149,509 research outputs found
A conjecture on Hubbard-Stratonovich transformations for the Pruisken-Sch\"afer parameterisations of real hyperbolic domains
Rigorous justification of the Hubbard-Stratonovich transformation for the
Pruisken-Sch\"afer type of parameterisations of real hyperbolic
O(m,n)-invariant domains remains a challenging problem. We show that a naive
choice of the volume element invalidates the transformation, and put forward a
conjecture about the correct form which ensures the desired structure. The
conjecture is supported by complete analytic solution of the problem for groups
O(1,1) and O(2,1), and by a method combining analytical calculations with a
simple numerical evaluation of a two-dimensional integral in the case of the
group O(2,2).Comment: Published versio
Windowed Green function method for wave scattering by periodic arrays of 2D obstacles
This paper introduces a novel boundary integral equation (BIE) method for the
numerical solution of problems of planewave scattering by periodic line arrays
of two-dimensional penetrable obstacles. Our approach is built upon a direct
BIE formulation that leverages the simplicity of the free-space Green function
but in turn entails evaluation of integrals over the unit-cell boundaries. Such
integrals are here treated via the window Green function method. The windowing
approximation together with a finite-rank operator correction -- used to
properly impose the Rayleigh radiation condition -- yield a robust second-kind
BIE that produces super-algebraically convergent solutions throughout the
spectrum, including at the challenging Rayleigh-Wood anomalies. The corrected
windowed BIE can be discretized by means of off-the-shelf Nystr\"om and
boundary element methods, and it leads to linear systems suitable for iterative
linear-algebra solvers as well as standard fast matrix-vector product
algorithms. A variety of numerical examples demonstrate the accuracy and
robustness of the proposed methodolog
Simple and Efficient Numerical Evaluation of Near-Hypersingular Integrals
Recently, significant progress has been made in the handling of singular and nearly-singular potential integrals that commonly arise in the Boundary Element Method (BEM). To facilitate object-oriented programming and handling of higher order basis functions, cancellation techniques are favored over techniques involving singularity subtraction. However, gradients of the Newton-type potentials, which produce hypersingular kernels, are also frequently required in BEM formulations. As is the case with the potentials, treatment of the near-hypersingular integrals has proven more challenging than treating the limiting case in which the observation point approaches the surface. Historically, numerical evaluation of these near-hypersingularities has often involved a two-step procedure: a singularity subtraction to reduce the order of the singularity, followed by a boundary contour integral evaluation of the extracted part. Since this evaluation necessarily links basis function, Green s function, and the integration domain (element shape), the approach ill fits object-oriented programming concepts. Thus, there is a need for cancellation-type techniques for efficient numerical evaluation of the gradient of the potential. Progress in the development of efficient cancellation-type procedures for the gradient potentials was recently presented. To the extent possible, a change of variables is chosen such that the Jacobian of the transformation cancels the singularity. However, since the gradient kernel involves singularities of different orders, we also require that the transformation leaves remaining terms that are analytic. The terms "normal" and "tangential" are used herein with reference to the source element. Also, since computational formulations often involve the numerical evaluation of both potentials and their gradients, it is highly desirable that a single integration procedure efficiently handles both
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