10 research outputs found
High-frequency asymptotic compression of dense BEM matrices for general geometries without ray tracing
Wave propagation and scattering problems in acoustics are often solved with
boundary element methods. They lead to a discretization matrix that is
typically dense and large: its size and condition number grow with increasing
frequency. Yet, high frequency scattering problems are intrinsically local in
nature, which is well represented by highly localized rays bouncing around.
Asymptotic methods can be used to reduce the size of the linear system, even
making it frequency independent, by explicitly extracting the oscillatory
properties from the solution using ray tracing or analogous techniques.
However, ray tracing becomes expensive or even intractable in the presence of
(multiple) scattering obstacles with complicated geometries. In this paper, we
start from the same discretization that constructs the fully resolved large and
dense matrix, and achieve asymptotic compression by explicitly localizing the
Green's function instead. This results in a large but sparse matrix, with a
faster associated matrix-vector product and, as numerical experiments indicate,
a much improved condition number. Though an appropriate localisation of the
Green's function also depends on asymptotic information unavailable for general
geometries, we can construct it adaptively in a frequency sweep from small to
large frequencies in a way which automatically takes into account a general
incident wave. We show that the approach is robust with respect to non-convex,
multiple and even near-trapping domains, though the compression rate is clearly
lower in the latter case. Furthermore, in spite of its asymptotic nature, the
method is robust with respect to low-order discretizations such as piecewise
constants, linears or cubics, commonly used in applications. On the other hand,
we do not decrease the total number of degrees of freedom compared to a
conventional classical discretization. The combination of the ...Comment: 24 pages, 13 figure
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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
A Fully discrete Galerkin method for high frequency exterior acoustic scattering in three dimensions
Standard Galerkin discretization techniques (with locally- or globally-supported basis functions) for boundary integral equations are inefficient for high frequency three dimensional exterior scattering simulations because they require a fixed number of unknowns per wavelength in each dimension, leading to large CPU time and memory requirements to set up the dense Galerkin matrix, with each entry requiring evaluation of multi-dimensional highly oscillatory integrals. In this work, using globally-supported basis functions, we describe an efficient fully discrete Galerkin surface integral equation algorithm for simulating high frequency acoustic scattering by three dimensional convex obstacles that includes a powerful integration scheme for evaluation of four dimensional Galerkin integrals with high-order accuracy. Such high-order order accuracy for various practically relevant frequencies (k∈[1,. 100,000]) substantially improves on approximations based on standard asymptotic techniques. We demonstrate the efficiency of our algorithm for spherical and non-spherical convex scattering for several wavenumbers 1≤k≤100,000 for low to high order prescribed tolerance. Our fully discrete algorithm requires only mild growth in the number of unknowns and CPU time as the frequency increases.22 page(s