Higdon Absorbing Boundary Condition Research Articles

Second-order symplectic PRK scheme for ground-penetrating radar simulations in dispersive materials

An accurate and effective forward model is helpful in interpreting measured ground-penetrating radar (GPR) data. In this paper, an explicit second-order symplectic partitioned Runge–Kutta scheme (SPRK) with Higdon's absorbing boundary condition is developed to simulate GPR wave propagation in dispersive media. For this purpose, the first-order Debye dispersion model is successfully applied to the second-order SPRK method. A single-layer dispersive medium model and a three-layer pavement structure model are simulated, in order to verify the accuracy and efficiency of the second-order SPRK method, as well as to clarify the effect of dispersion on the GPR echo signal. The numerical simulation results demonstrate that the proposed algorithm not only can maintain the simulation accuracy, but can also save about 19% of the calculation time, when compared with the FDTD algorithm, in a dispersive medium. We also obtain the GPR profile of a pavement structure with a circular cavity disease and an underground pipe structure in a dispersive medium, and the simulation results show that the dispersion characteristics of the materials lead to high absorption of the GPR electromagnetic waves. Finally, field experiment data are used to further verify the applicability of the algorithm. The numerical simulations and experiments indicate that the dispersion of the medium must be considered in GPR numerical simulation, and the authenticity of the simulated image must be ensured.

The Theory and New Unified Formulas of Displacement-Type Local Absorbing Boundary Conditions

ABSTRACTA series of classical absorbing boundary conditions (ABCs), including paraxial-approximation ABCs, Liao’s multi-transmitting formula (MTF), Higdon ABCs, and some other related techniques, have the common feature that the motion of an arbitrary artificial boundary node at each timestep is directly predicted from the motions of some adjacent nodes at several previous timesteps. They are expressed in somewhat equivalent forms, contain similar control parameters, and have comparable accuracy and stability in numerical simulations. This study develops a theoretical framework called displacement-type (a more exact name would be “prediction-type” or “extrapolation-type”) local ABCs to merge these boundary conditions. The idea of this theory mainly originates from the versatility of MTF, which uses a unified formula to approximate the propagation of outgoing waves through each boundary node. This idea can be generalized to other displacement-type local ABCs to unify their expressions and to optimize their applications. These ABCs have two basic control parameters; one is the boundary order, and the other is adjustable computational wave velocities. Considering the poor performance of paraxial ABCs and the slight imperfections in MTF and Higdon ABCs, we propose two new unified formulas to be the starting points of expressing, evaluating, and applying displacement-type local ABCs. One formula is an optimized MTF by introducing various computational wave velocities. The other formula is a generalized Higdon boundary formula, which is established in a unified local coordinate and uses the adjustable computational wave velocities. The rule of choosing boundary parameters for the absorption of acoustic and elastic waves is discussed in detail. Numerical tests validate the proposed theory and formulas. Issues on numerical stability are briefly reviewed and tested in simulation examples. This is still an active research topic related to displacement-type local ABCs.

Cosine-type weighted hybrid absorbing boundary based on the second-order Higdon boundary condition and its GPU implementation

AbstractIn this paper, we analyze the absorption effect of the hybrid absorbing boundary condition (ABC) with various hybrid modes, and propose a cosine-type optimized weighted hybrid mode taking into account the boundary reflection intensity of the inner and outer boundaries of transition area. Additionally, we derive a new finite-difference scheme of the second-order Higdon ABC and the corner equation for Graphic Processing Unit (GPU) acceleration. On this basis, a new type of second-order Higdon hybrid ABC applicable for GPU acceleration is established in the acoustic finite-difference modeling. Numerical experiments demonstrate that the proposed cosine-type weighted hybrid mode can achieve a better absorption effect compared with other weighted hybrid modes; the second-order Higdon ABC based on the proposed new finite-difference scheme can effectively improve the GPU speed-up ratio and is more effective in large-scale wavefield high-precision simulation based on GPU acceleration.

A study on lateral transient vibration of large diameter piles considering pile-soil interaction

This study focuses on the three-dimensional (3-D) characteristics of lateral transient vibration of large diameter piles. Firstly, a 3-D pile-soil model in Cartesian coordinate system is established. Then, the governing equations are established. To eliminate the reflected waves from artificial boundaries, the second-order Higdon absorbing boundary condition is applied herein. Based on the boundary and initial conditions, the numerical solution is obtained using staggered grid finite difference method. The reliability of the numerical simulation is corroborated by comparing calculation results with measured data. It is shown that the optimal sensor location for receiving signal is the center of pile top, which is subjected to the minimum 3-D interference. Dynamic stiffness and damping at pile top are investigated by changing the parameters of pile-soil system.

A hybrid absorbing boundary condition for frequency-domain finite-difference modelling

Liu and Sen (2010 Geophysics 75 A1?6; 2012 Geophys. Prospect. 60 1114?32) proposed an efficient hybrid scheme to significantly absorb boundary reflections for acoustic and elastic wave modelling in the time domain. In this paper, we extend the hybrid absorbing boundary condition (ABC) into the frequency domain and develop specific strategies for regular-grid and staggered-grid modelling, respectively. Numerical modelling tests of acoustic, visco-acoustic, elastic and vertically transversely isotropic (VTI) equations show significant absorptions for frequency-domain modelling. The modelling results of the Marmousi model and the salt model also demonstrate the effectiveness of the hybrid ABC. For elastic modelling, the hybrid Higdon ABC and the hybrid Clayton and Engquist (CE) ABC are implemented, respectively. Numerical simulations show that the hybrid Higdon ABC gets better absorption than the hybrid CE ABC, especially for S-waves. We further compare the hybrid ABC with the classical perfectly matched layer (PML). Results show that the two ABCs cost the same computation time and memory space for the same absorption width. However, the hybrid ABC is more effective than the PML for the same small absorption width and the absorption effects of the two ABCs gradually become similar when the absorption width is increased.

The First-Order Symplectic Euler Method for Simulation of GPR Wave Propagation in Pavement Structure

Construction of electromagnetic wave propagation model in layered pavement structure is a key problem for applying ground penetrating radar (GPR) to the road quality detection. A first-order explicit symplectic Euler method with Higdon absorbing boundary condition is presented to simulate GPR wave propagation in 2-D pavement structure. The incident wave is considered as line source and plane wave source, respectively. The total-field/scatter-field technique is used to simulate plane wave excitation. Numerical examples are provided to verify the accuracy and efficiency of the proposed algorithm. It can be observed that the symplectic Euler method achieves almost the same level of accuracy as the finite-difference time-domain scheme, while saving CPU time considerably.

Efficient LOD-BOR-FDTD Implementation Based on a Fundamental Scheme

An implicit locally 1-D body-of-revolution finite-difference time-domain method is efficiently implemented using a fundamental scheme. The formulation with dispersion control parameters is performed in convenient matrix-operator-free forms in the right-hand sides of resultant basic equations. The Higdon absorbing boundary condition is also incorporated. Modification to the calculation procedure reduces memory requirements for auxiliary variables, compared with the original fundamental scheme. The usefulness is discussed through the analysis of a fiber Bragg grating.

A hybrid absorbing boundary condition for elastic staggered‐grid modelling

ABSTRACTWe recently proposed an efficient hybrid scheme to absorb boundary reflections for acoustic wave modelling that could attain nearly perfect absorptions. This scheme uses weighted averaging of wavefields in a transition area, between the inner area and the model boundaries. In this paper we report on the extension of this scheme to 2D elastic wave modelling with displacement‐stress formulations on staggered grids using explicit finite‐difference, pseudo‐implicit finite‐difference and pseudo‐spectral methods. Numerical modelling results of elastic wave equations with hybrid absorbing boundary conditions show great improvement for modelling stability and significant absorption for boundary reflections, compared with the conventional Higdon absorbing boundary conditions, demonstrating the effectiveness of this scheme for elastic wave modelling. The modelling results also show that the hybrid scheme works well in 2D rotated staggered‐grid modelling for isotropic medium, 2D staggered‐grid modelling for vertically transversely isotropic medium and 2D rotated staggered‐grid modelling for tilted transversely isotropic medium.

On the stability of the high-order Higdon Absorbing Boundary Conditions

The Higdon sequence of Absorbing Boundary Conditions (ABCs) for the linear wave equation is considered. Building on a previous work of Ha-Duong and Joly, which related to other forms of boundary conditions, the Higdon ABCs are proved to be energy-stable (on the continuous level) up to any order. This type of stability is stronger than the more standard notion of stability of initial boundary value problems in the sense of Kreiss; in particular it leads to stability estimates which are uniform in time. In consequence to the theorem proved here, energy-stability is immediately implied for the high-order Givoli–Neta and Hagstrom–Warburton ABCs, which are reformulations of the Higdon ABCs using auxiliary variables. A weakness of this theory is that it requires sufficiently smooth data, and that the required smoothness level increases with the order of the ABC. This issue and its implications are discussed.

Finite element formulation with high-order absorbing boundary conditions for time-dependent waves

The Hagstrom–Warburton high-order absorbing boundary conditions (ABCs) are considered. They are based on a high-order form of the Higdon ABCs using auxiliary variables and constitute a modification of the previously proposed Givoli–Neta ABCs. Here the Hagstrom–Warburton ABCs, which were originally used in a finite difference scheme, are incorporated into a finite element formulation. Exterior time-dependent problems are considered with rectangular computational domains. Special corner conditions are used in conjunction with the ABCs to make the truncated problem well-posed. The properties of the Hagstrom–Warburton and Givoli–Neta formulations are compared, and the relations between the two formulations are established. Numerical examples demonstrate the performance of the Hagstrom–Warburton finite element scheme.

Alternatives to the Perfectly Matched Layer for Waveguide Simulation Using the FDFD Method

One of the most widely utilized absorbing boundaries for the finite difference frequency domain (FDFD) method is the perfectly matched layer (PML); however, this boundary is not ideal, as it significantly increases the computational complexity of a simulation. We compare three alternatives to the PML for modal solution using the FDFD method: the Higdon absorbing boundary condition, the complementary operators method, and the transparent absorbing boundary. These methods are integrated into an FDFD algorithm and applied to a well-known ridge waveguide test structure. The errors introduced by the absorbing boundaries are quantified in terms of propagation constant error and modal distortion, and compared to reported results using the PML.

Seismic propagation simulation in complex media with non-rectangular irregular-grid finite-difference

This paper presents a finite-difference (FD) method with spatially non-rectangular irregular grids to simulate the elastic wave propagation. Staggered irregular grid finite difference operators with a second-order time and spatial accuracy are used to approximate the velocity-stress elastic wave equations. This method is very simple and the cost of computing time is not much. Complicated geometries like curved thin layers, cased borehole and nonplanar interfaces may be treated with non-rectangular irregular grids in a more flexible way. Unlike the multi-grid scheme, this method requires no interpolation between the fine and coarse grids and all grids are computed at the same spatial iteration. Compared with the rectangular irregular grid FD, the spurious diffractions from “staircase” interfaces can easily be eliminated without using finer grids. Dispersion and stability conditions of the proposed method can be established in a similar form as for the rectangular irregular grid scheme. The Higdon's absorbing boundary condition is adopted to eliminate boundary reflections. Numerical simulations show that this method has satisfactory stability and accuracy in simulating wave propagation near rough solid-fluid interfaces. The computation costs are less than those using a regular grid and rectangular grid FD method.

Effects of weighting and damping coefficients on the accuracy of Higdon‐based absorbing boundary conditions

AbstractIn this work, the effects of weighted difference schemes and damping factor on the spurious errors arising from the application of Higdon's absorbing boundary condition and the concurrent complementary operators method (C‐COM) are investigated. An optimized combination of the weighted difference scheme for boundary conditions and the damping term used to stabilize the boundary scheme are proposed to minimize the reflections. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 41: 446–448, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20166

An absorbing boundary condition for FDTD truncation using multiple absorbing surfaces

An absorbing boundary condition (ABC) is developed using multiple absorbing surfaces. This ABC is stable and efficient. It can be easily added in the finite-difference time-domain computer program. It also produces the same reflection as a higher order Higdon ABC and it is therefore comparable to the perfectly matched layer method.

A Z-transform-based absorbing boundary conditions for 3-D TLM-SCN method

In this paper, an efficient absorbing boundary condition (ABC) constructed in the Z-transform domain [i.e., Z-transform-based absorbing boundary conditions (Z-ABCs)] for the three-dimensional symmetrical condensed-node transmission-line matrix method is presented. Numerical results indicate that the Z-ABCs show better performance than the conventional Higdon's ABC in suppressing instability caused by spurious modes.

Comparison of highly efficient absorbing boundary conditions for the beam propagation method.

Numerical experiments using the paraxial finite-difference beam propagation method have been performed with the following boundary conditions: perfectly matched layer, Higdon absorbing boundary conditions, complementary operators method, and extended complementary operators method. We have shown that Higdon operators must be modified for the paraxial wave equation to take into account the spectrum of incident rays on the boundaries of the computational domain. Reflection coefficients, accuracy, numerical dissipation/ gain, memory requirements, and time computation are compared and discussed for these absorbing techniques.

An explicit 2D finite element time domain scheme for electromagnetic wave propagation

An explicit 2D FETD formulation for electromagnetic wave propagation is presented. The space discretization is obtained using finite element triangles. The domain is truncated using a Higdon absorbing boundary condition. The time discretization is performed using a four level scheme. Mass lumping technique makes it explicit. Several numerical parameters are studied, such as the mesh density, the order of the ABC and the numerical stability.

Experiments with Higdon's absorbing boundary conditions for a number of wave equations

Simulation of wave propagation for seismic purposes is usually restricted to a small portion of the earth. Artificial boundary conditions are required where the subsurface model is truncated. Absorbing boundaries should ensure that waves hitting the artificial boundaries are not reflected. The vast amount of literature on the subject suggests that “good” conditions have not been found, and only “reasonable” solutions exist. A cursory overview of existing and a few new ideas is presented that may guide the construction of suitable boundary conditions. Because the intended application of the boundary conditions was a high-order finite-difference code that runs on a parallel computer, we have restricted our attention to local boundary conditions. A fundamental problem in the design of accurate local boundary conditions is pointed out: accuracy is required to keep the amount of reflected energy small, but at the same time allows for growing low-frequency modes. We have settled for Higdon’s boundary conditions. Higdon proposes to include some damping to suppress the growing low-frequency modes. We show that third-order conditions provide acceptable results for the simple scalar wave equation and the acoustic equation. In the elastic case, an additional low-frequency growing mode may occur. This mode can be suppressed by using a dissipative boundary scheme and by increasing the amount of damping. The increase in damping results in an increase in the amount of reflected energy, which is larger than in the scalar case. Numerical experiments exhibit a reasonable performance, although some improvement would be useful, particularly in the anisotropic elastic case.