29,276 research outputs found
A fast immersed boundary method for external incompressible viscous flows using lattice Green's functions
A new parallel, computationally efficient immersed boundary method for
solving three-dimensional, viscous, incompressible flows on unbounded domains
is presented. Immersed surfaces with prescribed motions are generated using the
interpolation and regularization operators obtained from the discrete delta
function approach of the original (Peskin's) immersed boundary method. Unlike
Peskin's method, boundary forces are regarded as Lagrange multipliers that are
used to satisfy the no-slip condition. The incompressible Navier-Stokes
equations are discretized on an unbounded staggered Cartesian grid and are
solved in a finite number of operations using lattice Green's function
techniques. These techniques are used to automatically enforce the natural
free-space boundary conditions and to implement a novel block-wise adaptive
grid that significantly reduces the run-time cost of solutions by limiting
operations to grid cells in the immediate vicinity and near-wake region of the
immersed surface. These techniques also enable the construction of practical
discrete viscous integrating factors that are used in combination with
specialized half-explicit Runge-Kutta schemes to accurately and efficiently
solve the differential algebraic equations describing the discrete momentum
equation, incompressibility constraint, and no-slip constraint. Linear systems
of equations resulting from the time integration scheme are efficiently solved
using an approximation-free nested projection technique. The algebraic
properties of the discrete operators are used to reduce projection steps to
simple discrete elliptic problems, e.g. discrete Poisson problems, that are
compatible with recent parallel fast multipole methods for difference
equations. Numerical experiments on low-aspect-ratio flat plates and spheres at
Reynolds numbers up to 3,700 are used to verify the accuracy and physical
fidelity of the formulation.Comment: 32 pages, 9 figures; preprint submitted to Journal of Computational
Physic
Trapped modes in finite quantum waveguides
The Laplace operator in infinite quantum waveguides (e.g., a bent strip or a
twisted tube) often has a point-like eigenvalue below the essential spectrum
that corresponds to a trapped eigenmode of finite L2 norm. We revisit this
statement for resonators with long but finite branches that we call "finite
waveguides". Although now there is no essential spectrum and all eigenfunctions
have finite L2 norm, the trapping can be understood as an exponential decay of
the eigenfunction inside the branches. We describe a general variational
formalism for detecting trapped modes in such resonators. For finite waveguides
with general cylindrical branches, we obtain a sufficient condition which
determines the minimal length of branches for getting a trapped eigenmode.
Varying the branch lengths may switch certain eigenmodes from non-trapped to
trapped states. These concepts are illustrated for several typical waveguides
(L-shape, bent strip, crossing of two stripes, etc.). We conclude that the
well-established theory of trapping in infinite waveguides may be incomplete
and require further development for being applied to microscopic quantum
devices
Harmonic density interpolation methods for high-order evaluation of Laplace layer potentials in 2D and 3D
We present an effective harmonic density interpolation method for the
numerical evaluation of singular and nearly singular Laplace boundary integral
operators and layer potentials in two and three spatial dimensions. The method
relies on the use of Green's third identity and local Taylor-like
interpolations of density functions in terms of harmonic polynomials. The
proposed technique effectively regularizes the singularities present in
boundary integral operators and layer potentials, and recasts the latter in
terms of integrands that are bounded or even more regular, depending on the
order of the density interpolation. The resulting boundary integrals can then
be easily, accurately, and inexpensively evaluated by means of standard
quadrature rules. A variety of numerical examples demonstrate the effectiveness
of the technique when used in conjunction with the classical trapezoidal rule
(to integrate over smooth curves) in two-dimensions, and with a Chebyshev-type
quadrature rule (to integrate over surfaces given as unions of non-overlapping
quadrilateral patches) in three-dimensions
On the evaluation of quasi-periodic Green functions and wave-scattering at and around Rayleigh-Wood anomalies
This article presents full-spectrum, well-conditioned, Green-function methodologies for evaluation of scattering by general periodic structures, which remains applicable on a set of challenging singular configurations, usually called Rayleigh-Wood (RW) anomalies (at which the quasi-periodic Green function ceases to exist), where most existing quasi-periodic solvers break down. After reviewing a variety of existing fast-converging numerical procedures commonly used to compute the classical quasi-periodic Green-function, the present work explores the difficulties they present around RW-anomalies and introduces the concept of hybrid âspatial/spectralâ representations. Such expressions allow both the modification of existing methods to obtain convergence at RW-anomalies as well as the application of a slight generalization of the Woodbury-Sherman-Morrison formulae together with a limiting procedure to bypass the singularities. (Although, for definiteness, the overall approach is applied to the scalar (acoustic) wave-scattering problem in the frequency domain, the approach can be extended in a straightforward manner to the harmonic Maxwell's and elasticity equations.) Ultimately, this thorough study of RW-anomalies yields fast and highly-accurate solvers, which are demonstrated with a variety of simulations of wave-scattering phenomena by arrays of particles, crossed impenetrable and penetrable diffraction gratings and other related structures. In particular, the methods developed in this article can be used to âupgradeâ classical approaches, resulting in algorithms that are applicable throughout the spectrum, and it provides new methods for cases where previous approaches are either costly or fail altogether. In particular, it is suggested that the proposed shifted Green function approach may provide the only viable alternative for treatment of three-dimensional high-frequency configurations with either one or two directions of periodicity. A variety of computational examples are presented which demonstrate the flexibility of the overall approach
Reconstruction of a function from its spherical (circular) means with the centers lying on the surface of certain polygons and polyhedra
We present explicit filtration/backprojection-type formulae for the inversion
of the spherical (circular) mean transform with the centers lying on the
boundary of some polyhedra (or polygons, in 2D). The formulae are derived using
the double layer potentials for the wave equation, for the domains with certain
symmetries. The formulae are valid for a rectangle and certain triangles in 2D,
and for a cuboid, certain right prisms and a certain pyramid in 3D. All the
present inversion formulae yield exact reconstruction within the domain
surrounded by the acquisition surface even in the presence of exterior sources.Comment: 9 figure
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