High-order divergence-free velocity reconstruction for free surface flows on unstructured Voronoi meshes

In this paper, we present an efficient semi-implicit scheme for the solution of the Reynolds-averaged Navier-Stokes equations for the simulation of hydrostatic and nonhydrostatic free surface flow problems. A staggered unstructured mesh composed by Voronoi polygons is used to pave the horizontal domain, whereas parallel layers are adopted along the vertical direction. Pressure, velocity, and vertical viscosity terms are taken implicitly, whereas the nonlinear convective terms as well as the horizontal viscous terms are discretized explicitly by using a semi-Lagrangian approach, which requires an interpolation of the three-dimensional velocity field to integrate the flow trajectories backward in time. To this purpose, a high-order reconstruction technique is proposed, which is based on a constrained least squares operator that guarantees a globally and pointwise divergence-free velocity field. A comparison with an analogous reconstruction, which is not divergence-free preserving, is also presented to give evidence of the new strategy. This allows the continuity equation to be satisfied up to machine precision even for high-order spatial discretizations. The reconstructed velocity field is then used for evaluating high-order terms of a Taylor method that is here adopted as ODE integrator for the flow trajectories. The proposed semi-implicit scheme is validated against a set of academic test problems, and proof of convergence up to fourth-order of accuracy in space is shown.     

A Free-Lagrange method for unsteady compressible flow: simulation of a confined cylindrical blast wave

A Free-Lagrange numerical procedure for the simulation of two-dimensional inviscid compressible flow is described in detail. The unsteady Euler equations are solved on an unstructured Lagrangian grid based on a density-weighted Voronoi mesh. The flow solver is of the Godunov type, utilising either the HLLE (2 wave) approximate Riemann solver or the more recent HLLC (3 wave) variant, each adapted to the Lagrangian frame. Within each mesh cell, conserved properties are treated as piece-wise linear, and a slope limiter of the MUSCL type is used to give non-oscillatory behaviour with nominal second order accuracy in space. The solver is first order accurate in time. Modifications to the slope limiter to minimise grid and coordinate dependent effects are described. The performances of the HLLE and HLLC solvers are compared for two test problems; a one-dimensional shock tube and a two-dimensional blast wave confined within a rigid cylinder. The blast wave is initiated by impulsive heating of a gas column whose centreline is parallel to, and one half of the cylinder radius from, the axis of the cylinder. For the shock tube problem, both solvers predict shock and expansion waves in good agreement with theory. For the HLLE solver, contact resolution is poor, especially in the blast wave problem. The HLLC solver achieves near-exact contact capture in both problems. Received May 25, 1995 / Accepted September 11, 1995     

On regular 4-coverings and their application for lattice coverings in normed planes

It is well known that the famous covering problem of Hadwiger is completely solved only in the planar case, i.e.: any planar convex body can be covered by four smaller homothetical copies of itself. Lassak derived the smallest possible ratio of four such homothets (having equal size), using the notion of regular 4-covering. We will continue these investigations, mainly (but not only) referring to centrally symmetric convex plates. This allows to interpret and derive our results in terms of Minkowski geometry (i.e., the geometry of finite dimensional real Banach spaces). As a tool we also use the notion of quasi-perfect and perfect parallelograms of normed planes, which do not differ in the Euclidean plane. Further on, we will use Minkowskian bisectors, different orthogonality types, and further notions from the geometry of normed planes, and we will construct lattice coverings of such planes and study related Voronoi regions and gray areas. Discussing relations to the known bundle theorem, we also extend Miquel’s six-circles theorem from the Euclidean plane to all strictly convex normed planes.     

Computing generalized higher-order Voronoi diagrams on triangulated surfaces

We present an algorithm for computing exact shortest paths, and consequently distance functions, from a generalized source (point, segment, polygonal chain or polygonal region) on a possibly non-convex triangulated polyhedral surface. The algorithm is generalized to the case when a set of generalized sites is considered, providing their distance field that implicitly represents the Voronoi diagram of the sites. Next, we present an algorithm to compute a discrete representation of the distance function and the distance field. Then, by using the discrete distance field, we obtain the Voronoi diagram of a set of generalized sites (points, segments, polygonal chains or polygons) and visualize it on the triangulated surface. We also provide algorithms that, by using the discrete distance functions, provide the closest, furthest and k-order Voronoi diagrams and an approximate 1-Center and 1-Median.     

Free and nonfree Voronoi polyhedra

The Voronoi polyhedron of some point v of a translation lattice is the closure of the set of points in space that are closer to v than to any other lattice points. Voronoi polyhedra are a special case of parallelohedra, i.e., polyhedra whose parallel translates can fill the entire space without gaps and common interior points. The Minkowski sum of a parallelohedron with a segment is not always a parallelohedron. A parallelohedron P is said to be free along a vector e if the sum of P with a segment of the line spanned by e is a parallelohedron. We prove a theorem stating that if the Voronoi polyhedron P _v (f) of a quadratic form f is free along some vector, then the Voronoi polyhedron P _v (g) of each form g lying in the closure of the L-domain of f is also free along some vector. For the dual root lattice E ₆*, and the infinite series of lattices D _2m ⁺ , m ≥ 4, we prove that their Voronoi polyhedra are nonfree in all directions.     

Solving the continuous space p-centre problem: planning application issues

^** Email: wei.97{at}osu.edu^*** Email: murray.308{at}osu.edu^**** Email: xiao.37{at}osu.edu The Voronoi diagram heuristic has been proposed for solvingthe p-centre problem in continuous space. However, importantassumptions underlie this heuristic and may be problematic forpractical applications. These simplifying assumptions includeuniformly distributed demand, representing a region as a rectangle;analysis of a simple Voronoi polygon in solving associated one-centreproblems and no restrictions on potential facility locations.In this paper, we explore the complexity of solving the continuousspace p-centre problem in location planning. Considering theissue of solution space feasibility, we present a spatiallyrestricted version of this problem and propose methods for solvingit heuristically. Theoretical and empirical results are provided.     

General framework for testing Poisson‐Voronoi assumption for real microstructures

Modeling microstructures is an interesting problem not just in materials science, but also in mathematics and statistics. The most basic model for steel microstructure is the Poisson‐Voronoi diagram. It has mathematically attractive properties and it has been used in the approximation of single‐phase steel microstructures. The aim of this article is to develop methods that can be used to test whether a real steel microstructure can be approximated by such a model. Therefore, a general framework for testing the Poisson‐Voronoi assumption based on images of two‐dimension sections of real metals is set out. Following two different approaches, according to the use or not of periodic boundary conditions, three different model tests are proposed. The first two are based on the coefficient of variation and the cumulative distribution function of the cells area. The third exploits tools from to topological data analysis, such as persistence landscapes.     

单轴拉伸过程中双相不锈钢局部微观变形行为的细观力学有限元分析

针对双相不锈钢中奥氏体相和铁素体相分别展开了纳米压痕实验,并通过有限元反演得到两相各自的拉伸应力-应变关系,利用Voronoi Tessellation法生成代表性的微结构体积单元,对双相不锈钢的单轴拉伸行为进行了有限元仿真和模拟,研究了双相不锈钢在拉伸过程中的局部应力、应变分布和演化规律．结果表明,利用Voronoi Tessellation法建立单元模型,结合本文通过纳米压痕实验获取的两相力学性能参数,可以很好地模拟双相不锈钢的整体单拉行为,奥氏体比铁素体软,拉伸载荷下双相不锈钢的应变集中在奥氏体中,应力集中在铁素体中;局部应力应变的分布特征与两相分布特征和晶粒形状有关,最大应变值主要集中在奥氏体晶粒狭长且尖锐的区域,而最大应力则主要发生在铁素体晶粒狭长和尖锐的区域;对于奥氏体和铁素体晶粒占比相当的双相不锈钢,其虽然可以具有较为综合的宏观力学性能,但是其微观应力集中的区域和应力最大值相对较大．研究成果为进一步揭示双相不锈钢局部失效机理奠定了基础．     

A heuristic algorithm for minimax sensor location in the plane

This paper addresses the problem of locating a finite number of sensors to detect an event in a given planar region. The objective is to minimize the maximum probability of non-detection where the underlying region consists of a convex polygon. The sensor location problem has a multitude of applications, including the location of aircraft detection sensors, the placement of sentries along a border to detect enemy penetration, the detection of nuclear tests, and the detection of natural and hazardous man-made events. The problem is a difficult nonlinear nonconvex programming problem even in the case of two sensors. A fast heuristic based on Voronoi polygons is developed in this paper. The algorithm can quickly generate high-quality solutions. Computational experience is provided.     

Statistical model of the habit and arrangement of mineral crystals in the collagen of bone

Randomly colored space tesselations are considered as models for the mineral/organic structure of bone. First, it is shown that the structure function for such models is always proportional to the average form factor of the individual tiles and hence independent of the mineral density in the sample. Then the structure function is calculated for three such models: for model I, based on a hexagonal, and model 2, on a Poisson-Voronoi tesselation of the plane and for model 3, based on a random tesselation of the line. These results are compared to experimental structure functions measured by small-angle scattering and excellent agreement is obtained between model 2 and the bone from mice and rats, as well as between model 3 and calcified turkey leg tendon. Divergent conclusions following recent experiments by small-angle x-ray scattering and by electron microscopy are discussed in the light of these structural models and an explanation is proposed which might remove the discrepancy.Dedicated to Prof. Oliver Penrose on the occasion of his 65th birthday.