Absorbing Boundary Research Articles

A frequency-independent absorption function surrogate for perfectly matched layer in exterior acoustics

In many engineering applications, the solution of acoustic wave problems in the infinite domain is required over a broad frequency range with densely sampled increments. In order to achieve efficient numerical simulations via a spatial discretization, e.g. finite element method, additional artificial absorbing boundaries are necessary to truncate the computational domain into appropriate bounded sizes. One of the most commonly used non-reflecting techniques to attenuate propagating waves is known as the perfectly matched layer. However, the system matrices arising from the finite element treatment of the Helmholtz equation in the absorbing layers are frequency-dependent, implying that they must be formed and inverted at each frequency of interest. Such a procedure is rather troublesome for frequency sweeps. To address this, a surrogate of perfectly matched layers is proposed, which enables the corresponding system matrices to be independent of the frequency. Moreover, it avoids the use of a relatively large computational domain and relatively thick enclosed layers at low frequencies, thus improving the ability of perfectly matched layers across the entire frequency range. After that, an adaptive projection-based model order reduction scheme is further developed to reduce the computational complexity of exterior acoustic systems. A robust error indicator based on the relative error of two constructed reduced order models is accordingly introduced. The performance of the present solution framework is discussed and compared with other implementation strategies, in the context of multi-frequency solution of two-dimensional test models with single or multiple scatterers.

The Method of Fictitious Negative Sources to Model Diffusive Channels With Absorbing Boundaries

The Method of Fictitious Negative Sources to Model Diffusive Channels With Absorbing Boundaries

A fast second-order absorbing boundary condition for the linearized Benjamin-Bona-Mahony equation

A fast second-order absorbing boundary condition for the linearized Benjamin-Bona-Mahony equation

Relating absorbing and hard wall boundary conditions for a one-dimensional run-and-tumble particle

The connection between absorbing boundary conditions and hard walls is well established in the mathematical literature for a variety of stochastic models, including for instance the Brownian motion. In this paper we explore this duality for a different type of process which is of particular interest in physics and biology, namely the run-tumble-particle, a toy model of active particle. For a one-dimensional run-and-tumble particle (RTP) subjected to an arbitrary external force, we provide a duality relation between the exit probability, i.e. the probability that the particle exits an interval from a given boundary before a certain time t, and the cumulative distribution of its position in the presence of hard walls at the same time t. We show this relation for a RTP in the stationary state by explicitly computing both quantities. At finite time, we provide a derivation using the Fokker–Planck equation. All the results are confirmed by numerical simulations.

A new variable-order fractional momentum operator for wave absorption when solving Schrödinger equations

Absorbing boundary conditions are often needed for solving Schrödinger equations when the analytical solution is unknown. This article presents a new wave-absorption method by unifying the Schrödinger equations with associated one-way wave equations via a fractional-order momentum operator pˆα+1 where 0≤α≤1. By gradually varying the fractional order α, the computational boundary is made to be transparent. This method allows for incident waves to propagate freely and forces reflected waves to decay exponentially with time. The unified equation is then solved using the Finite-Difference Time-Domain (FDTD) method so that the scheme is iteratively explicit and can be parallelized easily. Finally, the obtained fractional FDTD scheme is tested to simulate soliton and particle propagations and is compared with the existing Perfectly Matched Layer (PML) method. Results show that the new method is promising.

A generating absorbing boundary condition for simulating wave interaction with maritime structures in current or at forward speed

The lack of suitable boundary conditions in practical surface wave simulations with maritime structures in current or at forward speed may cause energy in the computational domain to accumulate due to spurious wave reflection. The common way to prevent wave reflection is to use passive wave absorbers, such as damping zones or relaxation zones, which requires larger domains at the cost of computational effort. Our goal is to derive a local generating absorbing boundary condition (GABC) for long-crested irregular waves on top of a mean flow, using the flow to model the forward speed of a structure such as a ship. Earlier work has demonstrated that a local GABC for free surface waves has a performance similar to passive wave absorbers, but at a reduced computational effort. New in the present work is that we extend, verify and validate the GABC in the presence of a nonzero mean flow. The GABC is designed to be accurate for a range of wave components in irregular sea states, with the resulting reflection coefficients for each component lower than a chosen value, say 5%. Having used potential flow theory for its derivation means that the boundary should not be placed at the exact location where wave breaking is expected, such as very close to the structure in the domain, or in the surf zone in coastal modeling. For the application with ships in this article that does not pose a limitation. The performance is demonstrated for a range of dimensionless wave number between 0 and 6. Such a boundary condition is obtained through a rational approximation of the linear dispersion relation with a mean flow, in combination with vertical derivatives of the solution variables along the boundary. Local linearization means that the GABC incorrectly considers bound, nonlinear wave components to be freely propagating wave components. Bound components, however, tend to have smaller amplitudes and do not appear to affect performance for the considered cases. Results of simulations with regular and irregular waves, on top of flows with different magnitudes and directions, are found to agree with the theory. The main source of differences is the implementation of the second derivate in the GABC near the free surface. Simulations of a Wigley hull at forward speed in irregular waves are compared to an experiment that was conducted specifically for validating the ABC. The data of the experiment are available as open data through doi: 10.4121/21320604. The comparison between simulation and experiment demonstrates that the GABC with a mean flow can be applied not only for theoretical simulations with propagating waves, but also for more practical applications with a structure in the domain.

Advanced absorbing boundaries for elastodynamic finite element analysis: The added degree of freedom method

Absorbing boundaries are essential in engineering simulations, especially for elastodynamic problems, to ensure accuracy, reliability and numerical stability. In this paper, the added degree of freedom method (ADM) is developed as a novel approach to address absorbing boundary challenges in finite element analysis. In ADM, additional degrees of freedom (DOFs) are introduced within the absorbing domain to attenuate outgoing elastic waves. For the proposed ADM to implement the absorbing boundary, the stiffness and mass properties of both the added DOFs and conventional DOFs within the absorbing domain are adjusted. This adjustment aims to reduce the propagation speed of elastic waves within the medium, and to prolong the duration of interaction between elastic waves and the surrounding medium. Consequently, the vibrations of nodes can be effectively attenuated by applying damping forces to the added DOFs. The numerical results show that the ADM has the ability to absorb one-dimensional and two-dimensional elastic waves across a broad range of frequencies. Therefore, the proposed ADM offers an innovative solution for modeling absorbing boundaries in various scientific and engineering applications, addressing the challenges of simulating wave propagation within finite computational domains.

Joint Localization and Clock Synchronization in Cuboid Bounded Diffusive Channel with Absorbing and Reflecting Boundaries

Joint Localization and Clock Synchronization in Cuboid Bounded Diffusive Channel with Absorbing and Reflecting Boundaries

Задача о колебаниях в цепи связанных линейных осцилляторов с демпфирующей и антидемпфирующей границами

The problem of oscillations in a finite homogeneous chain of coupled harmonic oscillators is considered under special boundary conditions that ensure a stable flow of energy from one end of the chain to the other. The problem covers, as special cases, the classical problem of oscillations in a chain with free and fixed ends, as well as the problem of oscillations in a chain with absorbing and anti-absorbing boundaries. Absorbing boundary conditions are used in numerical simulation of wave propagation to minimize the influence of non-physical boundaries. An exact analytical solution to the considered problem is obtained. The dynamical system of the problem is studied. In particular, a description of the invariant subspaces of the system is given. The oscillatory properties of the system are investigated. The phenomenon of a significant increase in the amplitude of low-frequency oscillations has been discovered and studied. The problem is solved in Schrodinger variables. The solution is presented both in terms of infinite series of Bessel functions and in terms of natural oscillations (eigenmodes) of the system.

A high-order explicit time-domain FEM using 15-node tetrahedral elements for room acoustics modeling: Basic performance

Time-domain room acoustic modeling using mass-lumped higher-order tetrahedral finite elements with an explicit time-marching scheme is highly attractive because of its excellent geometrical flexibility with unstructured meshing and applicability into massively parallel computing. However, the standard mass-lumped tetrahedral elements yield an unstable scheme, and only implicit time-marching schemes are available. This study proposes a novel wave-based room acoustics solver based on a high-order explicit time-domain FEM. The present solver uses mass-lumped 15-node tetrahedral elements for spatial discretization and a dissipation-free two-stage partitioned Runge-Kutta time integration for time discretization. The 15-node tetrahedral elements had been developed to apply mass-lumping methods to 10-node tetrahedral elements, which are the most common finite element in engineering applications, but its applicability in room acoustics problems is unknown. Higher accuracy and efficiency of the proposed method over the standard method using 10-node tetrahedral elements is presented through an eigenvalue analysis of a long duct model and acoustic simulations in a small room. Additionally, frequency-dependent sound absorbing boundary is implemented by revising an auxiliary differential equation method.

An accelerated novel meshless coupled algorithm for Non-local nonlinear behavior in 2D/3D space-fractional GPEs

In this work, an accelerated novel meshless coupled algorithm is first presented for the numerical study of high-dimensional space-fractional nonlinear Schrödinger or Gross-Pitaevskii equations (SFNLS/GPEs). The proposed meshless method is motivated by the novel coupled concepts: (a) a time splitting-step is adopted to decompose the NLSE; (b) the spatial Riesz fractional derivative under the Riemann-Liouville operator is discretized by the combination of numerical integration and meshless weighted-least-square schemes; (c) an accurate absorbing boundary condition is employed to treat the unbounded domain. Meanwhile, a parallel algorithm based on the above approximated schemes is designed to reduce the computing cost on a CUDA-program-based GPU card. Subsequently, the validation and numerical convergence rate of the proposed method for SFNLSE are discussed and illustrated by solving two 2D/3D examples. Furthermore, the proposed parallel meshless algorithm is extended to predict the nonlinear dynamic behaviors dominated by the space-fractional Gross-Pitaevskii equations (SFGPEs), and the influence of nonlocal effect on the nonlinear phenomena is also discussed. The presented numerical results illustrate that the proposed algorithm for multi-dimensional SFNLSEs is reliability and high-efficient.

Adaptive Absorbing Boundary Layer for the Nonlinear Schrödinger Equation

Abstract We present an adaptive absorbing boundary layer technique for the nonlinear Schrödinger equation that is used in combination with the Time-splitting Fourier spectral method (TSSP) as the discretization for the NLS equations. We propose a new complex absorbing potential (CAP) function based on high order polynomials, with the major improvement that an explicit formula for the coefficients in the potential function is employed for adaptive parameter selection. This formula is obtained by an extension of the analysis in [R. Kosloff and D. Kosloff, Absorbing boundaries for wave propagation problems, J. Comput. Phys. 63 1986, 2, 363–376]. We also show that our imaginary potential function is more efficient than what is used in the literature. Numerical examples show that our ansatz is significantly better than existing approaches. We show that our approach can very accurately compute the solutions of the NLS equations in one dimension, including in the case of multi-dominant wave number solutions.

Construction and analysis of discretization schemes for one-dimensional nonlocal Schrödinger equations with exact absorbing boundary conditions

The construction of exact absorbing boundary conditions (ABCs) for the one-dimensional nonlocal fully-discrete Schrödinger equation is proposed. An efficient computation of the Green’s functions for the discrete nonlocal Schrödinger equation is stated and used to build the ABCs. Numerical error estimates are then proved for the case of a singular interaction kernel through a novel way. The theory and numerical analysis is supported by numerical examples.

A stable staggered-grid finite-difference scheme for acoustic modeling beyond conventional stability limit

Staggered-grid finite-difference (SGFD) schemes have been widely used in acoustic wave modeling for geophysical problems. Many improved methods are proposed to enhance the accuracy of numerical modeling. However, these methods are inevitably limited by the maximum Courant-Friedrichs-Lewy (CFL) numbers, making them unstable when modeling with large time sampling intervals or small grid spacings. To solve this problem, we extend a stable SGFD scheme by controlling SGFD dispersion relations and maximizing the maximum CFL numbers. First, to improve modeling stability, we minimize the error between the FD dispersion relation and the exact relation in the given wave-number region, and make the FD dispersion approach a given function outside the given wave-number area, thus breaking the conventional limits of the maximum CFL number. Second, to obtain high modeling accuracy, we use the SGFD scheme based on the Remez algorithm to compute the FD coefficients. In addition, the hybrid absorbing boundary condition is adopted to suppress boundary reflections and we find a suitable weighting coefficient for the proposed scheme. Theoretical derivation and numerical modeling demonstrate that the proposed scheme can maintain high accuracy in the modeling process and the value of the maximum CFL number of the proposed scheme can exceed that of the conventional SGFD scheme when adopting a small maximum effective wavenumber, indicating that the proposed scheme improves stability during the modeling.

Accurate Absorbing Boundary Conditions for the Two-Dimensional Nonlocal Schrödinger Equations

Accurate Absorbing Boundary Conditions for the Two-Dimensional Nonlocal Schrödinger Equations

Analysis of the anomalous diffusion in comb structure with absorbing boundary conditions

The diffusion in comb structure is an important kind of anomalous diffusion with widespread applications. The special structure corresponds to a novel characteristic of anomalous diffusion, which is characterised by the Dirac delta function in the governing equation. By considering the memory characteristic, the fractional derivative is introduced into the constitutive relation, and a new fractional governing equation in the infinite regions is constructed. Instead of simply truncating for the infinite regions, the exact absorbing boundary conditions are deduced by using the (inverse) Laplace transform technique and the stability is analysed. To deal with the governing equation containing the Dirac function, the finite difference method is proposed and the term with the Dirac function is handled using an integration method. The stability and convergence of the numerical scheme are discussed in detail. A fast algorithm is presented that the normal L1-scheme is approximated via a sum-of-exponentials approximation. Three examples are conducted, in which the particle distributions and the mean square displacement for the anomalous diffusion in comb structure are discussed. The computational time between the normal numerical scheme and the fast numerical scheme is compared and the rationality and validity of absorbing boundary conditions are analysed. An important finding is that the distribution of the mean square displacement with the absorbing boundary conditions can match the exact one accurately, which demonstrates the effectiveness of the method.

Analysis of absorbing boundary conditions for the anomalous diffusion in comb model on unbounded domain by finite volume method

The diffusion in the comb model is an important kind of anomalous diffusion which is described by a governing equation containing the Dirac delta function and the solution’s domain is infinite. Two key problems are solved to analyze the mass transfer mechanism in the comb model. One is to construct the appropriate and reasonable boundaries for treating the infinite boundaries. The exact absorbing boundary conditions with the Caputo’s fractional derivative are deduced by the Laplace transform technique. The other one is applying the finite volume method by integrating the governing equation over a governed volume to obtain the numerical solution with the advantage that the effects of the singular Dirac functions disappear. In addition, by introducing a source term, the exact solution is deduced and the comparison between the numerical solution and the exact solution is presented, which verifies the effectiveness and reasonableness of the numerical method. The particle distributions with different parameters are analyzed and discussed. By comparing with the traditional truncated boundary method, it is noteworthy that the mean square displacement for the absorbing boundary conditions has a good agreement with the exact expression.

Hybrid absorbing boundary condition based on transmitting boundary and its application in 3D fractional viscoacoustic modeling

An accurate numerical simulation for wave equations is essential for understanding of wave propagation in the earth's interior as well as full waveform inversion and reverse time migration. However, due to computational cost and hardware capability limitations, numerical simulations are often performed within a finite domain. Thus, an adequate absorbing boundary condition (ABC) is indispensable for obtaining accurate numerical simulation results. In this study, we develop a hybrid ABC based on a transmitting boundary, which is referred to as THABC, to eliminate artificial boundary reflections in 3D second-order fractional viscoacoustic numerical simulations. Furthermore, we propose an adaptive weighted coefficient to reconcile the transmitting and viscoacoustic wavefields in THABC. Through several numerical examples, we determine that the proposed THABC approach is characterized by the following benefits. First, with the same number of absorbing layers, THABC exhibits a better ability in eliminating boundary reflection than traditional ABC schemes. Second, THABC is more effective in computation, since it only requires the wavefields at the current and last time steps to solve the transmitting formula within the absorbing layers. Benefiting from a simple but effective combination between the transmitting equation and the second-order wave equation, our scheme performs well in the 3D fractional Laplacian viscoacoustic numerical simulation.

Phase shift migration with absorbing boundary condition for full-matrix imaging heterogenous medium with high quality

Ultrasonic imaging with phased array has been widely employed for nondestructive testing in various fields. Phase shift migration (PSM) is a significant imaging method and shows great advantages for multilayer structures. However, the artifact reflections at the edges of the calculation region emerge in the wavefield reconstruction process, which greatly deteriorate the imaging quality of PSM. To overcome this problem, two absorbing boundary conditions for the wavefield extrapolation of PSM are proposed in this paper. The first absorbing boundary condition renders elements in the boundary regions lower weights in frequency-space domain. The second one follows a nonstationary phase shift scheme and filters the incident waves in the boundary regions in frequency-wavenumber domain. The implementing schemes of the absorbing boundary conditions for both homogenous and heterogenous medium are developed, and the full-matrix imaging method with absorbing boundary conditions is proposed. According to the simulation and experiment, the proposed boundary conditions have strong ability to suppress the background artifacts, and they can improve the signal-to-noise ratio (SNR) and resolution by 2 times when imaging heterogenous medium. Further experiments indicate that the boundary conditions are insensitive to the window functions, and a width of one-sixteenth the length of the scan line is recommended for the absorbing boundary conditions.

A two-dimensional boundary element method with generating absorbing boundary condition for floating bodies of arbitrary shape in the frequency domain

A two-dimensional (2D) boundary element method is developed for the rapid assessment of the hydrodynamic performance of floating structures in waves. The boundary element method is based on potential flow and has panels along all boundaries of the fluid domain – not only along the boundary of the floater – to make the extension to second order feasible. Panels along all boundaries requires the development of generating absorbing boundary conditions for use at radiation boundaries to send incident waves into the domain while absorbing waves originating from the floating body at the same boundary, at the same time. The model is verified by means of conservation of energy of a heaving wave energy converter, and by means of the propagation of second-order waves. The performance in terms of conservation of energy with 12 panels per wave length is good, the generating absorbing boundary condition works according to expectation and the second-order wave propagation corresponds to theory.