On the computation of zone and double zone diagrams

Classical objects in computational geometry are defined by explicit relations. Several years ago the pioneering works of T. Asano, J. Matousek and T. Tokuyama introduced "implicit computational geometry", in which the geometric objects are defined by implicit relations involving sets. An important member in this family is called "a zone diagram". The implicit nature of zone diagrams implies, as already observed in the original works, that their computation is a challenging task. In a continuous setting this task has been addressed (briefly) only by these authors in the Euclidean plane with point sites. We discuss the possibility to compute zone diagrams in a wide class of spaces and also shed new light on their computation in the original setting. The class of spaces, which is introduced here, includes, in particular, Euclidean spheres and finite dimensional strictly convex normed spaces. Sites of a general form are allowed and it is shown that a generalization of the iterative method suggested by Asano, Matousek and Tokuyama converges to a double zone diagram, another implicit geometric object whose existence is known in general. Occasionally a zone diagram can be obtained from this procedure. The actual (approximate) computation of the iterations is based on a simple algorithm which enables the approximate computation of Voronoi diagrams in a general setting. Our analysis also yields a few byproducts of independent interest, such as certain topological properties of Voronoi cells (e.g., that in the considered setting their boundaries cannot be "fat").Comment: Very slight improvements (mainly correction of a few typos); add DOI; Ref [51] points to a freely available computer application which implements the algorithms; to appear in Discrete & Computational Geometry (available online

A Characterization Theorem and An Algorithm for A Convex Hull Problem

Given $S= \{v_1, \dots, v_n\} \subset \mathbb{R} ^m$ and $p \in \mathbb{R} ^m$, testing if $p \in conv(S)$, the convex hull of $S$, is a fundamental problem in computational geometry and linear programming. First, we prove a Euclidean {\it distance duality}, distinct from classical separation theorems such as Farkas Lemma: $p$ lies in $conv(S)$ if and only if for each $p' \in conv(S)$ there exists a {\it pivot}, $v_j \in S$ satisfying $d(p',v_j) \geq d(p,v_j)$. Equivalently, $p \not \in conv(S)$ if and only if there exists a {\it witness}, $p' \in conv(S)$ whose Voronoi cell relative to $p$ contains $S$. A witness separates $p$ from $conv(S)$ and approximate $d(p, conv(S))$ to within a factor of two. Next, we describe the {\it Triangle Algorithm}: given $\epsilon \in (0,1)$, an {\it iterate}, $p' \in conv(S)$, and $v \in S$, if $d(p, p') < \epsilon d(p,v)$, it stops. Otherwise, if there exists a pivot $v_j$, it replace $v$ with $v_j$ and $p'$ with the projection of $p$ onto the line $p'v_j$. Repeating this process, the algorithm terminates in $O(mn \min \{\epsilon^{-2}, c^{-1}\ln \epsilon^{-1} \})$ arithmetic operations, where $c$ is the {\it visibility factor}, a constant satisfying $c \geq \epsilon^2$ and $\sin (\angle pp'v_j) \leq 1/\sqrt{1+c}$, over all iterates $p'$. Additionally, (i) we prove a {\it strict distance duality} and a related minimax theorem, resulting in more effective pivots; (ii) describe $O(mn \ln \epsilon^{-1})$-time algorithms that may compute a witness or a good approximate solution; (iii) prove {\it generalized distance duality} and describe a corresponding generalized Triangle Algorithm; (iv) prove a {\it sensitivity theorem} to analyze the complexity of solving LP feasibility via the Triangle Algorithm. The Triangle Algorithm is practical and competitive with the simplex method, sparse greedy approximation and first-order methods.Comment: 42 pages, 17 figures, 2 tables. This revision only corrects minor typo

The MAPPINGS III Library of Fast Radiative Shock Models

We present a new library of fully-radiative shock models calculated with the MAPPINGS III shock and photoionization code. The library consists of grids of models with shock velocities in the range v=100-1000 km/s and magnetic parameters B/sqrt(n) of 10^-4 - 10 muG cm^(3/2) for five different atomic abundance sets, and for a pre-shock density of 1.0 cm^(-3). Additionally, Solar abundance model grids have been calculated for densities of 0.01, 0.1, 10, 100, and 1000 cm^(-3) with the same range in v and B/sqrt(n). Each model includes components of both the radiative shock and its photoionized precursor, ionized by the EUV and soft X-ray radiation generated in the radiative gas. We present the details of the ionization structure, the column densities, and the luminosities of the shock and its precursor. Emission line ratio predictions are separately given for the shock and its precursor as well as for the composite shock+precursor structure to facilitate comparison with observations in cases where the shock and its precursor are not resolved. Emission line ratio grids for shock and shock+precursor are presented on standard line ratio diagnostic diagrams, and we compare these grids to observations of radio galaxies and a sample of AGN and star forming galaxies from the Sloan Digital Sky Survey. This library is available online, along with a suite of tools to enable the analysis of the shocks and the easy creation of emission line ratio diagnostic diagrams. These models represent a significant increase in parameter space coverage over previously available models, and therefore provide a unique tool in the diagnosis of emission by shocks.Comment: 39 pages, 34 figures, accepted for publication in ApJS, April 200

Zone diagrams in Euclidean spaces and in other normed spaces

Zone diagrams are a variation on the classical concept of Voronoi diagrams. Given n sites in a metric space that compete for territory, the zone diagram is an equilibrium state in the competition. Formally it is defined as a fixed point of a certain "dominance” map. Asano, Matoušek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in the Euclidean plane, and Reem and Reich showed existence for two arbitrary sites in an arbitrary metric space. We establish existence and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary (finite) dimension, and more generally, in a finite-dimensional normed space with a smooth and rotund norm. The proof is considerably simpler than that of Asano etal. We also provide an example of non-uniqueness for a norm that is rotund but not smooth. Finally, we prove existence and uniqueness for two point sites in the plane with a smooth (but not necessarily rotund) nor

Blockspin transformation for finite temperature field theories with gauge fields

A procedure is proposed to study QFT at zero or at finite temperature by a sequence of real space renormalization group (RG) or blockspin transformations. They transform to effective theories on coarser and coarser lattices.The ultimate aim is to compute constraint effective potentials, i.e. the free energy as a function of suitable order parameters.From the free energy one can read off the thermodynamic behaviour of the theory, in particular the existence and nature of phase transitions. In a finite temperature QFT one begins with either one or a sequence of transformations which transform the original theory into an effective theory on a three-dimensional lattice. Its effective action has temperature dependent coefficients. Thereafter one may proceed with further blockspin transformations of the three-dimensional theory. Assuming a finite volume, this can in principle be continued until one ends with a lattice with a single site. Its effective action is the constraint effective potential. In each RG-step, an integral over the high frequency part of the field has to be performed. This is done perturbatively and requires the knowledge of bare fluctuation field propagators and of interpolation operators which enter into the vertices. A detailed examination of these quantities is presented for scalar fields, abelian gauge fields and for Higgs fields, finite temperature is admitted.Comment: 88 pages, latex, 17 figures appende

Development of a GPU-accelerated flow simulation method for wind turbine applications

A new and novel GPU accelerated method has been developed for solving the Navier-Stokes equations for bodies of arbitrary geometry in both 2D and 3D. The present method utilises the vortex particles to discretize the governing equations in the Lagrangian frame. Those particles act as vorticity carriers which translate in accordance with the local velocity field. Vorticity information is thus propagated from the vorticity source to the rest of the flow domain in mimicking the advection and diffusion processes of the real flow. In the high-fidelity method, vorticity generation can take place around the bodies. The no-slip condition produces a boundary flux which is subsequently diffused to the neighbouring particles. The new method has been successfully validated by simulating the flow field of an impulsively started cylinder. The calculated drag curve matches well with the theoretical prediction and other numerical results in the literature. To extend the applicability of the code to wind-turbine applications, a simplified re-meshing strategy is adopted which is found to produce small numerical inaccuracies. In the engineering method, a simplified hybrid approach has been developed which decouples the advection and diffusion processes. The viscous effects are ignored on the bodies and are recovered in the wake. For this purpose, the Laplace equation that resulted from the irrotational assumption of the flow has been solved using the boundary element method. The solution produces a dipole distribution that is subsequently converted to viscous particles by employing the Hess’ equivalence principle. In addition, an accurate interpolation scheme has been developed to evaluate the dipole gradient across the distorted wake geometry. To reduce the simulation time, the fast multipole method has been implemented on the GPU in 2D and 3D. To parallelize the implementation, a novel data construction algorithm has been proposed. Furthermore, an analytical expression for the velocity strain has been derived. The new developed methods have been applied to problems involving aerofoils and vertical axis wind turbines. Comparisons with experimental data have shown that the new techniques are accurate and can be used with confidence for a wide variety of wind turbine applications

Interactive buckling in thin-walled rectangular hollow section struts

Thin-walled rectangular hollow section (RHS) struts are widely used in current structural engineering practice due to their mass efficiency and relative ease of manufacture. Owing to their optimized geometric properties, they are vulnerable to local--global interactive buckling and exhibit highly unstable post-buckling behaviour with severe imperfection sensitivity when the local buckling load is close to the global buckling load. The current work investigates the underlying mechanism of local–global interactive buckling of RHS struts using both analytical and finite element (FE) approaches. Variational models formulated using analytical techniques, describing the nonlinear local–global mode interaction in thin-walled RHS struts with varying flange–web joint rigidity, different strut lengths and geometric imperfections under pure compression, are presented. A system of nonlinear differential and integral equations subject to boundary conditions is formulated and solved using numerical continuation techniques. For the first time, the equilibrium behaviour of such struts with different cross-section joint rigidities is highlighted with characteristically unstable interactive buckling paths and a progressive change in the local buckling wavelength. Studies on the effects of strut length identify the boundaries for the four distinct length-dependent zones, where different characteristic post-buckling behaviour are exhibited. The most unstable zone is demonstrated to have a considerably narrower range than previously determined owing to the consideration of more realistic corner boundary conditions within the cross-section. Imperfection sensitivity studies identify the high degree of sensitivity of struts exhibiting mode interaction. They also reveal that local and global imperfections are relatively more significant where global and local buckling are critical respectively. Moreover, a unified local geometric imperfection measurement based on equal local bending energy is proposed to determine the most severe local imperfection profile. It reveals that the most severe profile is modulated rather than periodically distributed along the strut length for purely elastic case. For verification and extensive parametric study purposes, a nonlinear FE model, which considers material nonlinearity, geometric imperfections, and residual stresses, is developed within the commercial package Abaqus. The classical solutions and experimental results from two independent studies are used to verify and validate the FE model, both of which show excellent comparisons. The validated FE model is then used to verify the variational model, which also shows excellent comparisons in local buckling wavelengths, cross-section deformation profile, ultimate load and the mechanical behaviour. Finally, parametric studies on geometric properties, material nonlinearity and residual stresses are conducted using the developed FE model to understand the behaviour of RHS struts exhibiting mode interaction in more practically realistic scenarios. Based on the numerical results and existing experimental results from the literature, the current design rules for thin-walled welded RHS struts are assessed by means of reliability analysis in accordance with Annex D of EN1990. A modified Direct Strength Method (DSM) equation has been proposed and it is shown to provide a better ultimate load prediction than it does presently.Open Acces

Discrete kinetic and stochastic game theory for vehicular traffic: Modeling and mathematical problems

n this thesis we are concerned with the mathematical modeling of vehicular traffic atthe kinetic scale. In more detail, starting from the general structures proposed by Arlottiet al. and by Bellomo, we develop a discrete kinetic framework in which thevelocity of the vehicles is not regarded as a continuous variable but can take a finite number of values only. Discrete kinetic models have historically been conceived in connection with the celebrated Boltzmann equation, primarily as mathematical tools to reduce the analytical complexity of the latter (see e.g., Bellomo and Gatignol, Gatignol): The Boltzmann’s integro-differential equation is converted into a set of partial differential equations in time and space, which share with the former some good mathematical properties being at the same time easier to deal with. In the present context, however, the discretization of the velocity plays a specific role in modeling the system rather than being simply a mathematical simplification, because it allows one to relax the continuum hypothesis for the velocity variable and to include, at least partially, the strongly granular nature of the flow of cars in the kinetic theory of vehicular traffic. The discrete velocity framework also gives rise to an interesting structure of the interaction terms of the kinetic equations, which are inspired by the stochastic game theory