Polygonal finite elements for incompressible fluid flow

We discuss the use of polygonal finite elements for analysis of incompressible flow problems. It is well‐known that the stability of mixed finite element discretizations is governed by the so‐called inf‐sup condition, which, in this case, depends on the choice of the discrete velocity and pressure spaces. We present a low‐order choice of these spaces defined over convex polygonal partitions of the domain that satisfies the inf‐sup condition and, as such, does not admit spurious pressure modes or exhibit locking. Within each element, the pressure field is constant while the velocity is represented by the usual isoparametric transformation of a linearly‐complete basis. Thus, from a practical point of view, the implementation of the method is classical and does not require any special treatment. We present numerical results for both incompressible Stokes and stationary Navier–Stokes problems to verify the theoretical results regarding stability and convergence of the method. Copyright © 2013 John Wiley & Sons, Ltd.     

Bernoulli Cluster Field: Voronoi Tessellations

A new point process is proposed which can be viewed either as a Boolean cluster model with two cluster modes or as a p-thinned Neyman-Scott cluster process with the retention of the original parent point. Voronoi tessellation generated by such a point process has extremely high coefficients of variation of cell volumes as well as of profile areas and lengths in the planar and line induced tessellations. An approximate numerical model of tessellation characteristics is developed for the case of small cluster size; its predictions are compared with the results of computer simulations. Tessellations of this type can be used as models of grain structures in steels.     

Hot ice computer

We experimentally demonstrate that supersaturated solution of sodium acetate, commonly called ‘hot ice’, is a massively-parallel unconventional computer. In the hot ice computer data are represented by a spatial configuration of crystallization induction sites and physical obstacles immersed in the experimental container. Computation is implemented by propagation and interaction of growing crystals initiated at the data-sites. We discuss experimental prototypes of hot ice processors which compute planar Voronoi diagram, shortest collision-free paths and implement and and or logical gates.     

A swept‐intersection‐based remapping method in a ReALE framework

A complete reconnection‐based arbitrary Lagrangian–Eulerian (ReALE) strategy devoted to the computation of hydrodynamic applications for compressible fluid flows is presented here. In ReALE, we replace the rezoning phase of classical ALE method by a rezoning where we allow the connectivity between cells of the mesh to change. This leads to a polygonal mesh that recovers the Lagrangian features in order to follow more efficiently the flow. Those reconnections allow to deal with complex geometries and high vorticity problems contrary to ALE method. For optimizing the remapping phase, we have modified the idea of swept‐integration‐based. The new method is called swept‐intersection‐based remapping method. We demonstrate that our method can be applied to several numerical examples representative of hydrodynamic experiments.Copyright © 2012 John Wiley & Sons, Ltd.     

A semi‐implicit scheme for 3D free surface flows with high‐order velocity reconstruction on unstructured Voronoi meshes

In this paper, we present a computationally efficient semi‐implicit scheme for the simulation of three‐dimensional hydrostatic free surface flow problems on staggered unstructured Voronoi meshes. For each polygonal control volume, the pressure is defined in the cell center, whereas the discrete velocity field is given by the normal velocity component at the cell faces. A piecewise high‐order polynomial vector velocity field is then reconstructed from the scalar normal velocities at the cell faces by using a new high‐order constrained least‐squares reconstruction operator. The reconstructed high‐order piecewise polynomial velocity field is used for trajectory integration in a semi‐Lagrangian approach to discretize the nonlinear convective terms in the governing PDE. For that purpose, a high‐order Taylor method is used as ODE integrator. The resulting semi‐implicit algorithm is extensively validated on a large set of different academic test problems with exact analytical solution and is finally applied to a real‐world engineering problem consisting of a curved channel upstream of two micro‐turbines of a hydroelectric power plant. For this realistic case, some experimental reference data are available from field measurements. Copyright © 2012 John Wiley & Sons, Ltd.     

The Becke Fuzzy Cells Integration Scheme in the Amsterdam Density Functional Program Suite

In this article, we document a new implementation of the fuzzy cells scheme for numerical integration in polyatomic systems [Becke, J. Chem. Phys. 1998, 88, 2547] and compare its efficiency and accuracy with respect to an integration scheme based on the Voronoi space partitioning. We show that the accuracy of both methods is comparable, but that the fuzzy cells scheme is better suited for geometry optimization. For this method, we also introduce the locally dense grid concept and present a proof‐of‐concept application. © 2013 Wiley Periodicals, Inc.     

Advances in Studies and Applications of Centroidal Voronoi Tessellations

<正>Centroidal Voronoi tessellations(CVTs) have become a useful tool in many applications ranging from geometric modeling,image and data analysis,and numerical partial differential equations,to problems in physics,astrophysics,chemistry,and biology. In this paper,we briefly review the CVT concept and a few of its generalizations and well-known properties.We then present an overview of recent advances in both mathematical and computational studies and in practical applications of CVTs.Whenever possible,we point out some outstanding issues that still need investigating.     

Some theoretical challenges in digital geometry: A perspective

In recent years image analysis has become a research field of exceptional significance, due to its relevance to real life problems in important societal and governmental sectors, such as medicine, defense, and security. The explicit purpose of the present Perspective is to suggest a number of strategic objectives for theoretical research, with an emphasis on the combinatorial approach in image analysis. Most of the proposed objectives relate to the need to make the theoretical foundations of combinatorial image analysis better integrated within a number of well-established subjects of theoretical computer science and discrete applied mathematics, such as the theory of algorithms and problem complexity, combinatorial optimization and polyhedral combinatorics, integer and linear programming, and computational geometry.