Wave Problems In Unbounded Domains Research Articles

Scaled Boundary Perfectly Matched Layer (SBPML): A novel 3D time-domain artificial boundary method for wave problem in general-shaped and heterogeneous infinite domain

Artificial boundary method is widely used in the numerical modeling of unbounded wave problem. However, the accurate modeling of truncated infinite domain with general geometry and heterogeneous materials is still a challenging task, especially for direct time-domain analysis in three dimensions (3D). In this paper, a novel 3D time-domain artificial boundary method, called Scaled Boundary Perfectly Matched Layer (SBPML), is proposed. This method is a generalization of the Perfectly Matched Layer (PML) based on a scaled boundary coordinates transformation inspired by the Scaled Boundary Finite Element Method (SBFEM), which is capable of using artificial boundary of general geometry (not necessarily convex) and considering plane physical surfaces and interfaces extending to infinity in the truncated infinite domain. Local scaled boundary coordinates are firstly introduced on the element-level into the truncated infinite domain to describe general geometry properties of the infinite domain. Then, a complex stretching function from PML is applied to radial direction of the local scaled boundary coordinates to map the physical space onto the complex space, resulting in a SBPML domain. The spatial discretization of the SBPML domain produces semi-discrete mixed displacement–stress​ unsplit-field formulations of third orders in time. The order of the obtained formulation can be reduced by one, enabling a seamless coupling with the standard displacement-based finite element formulation of the interior domain. The coupled system can be solved by an explicit time integration algorithm efficiently. The validation of the SBPML is demonstrated through several benchmark tests, including wave problems in unbounded domains with general geometries and heterogeneous material properties. Furthermore, the application of the SBPML in dynamic soil–structure interaction (SSI) is demonstrated using two engineering problems, including an impact analysis of soft rock-nuclear island system and a vibration analysis of soil-lined tunnel system.

The Double Absorbing Boundary Method Incorporated in a High-Order Spectral Element Formulation

In this paper, we consider the numerical solution of the time-dependent wave equation in a semi-infinite waveguide. We employ the Double Absorbing Boundary (DAB) method, by introducing two parallel artificial boundaries on the side where waves are outgoing. In contrast to the original implementation of the DAB, where the numerical solution involved either a low-order finite difference scheme or a low-order finite element scheme, here we incorporate the DAB into a high-order spectral element formulation, which provides us with very accurate solutions of wave problems in unbounded domains. This is demonstrated by numerical experiments. While the method is highly accurate, it suffers from long-time instability. We show how to postpone the onset of the instability by a prudent choice of the computational parameters.

Overresolving in the Laplace Domain for Convolution Quadrature Methods

Convolution quadrature (CQ) methods have enjoyed tremendous interest in recent years as an efficient tool for solving time-domain wave problems in unbounded domains via boundary integral equation techniques. In this paper we consider CQ type formulations for the parallel space-time evaluation of multistep or stiffly accurate Runge--Kutta rules for the wave equation. In particular, we decouple the number of Laplace domain solves from the number of time steps. This allows us to overresolve in the Laplace domain by computing more Laplace domain solutions than there are time steps. We use techniques from complex approximation theory to analyze the error of the CQ approximation of the underlying time-stepping rule when overresolving in the Laplace domain and show that the performance is intimately linked to the location of the poles of the solution operator. Several examples using boundary integral equation formulations in the Laplace domain are presented to illustrate the main results.

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.

The double absorbing boundary method for elastodynamics in homogeneous and layered media

Abstract Background Recently the Double Absorbing Boundary (DAB) method was introduced as a new approach for solving wave problems in unbounded domains. It has common features to each of two types of existing techniques: local high-order Absorbing Boundary Conditions (ABC) and Perfectly Matched Layers (PML). However, it is different from both and enjoys relative advantages with respect to both. Methods The DAB method is based on truncating the unbounded domain to produce a finite computational domain, and on applying a local high-order ABC on two parallel artificial boundaries, which are a small distance apart, and thus form a thin non-reflecting layer. Auxiliary variables are defined on the two boundaries and within the layer, and participate in the numerical scheme. In previous studies DAB was developed for acoustic waves which are solutions to the scalar wave equation. Here the approach is extended to time-dependent elastic waves in homogeneous and layered media. The equations are written in second-order form in space and time. Standard Finite Elements (FE) are used for space discretization and the damped Newmark scheme is used for time discretization. Results The performance of the scheme is demonstrated via numerical examples. The DAB was applied to elastodynamics problems in conjunction with the FE method to demonstrate the performance of the method. Conclusions DAB is a viable method for solving wave problems in unbounded domains.

Double Absorbing Boundary Formulations for Acoustics and Elastodynamics

Wave problems in unbounded domains are often treated numerically by truncating the domain to produce a finite computational domain. The double absorbing boundary (DAB) method, which was invented recently as an alternative to methods of high-order absorbing boundary conditions and to the perfectly matched layer, is investigated here for problems in acoustics and elastodynamics. The paper offers two main contributions. The first one pertains to the well-posedness of the DAB scheme for the acoustics problem written in second-order form. The energy method is employed to obtain uniform-in-time estimates of the norm of the solution and the auxiliary functions, thus establishing the well-posedness and asymptotic stability of the DAB formulation. The second part pertains to the extension of the DAB to isotropic elastodynamics, written in first-order conservation form. Numerical experiments for an elastic wave guide demonstrate the performance of the scheme.

The Double Absorbing Boundary method

A new approach is devised for solving wave problems in unbounded domains. It has common features to each of two types of existing techniques: local high-order Absorbing Boundary Conditions (ABC) and Perfectly Matched Layers (PML). However, it is different from both and enjoys relative advantages with respect to both. The new method, called the Double Absorbing Boundary (DAB) method, is based on truncating the unbounded domain to produce a finite computational domain Ω, and on applying a local high-order ABC on two parallel artificial boundaries, which are a small distance apart, and thus form a thin non-reflecting layer. Auxiliary variables are defined on the two boundaries and inside the layer bounded by them, and participate in the numerical scheme. The DAB method is first introduced in general terms, using the 2D scalar time-dependent wave equation as a model. Then it is applied to the 1D Klein–Gordon equation, using finite difference discretization in space and time, and to the 2D wave equation in a wave guide, using finite element discretization in space and dissipative time stepping. The computational aspects of the method are discussed, and numerical experiments demonstrate its performance.

On the Linear Water Wave Problem in the Presence of a Critically Submerged Body

We study the two-dimensional problem of propagation of linear water waves in deep water in the presence of a critically submerged body (i.e., the body touching the water surface). Assuming uniqueness of the solution in the energy space, we prove the existence of a solution which satisfies the radiation conditions at infinity as well as at the cusp point where the body touches the water surface. This solution is obtained by the limiting absorption procedure. Next we introduce a relevant scattering matrix and analyze its properties. Under a geometric condition introduced by V. Maz'ya in 1978, we prove an important property of the scattering matrix, which may be interpreted as the absence of total internal reflection. This property also allows us to obtain uniqueness and existence of a solution in some function spaces (e.g., $H^2_{loc}\cap L^\infty$) without use of the radiation conditions and the limiting absorption principle, provided a spectral parameter in the boundary conditions on the surface of the water is large enough. The fact that the existence and uniqueness result does not rely on either the radiation conditions or the limiting absorption principle is the first result of this type known to us in the theory of linear wave problems in unbounded domains.

Numerical comparison of high-order absorbing boundary conditions and perfectly matched layers for a dispersive one-dimensional medium

High-order absorbing boundary conditions (ABC) and perfectly matched layers (PML) are two powerful methods to numerically solve wave problems in unbounded domains. The aim of the proposed study is to analyze and compare the performance of these methods in the one-dimensional problem governed by the dispersive wave equation. The PMLs proposed in literature for time-harmonic dynamics are applied to time-dependent wave problems, and linear, quadratic and cubic polynomial stretching functions are considered. The resulting PMLs exhibit a double absorbing action: (i) they reduce the amplitude of the incident waves and (ii) they delay the wave propagation. Then the ABC proposed by Givoli and Neta and those proposed by Hagstrom, Mar-Or and Givoli are considered. The former are a reformulation of the Higdon high-order non-reflecting boundary conditions, the latter improve the Higdon conditions and extend them to take into account evanescent waves. The accuracy of the PMLs and ABCs, implemented in a finite element code, is first investigated with respect to the frequency of the incident wave, being it progressive or evanescent. Then the response to a wave train characterized by a broad frequency spectrum, resulting from an impulsive force, is studied. A detailed analysis is performed to detect the influence of the parameters of both the ABCs and the PMLs on the absorption of waves. The performances of PMLs and ABCs are compared, and merits and drawbacks of the two methods are pointed out.

High‐order absorbing boundary conditions incorporated in a spectral element formulation

AbstractThe solution of the time‐dependent wave equation in a semi‐infinite wave guide is considered. An artificial boundary ℬ︁ is introduced, which encloses a finite computational domain. On ℬ︁ the Hagstrom–Warburton (H–W) local high‐order absorbing boundary condition (ABC) is imposed. In contrast to previous studies, which involved either finite difference schemes or low‐order finite element schemes, here the way to incorporate the H–W ABC into a spectral element formulation is shown. To this end, the Seriani–Priolo spectral element is used in space and a 4th‐order Runge–Kutta scheme is employed in time. This leads to exponential convergence in both the polynomial order of the spectral element, and in the order of the ABC, and thus avoids the convergence rate inconsistency which is otherwise present. The combined ABC‐spectral formulation enables one to obtain extremely accurate solutions of wave problems in unbounded domains. This is demonstrated via a number of numerical examples. Copyright © 2008 John Wiley & Sons, Ltd.

High‐order Higdon‐like boundary conditions for exterior transient wave problems

AbstractRecently developed non‐reflecting boundary conditions are applied for exterior time‐dependent wave problems in unbounded domains. The linear time‐dependent wave equation, with or without a dispersive term, is considered in an infinite domain. The infinite domain is truncated via an artificial boundary ℬ︁, and a high‐order non‐reflecting boundary condition (NRBC) is imposed on ℬ︁. Then the problem is solved numerically in the finite domain bounded by ℬ︁. The new boundary scheme is based on a reformulation of the sequence of NRBCs proposed by Higdon. We consider here two reformulations: one that involves high‐order derivatives with a special discretization scheme, and another that does not involve any high derivatives beyond second order. The latter formulation is made possible by introducing special auxiliary variables on ℬ︁. In both formulations the new NRBCs can easily be used up to any desired order. They can be incorporated in a finite element or a finite difference scheme; in the present paper the latter is used. In contrast to previous papers using similar formulations, here the method is applied to a fully exterior two‐dimensional problem, with a rectangular boundary. Numerical examples in infinite domains are used to demonstrate the performance and advantages of the new method. In the auxiliary‐variable formulation long‐time corner instability is observed, that requires special treatment of the corners (not addressed in this paper). No such difficulties arise in the high‐derivative formulation. Published in 2005 by John Wiley & Sons, Ltd.

High-order local non-reflecting boundary conditions: a review

A common method for numerically solving wave problems in unbounded domains is based on truncating the infinite domain via an artificial boundary B, thus defining a finite computational domain, and using a special non-reflecting boundary condition (NRBC) on B. Low-order local NRBCs have been constructed and practiced since the 1970s. Exact non-local NRBCs were introduced in the 1980s. Only recently high-order local NRBCs have been devised. These NRBCs, despite being of an arbitrarily high-order, do not involve high derivatives owing to the use of specially defined auxiliary variables. This paper reviews the latter approach, explains its advantages compared to previous approaches, and discusses the different schemes which have been proposed in this context.

High-order non-reflecting boundary scheme for time-dependent waves

A new non-reflecting boundary scheme is proposed for time-dependent wave problems in unbounded domains. The linear time-dependent wave equation, with or without a dispersive term, is considered in a semi-infinite wave guide. The infinite domain is truncated via an artificial boundary B , and a high-order non-reflecting boundary condition (NRBC) is imposed on B . Then the problem is solved numerically in the finite domain bounded by B . The new boundary scheme is based on a reformulation of the sequence of NRBCs proposed by Higdon. In contrast to the original formulation of the Higdon conditions, the scheme constructed here does not involve any high derivatives beyond second order. This is made possible by introducing special auxiliary variables on B . As a result, the new NRBCs can easily be used up to any desired order. They can be incorporated in a finite element or a finite difference scheme; in the present paper the latter is used. The parameters appearing in the NRBC are chosen automatically via a special procedure. Numerical examples concerning a semi-infinite wave guide are used to demonstrate the performance of the new method.

OPTIMAL LOCAL NONREFLECTING BOUNDARY CONDITIONS FOR TIME-DEPENDENT WAVES

Nonreflecting Boundary Conditions (NRBCs) are often used on artificial boundaries as a method for the numerical solution of wave problems in unbounded domains. Recently, a two-parameter hierarchy of optimal local NRBCs of increasing order has been developed for elliptic problems, including the problem of time-harmonic acoustic waves. The optimality is in the sense that the local NRBC best approximates the exact nonlocal Dirichlet-to-Neumann (DtN) boundary condition in the L2 norm for functions which can be Fourier-decomposed. The optimal NRBCs are combined with finite element discretization in the computational domain. Here this approach is extended to time-dependent acoustic waves. In doing this, the Semi-Discrete DtN approach is used as the starting point. Numerical examples involving propagating disturbances in two dimensions are given.

Optimal Local Nonreflecting Boundary Conditions for Time-Dependent Waves

Journal of Computational AcousticsVol. 08, No. 01, pp. 157-170 (2000) No AccessOPTIMAL LOCAL NONREFLECTING BOUNDARY CONDITIONS FOR TIME-DEPENDENT WAVESIGOR PATLASHENKO and DAN GIVOLIIGOR PATLASHENKOHummingbird Communications LTD., 1 Sparks Avenue, North York, Ontario, Canada M2H 2W1, Canada Search for more papers by this author and DAN GIVOLIDepartment of Aerospace Engineering, and Asher Center for Space Research, Technion — Israel Institute of Technology, Haifa 32000, Israel Search for more papers by this author https://doi.org/10.1142/S0218396X00000108Cited by:2 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AbstractNonreflecting Boundary Conditions (NRBCs) are often used on artificial boundaries as a method for the numerical solution of wave problems in unbounded domains. Recently, a two-parameter hierarchy of optimal local NRBCs of increasing order has been developed for elliptic problems, including the problem of time-harmonic acoustic waves. The optimality is in the sense that the local NRBC best approximates the exact nonlocal Dirichlet-to-Neumann (DtN) boundary condition in the L2 norm for functions which can be Fourier-decomposed. The optimal NRBCs are combined with finite element discretization in the computational domain. Here this approach is extended to time-dependent acoustic waves. In doing this, the Semi-Discrete DtN approach is used as the starting point. Numerical examples involving propagating disturbances in two dimensions are given. FiguresReferencesRelatedDetailsCited By 2An optimal high-order non-reflecting finite element scheme for wave scattering problemsDan Givoli and Igor Patlashenko1 January 2002 | International Journal for Numerical Methods in Engineering, Vol. 53, No. 10Nonreflecting boundary condition for the wave equationUsaf E. Aladl, A.S. Deakin and H. Rasmussen1 Jan 2002 | Journal of Computational and Applied Mathematics, Vol. 138, No. 2 Recommended Vol. 08, No. 01 Metrics History Received 5 July 1999 Revised 5 July 1999 PDF download