239 research outputs found
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
On-surface radiation condition for multiple scattering of waves
The formulation of the on-surface radiation condition (OSRC) is extended to
handle wave scattering problems in the presence of multiple obstacles. The new
multiple-OSRC simultaneously accounts for the outgoing behavior of the wave
fields, as well as, the multiple wave reflections between the obstacles. Like
boundary integral equations (BIE), this method leads to a reduction in
dimensionality (from volume to surface) of the discretization region. However,
as opposed to BIE, the proposed technique leads to boundary integral equations
with smooth kernels. Hence, these Fredholm integral equations can be handled
accurately and robustly with standard numerical approaches without the need to
remove singularities. Moreover, under weak scattering conditions, this approach
renders a convergent iterative method which bypasses the need to solve single
scattering problems at each iteration.
Inherited from the original OSRC, the proposed multiple-OSRC is generally a
crude approximate method. If accuracy is not satisfactory, this approach may
serve as a good initial guess or as an inexpensive pre-conditioner for Krylov
iterative solutions of BIE
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
Acoustic Saturation in Bubbly Cavitating Flow Adjacent to an Oscillating Wall
Bubbly cavitating flow generated by the normal oscillation of a wall bounding a semi-infinite domain of fluid is computed using a continuum two-phase flow model. Bubble dynamics are computed, on the microscale, using the Rayleigh-Plesset equation. A Lagrangian finite volume
scheme and implicit adaptive time marching are employed to accurately resolve bubbly shock waves and other steep gradients in the flow. The one-dimensional, unsteady computations show that when the wall oscillation frequency is much smaller than the bubble natural frequency, the power radiated away from the wall is limited by an acoustic saturation effect (the radiated power becomes
independent of the amplitude of vibration), which is similar to that found in a pure gas. That is, for large enough vibration amplitude, nonlinear steepening of the generated waves leads to shocking of the wave train, and the dissipation associated with the jump conditions across each shock limits the radiated power. In the model, damping of the bubble volume oscillations is restricted to a simple "effective" viscosity. For wall oscillation frequency less than the bubble natural frequency, the saturation amplitude of the radiated field is nearly independent of any specific damping mechanism. Finally, implications for noise radiation from cavitating flows are discussed
Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics
International audienceThe fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is now inevitable to increase the size of the problems that can be considered. The main constraint in the context of the FM-BEM is that the complete system is not assembled to reduce computational times and memory requirements. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to combine an approximate adjoint Dirichlet to Neumann (DtN) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet exterior scattering problems in 3D elasticity. The approximations of the adjoint DtN map are derived using tools proposed in [40]. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). We provide various numerical illustrations of the efficiency of the method for different smooth and non smooth geometries. In particular, the number of iterations is shown to be completely independent of the number of degrees of freedom and of the frequency for convex obstacles
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