High-order Absorbing Boundary Conditions Research Articles

Padé-type high-order absorbing boundary condition for a coupled hydrodynamic wave model with surface tension effect

Padé-type high-order absorbing boundary condition for a coupled hydrodynamic wave model with surface tension effect

Dispersive perfectly matched layers and high-order absorbing boundary conditions for electromagnetic quasinormal modes.

Resonances, also known as quasinormal modes (QNMs) in the non-Hermitian case, play a ubiquitous role in all domains of physics ruled by wave phenomena, notably in continuum mechanics, acoustics, electrodynamics, and quantum theory. The non-Hermiticity arises from the system losses, whether they are material (Joule losses in electromagnetism) or linked to the openness of the problem (radiation losses). In this paper, we focus on the latter delicate matter when considering bounded computational domains mandatory when using, e.g., finite elements. We address the important question of whether dispersive perfectly matched layer (PML) and high-order absorbing boundary conditions offer advantages in QNM computation and modal expansion of the optical responses compared with nondispersive PMLs.

High-order Absorbing Boundary Condition, Domain Decomposition Method and Stratified Dispersive Wave Model

The high-order Absorbing Boundary Condition proposed by Hagstrom and Warburton was applied to various models as the wave equation, dispersive, convected with stratified materials. We propose an other way to expand the use of this condition with Domain Decomposition Method. We apply it here to a Stratified Dispersive Wave Model not just to limit the real unbounded domain but to use a non-overlapping domain decomposition method with the classical Schwarz Waveform Relaxation method using the Absorbing Boundary Condition taken as an approximation of the Transparent Operator for the interface condition. We must add terms to enrich the equations associated to the Absorbing Boundary Condition and compute them with a cumulative process. Numerical examples are used to show the performance of this Domain Decomposition Method.

A high-order absorbing boundary condition for scalar wave propagation simulation in viscoelastic multilayered medium

PurposeThe finite element method (FEM) is used to calculate the two-dimensional anti-plane dynamic response of structure embedded in D’Alembert viscoelastic multilayered soil on the rigid bedrock. This paper aims to research a time-domain absorbing boundary condition (ABC), which should be imposed on the truncation boundary of the finite domain to represent the dynamic interaction between the truncated infinite domain and the finite domain.Design/methodology/approachA high-order ABC for scalar wave propagation in the D’Alembert viscoelastic multilayered media is proposed. A new operator separation method and the mode reduction are adopted to construct the time-domain ABC.FindingsThe derivation of the ABC is accurate for the single layer but less accurate for the multilayer. To achieve high accuracy, therefore, the distance from the truncation boundary to the region of interest can be zero for the single layer but need to be about 0.5 times of the total layer height of the infinite domain for the multilayer. Both single-layered and multilayered numerical examples verify that the accuracy of the ABC is almost the same for both cases of only using the modal number excited by dynamic load and using the full modal number of infinite domain. Using the ABC with reduced modes can not only reduce the computation cost but also be more friendly to the stability. Numerical examples demonstrate the superior properties of the proposed ABC with stability, high accuracy and remarkable coupling with the FEM.Originality/valueA high-order time-domain ABC for scalar wave propagation in the D’Alembert viscoelastic multilayered media is proposed. The proposed ABC is suitable for both linear elastic and D’Alembert viscoelastic media, and it can be coupled seamlessly with the FEM. A new operator separation method combining mode reduction is presented with better stability than the existing methods.

A local time-domain absorbing boundary condition for scalar wave propagation in a multilayered medium

An absorbing boundary condition (ABC) is particularly important for finite element simulation of wave propagation in a multilayered medium. In this paper, a spatially and temporally local high-order absorbing boundary condition is proposed for scalar wave propagation in semi-infinite multilayered media. A semi-discrete motion equation is derived by discretizing the truncation boundary of the semi-infinite domain along the vertical direction. The scalar dynamic stiffness in the frequency domain for a single degree of freedom (DOF) on the truncation boundary is obtained by only considering the first mode of the semi-infinite domain. The scalar dynamic stiffness is expressed as a continued fraction expansion that is stable and converges exponentially to the exact solution. The ABC based on the continued fraction for a single DOF on the truncation boundary is established by introducing auxiliary variables. The proposed ABC is always stable and it can be coupled straightforwardly with the existing finite element method. Since it is spatially decoupled and the coefficients in it are easy to obtain, the proposed ABC is convenient to apply in engineering. Numerical examples demonstrate the superior properties of the proposed method with high accuracy, high efficiency, and good stability.

A non-overlapping domain decomposition method with high-order transmission conditions and cross-point treatment for Helmholtz problems

A non-overlapping domain decomposition method (DDM) is proposed for the parallel finite-element solution of large-scale time-harmonic wave problems. It is well-known that the convergence rate of this kind of method strongly depends on the transmission condition enforced on the interfaces between the subdomains. Local conditions based on high-order absorbing boundary conditions (HABCs) have proved to be well-suited, as a good compromise between basic impedance conditions, which lead to suboptimal convergence, and conditions based on the exact Dirichlet-to-Neumann (DtN) map related to the complementary of the subdomain — which are too expensive to compute. However, a direct application of this approach for configurations with interior cross-points (where more than two subdomains meet) and boundary cross-points (points that belong to both the exterior boundary and at least two subdomains) is suboptimal and, in some cases, can lead to incorrect results.In this work, we extend a non-overlapping DDM with HABC-based transmission conditions approach to efficiently deal with cross-points for lattice-type partitioning. We address the question of the cross-point treatment when the HABC operator is used in the transmission condition, or when it is used in the exterior boundary condition, or both. The proposed cross-point treatment relies on corner conditions developed for Padé-type HABCs. Two-dimensional numerical results with a nodal finite-element discretization are proposed to validate the approach, including convergence studies with respect to the frequency, the mesh size and the number of subdomains. These results demonstrate the efficiency of the cross-point treatment for settings with regular partitions and homogeneous media. Numerical experiments with distorted partitions and smoothly varying heterogeneous media show the robustness of this treatment.

Corner treatments for high-order local absorbing boundary conditions in high-frequency acoustic scattering

This paper deals with the design and validation of accurate local absorbing boundary conditions set on convex polygonal and polyhedral computational domains for the finite element solution of high-frequency acoustic scattering problems. While high-order absorbing boundary conditions (HABCs) are accurate for smooth fictitious boundaries, the precision of the solution drops in the presence of corners if no specific treatment is applied. We present and analyze two strategies to preserve the accuracy of Padé-type HABCs at corners: first by using compatibility relations (derived for right angle corners) and second by regularizing the boundary at the corner. Exhaustive numerical results for two- and three-dimensional problems are reported in the paper. They show that using the compatibility relations is optimal for domains with right angles. For the other cases, the error still remains acceptable, but depends on the choice of the corner treatment according to the angle.

Application of a complete radiation boundary condition for the Helmholtz equation in locally perturbed waveguides

This paper deals with an application of a complete radiation boundary condition (CRBC) for the Helmholtz equation in locally perturbed waveguides. The CRBC, one of efficient high-order absorbing boundary conditions, has been analyzed in straight waveguides in Hagstrom and Kim (2019). In this paper, we apply CRBC to the Helmholtz equation posed in locally perturbed waveguides and establish the well-posedness of the problem and convergence of CRBC approximate solutions. The new CRBC proposed in this paper improves the one studied in Hagstrom and Kim (2019) in two aspects. The first one is that the new CRBC involves more damping parameters with the same computational cost as that of CRBC in Hagstrom and Kim (2019), which results in 50% smaller reflection errors. The second one is that the new CRBC takes a Neumann terminal condition of three term recurrence relations of auxiliary variables instead of a Dirichlet terminal condition used in Hagstrom and Kim (2019) so that it can treat cutoff modes effectively. Finally, we present numerical experiments illustrating the convergence theory.

A stable high-order absorbing boundary based on continued fraction for scalar wave propagation in 2D and 3D unbounded layers

PurposeThe purpose of this paper is to propose a stable high-order absorbing boundary condition (ABC) based on new continued fraction for scalar wave propagation in 2D and 3D unbounded layers.Design/methodology/approachThe ABC is obtained based on continued fraction (CF) expansion of the frequency-domain dynamic stiffness coefficient (DtN kernel) on the artificial boundary of a truncated infinite domain. The CF which has been used to the thin layer method in [69] will be applied to the DtN method to develop a time-domain high-order ABC for the transient scalar wave propagation in 2D. Furthermore, a new stable composite-CF is proposed in this study for 3D unbounded layers by nesting the above CF for 2D layer and another CF.FindingsThe ABS has been transformed from frequency to time domain by using the auxiliary variable technique. The high-order time-domain ABC can couple seamlessly with the finite element method. The instability of the ABC-FEM coupled system is discussed and cured.Originality/valueThis manuscript establishes a stable high-order time-domain ABC for the scalar wave equation in 2D and 3D unbounded layers, which is based on the new continued fraction. The high-order time-domain ABC can couple seamlessly with the finite element method. The instability of the coupled system is discussed and cured.

Well-posedness of the Westervelt equation with higher order absorbing boundary conditions

The focus of this work is on the analysis of the Westervelt equation modeling nonlinear propagation of high intensity ultrasound, in the practically relevant setting of a truncated computational domain with absorbing boundary conditions. We especially consider the zero and first order nonlinear absorbing boundary conditions devised in [38] in one and two space dimensions. As a matter of fact, the energy identities and estimates presented here were crucial for designing these absorbing boundary conditions in such a way that the desired energy dissipation through the boundary is guaranteed. Under the hypothesis of small initial data, we establish local well-posedness and provide higher order energy estimates, that we expect to be of additional use in boundary control and stabilization.

Optimized first-order absorbing boundary conditions for anisotropic elastodynamics

We propose new forms of low-order absorbing boundary conditions (ABC) for time-dependent elastic waves in isotropic and anisotropic media. The configuration considered is that of a two-dimensional elastic waveguide. The ABC shares with the widely-known Lysmer–Kuhlemeyer (LK) boundary condition the ease of implementation, especially in a finite-element (FE) setting, while offering gains in accuracy over the LK-like condition (or the LK condition in the isotropic case). The new conditions are proved to be energy-stable, even in the case when inverse modes are present, for which the normal components of the phase and group velocities have an opposite direction. The developed ABCs are particularly convenient to use in a Finite Element setting as they preserve the symmetry and positivity of the formulation and are simple to implement. The ABC features several parameters, which are optimized for accuracy. The optimization criterion is based on the notion of the energy-rate reflection coefficient. We test the performance of the scheme via a numerical example. While the gain in accuracy is found to be moderate (around 20% relative to the LK condition), the main result of this investigation lies in the development of ABCs for anisotropic elastodynamics that are proved to be stable from the outset, and are amenable for accuracy optimization. We plan to design in a future study analogous stable and optimized high-order ABCs using the methodology developed here.

Performance and stability of the double absorbing boundary method for acoustic-wave propagation

The double absorbing boundary (DAB) is a novel extension to the family of high-order absorbing boundary condition operators. It uses auxiliary variables in a boundary layer to set up cancellation waves that suppress wavefield energy at the computational-domain boundary. In contrast to the perfectly matched layer (PML), the DAB makes no assumptions about the incoming wavefield and can be implemented with a boundary layer as thin as three computational grid-point cells. Our implementation incorporates the DAB into the boundary cell layer of high-order finite-difference (FD) techniques, thus avoiding the need to specify a padding region within the computational domain. We tested the DAB by propagating acoustic waves through homogeneous and heterogeneous 3D earth models. Measurements of the spectral response of energy reflected from the DAB indicate that it reflects approximately 10–15 dB less energy for heterogeneous models than a convolutional PML of the same computational memory complexity. The same measurements also indicate that a DAB boundary layer implemented with second-order FD operators couples well with higher-order FD operators in the computational domain. Long-term stability tests find that the DAB and CPML methods are stable for the acoustic-wave equation. The DAB has promise as a robust and memory-efficient absorbing boundary for 3D seismic imaging and inversion applications as well as other wave-equation applications in applied physics.

Root-finding absorbing boundary conditions for scalar and elastic waves in infinite media

Root-Finding Absorbing Boundary Conditions (RFABCs) for scalar and elastic waves in infinite media are developed. First, the existing RFABC formulation for scalar-wave propagation problems is refined. Specifically, a Fourier series expansion or an eigenfunction expansion, as in Sturm–Liouville problems, is applied to general scalar-wave fields and a consistent nodal flux is derived for eventual combination with the discrete representation of the problem under consideration. This newly-proposed approach for scalar waves is extended to elastic waves, governed by two scalar-wave equations for pressure and shear waves. Stability of the refined RFABCs can be proven at both the continuous and discrete levels. The newly-developed RFABCs for scalar and elastic waves are applied to various wave-propagation problems. It is verified that they produce accurate and stable results. Perfectly matched layers and high-order absorbing boundary conditions are the most popular of the available tools for wave-propagation problems. But, their stability properties have not been established in applications to problems of elastic waves in waveguides. It is expected that existing high-order absorbing boundaries will lead to stable results for elastic waves in waveguides, when implemented using the technique proposed in the present study.

A Trefftz Discontinuous Galerkin method for time-harmonic waves with a generalized impedance boundary condition

ABSTRACTWe show how a Trefftz Discontinuous Galerkin (TDG) method for the displacement form of the Helmholtz equation can be used to approximate problems having a generalized impedance boundary condition (GIBC) involving surface derivatives of the solution. Such boundary conditions arise naturally when modeling scattering from a scatterer with a thin coating. The thin coating can then be approximated by a GIBC. A second place GIBCs arise is as higher order absorbing boundary conditions. This paper also covers both cases. Because the TDG scheme has discontinuous elements, we propose to couple it to a surface discretization of the GIBC using continuous finite elements. We prove convergence of the resulting scheme and demonstrate it with two numerical examples.

A high-order absorbing boundary condition for 2D time-harmonic elastodynamic scattering problems

In this paper, we are concerned with the construction of a new high-order Absorbing Boundary Condition (ABC) for 2D-elastic scattering problems. It is defined by an approximate local Dirichlet-to-Neumann (DtN) map. First, we explain the derivation of this approximation. Next, a detailed analytical study in terms of Hankel functions in the circular case is addressed. The new ABC is compared with the standard low-order Lysmer–Kuhlemeyer ABC. Finally, its accuracy and efficiency are investigated for various numerical examples, particularly at high frequencies.

Stable high-order absorbing boundary condition based on new continued fraction for scalar wave propagation in unbounded multilayer media

A long-time stable high-order absorbing boundary condition (ABC) is developed for the finite element simulation of time-dependent scalar wave propagation in unbounded multilayer media that is a multilayer waveguide. The ABC is obtained based on a new continued fraction (CF) expansion of the frequency-domain dynamic stiffness matrix on the artificial boundary of a truncated infinite domain. The dynamic stiffness in scalar CF form is first proposed in the modal space for a single-layered medium and is subsequently extended to a matrix CF form for multilayer media. The new CF converges to dynamic stiffness over the whole frequency range as its order increases for the single-layered medium, and has high accuracy for the multilayer media. The CF-based high-order ABC is therefore accurate and stable in the time domain. After being coupled seamlessly with finite element method (FEM), instability phenomena are observed during long-time computation. In order to eliminate the instability of the coupled ABC and FEM system, a straightforward and effective method is proposed by introducing damping proportional to the stiffness matrix of spatially discretised finite domain. Numerical examples demonstrate the superior properties of the proposed ABC with high accuracy and long-time stability as well as remarkable coupling with FEM.

The Double Absorbing Boundary method for a class of anisotropic elastic media

The Double Absorbing Boundary (DAB) method was recently introduced as a new approach for solving wave problems in unbounded domains. It has common features to local high-order Absorbing Boundary Conditions (ABC) on one hand, and to Perfectly Matched Layers (PML) on the other, and has relative advantages with respect to both. In the DAB method, the unbounded domain is truncated to produce a finite computational domain, which is then enclosed by a thin layer, as in the PML. Local high-order ABC are then imposed on both the inner and outer boundaries of the layer. Auxiliary variables are defined on the two boundaries as well as inside the layer, and participate in the numerical scheme. Only low order derivatives appear in the formulation, which allows the use of standard discretization methods in space and time. In previous studies, the DAB was developed for acoustic waves which are solutions to the scalar wave equation, and for elastic waves in isotropic media. Here the approach is extended to time-dependent elastic waves in anisotropic media. The geometrical configuration assumed is that of a semi-infinite wave guide, truncated via the DAB layer. Standard Finite Elements (FE) are used for space discretization and the damped Newmark scheme is used for time discretization. The problems considered here are restricted to those with periodic lateral boundary conditions, and with anisotropic media which do not support inverse modes, since removing these limitations may lead to a numerical instability. The performance of the scheme is demonstrated via numerical examples, including uniform orthotropic and layered orthotropic media.

Optimized double sweep Schwarz method by complete radiation boundary conditions

We present an optimized double sweep nonoverlapping Schwarz method for solving the Helmholtz equation in semi-infinite waveguides. The domain is decomposed into nonoverlapped layered subdomains along the axis of the waveguide and local wave propagation problems equipped with complete radiation conditions for high-order absorbing boundary conditions are solved forward and backward sequentially. For communication between subdomains, Neumann data of local solutions in one domain are transferred to the neighboring subdomain in the forward direction and Dirichlet data are exploited in the backward direction. The complete radiation boundary conditions enable us to not only minimize reflection coefficients for most important modes in an optimal way but also find Neumann data without introducing errors that would be produced if finite difference formulas were used for computing Neumann data. The convergence of the double sweep Schwarz method is proved and numerical experiments using it as a preconditioner are presented to confirm the convergence theory.

Time exponential splitting technique for the Klein–Gordon equation with Hagstrom–Warburton high-order absorbing boundary conditions

Klein-Gordon equations on an unbounded domain are considered in one dimensional and two dimensional cases. Numerical computation is reduced to a finite domain by using the Hagstrom-Warburton (H-W) high-order absorbing boundary conditions (ABCs). Time integration is made by means of exponential splitting schemes that are efficient and easy to implement. In this way, it is possible to achieve a negligible error due to the time integration and to study the behavior of the absorption error. Numerical experiments displaying the accuracy of the numerical solution for the two dimensional case are provided. The influence of the dispersion coefficient on the error is also studied.

Performance of the Double Absorbing Boundary Method when Applied to the 3D Acoustic Wave Equation

SUMMARY The double absorbing boundary (DAB) is a new highorder absorbing boundary condition for the scalar acoustic wave equation. It suppresses scattered waves at the edge of a boundary layer in computational domain boundary by using destructive interference analogous to a noise-cancelling headphone. This method has advantages in that it addresses some of the shortfalls in existing boundary conditions, such as the need for tuning in Perfectly Matched Layers or complex formulations at corners such as in high-order absorbing boundary conditions. We extend the original formulation of the DAB to three dimensions and higher-order stencils. Through numerical simulation we test the performance of the DAB by comparison with a reflecting boundary. We find that the DABC is a broadband attenuator with a power attenuation of 20-30dB using only six boundary cells. Increasing the order of the method improves accuracy for wavelengths less than 10 cells, whereas increasing the layer width does not improve accuracy. The method shows promise as a robust and computationally efficient boundary condition for seismic applications.