Fifth-generation (5G) networks providing much higher bandwidth and faster data rates will allow connecting vast number of stationary and mobile devices, sensors, agents, users, machines, and ...vehicles, supporting Internet-of-Things (IoT), real-time dynamic networks of mobile things. Positioning and location awareness will become increasingly important, enabling deployment of new services and contributing to significantly improving the overall performance of the 5G system. Many of the currently talked about solutions to positioning in 5G are centralized, mostly requiring direct communication to the access nodes (or anchors, i.e., nodes with known locations), which in turn requires a high density of anchors. But such centralized positioning solutions may become unwieldy as the number of users and devices continues to grow without limit in sight. As an alternative to the centralized solutions, this paper discusses distributed localization in a 5G-enabled IoT environment where many low power devices, users, or agents are to locate themselves without a direct access to anchors. Even though positioning is essentially a nonlinear problem (solving circle equations by trilateration or triangulation), we discuss a cooperative linear distributed iterative solution with only local measurements, local communication, and local computation needed at each agent. Linearity is obtained by reparametrization of the agent location through barycentric coordinate representations based on local neighborhood geometry that may be computed in terms of certain Cayley-Menger determinants involving relative local inter-agent distance measurements. After a brief introduction to the localization problem, and other available distributed solutions primarily based on directly addressing the nonlinear formulation, we present the distributed linear solution for stationary agent networks and study its convergence, its robustness to noise, and extensions to mobile scenarios, in which agents, users, and (possibly) anchors are dynamic.
Flame particles (FP) are massless, virtual particles which follow material points on the flame surface. This work presents a tracking algorithm for FPs which utilizes barycentric coordinates. The ...methodology can be used with any cell shape in the computational mesh and allows computationally fast spatial interpolation as well as efficient determination of the intersection of FP trajectories with iso-surfaces. In contrast to previous flame particle tracking (FPT) approaches, the code is fully parallelized and can therefore be used in-situ during the simulation. It also includes fully parallelized computation of flame consumption speed by integrating reaction rates along a line normal to the flame surface at each FP position. Direct numerical simulations of laminar pulsating premixed hydrogen–air Bunsen flames serve as validation cases and showcase the added value of tracking material points for studying local flame dynamics. Exciting the inlet flow harmonically with frequencies equal to the inverse flame time scale leads to a pulsating mode where the flame front is corrugated. Ten times higher frequencies nearly resemble the steady state solution. The FPs are seeded along the flame surface and are used to track the unsteady diffusive, convective and chemical contributions at arbitrary points on the flame front over time. Their trajectories reveal a phase shift between the unsteady flame stretch rate and local flame speed of the order of 0.1 flame time scales for rich hydrogen flames. This is caused by a time delay between straining and stretch due to curvature. The reason is that diffusive processes follow the time signal of curvature while chemical processes are most strongly affected by the straining rate, which dominates the high Lewis number hydrogen flames investigated. This time history effect may help to explain the large scattering in the correlation of local flame speed with flame stretch found in turbulent flames.
•Non-linear objective functions are considered.•Intuitionistic fuzzy TOPSIS approach is proposed.•Linear membership and non-membership functions are used.•Defuzzification based upon barycentric ...coordinates of a triangle is proposed.•Two different types of problems are solved with the proposed methodology.
The given study intends to propose a new approach, named intuitionistic fuzzy TOPSIS, to solve a non-linear multi-objective intuitionistic fuzzy problem, whose application is not confined to only one type but has a wider scope. Considering the distance of positive and negative ideal solutions from the optimal values directly in terms of triangular intuitionistic fuzzy numbers makes the study unique. Along with this, a new technique, based upon barycentric coordinates of a triangle, to defuzzify triangular intuitionistic fuzzy numbers has been proposed. Two different types of substantive illustrations viz., non-linear multi-objective intuitionistic fuzzy transportation problem and non-linear multi-objective intuitionistic fuzzy manufacturing problem, are solved and compared with the paradigmatic approaches existing in the literature.
In this paper we propose a simple procedure for numerically computing the Lagrange interpolation polynomial on a unisolvent set of points in the plane. We suggest the use of the canonical polynomial ...basis centered at the barycenter of the set of points and the PA=LU decomposition for solving the associated Vandermonde system to compute the coefficients of the Taylor polynomial. We show that the 1-norm condition number of the Vandermonde matrix is an upper bound for the Lebesgue constant of the interpolation node set in the unit disk. Therefore, the analysis of the condition number can be useful to select the unisolvent set of nodes in a set of scattered nodes. Numerical experiments show the efficiency and accuracy of the proposed method.
•Use polynomials of Wachspress GBCs to construct smooth vertex splines over convex quadrilaterals.•The derivation of the construction is aided by MATHEMATICA and MATLAB to ensure the ...correctness.•These smooth vertex splines are locally supported which can be used to adjust and mend a surface.•Quasi-interpolatory operators are constructed and their approximation properties are shown.•These vertex splines are used to construct smooth surfaces such as suitcase corners.
In this paper we construct smooth bivariate spline functions over a polygonal partition, e.g. a convex quadrilateral partition by using vertex spline techniques. Vertex splines, introduced in Chui and Lai (1985) are smooth piecewise polynomial functions supported over a collection of triangles sharing a vertex. In this paper, we extend the concept of vertex splines to the partition of polygons and describe how to construct C1 vertex polygonal splines over a collection of quadrilaterals. We begin with our construction of C1 vertex splines over a collection of parallelograms, although they may not be axis-orientated. Then the construction is generalized to the setting of general convex quadrilaterals. We will use various monomials of Wachspress GBC functions of degrees 5 and 7 to explain how to construct C1 vertex splines together with additional special splines called edge and face splines. With these splines at hand, we construct quasi-interpolatory formulas, whose approximation properties will be shown. Numerical interpolation and approximation results will be presented. Finally, three applications of these splines are explained: the first one is to form smooth locally supported GBC functions, the second one is to construct smooth suitcase corners, and the third one to construct C1 surfaces over quadrilateral partitions with extraordinary points (EP). Several examples will be demonstrated to show the convenience of using these splines.
In recent years, there has been growing interest in the representation of volumes within the field of geometric modeling (GM). While polygonal patches for surface modeling have been extensively ...studied, there has been little focus on the representation of polyhedral volumes. Inspired by the polygonal representation of the Generalized Bézier (GB) patch proposed by Várady et al. (2016), this paper introduces a novel method for polyhedral volumetric modeling called the Generalized Bézier (GB) volume.
GB volumes are defined over simple convex polyhedra using generalized barycentric coordinates (GBCs), with the control nets which are a direct generalization of those of tensor-product Bézier volumes. GB volumes can be smoothly connected to adjacent tensor-product Bézier or GB volumes with G1 or G2 continuity. Besides, when the parametric polyhedron becomes a prism, the GB volume also degenerates into a tensor-product form. We provide some practical examples to demonstrate the advantages of GB volumes. Suggestions for future work are also discussed.
•We propose a novel polyhedral volumetric representation method, called Generalized Bézier (GB) volume.•A GB volume behaves as a tensor-product Bézier volume along each quadrilateral boundary surface and a tensor-product GB volume along each multi-sided boundary surface.•A GB volume can be easily connected to adjacent tensor-product Bézier or (tensor-product) GB volume with high order geometric continuity (G1 or G2).•A GB volume will reproduce a tensor-product GB volume when the domain polyhedron becomes a prism.•We also present a polyhedral Bézier extraction algorithm that enables the construction of a globally smooth volume from any input hexmesh with arbitrary irregularities or from input simple-convex-polyhedral meshes.
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