384 research outputs found

    On Computing the Vertex Centroid of a Polyhedron

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    Let P\mathcal{P} be an H\mathcal{H}-polytope in Rd\mathbb{R}^d with vertex set VV. The vertex centroid is defined as the average of the vertices in VV. We prove that computing the vertex centroid of an H\mathcal{H}-polytope is #P-hard. Moreover, we show that even just checking whether the vertex centroid lies in a given halfspace is already #P-hard for H\mathcal{H}-polytopes. We also consider the problem of approximating the vertex centroid by finding a point within an ϵ\epsilon distance from it and prove this problem to be #P-easy by showing that given an oracle for counting the number of vertices of an H\mathcal{H}-polytope, one can approximate the vertex centroid in polynomial time. We also show that any algorithm approximating the vertex centroid to \emph{any} ``sufficiently'' non-trivial (for example constant) distance, can be used to construct a fully polynomial approximation scheme for approximating the centroid and also an output-sensitive polynomial algorithm for the Vertex Enumeration problem. Finally, we show that for unbounded polyhedra the vertex centroid can not be approximated to a distance of d1/2δd^{{1/2}-\delta} for any fixed constant δ>0\delta>0

    Software for Exact Integration of Polynomials over Polyhedra

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    We are interested in the fast computation of the exact value of integrals of polynomial functions over convex polyhedra. We present speed ups and extensions of the algorithms presented in previous work. We present the new software implementation and provide benchmark computations. The computation of integrals of polynomials over polyhedral regions has many applications; here we demonstrate our algorithmic tools solving a challenge from combinatorial voting theory.Comment: Major updat

    Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

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    One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology

    Complexity of some polyhedral enumeration problems

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    In this thesis we consider the problem of converting the halfspace representation of a polytope to its vertex representation - the Vertex Enumeration problem - and various other basic and closely related computational problems about polytopes. The problem of converting the vertex representation to halfspace representation - the Convex Hull problem - is equivalent to vertex enumeration. In chapter 3 we prove that enumerating the vertices of an unbounded H-polyhedron P is NP-hard even if P has only 0=1 vertices. This strengthens a previous result of Khachiyan et. al. [KBB+06]. In chapters 4 to 6 we prove that many other operations on polytopes like computing the Minkowski sum, intersection, projection, etc. are NP-hard for many representations and equivalent to vertex enumeration in many others. In chapter 7 we prove various hardness results about a cone covering problem where one wants to check whether a given set of polyhedral cones cover another given set. We prove that in general this is an NP-complete problem and includes important problems like vertex enumeration and hypergraph transversal as special cases. Finally, in chapter 8 we relate the complexity of vertex enumeration to graph isomorphism by proving that a certain graph isomorphism hard problem is graph isomorphism easy if and only if vertex enumeration is graph isomorphism easy. We also answer a question of Kaibel and Schwartz about the complexity of checking self-duality of a polytope.In dieser Arbeit betrachten wir das Problem, die Halbraumdarstellung eines Polytops in seine Eckendarstellung umzuwandeln, - das sogenannte Problem der Eckenaufzählung - sowie viele andere grundlegende und eng verwandte Berechnungsprobleme für Polytope. Das Problem, die Eckendarstellung in die Halbraumdarstellung umzuwandeln - das sogenannte Konvexe-Hüllen Problem - ist äquivalent zum Problem der Eckenaufzählung. In Kapitel 3 zeigen wir, dass Eckenaufzählung für ein unbeschränktes H-Polyeder P selbst dann NP-schwer ist, wenn P nur 0=1-Ecken hat. Das verbessert ein Ergebnis von Khachiyan et. al. [KBB+06]. In den Kapiteln 4 bis 6 zeigen wir, dass viele andere Operationen auf Polytopen, wie Berechnung von Minkowski-Summe, Durchschnitt, Projektion usw., für viele Darstellungen NP-schwer sind und für viele weitere äquivalent zu Eckenaufzählung sind. In Kapitel 7 beweisen wir Härteresultate über ein Kegelüberdeckungsproblem, das danach fragt, ob eine gegebene Menge polyedrischer Kegel eine andere gegebene Menge überdeckt. Wir zeigen, dass dies im Allgemeinen ein NP-vollständiges Problem ist und wichtige Probleme wie Eckenaufzählung und Hypergraphentraversierung als Spezialfälle umfasst. Schließlich stellen wir in Kapitel 8 einen Zusammenhang zwischen Eckenaufzählung und Graphisomorphie her, indem wir beweisen, dass ein bestimmtes Graphisomorphie-schweres Problem genau dann Graphisomorphie-leicht ist, wenn Eckenaufzählung Graphisomorphie-leicht ist. Außerdem beantworten wir eine Frage von Kaibel und Schwartz über das Testen der Selbst-Dualität von Polytopen

    Unstructured un-split geometrical Volume-of-Fluid methods -- A review

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    Geometrical Volume-of-Fluid (VoF) methods mainly support structured meshes, and only a small number of contributions in the scientific literature report results with unstructured meshes and three spatial dimensions. Unstructured meshes are traditionally used for handling geometrically complex solution domains that are prevalent when simulating problems of industrial relevance. However, three-dimensional geometrical operations are significantly more complex than their two-dimensional counterparts, which is confirmed by the ratio of publications with three-dimensional results on unstructured meshes to publications with two-dimensional results or support for structured meshes. Additionally, unstructured meshes present challenges in serial and parallel computational efficiency, accuracy, implementation complexity, and robustness. Ongoing research is still very active, focusing on different issues: interface positioning in general polyhedra, estimation of interface normal vectors, advection accuracy, and parallel and serial computational efficiency. This survey tries to give a complete and critical overview of classical, as well as contemporary geometrical VOF methods with concise explanations of the underlying ideas and sub-algorithms, focusing primarily on unstructured meshes and three dimensional calculations. Reviewed methods are listed in historical order and compared in terms of accuracy and computational efficiency

    Collection of abstracts of the 24th European Workshop on Computational Geometry

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    International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop

    A new rock slicing method based on linear programming

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    One of the important pre-processing stages in the analysis of jointed rock masses is the identification of rock blocks from discontinuities in the field. In 3D, the identification of polyhedral blocks usually involve tedious housekeeping algorithms, because one needs to establish their vertices, edges and faces, together with a hierarchical data structure: edges by pairs of vertices, faces by bounding edges, polyhedron by bounding faces. In this paper, we present a novel rock slicing method, based on the subdivision approach and linear programming optimisation, which requires only a single level of data structure rather than the current 2 or 3 levels presented in the literature. This method exploits the novel mathematical framework for contact detection introduced in Boon et al. (2012). In the proposed method, it is not necessary to calculate the intersections between a discontinuity and the block faces, because information on the block vertices and edges is not needed. The use of a simpler data structure presents obvious advantages in terms of code development, robustness and ease of maintenance. Non-persistent joints are also introduced in a novel way within the framework of linear programming. Advantages and disadvantages of the proposed modelling of non-persistent joints are discussed in this paper. Concave blocks are generated using established methods in the sequential subdivision approach, i.e. through fictitious joints

    Planar hexagonal meshing for architecture

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