SUPERCONVERGENCE OF LEAST-SQUARES MIXED FINITE ELEMENTS FOR ELLIPTIC PROBLEMS ON TRIANGULATION

In this paper, we present the least-squares mixed finite element method and investigate superconvergence phenomena for the second order elliptic boundary-value problems over triangulations. On the basis of the L~2-projection and some mixed finite element projections, we obtain the superconvergence result of least-squares mixed finite     

N(o)ther-type theorem of piecewise algebraic curves on triangulation

The piecewise algebraic curve is a kind generalization of the classical algebraic curve.N(o)ther-type theorem of piecewise algebraic curves on the cross-cut partition is very important to construct the Lagrange interpolation sets for a bivariate spline space. In this paper, using the properties of bivariate splines, the N(o)ther-type theorem of piecewise algebraic curves on the arbitrary triangulation is presented.     

Nöther-type theorem of piecewise algebraic curves on triangulation

The piecewise algebraic curve is a kind generalization of the classical algebraic curve. Nöther-type theorem of piecewise algebraic curves on the cross-cut partition is very important to construct the Lagrange interpolation sets for a bivariate spline space. In this paper, using the properties of bivariate splines, the Nöther-type theorem of piecewise algebraic curves on the arbitrary triangulation is presented.     

Triangular embeddings of complete graphs (neighborly maps) with 12 and 13 vertices

In this paper, we describe the generation of all nonorientable triangular embeddings of the complete graphs K₁₂ and K₁₃. (The 59 nonisomorphic orientable triangular embeddings of K₁₂ were found in 1996 by Altshuler, Bokowski, and Schuchert, and K₁₃ has no orientable triangular embeddings.) There are 182,200 nonisomorphic nonorientable triangular embeddings for K₁₂, and 243,088,286 for K₁₃. Triangular embeddings of complete graphs are also known as neighborly maps and are a type of twofold triple system. We also use methods of Wilson to provide an upper bound on the number of simple twofold triple systems of order n, and thereby on the number of triangular embeddings of K_n. We mention an application of our results to flexibility of embedded graphs. © 2005 Wiley Periodicals, Inc. J Combin Designs     

A particle finite element method applied to long wave run‐up

This paper presents a Lagrangian–Eulerian finite element formulation for solving fluid dynamics problems with moving boundaries and employs the method to long wave run‐up. The method is based on a set of Lagrangian particles which serve as moving nodes for the finite element mesh. Nodes at the moving shoreline are identified by the alpha shape concept which utilizes the distance from neighbouring nodes in different directions. An efficient triangulation technique is then used for the mesh generation at each time step. In order to validate the numerical method the code has been compared with analytical solutions and a preexisting finite difference model. The main focus of our investigation is to assess the numerical method through simulations of three‐dimensional dam break and long wave run‐up on curved beaches. Particularly the method is put to test for cases where different shoreline segments connect and produce a computational domain surrounding dry regions. Copyright © 2006 John Wiley & Sons, Ltd.     

Meshing skin surfaces with certified topology

Skin surfaces are used for the visualization of molecules. They form a class of tangent continuous surfaces defined in terms of a set of balls (the atoms of the molecule) and a shrink factor. More recently, skin surfaces have been used for approximation purposes.

We present an algorithm that approximates a skin surface with a topologically correct mesh. The complexity of the mesh is linear in the size of the Delaunay triangulation of the balls, which is worst case optimal.

We also adapt two existing refinement algorithms to improve the quality of the mesh and show that the same algorithm can be used for meshing a union of balls.     

Delaunay configurations and multivariate splines: A generalization of a result of B. N. Delaunay

In the 1920s, B. N. Delaunay proved that the dual graph of the Voronoi diagram of a discrete set of points in a Euclidean space gives rise to a collection of simplices, whose circumspheres contain no points from this set in their interior. Such Delaunay simplices tessellate the convex hull of these points. An equivalent formulation of this property is that the characteristic functions of the Delaunay simplices form a partition of unity. In the paper this result is generalized to the so-called Delaunay configurations. These are defined by considering all simplices for which the interiors of their circumspheres contain a fixed number of points from the given set, in contrast to the Delaunay simplices, whose circumspheres are empty. It is proved that every family of Delaunay configurations generates a partition of unity, formed by the so-called simplex splines. These are compactly supported piecewise polynomial functions which are multivariate analogs of the well-known univariate B-splines. It is also shown that the linear span of the simplex splines contains all algebraic polynomials of degree not exceeding the degree of the splines.

     

A framework for advancing front techniques of finite element mesh generation

Advancing front techniques are a family of methods for finite element mesh generation that are particularly effective in dealing with complicated boundary geometries. In the first part of this paper, conditions are presented which ensure that any planar aft algorithm that meets these conditions terminates in a finite number of steps with a valid triangulation of the input domain. These conditions are described by specifying a framework of subtasks that can accommodate many aft methods and by prescribing the minimal requirements on each subtask that ensure correctness of an algorithm that conforms to the framework.An important efficiency factor in implementing an aft is the data structure used to represent the unmeshed regions during the execution of the algorithm. In the second part of the paper, we discuss the use of the constrained Delaunay triangulation as an efficient abstract data structure for the unmeshed regions. We indicate how the correctness conditions of the first part of the paper can be met using this representation. In this case, we also discuss the additional requirements on the framework which ensure that the generated mesh is a constrained Delaunay triangulation for the original boundary.The first author has been supported by CERFACS, Toulouse, France. Support was provided to the second author by the Natural Sciences and Engineering Research Council of Canada, and by the Information Technology Research Centre of Ontario.     

Conforming Delaunay triangulations in 3D

We describe an algorithm which, for any piecewise linear complex (PLC) in 3D, builds a Delaunay triangulation conforming to this PLC. The algorithm has been implemented, and yields in practice a relatively small number of Steiner points due to the fact that it adapts to the local geometry of the PLC. It is, to our knowledge, the first practical algorithm devoted to this problem.     

关于图的L(d,1)-标号问题

图G的L(2,1)-标号是一个从顶点集V(G)到非负整数集的函数f(x),使得若d(x,y)=1,则|f(x)-f(y)|≥2;若d(x,y)=2,则|f(x)-f(y)|≥1.图G的L(2,1)-标号数λ(G)是使得G有max{f(v)v∈V(G)}=k的L(2,1)-标号中的最小数k.Griggs和Yeh猜想对最大度为△的一般图G,有λ(G)≤△2.此文研究了作为L(2,1)-标号问题的推广的L(d,1)-标号问题,并得出了平面三角剖分图、立体四面体剖分图、平面近四边形剖分图的L(d,1)-标号的上界,作为推论证明了对上述几类图该猜想成立.