayar Kangarlu
Keyword(s):  
Boundary Condition ◽  
Larmor Frequency ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary Condition ◽  
Rf Coils ◽  
Absorbing Boundary ◽  
Magnetic Resonance Imaging Mri ◽  
Experimental Findings ◽  
High Pass ◽  
Head Coil

Computational methods such as the finite difference time domain (FDTD) play an important role in simulating radiofrequency (RF) coils used in magnetic resonance imaging (MRI). The choice of absorbing boundary conditions affects the final outcome of such studies. We have used FDTD to assess the Berenger's perfectly matched layer (PML) as an absorbing boundary condition for computation of the resonance patterns and electromagnetic fields of RF coils. We first experimentally constructed a high-pass birdcage head coil, measured its resonance pattern, and used it to acquire proton phantom MRI images. We then computed the resonance pattern and field of the coil using FDTD with a PML as an absorbing boundary condition. We assessed the accuracy and efficiency of PML by adjusting the parameters of the PML and comparing the calculated results with measured ones. The optimal PML parameters that produce accurate (comparable to the experimental findings) FDTD calculations are then provided for the birdcage head coil operating at 127.72 MHz, the Larmor frequency of at 3 Tesla (T).

scholarly journals Frequency-domain elastic wavefield simulation with hybrid absorbing boundary conditions

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.

III‐posedness of absorbing boundary conditions for migration

Geophysics ◽  
1988 ◽  
Vol 53 (5) ◽  
pp. 593-603 ◽  
Author(s):  
Louis H. Howell ◽  
Lloyd N. Trefethen
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Absorbing Boundary Conditions ◽  
Order Condition ◽  
Third Order ◽  
Absorbing Boundary ◽  
Finite Speed ◽  
Ill Posed ◽  
Migration Equation ◽  
Well Posed

Absorbing boundary conditions for wave‐equation migration were introduced by Clayton and Engquist. We show that one of these boundary conditions, the B2 (second‐order) condition applied with the 45° (third‐order) migration equation, is ill‐posed. In fact, this boundary condition is subject to two distinct mechanisms of ill‐posedness: a Kreiss mode with finite speed at one boundary and another mode of a new kind involving wave propagation at unbounded speed back and forth between two boundaries. Unlike B2, the third‐order Clayton‐Engquist boundary condition B3 is well‐posed. However, we show that it is impossible for any boundary condition of Clayton‐Engquist type of order higher than one to be well‐posed with a migration equation whose order is higher than three.

Instabilities in applying absorbing boundary conditions to high‐order seismic modeling algorithms

Geophysics ◽  
1998 ◽  
Vol 63 (3) ◽  
pp. 1017-1023 ◽  
Author(s):  
Antonio Simone ◽  
Stig Hestholm
Keyword(s):  
Boundary Condition ◽  
Acoustic Wave ◽  
Elastic Wave ◽  
Wave Equations ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary Condition ◽  
Seismic Modeling ◽  
Absorbing Boundary ◽  
Numerical Grid ◽  
Numerical Discretization

The problem of artificial reflections from grid boundaries in the numerical discretization of elastic and acoustic wave equations has long plagued geophysicists. Even if modern computers have made it possible to extend the synthetics over more wavelengths (equivalent to larger propagation distances), efficient absorption methods are still needed to minimize interference from unwanted reflections from the numerical grid boundaries. In this study, we examine applicabilities and stabilities of the optimal absorbing boundary condition (OABC) of Peng and Toksöz (1994, 1995) for 2-D and 3-D acoustic and elastic wave modeling. As a basis for comparison, we use exponential damping (ED) (Cerjan et. al., 1985), in which velocities and stresses are multiplied by progressively decreasing terms when approaching the boundaries of the numerical grid.

scholarly journals A Generating - Absorbing Boundary Condition Applied to Wave - Current Interactions Using the Method of Fundamental Solutions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammed Loukili ◽  
Kamila Kotrasova ◽  
Amine Bouaine
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Numerical Solutions ◽  
Fundamental Solutions ◽  
Absorbing Boundary Conditions ◽  
Method Of Fundamental Solutions ◽  
Computational Domain ◽  
Numerical Wave Tank ◽  
Absorbing Boundary ◽  
Wave Properties

Abstract The purpose of this work is to study the feasibility and efficiency of Generating Absorbing Boundary Conditions (GABCs), applied to wave-current interactions using the Method of Fundamental Solutions (MFS) as radial basis function, the problem is solved by collocation method. The objective is modeling wave-current interactions phenomena applied in a Numerical Wave Tank (NWT) where the flow is described within the potential theory, using a condition without resorting to the sponge layers on the boundaries. To check the feasibility and efficiency of GABCs presented in this paper, we verify accurately the numerical solutions by comparing the numerical solutions with the analytical ones. Further, we check the accuracy of numerical solutions by trying a different number of nodes. Thereafter, we evaluate the influence of different aspects of current (coplanar current, without current, and opposing current) on the wave properties. As an application, we take into account the generating-absorbing boundary conditions GABCs in a computational domain with a wavy downstream wall to confirm the efficiency of the adopted numerical boundary condition.

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