Dan Givoli

Non-reflecting boundary conditions

Abstract Past and recent research on the use of non-reflecting boundary conditions in the numerical solution of wave problems is reviewed. Local and nonlocal boundary conditions are discussed, as well as special procedures which involve artificial boundaries. Various problems from different disciplines of applied mathematics and engineering are considered in a uniform manner. Future research directions are addressed.

Non-reflecting boundary conditions for elastic waves

Abstract An exact non-reflecting boundary condition is devised for time-harmonic two-dimensional elastodynamics in infinite domains. The domain is made finite by the introduction of a circular artificial boundary on which this exact condition is imposed. In the finite computational domain a finite element method is employed. Numerical examples are presented in which the accuracy and efficiency of the method using the exact non-local boundary condition are compared with those of methods based on approximate local boundary conditions. The method is also used to solve problems in large finite domains by reducing them to smaller domains. In addition, local boundary conditions are derived which are exact for waves with a limited number of angular Fourier components.

Recent advances in the DtN FE Method

SummaryThe Dirichlet-to-Neumann (DtN) Finite Element Method is a general technique for the solution of problems in unbounded domains, which arise in many fields of application. Its name comes from the fact that it involves the nonlocal Dirichlet-to-Neumann (DtN) map on an artificial boundary which encloses the computational domain. Originally the method has been developed for the solution of linear elliptic problems, such as wave scattering problems governed by the Helmholtz equation or by the equations of time-harmonic elasticity. Recently, the method has been extended in a number of directions, and further analyzed and improved, by the authors group and by others. This article is a state-of-the-art review of the method. In particular, it concentrates on two major recent advances: (a) the extension of the DtN finite element method tononlinear elliptic and hyperbolic problems; (b) procedures forlocalizing the nonlocal DtN map, which lead to a family of finite element schemes with local artificial boundary conditions. Possible future research directions and additional extensions are also discussed.

High-order local absorbing conditions for the wave equation: Extensions and improvements

The solution of the time-dependent wave equation in an unbounded domain is considered. An artificial boundary B is introduced which encloses a finite computational domain. On B an absorbing boundary condition (ABC) is imposed. A formulation of local high-order ABCs recently proposed by Hagstrom and Warburton and based on a modification of the Higdon ABCs, is further developed and extended in a number of ways. First, the ABC is analyzed in new ways and important information is extracted from this analysis. Second, The ABCs are extended to the case of a dispersive medium, for which the Klein-Gordon wave equation governs. Third, the case of a stratified medium is considered and the way to apply the ABCs to this case is explained. Fourth, the ABCs are extended to take into account evanescent modes in the exact solution. The analysis is applied throughout this paper to two-dimensional wave guides. Two numerical algorithms incorporating these ABCs are considered: a standard semi-discrete finite element formulation in space followed by time-stepping, and a high-order finite difference discretization in space and time. Numerical examples are provided to demonstrate the performance of the extended ABCs using these two methods.

High-order boundary conditions and finite elements for infinite domains

A finite element method for the solution of linear elliptic problems in infinite domains is proposed. The two-dimensional Laplace, Helmholtz and modified Helmholtz equations outside an obstacle and in a semi-infinite strip, are considered in detail. In the proposed method, an artificial boundary B is first introduced, to make the computational domain Ω finite. Then the exact nonlocal Dirichlet-to-Neumann (DtN) boundary condition is derived on B. This condition is localized, and a sequence of local boundary conditions on B, of increasing order, is obtained. The problem in Ω, with a localized DtN boundary condition on B, is then solved using the finite element method. The numerical stability of the scheme is discussed. A hierarchy of special conforming finite elements is developed and used in the layer adjacent to B, in conjunction with the local high-order boundary condition applied on B. An error analysis is given for both nonlocal and local boundary conditions. Numerical experiments are presented to demonstrate the performance of the method.

High-order Absorbing Boundary Conditions for anisotropic and convective wave equations

High-order Absorbing Boundary Conditions (ABCs), applied on a rectangular artificial computational boundary that truncates an unbounded domain, are constructed for a general two-dimensional linear scalar time-dependent wave equation which represents acoustic wave propagation in anisotropic and subsonically convective media. They are extensions of the construction of Hagstrom, Givoli and Warburton for the isotropic stationary case. These ABCs are local, and involve only low-order derivatives owing to the use of auxiliary variables on the artificial boundary. The accuracy and well-posedness of these ABCs is analyzed. Special attention is given to the issue of mismatch between the directions of phase and group velocities, which is a potential source of concern. Numerical examples for the anisotropic case are presented, using a finite element scheme.

Computational Absorbing Boundaries

The subject of this chapter is the treatment of artificial boundaries in wave problems. Artificial boundaries are introduced when the problem under study is associated with an unbounded medium, yet one is interested (or is forced) to solve the problem in a finite computational domain. In this context the artificial boundaries are often called absorbing boundaries, for reasons that will be explained. After discussing the difficulties involved, the major milestones that have been set in the development of absorbing boundaries are surveyed. These include the classical absorbing boundary conditions, exact nonlocal conditions, absorbing layers, perfectly matched layers and high–order local boundary conditions. Infinite elements and boundary element methods are also mentioned. Examples from previous publications are given.

High-order non-reflecting boundary conditions for dispersive waves

Problems of linear time-dependent dispersive waves in an unbounded domain are considered. 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 by a finite difference (FD) scheme in the finite domain bounded by B. The sequence of NRBCs proposed by Higdon is used. However, in contrast to the original low-order implementation of the Higdon conditions, a new scheme is devised which allows the easy use of a Higdon-type NRBC of any desired order. In addition, a procedure for the automatic choice of the parameters appearing in the NRBC is proposed. The performance of the scheme is demonstrated via numerical examples.

Radiation boundary conditions for time-dependent waves based on complete plane wave expansions

We develop complete plane wave expansions for time-dependent waves in a half-space and use them to construct arbitrary order local radiation boundary conditions for the scalar wave equation and equivalent first order systems. We demonstrate that, unlike other local methods, boundary conditions based on complete plane wave expansions provide nearly uniform accuracy over long time intervals. This is due to their explicit treatment of evanescent modes. Exploiting the close connection between the boundary condition formulations and discretized absorbing layers, corner compatibility conditions are constructed which allow the use of polygonal artificial boundaries. Theoretical arguments and simple numerical experiments are given to establish the accuracy and efficiency of the proposed methods.

A spatially exact non-reflecting boundary condition for time dependent problems

Abstract A special artificial boundary condition is devised and incorporated in a finite element scheme for use in the numerical solution of exterior two-dimensional time-dependent wave problems. The time-dependent wave equation is first discretized in time, leading to a sequence of boundary value problems in an unbounded domain. Then an artificial boundary is introduced, and a spatially exact boundary condition is derived on it. This boundary condition, which is non-local in space but local in time, is employed in conjunction with a finite element scheme in the interior domain. In the proposed method the artificial boundary may be set very close to the scatterer without reducing the accuracy of the numerical results. The spatially exact boundary condition is compared to two commonly used local boundary conditions.