Evaluation of the Perfectly Matched Layer for Computational Acoustics

doi:10.1006/jcph.1997.5868

Journal of Computational Physics

Volume 139, Issue 1, 1 January 1998, Pages 166-183

https://doi.org/10.1006/jcph.1997.5868 Get rights and content

Abstract

The perfectly matched layer (PML) recently formulated by Berenger for the absorption of radiated/scattered waves in computational electromagnetics is adapted to computational acoustics, and its effectiveness as a nonreflecting boundary is examined. The excellent absorbing ability of the PML is demonstrated by its small reflection coefficient for a plane wave incident on a plane interface. However, additional frequency-domain and time-domain solutions show that the PML may not be an appropriate computational boundary if the analyst is only interested in the response of the radiator/scatterer and/or the acoustic field in the vicinity of the radiator/scatterer.

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(1988)

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Cited by (107)

A hybrid PML formulation for the 2D three-field dynamic poroelastic equations
2023, Computer Methods in Applied Mechanics and Engineering
Simulation of wave propagation in poroelastic half-spaces presents a common challenge in fields like geomechanics and biomechanics, requiring Absorbing Boundary Conditions (ABCs) at the semi-infinite space boundaries. Perfectly Matched Layers (PML) are a popular choice due to their excellent wave absorption properties. However, PML implementation can lead to problems with unknown stresses or strains, time convolutions, or PDE systems with Auxiliary Differential Equations (ADEs), which increases computational complexity and resource consumption.
This article introduces a novel hybrid PML formulation for arbitrary poroelastic domains. Instead of using ADEs, this formulation utilizes time-history variables to reduce the number of unknowns and mathematical operations. The modification of the PDEs to account for the PML is limited to the PML domain only, resulting in smaller matrices while maintaining the governing equations in the interior domain and preserving the temporal structure of the problem. The hybrid approach introduces three scalar variables localized within the PML domain.
The proposed formulation was tested in three numerical experiments in geophysics using realistic parameters for soft sites with free surfaces. The results were compared with numerical solutions from extended domains and simpler ABCs, such as paraxial approximation, demonstrating the accuracy, efficiency, and precision of the proposed method. The article also discusses the applicability of the method to complex media and its extension to the Multiaxial PML formulation.
The codes used for the simulations are available for download from https://github.com/hmella/POROUS-HYBRID-PML.
Parametrization-free locally-conformal perfectly matched layer method for finite element solution of Helmholtz equation
2023, Computer Physics Communications
We present a novel Locally-Conformal Perfectly Matched Layer (LCPML) method for mesh truncation in the Finite Element Method (FEM), to solve wave equations derived from Maxwell's equations. The original LCPML method was proposed by Ozgun and Kuzuoglu in 2006 [16] by utilizing the concept of complex elements in which nodal coordinates of elements are replaced by complex coordinates obtained by using a complex coordinate-transformation. In the proposed LCPML method, named as LCPML-log method, a more efficient coordinate-transformation is used together with new FEM formulation and implementation. The proposed method offers the advantage that only a few (or even a single) PML layer is sufficient to get reliable results. In addition, it does not require any PML parameter tuning, and hence, the decaying wave behavior in PML is achieved self-adaptively. The numerical results of some detailed and systematic performance analyses are also presented.
Upwind finite element-PML approximation of a novel linear potential model for free surface flows produced by a floating rigid body
2022, Applied Mathematical Modelling
Citation Excerpt :
Currently, the Perfectly Matched Layer (PML) is a standard technique to model wave propagation phenomena in unbounded domains, in such a way that the computational domain of interest is surrounded by an absorbing artificial layer, which does not introduce any spurious reflections in the solution of the original problem. It was first proposed by Berenger [9] for electromagnetic problems but it is also widely used in other fields as acoustics (see Abarbanel et al. [10], Qi and Geers [11], Bermúdez et al. [12]), or linearised water waves (see Cohen and Imperiale [13]). In the present approach the two-dimensional Cartesian PML model has been extended to three dimensions in cylindrical coordinates with the aim of absorbing the outgoing wave pattern generating by the floating body (i.e., the Kelvin wake) and preserving the structure of the convected terms presented in the free boundary condition.
A novel linear potential model is presented to compute free surface flows of incompressible fluids produced by the motion of a floating rigid body in the presence of an underlying non-uniform flow. In particular, the proposed model enables the accurate numerical simulation of the Kelvin wake pattern in a computational domain of reduced size. The governing equations are obtained by using an Arbitrary Lagrangian Eulerian (ALE) formulation which involves the underlying velocity of the fluid around the floating body assuming flat free surface, and a non-dimensional analysis to derive the novel linear system of equations for free surface flows. The discretization of the proposed model is made by a standard Galerkin finite element method, where a SUPG-inspired upwinding strategy has been used in combination with a Perfectly Matched Layer technique which allows truncating the original unbounded fluid domain without introducing spurious reflections in the Kelvin wake pattern. The numerical simulations computed with the proposed approach are compared with the results obtained by the classical linear potential model with uniform underlying flow and also with those from the full incompressible Navier–Stokes equations equipped with the $k - ω$ SST turbulent model. This numerical comparison is discussed in terms of a classical hydrodynamic floating body benchmark involving the Wigley hull.
Krylov subspaces recycling based model order reduction for acoustic BEM systems and an error estimator
2020, Computer Methods in Applied Mechanics and Engineering
Boundary Element Method frequency sweep analyses in acoustics are usually accompanied by a vast numerical cost of assembling and solving numerous linear systems. In that context, this work proposes a model order reduction technique to mitigate the resulting computational cost of such analyses. First, a series expansion of the Green’s function BEM kernel is leveraged to construct a series of frequency independent matrices. Next, in a model order reduction way, the arising matrices are projected on a reduced basis utilizing a Galerkin projection. By this off-line matrix projection, both the assembly and the solution of the BEM full-size linear systems degenerate into assembling and solving a reduced system for all frequencies. Significant speed-up factors can, thus, be achieved for both operations. The projection basis employed in this model reduction scheme is developed through an Arnoldi algorithm for the BEM systems on a grid of master frequencies. The method is based on Krylov subspaces recycling, as the subspaces produced at master frequencies are recycled to approximate the surface distribution of the acoustic variables on the whole frequency range of interest. Utilizing Krylov subspaces facilitates as well the definition of a robust error estimator that indicates the quality of the reduced system. The performance of the proposed method is assessed for both an exterior and an interior problem for a simple and more complicated geometry respectively.
Comparative evaluation of foundation modeling for SSI analyses using two different ABC approaches: Applications to dams
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The numerical modeling of the unbounded domain is one of the most challenging issues both for site response and soil-structure interaction analyses. The viscous-type absorbing boundary conditions (ABCs) are commonly used for the modeling of the unbounded domain due to their low computational cost and simplicity. However, in cases, their application can lead to substantial errors. In this paper, a time-domain viscoelastic perfectly matched layer (PML) is implemented for modeling the unbounded domain in the dynamic response evaluation of a concrete gravity dam. The performance of the viscoelastic PML and the viscous ABCs are examined through numerical examples under different types of dynamic excitations including uniform and spatially variable seismic ground motions. The numerical results highlight the excellent performance of the PML approach to model the unbounded domain under all types of excitations. It is also shown that the viscous-type ABCs perform poorly under spatially variable excitations and reflected waves from the structures, as they cannot properly absorb inclined waves.
The nonlinear response of a concrete gravity dam subjected to deconvolved seismic excitations at depth resulting from five surface seismic records was then evaluated. The effect of the finite foundation depth on the dam response was also examined by considering a shallow (approximately equal to the height of the dam) and a deep (approximately double the height of the dam) finite foundation length. The results indicate that the PML is capable of maintaining its accuracy even if the size of the computational finite domain is reduced. On the other hand, it was shown that the shallow finite foundations modeled with viscous ABCs substantially underestimate the dam response under uniform excitations, whereas the use of a deep finite foundation model with viscous ABCs can lead to more conservative results. An important conclusion of this study is that the use of foundation models with viscous-type ABCs for the dynamic analysis of dams, and, consequently, other long structures, subjected to spatially variable excitations, should be avoided as it can significantly underestimate the crest displacement and the total damage in the dam.
A local collocation method to construct Dirichlet-type absorbing boundary conditions for transient scalar wave propagation problems
2019, Computer Methods in Applied Mechanics and Engineering
This paper introduces an effective way to equip the standard finite element method (FEM) for the solution of transient scalar wave propagation problems in unbounded domains. Similar to many other methods, we truncate the unbounded domain at an artificial boundary and convert the problem into a bounded one by prescribing appropriate absorbing boundary conditions (ABCs) at the truncating boundary. In the present method, the ABCs are time-dependent, and they are constructed by a simple collocation approach which is local in space and time. Therefore, the method does not make use of any routine schemes such as Fourier and Laplace transform. We shall show that the method is simple, and it can be easily applied to an explicit time domain FEM approach so that the sparsity of the FEM scheme (as well as its efficiency) can be preserved. The proposed method does not require any auxiliary variables as well as any approximating differential operators. This feature roots from the fact that here the ABCs are Dirichlet-type (or first-type) and thus they can be easily imposed to the corresponding boundaries. Therefore, the method shares some similarities with the conventional 1st and 2nd order ABC methods in terms of the simplicity of implementation. The method employs basis functions that exactly satisfy the governing and dispersion wave equations. The basis function can be easily adjusted to act as outgoing waves transmitting energy from the interior domain (near field) towards exterior domain (far field); i.e, they can cope with satisfaction of radiation boundary conditions. Several numerical examples are presented to evaluate the performance and to demonstrate the effectiveness of the approach. We shall show that the present method is capable of yielding results with a proper level of accuracy, similar to that of the perfectly matched layers method (PMLs), and it performs stably even in the case of long-term computations.

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Regular ArticleEvaluation of the Perfectly Matched Layer for Computational Acoustics

Abstract

J. Comput. Phys.

J. Comput. Phys.

Wave Motion

Regular Article
Evaluation of the Perfectly Matched Layer for Computational Acoustics