239 research outputs found
A frequency-independent boundary element method for scattering by two-dimensional screens and apertures
We propose and analyse a hybrid numerical-asymptotic boundary element method for time-harmonic scattering of an incident plane wave by an arbitrary collinear array of sound-soft two-dimensional screens. Our method uses an approximation space enriched with oscillatory basis functions, chosen to capture the high frequency asymptotics of the solution. Our numerical results suggest that fi�xed accuracy can be achieved at arbitrarily high frequencies with a frequency-independent computational cost. Our analysis does not capture this observed behaviour completely, but we provide a rigorous frequency-explicit error analysis which proves that the method converges exponentially as the number of degrees of freedom increases, and that to achieve any desired accuracy it is sufficient to increase in proportion to the square of the logarithm of the frequency as the frequency increases (standard boundary element methods require to increase at least linearly with frequency to retain accuracy). We also show how our method can be applied to the complementary "breakwater" problem of propagation through an aperture in an infinite sound-hard screen
A high frequency boundary element method for scattering by convex polygons
In this paper we propose and analyze a hybrid boundary element method for the solution of problems of high frequency acoustic scattering by sound-soft convex polygons, in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution. We demonstrate, both theoretically and via numerical examples, exponential convergence with respect to the order of the polynomials, moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern, in which the dependence on the frequency of all constants is explicit. Importantly, these estimates prove that, to achieve any desired accuracy in the computation of these quantities, it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods
Isogeometric FEM-BEM coupled structural-acoustic analysis of shells using subdivision surfaces
We introduce a coupled finite and boundary element formulation for acoustic
scattering analysis over thin shell structures. A triangular Loop subdivision
surface discretisation is used for both geometry and analysis fields. The
Kirchhoff-Love shell equation is discretised with the finite element method and
the Helmholtz equation for the acoustic field with the boundary element method.
The use of the boundary element formulation allows the elegant handling of
infinite domains and precludes the need for volumetric meshing. In the present
work the subdivision control meshes for the shell displacements and the
acoustic pressures have the same resolution. The corresponding smooth
subdivision basis functions have the continuity property required for the
Kirchhoff-Love formulation and are highly efficient for the acoustic field
computations. We validate the proposed isogeometric formulation through a
closed-form solution of acoustic scattering over a thin shell sphere.
Furthermore, we demonstrate the ability of the proposed approach to handle
complex geometries with arbitrary topology that provides an integrated
isogeometric design and analysis workflow for coupled structural-acoustic
analysis of shells
On stability of discretizations of the Helmholtz equation (extended version)
We review the stability properties of several discretizations of the
Helmholtz equation at large wavenumbers. For a model problem in a polygon, a
complete -explicit stability (including -explicit stability of the
continuous problem) and convergence theory for high order finite element
methods is developed. In particular, quasi-optimality is shown for a fixed
number of degrees of freedom per wavelength if the mesh size and the
approximation order are selected such that is sufficiently small and
, and, additionally, appropriate mesh refinement is used near
the vertices. We also review the stability properties of two classes of
numerical schemes that use piecewise solutions of the homogeneous Helmholtz
equation, namely, Least Squares methods and Discontinuous Galerkin (DG)
methods. The latter includes the Ultra Weak Variational Formulation
Sharp high-frequency estimates for the Helmholtz equation and applications to boundary integral equations
We consider three problems for the Helmholtz equation in interior and
exterior domains in R^d (d=2,3): the exterior Dirichlet-to-Neumann and
Neumann-to-Dirichlet problems for outgoing solutions, and the interior
impedance problem. We derive sharp estimates for solutions to these problems
that, in combination, give bounds on the inverses of the combined-field
boundary integral operators for exterior Helmholtz problems.Comment: Version 3: 42 pages; improved exposition in response to referee
comments and added several reference
Can coercive formulations lead to fast and accurate solution of the Helmholtz equation?
A new, coercive formulation of the Helmholtz equation was introduced in
[Moiola, Spence, SIAM Rev. 2014]. In this paper we investigate -version
Galerkin discretisations of this formulation, and the iterative solution of the
resulting linear systems. We find that the coercive formulation behaves
similarly to the standard formulation in terms of the pollution effect (i.e. to
maintain accuracy as , must decrease with at the same rate
as for the standard formulation). We prove -explicit bounds on the number of
GMRES iterations required to solve the linear system of the new formulation
when it is preconditioned with a prescribed symmetric positive-definite matrix.
Even though the number of iterations grows with , these are the first such
rigorous bounds on the number of GMRES iterations for a preconditioned
formulation of the Helmholtz equation, where the preconditioner is a symmetric
positive-definite matrix.Comment: 27 pages, 7 figure
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