Numerical computation of the cut locus via a variational approximation of the distance function

We propose a new method for the numerical computation of the cut locus of a compact submanifold of R3 without boundary. This method is based on a convex variational problem with conic constraints, with proven convergence. We illustrate the versatility of our approach by the approximation of Voronoi cells on embedded surfaces of R3.

Porous Structure of Cylindrical Particle Compacts

The porous compacts of non-spherical particles are frequently used in energy storage devices and other advanced applications. In the present work, the microstructures of compacts of monodisperse cylindrical particles are investigated. The cylindrical particles with various aspect ratios are generated using superquadrics, and the discrete element method was adopted to simulate the compacts formed under gravity deposition of randomly oriented particles. The Voronoi tessellation is then used to quantify the porous microstructure of compacts. With one exception, the median reduced free volume of Voronoi cells increases, and the median local packing density decreases for compacts composed of cylinders with a high aspect ratio, indicating a loose packing of long cylinders due to their mechanical interlocking during compaction. The obtained data are needed for further optimization of compact porous microstructure to improve the transport properties of compacts of non-spherical particles.

Cascade events in geographical space

In this paper, we present a simple discrete model of cascade behavior in an actual geographical space with built environments. By simultaneously triggering and relaxing random locations in a network of Voronoi cells interacting via the gravity model, we observe nontrivial statistics with heavy-tailed distributions of cells and actual area extents involved in the cascade. The distributions of these affected areas follow unimodal statistics, unlike the other externally-driven models operating over uniform neighborhoods that exhibit power-laws. Majority of the cascades are limited within the immediate neighborhoods of adjacent Voronoi cells, even for sufficiently large triggering magnitudes. The results are viewed from the perspective of inhomogeneous driving in sandpile-based models, and benchmarked with distributions obtained in other geographic datasets. The method offers a complexity perspective into the generation of large-scale events in physical and intangible flows, and explains their origins from cascaded accumulations of slow, random, and intermittent processes.

Predictive control strategy for multi-agent relay tracking systems with time delays

This paper investigates the multi-agent systems for target tracking applications, in which the target is cooperatively relay tracked by multiple agents based on the Voronoi diagram. The relay tracking scheme is effective and reduces the tracking time of a target. However, the existence of time delays results in inaccurate determination of boundaries of Voronoi cells. Thus, the replacement of tracking agent is no longer optimal, which consequently affects the tracking time. In order to solve this issue, a predictive controller is proposed for the relay tracking system with time delays. Moreover, the boundaries of Voronoi cells are determined with the predictive values of agents’ information. In order to analyze the stability of the system, an impulse-time-dependent Lyapunov function is designed. Then, the orthogonal polynomials based inequality is applied and the average dwell time switching technique is adopted to solve the difficulty of stability analysis of the overall relay tracking system caused by the agent replacements of time-delay agents. Finally, the effectiveness and advantages of the proposed predictive control method for relay target tracking system with time delays are illustrated by comparative simulations.

Tetrahedra of varying density and their applications

We propose concepts to utilize basic mathematical principles for computing the exact mass properties of objects with varying densities. For objects given as 3D triangle meshes, the method is analytically accurate and at the same time faster than any established approximation method. Our concept is based on tetrahedra as underlying primitives, which allows for the object’s actual mesh surface to be incorporated in the computation. The density within a tetrahedron is allowed to vary linearly, i.e., arbitrary density fields can be approximated by specifying the density at all vertices of a tetrahedral mesh. Involved integrals are formulated in closed form and can be evaluated by simple, easily parallelized, vector-matrix multiplications. The ability to compute exact masses and centroids for objects of varying density enables novel or more exact solutions to several interesting problems: besides the accurate analysis of objects under given density fields, this includes the synthesis of parameterized density functions for the make-it-stand challenge or manufacturing of objects with controlled rotational inertia. In addition, based on the tetrahedralization of Voronoi cells we introduce a precise method to solve $$L_{2|\infty }$$ L 2 | ∞ Lloyd relaxations by exact integration of the Chebyshev norm. In the context of additive manufacturing research, objects of varying density are a prominent topic. However, current state-of-the-art algorithms are still based on voxelizations, which produce rather crude approximations of masses and mass centers of 3D objects. Many existing frameworks will benefit by replacing approximations with fast and exact calculations. Graphic abstract

Representing hierarchies of visual regard in eye-tracking analysis

In the following study, the author developed a method for representing data from eye-tracking recordings. The study proposed a form of graphical analysis that illustrates hierarchical densities of visual regard without obscuring the original pictorial stimulus. Across three different case studies, subjects’ fixation patterns were used to propagate Voronoi generating points. Integrating both fixation locations and their respective dwell times, randomized Gaussian distribution provided a technique to augment Voronoi generating seeds and enhance graphical resolution. Color pixel values were then used to fill in resultant Voronoi cells, in relation to color values provided by the original stimulus. The study revealed a form of analysis that allowed for effective differentiation of viewing behaviors between different subjects, in which emphasis was placed on a subject's attentional distribution rather than on graphic icons.

A Computational Geometry Generation Method for Creating 3D Printed Composites and Porous Structures

A computational method for generating porous materials and composite structures was developed and implemented. The method is based on using 3D Voronoi cells to partition a defined space into segments. The topology of the segments can be controlled by controlling the Voronoi cell set. The geometries can be realized by additive manufacturing methods, and materials can be assigned to each segment. The geometries are generated and processed virtually. The macroscopic mechanical properties of the resulting structures can be tuned by controlling microstructural features. The method is implemented in generating porous and composite structures using polymer filaments i.e., polylactic acid (PLA), thermoplastic polyurethane (TPU) and nylon. The geometries are realized using commercially available double nozzle fusion deposition modelling (FDM) equipment. The compressive properties of the generated porous and composite configurations are tested quasi statically. The structures are either porous of a single material or composites of two materials that are geometrically intertwined. The method is used to produce and explore promising material combinations that could otherwise be difficult to mix. It is potentially applicable with a variety of additive manufacturing methods, size scales, and materials for a range of potential applications.