On Range Searching with Semialgebraic Sets II

Let $P$ be a set of $n$ points in $\R^d$. We present a linear-size data structure for answering range queries on $P$ with constant-complexity semialgebraic sets as ranges, in time close to $O(n^{1-1/d})$. It essentially matches the performance of similar structures for simplex range searching, and, for $d\ge 5$, significantly improves earlier solutions by the first two authors obtained in~1994. This almost settles a long-standing open problem in range searching. The data structure is based on the polynomial-partitioning technique of Guth and Katz [arXiv:1011.4105], which shows that for a parameter $r$, $1 < r \le n$, there exists a $d$-variate polynomial $f$ of degree $O(r^{1/d})$ such that each connected component of $\R^d\setminus Z(f)$ contains at most $n/r$ points of $P$, where $Z(f)$ is the zero set of $f$. We present an efficient randomized algorithm for computing such a polynomial partition, which is of independent interest and is likely to have additional applications

Design and Freeform Fabrication of Deployable Structures with Lattice Skins

Frontier environments—such as battlefields, hostile territories, remote locations, or outer space—drive the need for lightweight, deployable structures that can be stored in a compact configuration and deployed quickly and easily in the field. We introduce the concept of lattice skins to enable the design, solid freeform fabrication (SFF), and deployment of customizable structures with nearly arbitrary surface profile and lightweight multi-functionality. Using Duraform FLEX® material in a selective laser sintering machine, large deployable structures are fabricated in a nominal build chamber by either virtually collapsing them into a condensed form or decomposing them into smaller parts. Before fabrication, lattice sub-skins are added strategically beneath the surface of the part. The lattices provide elastic energy for folding and deploying the structure or constrain expansion upon application of internal air pressure. Nearly arbitrary surface profiles are achievable and internal space is preserved for subsequent usage. In this paper, we present the results of a set of experimental and computational models that are designed to provide proof of concept for lattice skins as a deployment mechanism in SFF and to demonstrate the effect of lattice structure on deployed shape.Mechanical Engineerin

Inflations of ideal triangulations

Starting with an ideal triangulation of the interior of a compact 3-manifold M with boundary, no component of which is a 2-sphere, we provide a construction, called an inflation of the ideal triangulation, to obtain a strongly related triangulations of M itself. Besides a step-by-step algorithm for such a construction, we provide examples of an inflation of the two-tetrahedra ideal triangulation of the complement of the figure-eight knot in the 3-sphere, giving a minimal triangulation, having ten tetrahedra, of the figure-eight knot exterior. As another example, we provide an inflation of the one-tetrahedron Gieseking manifold giving a minimal triangulation, having seven tetrahedra, of a nonorientable compact 3-manifold with Klein bottle boundary. Several applications of inflations are discussed.Comment: 48 pages, 45 figure

Stability and Vortex Shedding of Bluff Body Arrays

The primary purpose of this study was to develop an understanding of the stability of laminar flow through bluff body arrays, and investigate the nature of the unsteady vortex shedding regime that follows. The flow was numerically investigated using a specially developed multi-domain spectral element solver. Important criteria in the solver development were flexibility, efficiency, and accuracy. Flexibility was critical to the functionality of the code, as arrays of varying geometry were investigated. Efficiency with a high degree of accuracy was also of primary importance, with the code implemented to run efficiently on today's massively parallel architectures. Numerical two-dimensional stability analysis of the flow in several configurations of inline and staggered array geometries was performed. The growth rate, eigenfunction, and frequency of the disturbances were determined. The critical Reynolds number for flow transition in each case was identified and compared to that of flow over a single body. Based on the solutions of the laminar flow, a one-dimensional analytical analysis was performed on selected velocity profiles in the wake region. The results of this analysis were used to guide the interpretation of the two dimensional results and formulate a general theory of stability of inline and staggered bluff body arrays. The nature of the flow in the unsteady regime following the onset of instability was examined for an inline and a staggered arrangement. Particular attention was focused on the vortex shedding which was visualized and quantified through computation of the flow swirl, a quantity which identifies regions of rotary motion. The conditions required for the generation of leading edge vortex shedding were identified and discussed. Finally, a third geometry related to the inline and staggered arrays was considered. Flow solution data for this geometry is presented and its suitability as a model for louvered arrays was discussed.Air Conditioning and Refrigeration Project 11