Hyperorthogonal well-folded Hilbert curves

R-trees can be used to store and query sets of point data in two or more dimensions. An easy way to construct and maintain R-trees for two-dimensional points, due to Kamel and Faloutsos, is to keep the points in the order in which they appear along the Hilbert curve. The R-tree will then store bounding boxes of points along contiguous sections of the curve, and the efficiency of the R-tree depends on the size of the bounding boxes---smaller is better. Since there are many different ways to generalize the Hilbert curve to higher dimensions, this raises the question which generalization results in the smallest bounding boxes. Familiar methods, such as the one by Butz, can result in curve sections whose bounding boxes are a factor $\Omega(2^{d/2})$ larger than the volume traversed by that section of the curve. Most of the volume bounded by such bounding boxes would not contain any data points. In this paper we present a new way of generalizing Hilbert's curve to higher dimensions, which results in much tighter bounding boxes: they have at most 4 times the volume of the part of the curve covered, independent of the number of dimensions. Moreover, we prove that a factor 4 is asymptotically optimal.Comment: Manuscript submitted to Journal of Computational Geometry. An abstract appeared in the 31st Int Symp on Computational Geometry (SoCG 2015), LIPIcs 34:812-82

Constructing Delaunay triangulations along space-filling curves

Incremental construction con BRIO using a space-filling curve order for insertion is a popular algorithm for constructing Delaunay triangulations. So far, it has only been analyzed for the case that a worst-case optimal point location data structure is used which is often avoided in implementations. In this paper, we analyze its running time for the more typical case that points are located by walking. We show that in the worst-case the algorithm needs quadratic time, but that this can only happen in degenerate cases. We show that the algorithm runs in O(n logn) time under realistic assumptions. Furthermore, we show that it runs in expected linear time for many random point distributions. This research was supported by the Deutsche Forschungsgemeinschaft within the European graduate program ’Combinatorics, Geometry, and Computation’ (No. GRK 588/2) and by the Netherlands’ Organisation for Scientific Research (NWO) under BRICKS/FOCUS grant number 642.065.503 and project no. 639.022.707

A scalable parallel finite element framework for growing geometries. Application to metal additive manufacturing

This work introduces an innovative parallel, fully-distributed finite element framework for growing geometries and its application to metal additive manufacturing. It is well-known that virtual part design and qualification in additive manufacturing requires highly-accurate multiscale and multiphysics analyses. Only high performance computing tools are able to handle such complexity in time frames compatible with time-to-market. However, efficiency, without loss of accuracy, has rarely held the centre stage in the numerical community. Here, in contrast, the framework is designed to adequately exploit the resources of high-end distributed-memory machines. It is grounded on three building blocks: (1) Hierarchical adaptive mesh refinement with octree-based meshes; (2) a parallel strategy to model the growth of the geometry; (3) state-of-the-art parallel iterative linear solvers. Computational experiments consider the heat transfer analysis at the part scale of the printing process by powder-bed technologies. After verification against a 3D benchmark, a strong-scaling analysis assesses performance and identifies major sources of parallel overhead. A third numerical example examines the efficiency and robustness of (2) in a curved 3D shape. Unprecedented parallelism and scalability were achieved in this work. Hence, this framework contributes to take on higher complexity and/or accuracy, not only of part-scale simulations of metal or polymer additive manufacturing, but also in welding, sedimentation, atherosclerosis, or any other physical problem where the physical domain of interest grows in time

Faceting Transition in an Exactly Solvable Terrace-Ledge-Kink model

We solve exactly a Terrace-Ledge-Kink (TLK) model describing a crystal surface at a microscopic level. We show that there is a faceting transition driven either by temperature or by the chemical potential that controls the slope of the surface. In the rough phase we investigate thermal fluctuations of the surface using Conformal Field Theory.Comment: 27 pages, 18 EPS figure

Methods for Detection and Correction of Sudden Pixel Sensitivity Drops

PDC 8.0 includes implementation of a new algorithm to detect and correct step discontinuities appearing in roughly one of every twenty stellar light curves during a given quarter. An example of such a discontinuity in an actual light curve is shown in fig. 1. The majority of such discontinuities are believed to result from high-energy particles (either cosmic or solar in origin) striking the photometer and causing permanent local changes (typically -0.5% in summed apertures) in quantum efficiency, though a partial exponential recovery is often observed. Since these features, dubbed sudden pixel sensitivity dropouts (SPSDs), are uncorrelated across targets they cannot be properly accounted for by the current detrending algorithm. PDC de-trending is based on the assumption that features in flux time series are due either to intrinsic stellar phenomena or to systematic errors and that systematics will exhibit measurable correlations across targets. SPSD events violate these assumptions and their successful removal not only rectifies the flux values of affected targets, but demonstrably improves the overall performance of PDC de-trending