Iterative Universal Rigidity

A bar framework determined by a finite graph $G$ and configuration $\bf p$ in $d$ space is universally rigid if it is rigid in any ${\mathbb R}^D \supset {\mathbb R}^d$. We provide a characterization of universally rigidity for any graph $G$ and any configuration ${\bf p}$ in terms of a sequence of affine subsets of the space of configurations. This corresponds to a facial reduction process for closed finite dimensional convex cones.Comment: 41 pages, 12 figure

Application of local linearization and the transonic equivalence rule to the flow about slender analytic bodies at Mach numbers near 1.0

The local linearization method for axisymmetric flow is combined with the transonic equivalence rule to calculate pressure distribution on slender bodies at free-stream Mach numbers from .8 to 1.2. This is an approximate solution to the transonic flow problem which yields results applicable during the preliminary design stages of a configuration development. The method can be used to determine the aerodynamic loads on parabolic arc bodies having either circular or elliptical cross sections. It is particularly useful in predicting pressure distributions and normal force distributions along the body at small angles of attack. The equations discussed may be extended to include wing-body combinations

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 “spatial/spectral” 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 “upgrade” 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