Unbounded Direction Research Articles

Unbounded Regions of High-Order Voronoi Diagrams of Lines and Line Segments in Higher Dimensions

We study the behavior at infinity of the farthest and the higher-order Voronoi diagram of n line segments or lines in a d-dimensional Euclidean space. The unbounded parts of these diagrams can be encoded by a Gaussian map on the sphere of directions Sd-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {S}^{d-1}$$\\end{document}. We show that the combinatorial complexity of the Gaussian map for the order-k Voronoi diagram of n line segments and lines is O(min{k,n-k}nd-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\min \\{k,n-k\\}n^{d-1})$$\\end{document}, which is tight for n-k=O(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-k=O(1)$$\\end{document}. This exactly reflects the combinatorial complexity of the unbounded features of these diagrams. All the d-dimensional cells of the farthest Voronoi diagram are unbounded, its (d-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(d-1)$$\\end{document}-skeleton is connected, and it does not have tunnels. A d-cell of the Voronoi diagram is called a tunnel if the set of its unbounded directions, represented as points on its Gaussian map, is not connected. In a three-dimensional space, the farthest Voronoi diagram of n≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 2$$\\end{document} lines in general position has exactly n(n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n(n-1)$$\\end{document} three-dimensional cells. The Gaussian map of the farthest Voronoi diagram of line segments and lines can be constructed in O(nd-1α(n))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(n^{d-1} \\alpha (n))$$\\end{document} time, for d≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 4$$\\end{document}, while if d=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=3$$\\end{document}, the time drops to worst-case optimal Θ(n2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta (n^2)$$\\end{document}. We extend the obtained results to bounded polyhedra and clusters of points as sites.

Receptance of a semi-infinite periodic railway track and an equivalent multi-rigid body system for use in truncated track models

In many problems involving railway track dynamics, the infinitely long railway track has to be truncated, generating a finite track model. To account for high-frequency vibration, high train speed and the presence of multiple carriages, the track model may need to be very long and demand too great computational effort. Alternative modelling strategies are required so that the length of the track model is limited while it is still sufficiently able to exhibit dynamical characteristics of the infinite track. This may be achieved by dividing the track into three parts, two semi-infinite parts (two semi-infinite tracks) on either side of a central finite track model, treating the semi-infinite parts to be periodic in the unbounded direction and then replacing them with an equivalent multi-rigid body system. The multi-rigid body system may be determined based on the requirement that it provides the same receptance to the track model as the semi-infinite track. This paper is concerned with the calculation of the receptance and the determination of the equivalent system for vertical vibration of the track. The receptance is calculated based on the two semi-infinite tracks joined together as an infinitely long periodic structure subject to specific harmonic loads, in combination with the relationship between the internal forces and displacements of a Timoshenko beam. To estimate the parameters of the equivalent system, the structural layout of the system must be created first. The structural layout is created based on the characteristic frequencies and ‘modes’ of the semi-infinite track. The parameters of the equivalent system are estimated by letting the modal frequencies of the equivalent system be the same as the characteristic frequencies of the semi-infinite track and by minimising the relative difference in receptance between the multi-rigid body system and the semi-infinite track at a set of pre-defined frequencies, including the characteristic frequencies of the semi-infinite track. The above procedure is demonstrated for a typical high-speed slab track and a typical ballasted track.

Inhomogeneous Helmholtz equations in wave guides – existence and uniqueness results with energy methods

The Helmholtz equation $-\nabla\cdot (a\nabla u) - \omega^2 u = f$ is considered in an unbounded wave guide $\Omega := \mathbb{R} \times S \subset \mathbb{R}^d$ , $S\subset \mathbb{R}^{d-1}$ a bounded domain. The coefficient a is strictly elliptic and either periodic in the unbounded direction $x_1 \in \mathbb{R}$ or periodic outside a compact subset; in the latter case, two different periodic media can be used in the two unbounded directions. For non-singular frequencies $\omega$ , we show the existence of a solution u. While previous proofs of such results were based on analyticity arguments within operator theory, here, only energy methods are used.

Directional ellipticity on special domains: Weak Maximum and Phragmén–Lindelöf principles

We prove the validity of maximum principles for a class of fully nonlinear operators on unbounded subdomains Ω⊂Rn of cylindrical type. The main structural assumption is the uniform ellipticity of the operator along the bounded directions of Ω, with possible degeneracy along the unbounded directions.

On well-posedness of time-harmonic problems in an unbounded strip for a thin plate model

We study the propagation of elastic waves in the time-harmonic regime in a waveguide which is unbounded in one direction and bounded in the two other (transverse) directions. We assume that the waveguide is thin in one of these transverse directions, which leads us to consider a Kirchhoff-Love plate model in a locally perturbed 2D strip. For time harmonic scattering problems in unbounded domains, well-posedness does not hold in a classical setting and it is necessary to prescribe the behaviour of the solution at infinity. This is challenging for the model that we consider and constitutes our main contribution. Two types of boundary conditions are considered: either the strip is simply supported or the strip is clamped. The two boundary conditions are treated with two different methods. For the simply supported problem, the analysis is based on a result of Hilbert basis in the transverse section. For the clamped problem, this property does not hold. Instead we adopt the Kondratiev's approach, based on the use of the Fourier transform in the unbounded direction, together with techniques of weighted Sobolev spaces with detached asymptotics. After introducing radiation conditions, the corresponding scattering problems are shown to be well-posed in the Fredholm sense. We also show that the solutions are the physical (outgoing) solutions in the sense of the limiting absorption principle.

Local energy decay and diffusive phenomenon in a dissipative wave guide

We prove the local energy decay for the wave equation in a wave guide with dissipation at the boundary. It appears that for large times the dissipated wave behaves like a solution of a heat equation in the unbounded directions. The proof is based on resolvent estimates, in particular at the low and high frequency limits. Since the eigenvectors for the transverse operator do not form a Riesz basis, the spectral analysis does not reduce as usual to separate subquestions on compact and Euclidean domains. One of the difficulties is then to localize the problem with respect to the non-discrete spectrum of a non-selfadjoint operator.

A regularization method for solving the Poisson equation for mixed unbounded-periodic domains

Regularized Green's functions for mixed unbounded-periodic domains are derived. The regularization of the Green's function removes its singularity by introducing a regularization radius which is related to the discretization length and hence imposes a minimum resolved scale. In this way the regularized unbounded-periodic Green's functions can be implemented in an FFT-based Poisson solver to obtain a convergence rate corresponding to the regularization order of the Green's function. The high order is achieved without any additional computational cost from the conventional FFT-based Poisson solver and enables the calculation of the derivative of the solution to the same high order by direct spectral differentiation. We illustrate an application of the FFT-based Poisson solver by using it with a vortex particle mesh method for the approximation of incompressible flow for a problem with a single periodic and two unbounded directions.

Mixed boundary value problems on cylindrical domains

We study second-order divergence-form systems on half-infinite cylindrical domains with a bounded and possibly rough base, subject to homogeneous mixed boundary conditions on the lateral boundary and square integrable Dirichlet, Neumann, or regularity data on the cylinder base. Assuming that the coefficients $A$ are close to coefficients $A_0$ that are independent of the unbounded direction with respect to the modified Carleson norm of Dahlberg, we prove a priori estimates and establish well-posedness if $A_0$ has a special structure. We obtain a complete characterization of weak solutions whose gradient either has an $L^2$-bounded non-tangential maximal function or satisfies a Lusin area bound. To this end, we combine the first-order approach to elliptic systems with the Kato square root estimate for operators with mixed boundary conditions.

Electron energy probability function and L-p similarity in low pressure inductively coupled bounded plasma

Particle-In-Cell (PIC) simulations are carried out to investigate the effect of discharge length (L) and pressure (p) on Electron Energy Probability Function (EEPF) in a low pressure radio frequency (rf) inductively coupled plasma at 13.56 MHz. It is found that for both cases of varying L (0.1 - 0.5 m) and p (1 - 10 mTorr), the EEPF is a bi-Maxwellian with a step in the bounded direction (x) and in the symmetric unbounded directions (y, z) the EEPF are a Maxwellian with a hot tail. The plasma space potential decreases with increase in both L and p, the trapped electrons having energies in the range 0 to 20 eV. In a conventional discharge bounded in all directions, we infer that L and p are similarity parameters for low energy electrons trapped in the bulk plasma that have energies below the plasma space potential (eVp). The simulation results are consistent with a particle balance model.

Vortex Particle-Mesh with Immersed Lifting Lines for Aerospace and Wind Engineering

We present the treatment of lifting lines with a Vortex Particle-Mesh (VPM) methodology. The VPM method relies on the Lagrangian discretization of the Navier-Stokes equations in vorticity-velocity formulation. The use of this hybrid discretization offers several advantages. The particles are used solely for the advection, thereby waiving classical time stability constraints. They also exploit the compactness of vorticity support, leading to high computational gains for external flow simulations. The mesh, on the other hand, handles all the other computationally intensive tasks, such as the evaluation of the differential operators and the use of fast Fourier-based Poisson solvers, which allow the combination of unbounded directions and inlet/outlet boundaries. Both discretizations communicate through high order interpolation. The mesh and the interpolation also allow for additional advances; they are used to handle Lagrangian distortion by reinitializing the particle positions onto a regular grid. This crucial step, referred to as remeshing, guarantees the accuracy of the method. In addition, the resulting methodology provides computational efficiency and scalability to massively parallel architectures.Sources of vorticity are accounted for through a lifting line approach. This line handles the attached and shed vorticity contributions in a Lagrangian manner. Its immersed treatment efficiently captures the development of vorticity from thin sheets into a three-dimensional field. We apply this approach to the simulation of wake flows encountered in aeronautical and wind energy applications. An important aspect in these fields is the handling of turbulent inflows. We have developed a technique for the introduction of pre-computed or synthetic turbulent flow fields in vorticity form. Our treatment is based on particles as well and consistent with the Lagrangian character of the method. We apply here our method to the investigation of wind turbine wakes over very large distances, reaching cluster or wind farm sizes.

COMMUNITY COMPOSITION AFFECTS THE SHAPE OF MATE RESPONSE FUNCTIONS

The evolution of mate preferences can be critical for the evolution of reproductive isolation and speciation. Heterospecific interference may carry substantial fitness costs and result in preferences where females are most responsive to the mean conspecific trait with low response to traits that differ from this value. However, when male traits are unbounded by heterospecifics, there may not be selection against females that respond to extreme trait values in the unbounded direction. To test how heterospecifics affected the shape of female response functions, I presented female Oecanthus tree crickets with synthetic calls representing a range of male calls, then measured female phonotaxis to construct response functions. The species with the fastest pulse rates in the community consistently responded to pulse rates faster than those produced by their males, whereas in the intermediate and slowest pulse rate species there was no significant difference between the male trait and the female response. This work suggests that species with the most extreme signal in the community respond to a greater range of signals, potentially resulting in a higher probability of hybridization during secondary contact, and revealing interactions between mate recognition and other aspects of sexual selection.

Large Eddy Simulation of Wind Turbine Wakes

We present the coupling of a vortex particle-mesh method with immersed lifting lines for the Large Eddy Simulation of wind turbine wakes. The method relies on the Lagrangian discretization of the Navier–Stokes equations in vorticity-velocity formulation. Advection is handled by the particles while the mesh allows the evaluation of the differential operators and the use of fast Poisson solvers. We use a Fourier-based fast Poisson solver which simultaneously allows unbounded directions and inlet/outlet boundaries. The method also allows the feeding of a turbulent incoming flow. We apply this methodology to the study of large scale aerodynamics and wake behavior of tandem wind turbines. We analyze the generators performance, unsteady power, loads and aerodynamics they are subjected to. The average flow field of the wakes is also computed and turbulence statistics are extracted. In particular, we investigate the influence of the type of turbulent inflow used—precomputed or synthetic—, and study wake meandering.

ANALYSIS OF IMPOSSIBILITY OF TESTING SPATIAL NONCOMMUTATIVITY USING RYDBERG ATOMS IN THE VICINITY OF SADDLE-POINT IN CROSSED ELECTRIC AND MAGNETIC FIELDS

In relation to the current controversy over the need for an experimental test of spatial noncommutativity at the normal quantum scale, the spectra of a Rydberg electron in the vicinity of the saddle-point in a suitable combination of crossed electric and magnetic fields are investigated in both cases of commutative space and noncommutative space. Its local behavior around the saddle-point is clarified. An unbounded mode is revealed. It describes the unbounded direction along which the electron will escape the vicinity of the saddle-point, which means that it is impossible to locate electronic states. This finding indicates that the possibility of testing spatial noncommutativity in the vicinity of the saddle-point is ruled out. Because this unbounded mode applies only to states near the saddle-point, not to those from the outer well, it is suggested that if any test of spatial noncommutativity can be performed by using atoms in the crossed-field, then the test must be performed using states from the outer well, not those close to the saddle-point.

A Fourier-based elliptic solver for vortical flows with periodic and unbounded directions

We present a computationally efficient, adaptive solver for the solution of the Poisson and Helmholtz equation used in flow simulations in domains with combinations of unbounded and periodic directions. The method relies on using FFTs on an extended domain and it is based on the method proposed by Hockney and Eastwood for plasma simulations. The method is well-suited to problems with dynamically growing domains and in particular flow simulations using vortex particle methods. The efficiency of the method is demonstrated in simulations of trailing vortices.

Maximum disorientation angles between crystals of any point groups and their corresponding rotation axes

The symmetry-reduced misorientation,i.e.disorientation, between two crystals is represented in the angle–axis format, and the maximum disorientation angle between any two lattices of the 32 point groups is obtained by constructing the fundamental zone of the associated misorientation space (i.e.Rodrigues–Frank space) using quaternion algebra. A computer program based on vertex enumeration was designed to automatically calculate the vertices of these fundamental zones and to seek the maximum disorientation angles and respective rotation axes. Of the C_{32}^2 = 528 possible combinations of any two crystals, 129 pairs give rise to incompletely bounded fundamental zones (i.e.zones having at least one unbounded direction inR3); these correspond to a maximum disorientation angle of 180° (the trivial value). The other 399 pairs produce fully bounded fundamental zones that lead to nine different nontrivial maximum disorientation angles; these are 56.60, 61.86, 62.80, 90, 90.98, 93.84, 98.42, 104.48 and 120°. The associated rotation axes were obtained and are plotted in stereographic projection. These angles and axes are solely determined by the symmetries of the point groups under consideration, and the only input data needed are the symmetry operators of the lattices.

Representation of orientation relationships in Rodrigues–Frank space for any two classes of lattice

The fundamental zones of Rodrigues–Frank (R-F) space applicable to misorientations between crystals of any two Laue groups are constructed by using a unified formulation in terms of quaternion algebra. Some of these regions are fully bounded by planes that are determined solely by the symmetries of the groups, while others have at least one unbounded direction. Each of the bounded fundamental zones falls into one of nine geometrically distinct configurations. The maximum symmetry-reduced angles and the corresponding Rodrigues–Frank vectors for these fundamental zones are evaluated. The use of Rodrigues–Frank space for the representation of orientation relationships between crystals of any two symmetry groups is also addressed. Examples concerning the transition of phases of the same symmetry group,i.e.from face-centered cubic to body-centered cubic, and of different groups,i.e.from body-centered cubic to hexagonal close-packed, are given to illustrate the usefulness of this space for representing orientation relationships during phase transformation or precipitation.

Exponential averaging for traveling wave solutions in rapidly varying periodic media

AbstractReaction diffusion systems on cylindrical domains with terms that vary rapidly and periodically in the unbounded direction can be analyzed by averaging techniques. Here, using iterated normal form transformations and Gevrey regularity of bounded solutions, we prove a result on exponential averaging for such systems, i.e., we show that traveling wave solutions can be described by a spatially homogenous equation and exponentially small remainders. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Localized Thickening of a Compressed Elastic Band

An infinite elastic band is compressed along its unbounded direction, giving rise to a continuous family of homogeneous configurations that is parameterized by the compression rate β < 1 (β = 1 when there is no compression). It is assumed that, for some critical value β0, the compression force as a function of β has a strict local extremum and that the linearized equation around the corresponding homogeneous configuration is strongly elliptic. Under these conditions, there are nearby localized deformations that are asymptotically homogeneous. When the compression force reaches a strict local maximum at β0, they describe localized thickening and they occur for values of β slightly smaller than β0. Since the material is supposed to be hyperelastic, homogeneous and isotropic, the localized deformations are not due to localized imperfections. The method follows the one developed by A. Mielke for an elastic band under traction: interpretation of the nonlinear elliptic system as an infinite dimensional dynamical system in which the unbounded direction plays the role of time, its reduction to a center manifold and the existence of a homoclinic solution to the reduced finite dimensional problem in [A. Mielke, Hamiltonian and Lagrangian fiows on center manifolds, Lecture Notes in Mathematics 1489. Springer, Berlin Heidelberg New York, 1991]. The main difference lies in the fact that Agmon's condition does not hold anymore and therefore the linearized problem cannot be analyzed as in Mielke's work.

Propagating modes in optical waveguides terminated by perfectly matched layers

For numerical simulations of optical waveguides, perfectly matched layers (PMLs) are widely used to truncate the unbounded transverse directions. However, a PML induces some undesired side effects to the propagating modes in the waveguide. In general, the propagation constant becomes complex, so that the mode may grow or decay exponentially along the waveguide axis. Through a perturbation analysis, we obtain approximate formulas for the propagation constants of PML-terminated planar waveguides.

Dynamical Properties of Spatially Non-Decaying 2D Navier?Stokes Flows with Kolmogorov Forcing in an Infinite Strip

The main result of this paper is the global well-posedness of the Cauchy problem to the 2D Navier–Stokes system with the initial data $u_0 \in {\text{BUC}}(\mathbb{O})$ and the external force $F \in C([0,\infty ),L^\infty (\mathbb{O}))$ on the manifold $\mathbb{O} = \mathbb{R} \times \mathbb{S}^1 ,$ i.e., the fluid flow is supposed to be periodic in one of the spatial directions whereas in the unbounded direction only uniform boundedness is assumed. However, to obtain uniqueness we need to make assumptions which suppresses additional pressure gradients. For this aim Riesz operators on ${\text{L}}^\infty (\mathbb{O})$ are used to define $p(t) \in {\text{BMO}}(\mathbb{O}).$ For time-independent forces the solutions are shown to grow at most cubically in the time t.