374 research outputs found
Finite-Difference and Pseudospectral Time-Domain Methods Applied to Backwards-Wave Metamaterials
Backwards-wave (BW) materials that have simultaneously negative real parts of
their electric permittivity and magnetic permeability can support waves where
phase and power propagation occur in opposite directions. These materials were
predicted to have many unusual electromagnetic properties, among them
amplification of the near-field of a point source, which could lead to the
perfect reconstruction of the source field in an image [J. Pendry, Phys. Rev.
Lett. \textbf{85}, 3966 (2000)]. Often systems containing BW materials are
simulated using the finite-difference time-domain technique. We show that this
technique suffers from a numerical artifact due to its staggered grid that
makes its use in simulations involving BW materials problematic. The
pseudospectral time-domain technique, on the other hand, uses a collocated grid
and is free of this artifact.
It is also shown that when modeling the dispersive BW material, the linear
frequency approximation method introduces error that affects the frequency of
vanishing reflection, while the auxiliary differential equation, the Z
transform, and the bilinear frequency approximation method produce vanishing
reflection at the correct frequency. The case of vanishing reflection is of
particular interest for field reconstruction in imaging applications.Comment: 9 pages, 8 figures, accepted by IEEE Transactions on Antennas and
Propagatio
Nonreflecting boundary conditions for time-dependent wave propagation
Many problems in computational science arise in unbounded domains and
thus require an artificial boundary B, which truncates the unbounded exterior
domain and restricts the region of interest to a finite computational
domain,
. It then becomes necessary to impose a boundary condition at
B, which ensures that the solution in
coincides with the restriction to
of the solution in the unbounded region. If we exhibit a boundary condition,
such that the fictitious boundary appears perfectly transparent, we shall call
it exact. Otherwise it will correspond to an approximate boundary condition
and generate some spurious reflection, which travels back and spoils the
solution everywhere in the computational domain. In addition to the transparency
property, we require the computational effort involved with such a
boundary condition to be comparable to that of the numerical method used
in the interior. Otherwise the boundary condition will quickly be dismissed
as prohibitively expensive and impractical. The constant demand for increasingly
accurate, efficient, and robust numerical methods, which can handle a
wide variety of physical phenomena, spurs the search for improvements in
artificial boundary conditions.
In the last decade, the perfectly matched layer (PML) approach [16] has
proved a flexible and accurate method for the simulation of waves in unbounded
media. Standard PML formulations, however, usually require wave
equations stated in their standard second-order form to be reformulated as
first-order systems, thereby introducing many additional unknowns. To circumvent
this cumbersome and somewhat expensive step we propose instead
a simple PML formulation directly for the wave equation in its second-order
form. Our formulation requires fewer auxiliary unknowns than previous formulations
[23, 94].
Starting from a high-order local nonreflecting boundary condition (NRBC)
for single scattering [55], we derive a local NRBC for time-dependent multiple
scattering problems, which is completely local both in space and time. To do so, we first develop a high order exterior evaluation formula for a purely
outgoing wave field, given its values and those of certain auxiliary functions needed for the local NRBC on the artificial boundary. By combining that
evaluation formula with the decomposition of the total scattered field into
purely outgoing contributions, we obtain the first exact, completely local,
NRBC for time-dependent multiple scattering. Remarkably, the information
transfer (of time retarded values) between sub-domains will only occur
across those parts of the artificial boundary, where outgoing rays intersect
neighboring sub-domains, i.e. typically only across a fraction of the artificial
boundary. The accuracy, stability and efficiency of this new local NRBC is
evaluated by coupling it to standard finite element or finite difference methods
Fast and Accurate Computation of Time-Domain Acoustic Scattering Problems with Exact Nonreflecting Boundary Conditions
This paper is concerned with fast and accurate computation of exterior wave
equations truncated via exact circular or spherical nonreflecting boundary
conditions (NRBCs, which are known to be nonlocal in both time and space). We
first derive analytic expressions for the underlying convolution kernels, which
allow for a rapid and accurate evaluation of the convolution with
operations over successive time steps. To handle the onlocality in space,
we introduce the notion of boundary perturbation, which enables us to handle
general bounded scatters by solving a sequence of wave equations in a regular
domain. We propose an efficient spectral-Galerkin solver with Newmark's time
integration for the truncated wave equation in the regular domain. We also
provide ample numerical results to show high-order accuracy of NRBCs and
efficiency of the proposed scheme.Comment: 22 pages with 9 figure
Efficient PML for the wave equation
In the last decade, the perfectly matched layer (PML) approach has proved a
flexible and accurate method for the simulation of waves in unbounded media.
Most PML formulations, however, usually require wave equations stated in their
standard second-order form to be reformulated as first-order systems, thereby
introducing many additional unknowns. To circumvent this cumbersome and
somewhat expensive step, we instead propose a simple PML formulation directly
for the wave equation in its second-order form. Inside the absorbing layer, our
formulation requires only two auxiliary variables in two space dimensions and
four auxiliary variables in three space dimensions; hence it is cheap to
implement. Since our formulation requires no higher derivatives, it is also
easily coupled with standard finite difference or finite element methods.
Strong stability is proved while numerical examples in two and three space
dimensions illustrate the accuracy and long time stability of our PML
formulation.Comment: 16 pages, 6 figure
Boundary Conditions for Direct Computation of Aerodynamic Sound Generation
Accurate computation of the far-field sound along with the near-field source terms associated with a free shear flow requires that the Navier-Stokes equations be solved using accurate numerical differentiation and time-marching schemes, with nonreflecting boundary conditions. Nonreflecting boundary conditions have been developed for two-dimensional linearized Euler equations by Giles. These conditions are modified for use with nonlinear Navier-Stokes computations of open flow problems. At an outflow, vortical structures are found to produce large reflections due to nonlinear effects; these reflection errors cannot be improved by increasing the accuracy of the linear boundary conditions. An exit zone just upstream of an outflow where disturbances are significantly attenuated through grid stretching and filtering is developed for use with the nonreflecting boundary conditions; reflections from vortical structures are decreased by 3 orders of magnitude. The accuracy and stability of the boundary conditions are investigated in several model flows that include sound radiation by an energy source in a uniformly sheared viscous flow, the propagation of vortices in a uniform flow, and the spatial evolution of a compressible mixing layer
Periodic Time-Domain Nonlocal Nonreflecting Boundary Conditions for Duct Acoustics
Periodic time-domain boundary conditions are formulated for direct numerical simulation of acoustic waves in ducts without flow. Well-developed frequency-domain boundary conditions are transformed into the time domain. The formulation is presented here in one space dimension and time; however, this formulation has an advantage in that its extension to variable-area, higher dimensional, and acoustically treated ducts is rigorous and straightforward. The boundary condition simulates a nonreflecting wave field in an infinite uniform duct and is implemented by impulse-response operators that are applied at the boundary of the computational domain. These operators are generated by convolution integrals of the corresponding frequency-domain operators. The acoustic solution is obtained by advancing the Euler equations to a periodic state with the MacCormack scheme. The MacCormack scheme utilizes the boundary condition to limit the computational space and preserve the radiation boundary condition. The success of the boundary condition is attributed to the fact that it is nonreflecting to periodic acoustic waves. In addition, transient waves can pass rapidly out of the solution domain. The boundary condition is tested for a pure tone and a multitone source in a linear setting. The effects of various initial conditions are assessed. Computational solutions with the boundary condition are consistent with the known solutions for nonreflecting wave fields in an infinite uniform duct
- …