33 research outputs found
Uniqueness in Discrete Tomography of Delone Sets with Long-Range Order
We address the problem of determining finite subsets of Delone sets
with long-range order by -rays in prescribed
-directions, i.e., directions parallel to non-zero interpoint
vectors of . Here, an -ray in direction of a finite set
gives the number of points in the set on each line parallel to . For our
main result, we introduce the notion of algebraic Delone sets
and derive a sufficient condition for the determination
of the convex subsets of these sets by -rays in four prescribed
-directions.Comment: 15 pages, 2 figures; condensed and revised versio
Magic numbers in the discrete tomography of cyclotomic model sets
We report recent progress in the problem of distinguishing convex subsets of
cyclotomic model sets by (discrete parallel) X-rays in prescribed
-directions. It turns out that for any of these model sets
there exists a `magic number' such that any two
convex subsets of can be distinguished by their X-rays in any set
of prescribed -directions. In particular, for
pentagonal, octagonal, decagonal and dodecagonal model sets, the least possible
numbers are in that very order 11, 9, 11 and 13.Comment: 6 pages, 1 figure; based on the results of arXiv:1101.4149 [math.MG];
presented at Aperiodic 2012 (Cairns, Australia
Solution of a uniqueness problem in the discrete tomography of algebraic Delone sets
We consider algebraic Delone sets in the Euclidean plane and
address the problem of distinguishing convex subsets of by X-rays
in prescribed -directions, i.e., directions parallel to nonzero
interpoint vectors of . Here, an X-ray in direction of a finite
set gives the number of points in the set on each line parallel to . It is
shown that for any algebraic Delone set there are four prescribed
-directions such that any two convex subsets of can be
distinguished by the corresponding X-rays. We further prove the existence of a
natural number such that any two convex subsets of
can be distinguished by their X-rays in any set of
prescribed -directions. In particular, this
extends a well-known result of Gardner and Gritzmann on the corresponding
problem for planar lattices to nonperiodic cases that are relevant in
quasicrystallography.Comment: 21 pages, 1 figur
Discrete tomography: Magic numbers for -fold symmetry
We consider the problem of distinguishing convex subsets of -cyclotomic
model sets by (discrete parallel) X-rays in prescribed
-directions. In this context, a `magic number' has
the property that any two convex subsets of can be distinguished
by their X-rays in any set of prescribed
-directions. Recent calculations suggest that (with one exception
in the case ) the least possible magic number for -cyclotomic model
sets might just be , where .Comment: 5 pages, 2 figures; new computer calculations based on the results of
arXiv:1101.4149 and arXiv:1211.6318; presented at ICQ 12 (Cracow, Poland
Discrete Tomography of Icosahedral Model Sets
The discrete tomography of B-type and F-type icosahedral model sets is
investigated, with an emphasis on reconstruction and uniqueness problems. These
are motivated by the request of materials science for the unique reconstruction
of quasicrystalline structures from a small number of images produced by
quantitative high resolution transmission electron microscopy.Comment: 21 pages, 3 figures; revised version, figures adde
A Note on Affinely Regular Polygons
The affinely regular polygons in certain planar sets are characterized. It is
also shown that the obtained results apply to cyclotomic model sets and,
additionally, have consequences in the discrete tomography of these sets.Comment: 10 pages, 1 figur
Emergence of Structures in Particle Systems: Mechanics, Analysis and Computation
The meeting focused on the last advances in particle systems. The talks covered a broad range of topics, ranging from questions in crystallization and atomistic systems to mesoscopic models of defects to machine learning approaches and computational aspects
IST Austria Thesis
In this thesis we study persistence of multi-covers of Euclidean balls and the geometric structures underlying their computation, in particular Delaunay mosaics and Voronoi tessellations.
The k-fold cover for some discrete input point set consists of the space where at least k balls of radius r around the input points overlap. Persistence is a notion that captures, in some sense, the topology of the shape underlying the input. While persistence is usually computed for the union of balls, the k-fold cover is of interest as it captures local density,
and thus might approximate the shape of the input better if the input data is noisy. To compute persistence of these k-fold covers, we need a discretization that is provided by higher-order Delaunay mosaics.
We present and implement a simple and efficient algorithm for the computation of higher-order Delaunay mosaics, and use it to give experimental results for their combinatorial properties. The algorithm makes use of a new geometric structure, the rhomboid tiling. It contains the higher-order Delaunay mosaics as slices, and by introducing a filtration
function on the tiling, we also obtain higher-order α-shapes as slices. These allow us to compute persistence of the multi-covers for varying radius r; the computation for varying k is less straight-foward and involves the rhomboid tiling directly. We apply our algorithms to experimental sphere packings to shed light on their structural properties. Finally, inspired by periodic structures in packings and materials, we propose and implement an algorithm for periodic Delaunay triangulations to be integrated into the Computational Geometry Algorithms Library (CGAL), and discuss
the implications on persistence for periodic data sets
Composite Finite Elements for Trabecular Bone Microstructures
In many medical and technical applications, numerical simulations need to be performed for objects with interfaces of geometrically complex shape. We focus on the biomechanical problem of elasticity simulations for trabecular bone microstructures. The goal of this dissertation is to develop and implement an efficient simulation tool for finite element simulations on such structures, so-called composite finite elements. We will deal with both the case of material/void interfaces (complicated domains) and the case of interfaces between different materials (discontinuous coefficients). In classical finite element simulations, geometric complexity is encoded in tetrahedral and typically unstructured meshes. Composite finite elements, in contrast, encode geometric complexity in specialized basis functions on a uniform mesh of hexahedral structure. Other than alternative approaches (such as e.g. fictitious domain methods, generalized finite element methods, immersed interface methods, partition of unity methods, unfitted meshes, and extended finite element methods), the composite finite elements are tailored to geometry descriptions by 3D voxel image data and use the corresponding voxel grid as computational mesh, without introducing additional degrees of freedom, and thus making use of efficient data structures for uniformly structured meshes. The composite finite element method for complicated domains goes back to Wolfgang Hackbusch and Stefan Sauter and restricts standard affine finite element basis functions on the uniformly structured tetrahedral grid (obtained by subdivision of each cube in six tetrahedra) to an approximation of the interior. This can be implemented as a composition of standard finite element basis functions on a local auxiliary and purely virtual grid by which we approximate the interface. In case of discontinuous coefficients, the same local auxiliary composition approach is used. Composition weights are obtained by solving local interpolation problems for which coupling conditions across the interface need to be determined. These depend both on the local interface geometry and on the (scalar or tensor-valued) material coefficients on both sides of the interface. We consider heat diffusion as a scalar model problem and linear elasticity as a vector-valued model problem to develop and implement the composite finite elements. Uniform cubic meshes contain a natural hierarchy of coarsened grids, which allows us to implement a multigrid solver for the case of complicated domains. Besides simulations of single loading cases, we also apply the composite finite element method to the problem of determining effective material properties, e.g. for multiscale simulations. For periodic microstructures, this is achieved by solving corrector problems on the fundamental cells using affine-periodic boundary conditions corresponding to uniaxial compression and shearing. For statistically periodic trabecular structures, representative fundamental cells can be identified but do not permit the periodic approach. Instead, macroscopic displacements are imposed using the same set as before of affine-periodic Dirichlet boundary conditions on all faces. The stress response of the material is subsequently computed only on an interior subdomain to prevent artificial stiffening near the boundary. We finally check for orthotropy of the macroscopic elasticity tensor and identify its axes.Zusammengesetzte finite Elemente fĂŒr trabekulĂ€re Mikrostrukturen in Knochen In vielen medizinischen und technischen Anwendungen werden numerische Simulationen fĂŒr Objekte mit geometrisch komplizierter Form durchgefĂŒhrt. Gegenstand dieser Dissertation ist die Simulation der ElastizitĂ€t trabekulĂ€rer Mikrostrukturen von Knochen, einem biomechanischen Problem. Ziel ist es, ein effizientes Simulationswerkzeug fĂŒr solche Strukturen zu entwickeln, die sogenannten zusammengesetzten finiten Elemente. Wir betrachten dabei sowohl den Fall von Interfaces zwischen Material und Hohlraum (komplizierte Gebiete) als auch zwischen verschiedenen Materialien (unstetige Koeffizienten). In klassischen Finite-Element-Simulationen wird geometrische KomplexitĂ€t typischerweise in unstrukturierten Tetraeder-Gittern kodiert. Zusammengesetzte finite Elemente dagegen kodieren geometrische KomplexitĂ€t in speziellen Basisfunktionen auf einem gleichförmigen WĂŒrfelgitter. Anders als alternative AnsĂ€tze (wie zum Beispiel fictitious domain methods, generalized finite element methods, immersed interface methods, partition of unity methods, unfitted meshes und extended finite element methods) sind die zusammengesetzten finiten Elemente zugeschnitten auf die Geometriebeschreibung durch dreidimensionale Bilddaten und benutzen das zugehörige Voxelgitter als Rechengitter, ohne zusĂ€tzliche Freiheitsgrade einzufĂŒhren. Somit können sie effiziente Datenstrukturen fĂŒr gleichförmig strukturierte Gitter ausnutzen. Die Methode der zusammengesetzten finiten Elemente geht zurĂŒck auf Wolfgang Hackbusch und Stefan Sauter. Man schrĂ€nkt dabei ĂŒbliche affine Finite-Element-Basisfunktionen auf gleichförmig strukturierten Tetraedergittern (die man durch Unterteilung jedes WĂŒrfels in sechs Tetraeder erhĂ€lt) auf das approximierte Innere ein. Dies kann implementiert werden durch das Zusammensetzen von Standard-Basisfunktionen auf einem lokalen und rein virtuellen Hilfsgitter, durch das das Interface approximiert wird. Im Falle unstetiger Koeffizienten wird die gleiche lokale Hilfskonstruktion verwendet. Gewichte fĂŒr das Zusammensetzen erhĂ€lt man hier, indem lokale Interpolationsprobleme gelöst werden, wozu zunĂ€chst Kopplungsbedingungen ĂŒber das Interface hinweg bestimmt werden. Diese hĂ€ngen ab sowohl von der lokalen Geometrie des Interface als auch von den (skalaren oder tensorwertigen) Material-Koeffizienten auf beiden Seiten des Interface. Wir betrachten WĂ€rmeleitung als skalares und lineare ElastizitĂ€t als vektorwertiges Modellproblem, um die zusammengesetzten finiten Elemente zu entwickeln und zu implementieren. Gleichförmige WĂŒrfelgitter enthalten eine natĂŒrliche Hierarchie vergröberter Gitter, was es uns erlaubt, im Falle komplizierter Gebiete einen Mehrgitterlöser zu implementieren. Neben Simulationen einzelner LastfĂ€lle wenden wir die zusammengesetzten finiten Elemente auch auf das Problem an, effektive Materialeigenschaften zu bestimmen, etwa fĂŒr mehrskalige Simulationen. FĂŒr periodische Mikrostrukturen wird dies erreicht, indem man Korrekturprobleme auf der Fundamentalzelle löst. DafĂŒr nutzt man affin-periodische Randwerte, die zu uniaxialem Druck oder zu Scherung korrespondieren. In statistisch periodischen trabekulĂ€ren Mikrostrukturen lassen sich ebenfalls Fundamentalzellen identifizieren, sie erlauben jedoch keinen periodischen Ansatz. Stattdessen werden makroskopische Verschiebungen zu denselben affin-periodischen Randbedingungen vorgegeben, allerdings durch Dirichlet-Randwerte auf allen SeitenflĂ€chen. Die Spannungsantwort des Materials wird anschlieĂend nur auf einem inneren Teilbereich berechnet, um kĂŒnstliche Versteifung am Rand zu verhindern. SchlieĂlich prĂŒfen wir den makroskopischen ElastizitĂ€tstensor auf Orthotropie und identifizieren deren Achsen