Exact Nonreflecting Boundary Conditions Research Articles

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.

Interactive computing methods for aeroelastic analysis of turbomachinery

AbstractAn interaction viscous‐inviscid method for efficiently computing steady and unsteady viscous flows is presented. The inviscid domain is modeled using a finite element discretization of the full potential equation. The viscous region is modeled using a finite difference boundary layer technique. The two regions are simultaneously coupled using the transpiration approach. A time linearization technique is applied to this interactive method. For unsteady flows, the fluid is assumed to be composed of a mean or steady flow plus a harmonically varying small unsteady disturbance. Numerically exact nonreflecting boundary conditions are used for the far field conditions. Results for some steady and unsteady, laminar and turbulent flow problems are compared to linearized Navier‐Stokes or time‐marching boundary layer methods. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Dirichlet-to-Neumann boundary conditions for multiple scattering problems

A Dirichlet-to-Neumann (DtN) condition is derived for the numerical solution of time-harmonic multiple scattering problems, where the scatterer consists of several disjoint components. It is obtained by combining contributions from multiple purely outgoing wave fields. The DtN condition yields an exact non-reflecting boundary condition for the situation, where the computational domain and its exterior artificial boundary consist of several disjoint components. Because each sub-scatterer can be enclosed by a separate artificial boundary, the computational effort is greatly reduced and becomes independent of the relative distances between the different sub-domains. The DtN condition naturally fits into a variational formulation of the boundary-value problem for use with the finite element method. Moreover, it immediately yields as a by-product an exact formula for the far-field pattern of the scattered field. Numerical examples show that the DtN condition for multiple scattering is as accurate as the well-known DtN condition for single scattering problems [J. Comput. Phys. 82 (1989) 172; Numerical Methods for Problems in Infinite Domains, Elsevier, Amsterdam, 1992], while being more efficient due to the reduced size of the computational domain.

Finite element method for time dependent scattering: nonreflecting boundary condition, adaptivity, and energy decay

An adaptive finite element method is developed for acoustic wave propagation in unbounded media. The efficiency and high accuracy of the method are achieved by combining an exact nonreflecting boundary condition [SIAM J. Appl. Math. 55 (1995) 280; J. Comput. Phys. 127 (1996) 52] with space–time adaptivity [East–West J. Numer. Math. 7(4) (1999) 263]. Hence the computational effort is concentrated where needed, while the artificial boundary can be brought as close as desired to the scatterer. Both features combined yield high accuracy and keep the number of unknowns to a minimum. An energy inequality is derived for the initial-boundary value problem at the continuous level. Together with an implicit second order time discretization it guarantees unconditional stability of the semi-discrete system. The resulting fully discrete linear system that needs to be solved every time step is unsymmetric but can be transformed into an equivalent sequence of small nonsymmetric and large symmetric positive definite systems, which are efficiently solved by conjugate gradient methods. Numerical examples illustrate the high accuracy of the method, in particular in the presence of complex geometry.

Nonreflecting boundary conditions for the time-dependent convective wave equation in a duct

Solutions for the three-dimensional wave equation in an annular duct are represented in terms of Fourier–Bessel modes, each obeying a one-dimensional dispersive wave equation. An exact nonreflecting boundary condition, nonlocal in time but local in space, is derived for each mode. Since, in most applications, the number of propagating modes is a small finite subset of all the modes, the present condition yields a computationally efficient scheme. Convergence of the solution depends on the radial eigenvalues which characterize the dispersion of the propagating duct modes. For periodic forcing, convergence analysis shows that the solution tends toward its asymptotic limit as the reciprocal of the square root of time. Near a duct mode cut-on, the computational time required for convergence is proportional to the reciprocal of the square of the group velocity. The boundary condition is implemented numerically and tested by computing the propagation of dispersive waves with various frequencies and comparing the results with the analytic solution. Two local boundary conditions are also implemented and their performance relative to the nonlocal boundary condition is studied. Significant improvements in accuracy are observed by using the present boundary condition near cut-on. Numerical results also show that the rate of convergence of the solution to a time-periodic solution significantly decreases as the group velocity of the incident waves becomes small, consistent with the analytic results.

Fast evaluation of nonreflecting boundary conditions for the Schrödinger equation in one dimension

We present a fast algorithm for the evaluation of the exact nonreflecting boundary conditions for the Schrödinger equation in one dimension. The exact nonreflecting boundary condition contains a nonlocal term which is a convolution integral in time, with a kernel proportional to 1 √t. The key observation is that this integral can be split into two parts: a local part and a history part, each of which allows for separate treatment. The local part is computed by a quadrature suited for square-root singularities. For the history part, we approximate the convolution kernel uniformly by a sum of exponentials. The integral can then be evaluated recursively. As a result, the computation of the nonreflecting boundary conditions is both accurate and efficient.

Exact non-reflecting boundary conditions on general domains

The authors present a new method for the realization of exact non-reflecting (transparent) boundary conditions in two dimensional direct scattering problems. This work is an extension of Keller, Givoli, and Grote’s work on such conditions which required that the shape of the boundary be quite specific, i.e. circular or elliptical. The condition is enforced via the Dirichlet–Neumann operator (DNO) which, on general boundaries, presents the main difficulty in the method. The implementation is performed by one of two perturbative methods (where the perturbation parameter measures the deformation of the general geometry from a canonical one). A rigorous proof of the analyticity for the DNO with respect to this perturbation parameter is presented. Numerical results show both perturbative methods are fast and accurate, and can enable significant computational savings.

Non‐reflecting artificial boundaries for transient scalar wave propagation in a two‐dimensional infinite homogeneous layer

AbstractThis paper presents an exact non‐reflecting boundary condition for dealing with transient scalar wave propagation problems in a two‐dimensional infinite homogeneous layer. In order to model the complicated geometry and material properties in the near field, two vertical artificial boundaries are considered in the infinite layer so as to truncate the infinite domain into a finite domain. This treatment requires the appropriate boundary conditions, which are often referred to as the artificial boundary conditions, to be applied on the truncated boundaries. Since the infinite extension direction is different for these two truncated vertical boundaries, namely one extends toward x →∞ and another extends toward x→‐ ∞, the non‐reflecting boundary condition needs to be derived on these two boundaries. Applying the variable separation method to the wave equation results in a reduction in spatial variables by one. The reduced wave equation, which is a time‐dependent partial differential equation with only one spatial variable, can be further changed into a linear first‐order ordinary differential equation by using both the operator splitting method and the modal radiation function concept simultaneously. As a result, the non‐reflecting artificial boundary condition can be obtained by solving the ordinary differential equation whose stability is ensured. Some numerical examples have demonstrated that the non‐reflecting boundary condition is of high accuracy in dealing with scalar wave propagation problems in infinite and semi‐infinite media. Copyright © 2003 John Wiley & Sons, Ltd.

Exact boundary condition for time-dependent wave equation based on boundary integral

An exact non-reflecting boundary conditions based on a boundary integral equation or a modified Kirchhoff-type formula is derived for exterior three-dimensional wave equations. The Kirchhoff-type non-reflecting boundary condition is originally proposed by L. Ting and M.J. Miksis [J. Acoust. Soc. Am. 80 (1986) 1825] and numerically tested by D. Givoli and D. Cohen [J. Comput. Phys. 117 (1995) 102] for a spherically symmetric problem. The computational advantage of Ting–Miksis boundary condition is that its temporal non-locality is limited to a fixed amount of past information. However, a long-time instability is exhibited in testing numerical solutions by using a standard non-dissipative finite-difference scheme. The main purpose of this work is to present a new exact boundary condition and to eliminate the long-time instability. The proposed exact boundary condition can be considered as a limit case of Ting–Miksis boundary condition when the two artificial boundaries used in their method approach each other. Our boundary condition is actually a boundary integral equation on a single artificial boundary for wave equations, which is to be solved in conjunction with the interior wave equation. The new boundary condition needs only one artificial boundary, which can be of any shape, i.e., sphere, cubic surface, etc. It keeps all merits of the original Kirchhoff boundary condition such as restricting the temporal non-locality, free of numerical evaluation of any special functions and so on. Numerical approximation to the artificial boundary condition on cubic surface is derived and three-dimensional numerical tests are carried out on the cubic computational domain.

Dirichlet-to-Neumann map for three-dimensional elastic waves

An exact nonreflecting boundary condition is derived for time-harmonic elastic waves in three space dimensions. This condition holds on a spherical surface, B , which separates the computational domain from the surrounding unbounded region. It establishes an exact relation between the displacement field and its normal and tangential tractions, and thus defines a Dirichlet-to-Neumann (DtN) map on B . Then a modified DtN condition (MDtN) is derived to remove the difficulties that arise when the exact DtN condition is truncated for use in computation, by modifying the truncated condition. Uniqueness of the solution to the boundary-value problem with the MDtN condition imposed on B is proved. Both the DtN and the MDtN conditions naturally fit into the variational formulation for use with the finite element method. Numerical results with a finite difference method demonstrate the high improvement in accuracy over standard methods.

Nonreflecting Boundary Conditions for the Time-Dependent Wave Equation

We describe a new, efficient approach to the imposition of exact nonreflecting boundary conditions for the scalar wave equation. We compare the performance of our approach with that of existing methods by coupling the boundary conditions to finite-difference schemes. Numerical experiments demonstrate a significant gain in accuracy at no additional cost.

Computation of far-field solutions based on exact nonreflecting boundary conditions for the time-dependent wave equation

In this work we show how to combine in the exact nonreflecting boundary conditions (NRBC) first derived by Grote and Keller, the calculation of the exterior (far-field) solution for time-dependent radiation and scattering in an unbounded domain. At each discrete time step, radial modes computed on a spherical artificial boundary which drive the exact NRBC for the near-field solution, are imposed as Cauchy data for the radial wave equation in the far-field. Similar to the far-field computation scheme used by Wright, the radial modes in the exterior region are computed using an explicit finite difference solver. However, instead of using an `infinite grid', we truncate the exterior radial grid at the far-field point of interest, and for each harmonic, impose the same exact NRBC used for the near-field truncation boundary, here expressed in modal form. Using this approach, two different methods for extrapolating the near-field solution to the far-field are possible. In the first, the near-field solution is computed using the exact NRBC, then, based on the solution for the radial modes evaluated on the artificial boundary, the exterior solution may be computed as a post-process. In the second, we show how to compute the far-field solution concurrently with the near-field solution and the NRBC. Numerical studies demonstrate that the method is highly accurate and efficient for direct time-domain computations of far-field solutions.

Non‐reflecting boundary conditions for electromagnetic scattering

An exact non-reflecting boundary condition was derived previously for use with the time-dependent Maxwell equations in three space dimensions. Here it is shown how to combine that boundary condition with the variational formulation for use with the finite element method. The fundamental theory underlying the derivation of the exact boundary condition is reviewed. Numerical examples with the finite-difference time-domain method are presented which demonstrate the high accuracy and confirm the expected rate of convergence of the numerical method. Copyright © 2000 John Wiley & Sons, Ltd.

Nonreflecting Boundary Conditions for Elastodynamic Scattering

An exact nonreflecting boundary condition was derived previously for timedependent elastic waves in three space dimensions [SIAM J. Appl. Math.60, 803 (2000)]. It is local in time, nonlocal on the artificial boundary, and involves only first derivatives of the displacement. Here it is shown how to combine that boundary condition with finite difference and finite element methods. Stability issues are discussed. Numerical examples with a finite difference method demonstrate the high improvement in accuracy over standard methods.

Implementation of exact non-reflecting boundary conditions in the finite element method for the time-dependent wave equation

When solving the wave equation in infinite regions using finite element methods, the domain must be truncated. We investigate the accuracy of time-dependent non-reflecting boundary conditions (NRBC) derived in Grote, Keller (1995), when implemented in the finite element method. The NRBC annihilate the first N wave harmonics on a spherical truncation boundary. High-order temporal derivatives are formulated as a system of first-order ordinary differential equations. Several versions of implicit and explicit multi-step, time-integration schemes are presented for solution of the finite element equations concurrently with the first-order system appearing in the NRBC. An alternative scaling of the boundary variables is introduced which leads to a well-conditioned coefficient matrix. Although the boundary conditions are global over the boundary, when implemented in the finite element method, they only require inner products of spherical harmonics within the force vector, and as a result, they are easy to implement and do not disturb the banded/sparse structure of the matrix equations. Several numerical examples are presented which demonstrate the improvement in accuracy over standard finite element methods.

Non-reflecting boundary conditions for electromagnetic scattering

An exact non-reflecting boundary condition was derived previously for use with the time-dependent Maxwell equations in three space dimensions. Here it is shown how to combine that boundary condition with the variational formulation for use with the finite element method. The fundamental theory underlying the derivation of the exact boundary condition is reviewed. Numerical examples with the finite-difference time-domain method are presented which demonstrate the high accuracy and confirm the expected rate of convergence of the numerical method. Copyright © 2000 John Wiley & Sons, Ltd.

Exact Nonreflecting Boundary Condition For Elastic Waves

An exact nonreflecting boundary condition is derived for the time-dependent elastic wave equation in three space dimensions. This condition holds on a spherical surface $\mathcal B$, outside of which the medium is assumed to be linear, homogeneous, isotropic, and source-free. It is local in time, nonlocal on $\mathcal B$, and involves only first derivatives of the solution. Therefore it can be combined easily with any numerical method in the interior region.

Computation of transient radiation in semi-infinite regions based on exact nonreflecting boundary conditions and mixed time integration

Transient radiation in a semi-infinite region, bounded by a planar infinite baffle with a local acoustic source is considered. The numerical simulation of the transient radiation problem requires an artificial boundary Γ, here chosen to be a hemisphere, which separates the computational region from the surrounding unbounded acoustic medium. Inside the computational region we use a semidiscrete finite element method. On Γ, we apply the exact nonreflecting boundary condition (NRBC) first derived by Grote and Keller for the free-space problem. Since the problem is symmetric about the infinite planar surface, in order to satisfy the rigid baffle condition it is sufficient to restrict the indices in the spherical harmonic expansion which defines the NRBC and scale the radial harmonics which drive auxiliary equations on the boundary. The Fourier expansion in the circumferential angle appearing in the NRBC may be used to efficiently model axisymmetric problems in two dimensions. A new mixed explicit-implicit time integration method which retains the efficiency of explicit pressure field updates without the need for diagonal matrices in the auxiliary equations on Γ is presented. Here, the interior finite element equations are integrated explicitly in time while the auxiliary equations are integrated implicitly. The result is a very natural and highly efficient algorithm for large-scale wave propagation analysis. Numerical examples of fully transient radiation resulting from a piston transducer mounted in an infinite planar baffle are compared to analytical solutions to demonstrate the accuracy of the mixed time integration method with the NRBC for the half-space problem.

Finite element formulation of exact non-reflecting boundary conditions for the time-dependent wave equation

A modi ed version of an exact Non-re ecting Boundary Condition (NRBC) rst derived by Grote and Keller is implemented in a nite element formulation for the scalar wave equation. The NRBC annihilate the rst N wave harmonics on a spherical truncation boundary, and may be viewed as an extension of the secondorder local boundary condition derived by Bayliss and Turkel. Two alternative nite element formulations are given. In the rst, the boundary operator is implemented directly as a ‘natural’ boundary condition in the weak form of the initial–boundary value problem. In the second, the operator is implemented indirectly by introducing auxiliary variables on the truncation boundary. Several versions of implicit and explicit timeintegration schemes are presented for solution of the nite element semidiscrete equations concurrently with the rst-order di erential equations associated with the NRBC and an auxiliary variable. Numerical studies are performed to assess the accuracy and convergence properties of the NRBC when implemented in the nite element method. The results demonstrate that the nite element formulation of the (modi ed) NRBC is remarkably robust, and highly accurate. Copyright ? 1999 John Wiley & Sons, Ltd.

Fictitious Domain Methods for the Numerical Solution of Two-Dimensional Scattering Problems

Fictitious domain methods for the numerical solution of two-dimensional scattering problems are considered. The original exterior boundary value problem is approximated by truncating the unbounded domain and by imposing a nonreflecting boundary condition on the artificial boundary. First-order, second-order, and exact nonreflecting boundary conditions are tested on rectangular and circular boundaries. The finite element discretizations of the corresponding approximate boundary value problems are performed using locally fitted meshes, and the discrete equations are solved with fictitious domain methods. A special finite element method using nonmatching meshes is considered. This method uses the macro-hybrid formulation based on domain decomposition to couple polar and cartesian coordinate systems. A special preconditioner based on fictitious domains is introduced for the arising algebraic saddle-point system such that the subspace of constraints becomes invariant with respect to the preconditioned iterative procedure. The performance of the new method is compared to the fictitious domain methods both with respect to accuracy and computational cost.