Local Non-reflecting Boundary Conditions Research Articles

Study of Local Non-reflective Boundary Condition on Soil Domain

This paper presents a computational (finite element analysis) study for numerically simulating the elastic wave propagation problem in a two-dimensional homogeneous half space. The theoretical formulations of development and implementation of local non-reflecting boundary condition (NRBC) at the soil truncated boundaries has been discussed in detail. A dynamic analysis of the elastic soil domain using time marching scheme has been carried out and comparative studies of the responses from different boundary conditions at the truncated boundaries also has been provided to get the physical idea of the wave reflection phenomena and its absorption capability using NRBCs. The results justify the utility of implementing the local types of NRBC's on the truncated boundaries of the soil domain for solving coupled dynamic problems.

Finite Element Analysis of Dam-Foundation Coupled System Considering Cone-Type Local Non-Reflecting Boundary Condition

This article deals with the seismic analysis of the dam-foundation coupled system considering direct coupling approach using displacement based finite element method. A cone-type local non-reflecting boundary condition (NRBC) is adopted to model the semi-infinite soil domain. The reservoir effect adjacent to dam body has also been incorporated in the analysis. The results depict the superiority of the proposed NRBC in absorbing spurious wave reflections over the traditional viscous boundary conditions. The analysis results show that the dynamic response of the dam gets highly affected due to the consideration of the elastic foundation domain at its base.

Acoustic eigenanalysis of 2D open cavity with Vekua approximations and the method of particular solutions

This paper discusses the efficient extraction of acoustic resonances in 2D open cavities using a meshless method, the method of particular solutions. A first order, local non-reflecting boundary condition is chosen to account for the opening and Fourier-Bessel functions are employed to approximate the pressure at the borders of the cavity and inside. A minimization problem is then solved for the complex frequency range of interest. For the investigated cavity, the minimum values obtained match those found in previously published studies or by other numerical methods. But, unlike the perfectly matched layer absorbing boundary conditions now usually employed, this approach is free from spurious eigenfrequencies. Moreover, the specific treatments of the geometric singularities allow this method to be particularly efficient in the presence of corner singularities.

Application of H–W boundary condition in dam–reservoir interaction problem

One of the most efficient techniques for considering the unbounded media in a mathematical model is the local non-reflecting boundary conditions (NRBCs). A relatively recent approach in this context is the one proposed by Hagstrom and Warburton (H–W boundary condition). This approach is available for scalar wave equation. This equation governs the hydrodynamic pressure distribution inside a reservoir and hence, the H–W boundary condition may be used to solve the corresponding problems. A well-known case in which one has to deal with the dynamic analysis of unbounded reservoir is the dam–reservoir interaction problem. In this study the H–W NRBC has been applied to this problem and its harmonic response is calculated. By comparing the results with the exact solution, the performance and accuracy of this NRBC is examined. The numerical results confirm the very good behavior of the NRBC in the frequencies above the fundamental frequency of the reservoir. However, below this frequency range, this boundary condition does not perform very well, especially when it is applied in close distances from the dam.

Finite-Element Modeling of Continuous Surface Waves Tests: Numerical Accuracy with respect to Domain Size

Near-field effects are inherent in continuous surface wave (CSW) tests. Such effects on the dispersion curves can be studied using the finite-element (FE) method. However, near-field effects may be obscured by boundary effects in a FE model. Although the use of local nonreflecting boundary conditions (NRBCs) may alleviate the problem, these need to be placed sufficiently far away to be effective. Quantitative guidelines on domain size with NRBCs are still lacking. This paper provides specific guidelines to determine the domain size in terms of the wavelength-to-domain size ratio, λ*/l*. With local NRBCs, numerical errors were found to be negligible when the domain size (l*) was extended to twice the longest wavelength (λ*). However, in modeling soft soil deposits in which shear-wave velocity (Vs) is less than 100 m/s, a smaller domain size (e.g., λ*/l*=2) is sufficient.

Local nonreflecting boundary condition for time-dependent multiple scattering

Starting from a high-order local nonreflecting boundary condition (NBC) for single scattering [25], we derive a local NBC for time-dependent multiple scattering problems in three space dimensions, which is completely local both in space and time. To do so, we first develop an exterior evaluation formula for a purely outgoing wave field, given its values and those of certain auxiliary functions needed for the local NBC at the artificial boundary. By combining that evaluation formula with the decomposition of the total scattered field into purely outgoing contributions, we obtain a completely local NBC for time-dependent multiple scattering problems. The accuracy and stability of this new local NBC are evaluated by coupling it to a standard finite difference method.

On local nonreflecting boundary conditions for time dependent wave propagation

The simulation of wave phenomena in unbounded domains generally requires an artificial boundary to truncate the unbounded exterior and limit the computation to a finite region. At the artificial boundary a boundary condition is then needed, which allows the propagating waves to exit the computational domain without spurious reflection. In 1977, Engquist and Majda proposed the first hierarchy of absorbing boundary conditions, which allows a systematic reduction of spurious reflection without moving the artificial boundary farther away from the scatterer. Their pioneering work, which initiated an entire research area, is reviewed here from a modern perspective. Recent developments such as high-order local conditions and their extension to multiple scattering are also presented. Finally, the accuracy of high-order local conditions is demonstrated through numerical experiments.

Exact local nonreflecting boundary conditions for time‐dependent multiple scattering

AbstractIn [2, 3] a nonreflecting boundary condition(NBC) for time‐dependent multiple scattering was derived, which is local in time but nonlocal in space. Here, based on a high‐order local nonreflecting boundary condition (NBC) for single scattering [4], we seek a local NBC for time‐dependent multiple scattering, which is completely local both in space and time. To do so, we first develop a high order representation formula for a purely outgoing wave field, given its values and those of certain auxiliary functions needed for the artificial boundary condition. By combining that representation formula with a decomposition of the total scattered field into purely outgoing contributions, we obtain the first exact, completely local, NBC for time‐dependent multiple scattering. The accuracy and stability of this local NBC is evaluated by coupling it to standard finite element and finite difference methods. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

STABILIZED DISCONTINUOUS GALERKIN METHODS FOR FLOW–SOUND INTERACTION

The wave propagation in musical instruments is considered. A high-order discontinuous Galerkin method for solving the Euler equations is presented, and an alternative concept for supplying local non-reflecting boundary conditions is introduced. This concept is based on local extrapolation in conjunction with a stabilizing slope limiting. The practicability of this method is discussed via numerical tests, and the results are compared with classical characteristic boundary treatment. Investigations of wave propagation phenomena in musical instruments like the bassoon are performed, and the results are presented.

Local nonreflecting boundary condition for Maxwell’s equations

An exact nonreflecting boundary condition is derived for the time dependent Maxwell equations in three space dimensions. It is local in space and time, and it holds on a spherical surface B of radius R, outside of which the medium is assumed to be homogeneous, isotropic, and source-free. This boundary condition does not involve high-order derivatives, but instead an infinite sequence of auxiliary variables defined on B . In practice, only a finite number, P, of auxiliary variables is used. Then, the boundary condition remains exact for any combination of spherical harmonics up to order P, while the error introduced at B generally behaves like R −2( P+1) . Hence, P can always be chosen large enough to reduce the error introduced at B below the discretization error inside the computational domain, at any fixed R. Because it does not involve high-order derivatives, this local boundary condition is easily combined with standard numerical methods and enables arbitrarily high order implementations. Numerical examples with the FDTD method demonstrate the usefulness and high accuracy of this local nonreflecting boundary condition.

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.

An optimal high‐order non‐reflecting finite element scheme for wave scattering problems

AbstractA new finite element scheme is proposed for the numerical solution of time‐harmonic wave scattering problems in unbounded domains. The infinite domain in truncated via an artificial boundary ℬ︁ which encloses a finite computational domain Ω. On ℬ︁ a local high‐order non‐reflecting boundary condition (NRBC) is applied which is constructed to be optimal in a certain sense. This NRBC is implemented in a special way, by using auxiliary variables along the boundary ℬ︁, so that it involves no high‐order derivatives regardless of its order. The order of the scheme is simply an input parameter, and it may be arbitrarily high. This leads to a symmetric finite element formulation where standard C0 finite elements are used in Ω. The performance of the method is demonstrated via numerical examples, and it is compared to other NRBC‐based schemes. The method is shown to be highly accurate and stable, and to lead to a well‐conditioned matrix problem. Copyright © 2002 John Wiley & Sons, Ltd.

High-Order Nonreflecting Boundary Conditions without High-Order Derivatives

A wave problem in an unbounded domain is often treated numerically by truncating the infinite domain via an artificial boundary B, imposing a so-called nonreflecting boundary condition (NRBC) on B, and then solving the problem numerically in the finite domain bounded by B. A general approach is devised here to construct high-order local NRBCs with a symmetric structure and with only low (first- or second-) order spatial and/or temporal derivatives. This enables the practical use of NRBCs of arbitrarily high order. In the case of time-harmonic waves with finite element discretization, the approach yields a symmetric C0 finite element formulation in which standard elements can be employed. The general methodology is presented for both the time-harmonic case (Helmholtz equation) and the time-dependent case (the wave equation) and is demonstrated numerically in the former case.

Nonreflecting boundary conditions in elastodynamics for finite element methods based upon off-surface boundary integral equations

In this work, an off-surface boundary integral (OSBI) method is presented as a mesh termination scheme for solving large or infinite domain problems of elastodynamics. The boundary integral equation is discretized using finite element shape functions and the Neumann boundary condition term is solved for in terms of the Dirichlet boundary condition term. This expression is then substituted into associated finite element formulation for the interior problem. Comparisons are made using the new OSBI technique, the DtN method of Givoli and Keller and several other local nonreflecting boundary conditions. The proposed boundary condition is shown to be accurate, is well suited for use with finite element methods and is competitive with the DtN method.

Stability and accuracy of optimal local non-reflecting boundary conditions

Problems in unbounded domains are often solved numerically by truncating the infinite domain via an artificial boundary B and applying some boundary condition on B , which is called a Non-Reflecting Boundary Condition (NRBC). Recently, a two-parameter hierarchy of optimal local NRBCs of increasing order has been developed. The optimality is in the sense that the local NRBC best approximates the exact nonlocal Dirichlet-to-Neumann (DtN) boundary condition in the L 2 norm for functions in C ∞. The optimal NRBCs are combined with finite element discretization in the computational domain. Here the theoretical properties of the resulting class of schemes are examined. In particular, theorems are proved regarding the numerical stability of the schemes and their rates of convergence.

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

Infinite elements for wave problems: a review of current formulations and an assessment of accuracy

Infinite element schemes for unbounded wave problems are reviewed and a procedure is presented forassessing their performance. A general computational scheme is implemented in which orthogonal functions are used for the transverse interpolation within the infinite element region. This is used as a basis for numerical studies of the effectiveness of various combinations of the radial test and trial functions which give rise to different conjugated and unconjugated formulations. Results are presented for the test case of a spherical radiator to which infinite elements are directly attached. Accuracy of the various schemes is assessed for pure multipole solutions of arbitrary order. Previous studies which have indicated that the conjugated and unconjugated schemes are more effective in the far and near fields, respectively, are confirmed by the current results. All of the schemes tested converge to the exact solution as radial order increases. All are however susceptible to ill conditioning. This places practical restrictions on their effectiveness at high radial orders. A close relationship is demonstrated between the discrete equations which arise from first-order infiniteelement schemes and those derived from the application of more traditional, local non-reflecting boundary conditions. Copyright © 2000 John Wiley & Sons, Ltd.

Optimal local non-reflecting boundary conditions

A class of numerical methods to solve problems in unbounded domains is based on truncating the infinite domain via an artificial boundary β and applying some boundary condition on β, which is called a Non-Reflecting Boundary Condition (NRBC). In this paper a systematic way to derive optimal local NRBCs of given order is developed in various configurations. The optimality is in the sense that the local NRBC best approximates the exact nonlocal Dirichlet-to-Neumann (DtN) boundary condition for C ∞ functions in the L 2 norm. The optimal NRBC may be of low order but still represent high-order modes in the solution. It is shown that the previously derived localized DtN conditions are special cases of the new optimal conditions. The performance of the first-order optimal NRBC is demonstrated via numerical examples, in conjunction with the finite element method.

A High-Order Godunov Method for Multiple Condensed Phases

Presented in this work is a nonlinear adaptive nonreflecting boundary condition (NRBC) that, when compared with classical local NRBCs, further reduces the wave reflection error at far fields in the numerical simulation of wave dominated problems. A nonlinear procedure can considerably improve the accuracy of NRBCs even if the problem itself is linear as in the case of the 2D wave equation. In fact the first- and second-order NRBCs of Engquist and Majda can be modified by adding proper weights to the coefficients of the difference equations. The weights are adaptively determined in such a way that the angle of complete absorption is locally coincident with the outgoing wave direction. The theoretical reflection coefficients and the numerical experiments show that a considerable reduction of the numerical error due to wave reflections is achieved by applying the present adaptive algorithm.