42 research outputs found
On wave boundary elements for radiation and scattering problems with piecewise constant impedance
Discrete methods of numerical analysis have been used successfully for decades for the solution of problems involving wave diffraction, etc. However, these methods, including the finite element and boundary element methods, can require a prohibitively large number of elements as the wavelength becomes progressively shorter. In this paper, a new type of interpolation for the wave field is described in which the usual conventional shape functions are modified by the inclusion of a set of plane waves propagating in multiple directions. Including such a plane wave basis in a boundary element formulation is found in this paper to be highly successful. Results are shown for a variety of scattering/radiating problems from convex and nonconvex obstacles on which are prescribed piecewise constant Robin conditions. Notable results include a conclusion that, using this new formulation, only approximately three degrees of freedom per wavelength are required
Experimental observation of exceptional points in coupled pendulums
The concept of exceptional point (EP) is demonstrated experimentally in the
case of a simple mechanical system consisting of two coupled pendulums.
Exceptional points correspond to specific values of the system parameters that
yield defective eigenvalues. These spectral singularities which are typical of
non-Hermitian system means that both the eigenvalues and their associated
eigenvectors coalesce. The existence of an EP requires an adequate
parameterization of the dynamical system. For this aim, the experimental device
has been designed with two controllable parameters which are the length of one
pendulum and a viscous-like damping which is produced via electromagnetic
induction. Thanks to the observation of the free response of the coupled
pendulums, most EP properties are experimentally investigated, showing good
agreements with theoretical considerations. In contrast with many studies on
EPs, mainly in the field of physics, the novelty of the present work is that
controllable parameters are restricted to be real-valued, and this requires the
use of adequate search algorithms. Furthermore, it offers the possibility of
exploiting the existence of EPs in time-domain dynamic problems
A high-wavenumber boundary-element method for an acoustic scattering problem
In this paper we show stability and convergence for a novel Galerkin boundary element method approach to the impedance boundary value problem for the Helmholtz equation in a half-plane with piecewise constant boundary data. This problem models, for example, outdoor sound propagation over inhomogeneous flat terrain. To achieve a good approximation with a relatively low number of degrees of freedom we employ a graded mesh with smaller elements adjacent to discontinuities in impedance, and a special set of basis functions for the Galerkin method so that, on each element, the approximation space consists of polynomials (of degree ) multiplied by traces of plane waves on the boundary. In the case where the impedance is constant outside an interval , which only requires the discretization of , we show theoretically and experimentally that the error in computing the acoustic field on is , where is the number of degrees of freedom and is the wavenumber. This indicates that the proposed method is especially commendable for large intervals or a high wavenumber. In a final section we sketch how the same methodology extends to more general scattering problems
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
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Analytic mode-matching for accurate handling of exceptional points in a lined acoustic waveguide
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