Planar Riemann surfaces with uniformly distributed cusps: parabolicity and hyperbolicity

We consider a planar Riemann surface R made of a non‐compact simply connected plane domain from which an infinite discrete set of points is removed. We give several conditions for the collars of the cusps in R caused by these points to be uniformly distributed in R in terms of Euclidean geometry. Then we associate a graph G with R by taking the Voronoi diagram for the uniformly distributed cusps and show that G represents certain geometric and analytic properties of R .     

Jamming II: Edwards’ statistical mechanics of random packings of hard spheres

The problem of finding the most efficient way to pack spheres has an illustrious history, dating back to the crystalline arrays conjectured by Kepler and the random geometries explored by Bernal in the 1960s. This problem finds applications spanning from the mathematician’s pencil, the processing of granular materials, the jamming and glass transitions, all the way to fruit packing in every grocery. There are presently numerous experiments showing that the loosest way to pack spheres gives a density of ∼55% (named random loose packing, RLP) while filling all the loose voids results in a maximum density of ∼63%-64% (named random close packing, RCP). While those values seem robustly true, to this date there is no well-accepted physical explanation or theoretical prediction for them. Here we develop a common framework for understanding the random packings of monodisperse hard spheres whose limits can be interpreted as the experimentally observed RLP and RCP. The reason for these limits arises from a statistical picture of jammed states in which the RCP can be interpreted as the ground state of the ensemble of jammed matter with zero compactivity, while the RLP arises in the infinite compactivity limit. We combine an extended statistical mechanics approach ‘a la Edwards’ (where the role traditionally played by the energy and temperature in thermal systems is substituted by the volume and compactivity) with a constraint on mechanical stability imposed by the isostatic condition. We show how such approaches can bring results that can be compared to experiments and allow for an exploitation of the statistical mechanics framework. The key result is the use of a relation between the local Voronoi volumes of the constituent grains (denoted the volume function) and the number of neighbors in contact that permits us to simply combine the two approaches to develop a theory of volume fluctuations in jammed matter. Ultimately, our results lead to a phase diagram that provides a unifying view of the disordered hard sphere packing problem and further sheds light on a diverse spectrum of data, including the RLP state. Theoretical results are well reproduced by numerical simulations that confirm the essential role played by friction in determining both the RLP and RCP limits. The RLP values depend on friction, explaining why varied experimental results can be obtained.     

Truncated Newton-Based Multigrid Algorithm for Centroidal Voronoi Diagram Calculation

In a variety of modern applications there arises a need to tessellate the domain into representative regions, called Voronoi cells. A particular type of such tessellations, called centroidal Voronoi tessellations or CVTs, are in big demand due to their optimality properties important for many applications. The availability of fast and reliable algorithms for their construction is crucial for their successful use in practical settings. This paper introduces a new multigrid algorithm for constructing CVTs that is based on the MG/Opt algorithm that was originally designed to solve large nonlinear optimization problems. Uniform convergence of the new method and its speedup comparing to existing techniques are demonstrated for linear and nonlinear densities for several 1d and 2d problems, and $O(k)$ complexity estimation is provided for a problem with $k$ generators.     

PMMA/PS共混物薄膜中Voronoi结构的形成

Vomnoi结构与一些自然结构十分相象,早在17世纪,Descartes就在他的著作中采用了类似Vomnoi结构的图形来表示太阳系及其环境中的物质分布．Vomnoi图（如图1所示）的概念来自于计算几何,设有平面空间点集合A={a1,a2,……an},其中ai=（xi,yi）,对于任意一个ai,满足｜P-ai｜≤min ｜P-aj｜（j=1,2,……,n;j≠i）的点P=（x,y）的轨迹称为点集A的vomnoi图．     

明渠湍流边界层中颗粒的运动与分布

利用直接数值模拟、点球浸入边界法和颗粒离散元法相结合的方法, 模拟了颗粒在明渠湍流边界层中的运动, 并对颗粒的瞬时位置进行了Voronoi 分析, 定量研究了颗粒在湍流边界层中的运动和分布规律. 研究发现:颗粒的输运对湍流的统计特征有影响, 其运动与近壁区湍流拟序结构密切相关, 在"喷发"结构作用下被带离壁面, 在"扫掠" 结构和自身重力作用下回到壁面; 在湍流边界层中, 颗粒倾向于聚集在低流速带, 呈条带状分布;颗粒在大部分时间处于"簇"状态, 偶尔跳跃到"空" 状态, 但能够很快回到邻近低速区域.      

ReALE: A reconnection-based arbitrary-Lagrangian–Eulerian method

We present a new reconnection-based arbitrary-Lagrangian–Eulerian (ALE) method. The main elements in a standard ALE simulation are an explicit Lagrangian phase in which the solution and grid are updated, a rezoning phase in which a new grid is defined, and a remapping phase in which the Lagrangian solution is transferred (conservatively interpolated) onto the new grid. In standard ALE methods the new mesh from the rezone phase is obtained by moving grid nodes without changing connectivity of the mesh. Such rezone strategy has its limitation due to the fixed topology of the mesh. In our new method we allow connectivity of the mesh to change in rezone phase, which leads to general polygonal mesh and allows to follow Lagrangian features of the mesh much better than for standard ALE methods. Rezone strategy with reconnection is based on using Voronoi tessellation. We demonstrate performance of our new method on series of numerical examples and show it superiority in comparison with standard ALE methods without reconnection.     

Incipient plasticity and voids nucleation of nanocrystalline gold nanofilms using molecular dynamics simulation

In this study, the incipient plasticity and voids nucleation of nanocrystalline gold were investigated using a molecular dynamics simulation. The effects of mean grain size and temperature were evaluated in terms of the material's stress-strain diagram, Young's modulus, yield strength, common-neighbor analysis, slip vectors, and deformation behaviors. From the stress-strain diagram, at 300?K, the maximum stress value corresponding to a grain size of 3.2?nm was much lower and the stress curve was clearly different from those corresponding to other grain sizes. Young's modulus increased with increasing mean grain size. The inverse Hall–Petch relation was observed. The slip was the main deformation behavior at a mean grain size of 3.2?nm. Moreover, the internal stress was more pronounced with increasing temperature. At 700?K, the main deformation area range was concentrated in the lattice at the middle of the samples, resulting in an almost force–induced structural transformation phenomenon in the middle. Void damage occurred at the junction of three–grain boundaries during the tensile process. With decreasing mean grain size, the less internal differential slip was generated under the same temperature and strain conditions.