Antibandwidth of three-dimensional meshes

The antibandwidth problem is to label vertices of a graph G=(V,E) bijectively by 0,1,2,…,|V|−1 so that the minimal difference of labels of adjacent vertices is maximised. In this paper we prove an almost exact result for the antibandwidth of three-dimensional meshes. Provided results are extensions of the two-dimensional case and an analogue of the result for the bandwidth of three-dimensional meshes obtained by FitzGerald.     

Robust intersection of structured hexahedral meshes and degenerate triangle meshes with volume fraction applications

Two methods for calculating the volume and surface area of the intersection between a triangle mesh and a rectangular hexahedron are presented. The main result is an exact method that calculates the polyhedron of intersection and thereafter the volume and surface area of the fraction of the hexahedral cell inside the mesh. The second method is approximate, and estimates the intersection by a least squares plane. While most previous publications focus on non-degenerate triangle meshes, we here extend the methods to handle geometric degeneracies. In particular, we focus on large-scale triangle overlaps, or double surfaces. It is a geometric degeneracy that can be hard to solve with existing mesh repair algorithms. There could also be situations in which it is desirable to keep the original triangle mesh unmodified. Alternative methods that solve the problem without altering the mesh are therefore presented. This is a step towards a method that calculates the solid area and volume fractions of a degenerate triangle mesh including overlapping triangles, overlapping meshes, hanging nodes, and gaps. Such triangle meshes are common in industrial applications. The methods are validated against three industrial test cases. The validation shows that the exact method handles all addressed geometric degeneracies, including double surfaces, small self-intersections, and split hexahedra.     

Pseudo-geodesics on three-dimensional surfaces and pseudo-geodesic meshes

We present two numerical methods to approximate the shortest path or a geodesic between two points on a three-dimensional parametric surface. The first one consists of minimizing the path length, working in the parameter domain, where the approximation class is composed of Bézier curves. In the second approach, we consider Bézier surfaces and their control net. The numerical implementation is based on finding the shortest path on the successive control net subdivisions. The convergence property of the Bézier net to the surface gives an approximation of the required shortest path. These approximations, also called pseudo-geodesics, are then applied to the creation of pseudo-geodesic meshes. Experimental results are also provided.     

On MAC schemes on triangular delaunay meshes,their convergence and application to coupled flow problems

We study the convergence of two generalized marker‐and‐cell covolume schemes for the incompressible Stokes and Navier–Stokes equations introduced by Cavendish, Hall, Nicolaides, and Porsching. The schemes are defined on unstructured triangular Delaunay meshes and exploit the Delaunay–Voronoi duality. The study is motivated by the fact that the related discrete incompressibility condition allows to obtain a discrete maximum principle for the finite volume solution of an advection–diffusion problem coupled to the flow. The convergence theory uses discrete functional analysis and compactness arguments based on recent results for finite volume discretizations for the biharmonic equation. For both schemes, we prove the strong convergence in L² for the velocities and the discrete rotations of the velocities for the Stokes and the Navier–Stokes problem. Further, for one of the schemes, we also prove the strong convergence of the pressure in L². These predictions are confirmed by numerical examples presented in the article. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1397–1424, 2014     

Least-squares polynomial approximation on weakly admissible meshes: Disk and triangle

We construct symmetric polar WAMs (weakly admissible meshes) with low cardinality for least-squares polynomial approximation on the disk. These are then mapped to an arbitrary triangle. Numerical tests show that the growth of the least-squares projection uniform norm is much slower than the theoretical bound, and even slower than that of the Lebesgue constant of the best known interpolation points for the triangle. As opposed to good interpolation points, such meshes are straightforward to compute for any degree. The construction can be extended to polygons by triangulation.     

Probabilistic modeling of random variables with inconsistent data

The aim of the present paper was to formulate probabilistic modeling for random variables with inconsistent data to facilitate accurate reliability assessment. Traditionally, random variables have some outputs available, based on which, some distribution is identified. However, as will be illustrated, the data relevant to those extreme events might not necessarily follow the same distribution as well as the other part, but they generally have small weights in the definition of the distribution due to their small quantity. The adoption of one single probabilistic distribution to describe random variables with such inconsistent data might cause great errors in the reliability assessment, especially for extreme events. One new formulation of probabilistic modeling is proposed here for such type of random variables. The inconsistency within the data set is identified and based on how the set is divided. Each division is described by the respective distribution and finally they are unified into one framework. The relevant problems in the modeling (e.g., the identification of the boundary between the divisions, the definition of the probability distributions, and the unification of the distributions into one framework) are presented and solved. The realization of the proposed approach in the practical numerical analysis is further investigated afterwards. Finally, two examples are presented to illustrate the application from different perspectives.     

Generation of Delaunay meshes in implicit domains with edge sharpening

A variational algorithm for the construction of 3D Delaunay meshes in implicit domains with a nonsmooth boundary is proposed. The algorithm is based on the self-organization of an elastic network in which each Delaunay edge is interpreted as an elastic strut. The elastic potential is constructed as a combination of the repulsion potential and the sharpening potential. The sharpening potential is applied only on the boundary and is used to minimize the deviation of the outward normals to the boundary faces from the direction of the gradient of the implicit function. Numerical experiments showed that in the case when the implicit function specifying the domain is considerably different from the signed distance function, the use of the sharpening potential proposed by Belyaev and Ohtake in 2002 leads to the mesh instability. A stable version of the sharpening potential is proposed. The numerical experiments showed that acceptable Delaunay meshes for complex shaped domains with sharp curved boundary edges can be constructed.     

An iterative method for the creation of structured hexahedral meshes over complex orography

In this paper we propose a technique for measuring the quality of hexahedral Cartesian meshes used to model meso-scale atmospheric circulation in 3D. It is used to verify the progress of a novel method for satisfying the Delaunay criterion for structured hexahedral meshes over complex orography with high gradients and wide gradient variability. Based on a simile with potential energy, the iterative method of mesh smoothing is shown to improve mesh quality with logarithmic convergence. The method is evaluated in a practical application in a specific geographic location.     

The analysis of structured qualitative data

The aim of this paper is to give an overview of the methodological contribution given by Italian researchers in introducing a priori information into multidimensional data analysis techniques, paying special attention to categorical variables. The basic method is Non‐Symmetrical Correspondence Analysis, which enables the analysis of a contingency table when the behaviour of one variable is supposed to be dependent on the other cross‐classified variable. As usual correspondence analysis decomposes an association index (Pearson's Φ²), in a principal component sense, the proposed method is based on a decomposition of a predictability index (Goodman and Kruskal's τ_b). Non‐symmetrical correspondence analysis has been extended to more than one dependent/explanatory variable(s), by means of proper flattening procedures, i.e. by the use of multiple tables, and the decomposition of Gray and Williams' multiple and partial τ_b's. In doing so multiple and partial versions have been proposed. A forward selection procedure for choosing the variables with higher predictive power is presented. After a brief review of non‐symmetrical correspondence analysis confirmatory approach, the problem of validating results in terms of analytical stability and replication stability is faced by means of influence functions and resampling techniques. Copyright © 1999 John Wiley & Sons, Ltd.     

Generation of modes on the three-dimensional sphere

We use the method of Lie generation of tensor fields, which works for fields of different tensor structures, to construct the complete system of scalar, vector, symmetric tensor, and spinor fields on the three-dimensional sphere. We construct the Pauli operator explicitly. We demonstrate the role of spin in forming the mode series. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 3, pp. 417–440, March, 2007.     

Generation of three-dimensional random rotations in fitting and matching problems

An algorithm is developed to generate random rotations in three-dimensional space that follow a probability distribution arising in fitting and matching problems. The rotation matrices are orthogonally transformed into an optimal basis and then parameterized using Euler angles. The conditional distributions of the three Euler angles have a very simple form: the two azimuthal angles can be decoupled by sampling their sum and difference from a von Mises distribution; the cosine of the polar angle is exponentially distributed and thus straighforward to generate. Simulation results are shown and demonstrate the effectiveness of the method. The algorithm is compared to other methods for generating random rotations such as a random walk Metropolis scheme and a Gibbs sampling algorithm recently introduced by Green and Mardia. Finally, the algorithm is applied to a probabilistic version of the Procrustes problem of fitting two point sets and applied in the context of protein structure superposition.     

Generation of structured difference grids in two-dimensional nonconvex domains using mappings

Generation of structured difference grids in two-dimensional nonconvex domains is considered using a mapping of a parametric domain with a given nondegenerate grid onto a physical domain. For that purpose, a harmonic mapping is first used, which is a diffeomorphism under certain conditions due to Rado’s theorem. Although the harmonic mapping is a diffeomorphism, its discrete implementation can produce degenerate grids in nonconvex domains with highly curved boundaries. It is shown that the degeneration occurs due to approximation errors. To control the coordinate lines of the grid, an additional mapping is used and universal elliptic differential equations are solved. This makes it possible to generate a nondegenerate grid with cells of a prescribed shape.     

An algorithm for constructing initial three-dimensional structured grids for domains of revolution

An algorithm for constructing initial three-dimensional structured grids for geometrically complicated domains of revolution is described. The algorithm prevents the appearance of degenerate (not hexagonal) cells and cells of small volume around the axis of rotation. Examples of grids constructed by the suggested algorithm are given.     

PBIBDs from weakly divisible nearrings and related codes

In [5] the authors are able to give a method for the construction of a family of partially balanced incomplete block designs from a special class of wd-nearrings (wd-designs). In this paper the wd-design incidence matrix and the connected row and column codes are studied. The parameters of two special classes of wd-designs and those of the related row and column codes are calculated.     

Generation of large-scale vortex dislocations in a three-dimensional wake-type flow

Numerical study of three-dimensional evolution of wake-type flow and vortex dislocations is performed by using a compact finite diffenence-Fourier spectral method to solve 3-D incompressible Navier-Stokes equations. A local spanwise nonuniformity in momentum defect is imposed on the incoming wake-type flow. The present numerical results have shown that the flow instability leads to three-dimensional vortex streets, whose frequency, phase as well as the strength vary with the span caused by the local nonuniformity. The vortex dislocations are generated in the nonuniform region and the large-scale chain-like vortex linkage structures in the dislocations are shown. The generation and the characteristics of the vortex dislocations are described in detail.     

Scalar characterizations of weakly cone-convex and weakly cone-quasiconvex functions

The principal aim of this paper is to show that weakly cone-convex vector-valued functions, as well as weakly cone-quasiconvex vector-valued functions, can be characterized in terms of usual weakly convexity and weakly quasiconvexity of certain real-valued functions, defined by means of the extreme directions of the polar cone or by Gerstewitz’s scalarization functions.     

Mixed hp-FEM on anisotropic meshes II: Hanging nodes and tensor products of boundary layer meshes

Summary. The divergence stability of mixed hp Finite Element Methods for incompressible fluid flow is analyzed. A discrete inf-sup condition is proved for a general class of meshes. The meshes may be refined anisotropically, geometrically and may contain hanging nodes on geometric patches. The inf-sup constant is shown to be independent of the aspect ratio of the anisotropic elements and the dependence on the polynomial degree is analyzed. Numerical estimates of inf-sup constants confirm the theoretical results. Received October 13, 1997 / Revised version received June 8, 1998 / Published online July 28, 1999