generalised voronoi tessellation for generating microstructural finite element models with controllable grain size distributions and grain aspect ratios summary a generalised voronoi tessellation is proposed to create three dimensional microstructural finite element model which can effectively reproduce the grain size distribution and grain aspect ratio obtained from experiments this new approach consists of two steps the first step generates the desired lognormal grain size distribution with a given average grain volume and standard deviation the second step requires grouping meshed elements to create a specific grain aspect ratio using the voronoi generators from the first step a new concept is introduced to describe the transition from the poisson voronoi tessellation to the centroidal voronoi tessellation more importantly instead of using the conventional way where the voronoi cells are first generated and then meshed into finite elements this new approach discretises the pre meshed specimen with the voronoi generators this new technique prevents the presence of high density mesh at the vertices of voronoi cells and can tessellate irregular geometry much more easily examples of microstructures with different size distributions non equiaxed grains and complicated specimen geometries further demonstrate that the proposed approach can offer great flexibility to model various specimen geometries while keeping the process simple and efficient copyright 2015 john wiley sons ltd

riesz transforms of non integer homogeneity on uniformly disconnected sets in this paper we obtain precise estimates for the l norm of the s dimensional riesz transforms on very general measures supported on cantor sets in r with d 1 s d from these estimates we infer that for the so called uniformly disconnected compact sets the capacity s associated with the riesz kernel x x is comparable to the capacity 2 3 d s 3 2 from non linear potential theory

the mixed cell complex partition of unity method we present a galerkin method for solving partial differential equations which is a blend of ideas from particle based methods on the one side and traditional finite element methods on the other side the method is here named mixed cell complex partition of unity method mccpum it can be arbitrarily considered as being based on a set of scattered particles in the domain and on its boundary or on a delaunay cell decomposition of the domain in contrast to the element free galerkin method and other meshless techniques the partition of unity is not constructed on circular or rectangular supports but rather on overlapping polyhedral supports generated from a voronoi delaunay decomposition of space the mixed cell complex this approach inherits most of the advantages of truly meshless schemes while it greatly facilitates the numerical integration of the weak forms required in galerkin approximations the discretization is exclusively governed by the selection of nodes and the approximation orders associated to the nodes locally here legendre polynomials of arbitrary orders are used the mixed cell complex and the corresponding galerkin discretization are explained numerical examples for the poisson problem in two dimensions are presented and the efficiency of the method is discussed

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

an optimal algorithm for approximate nearest neighbor searching fixed dimensions consider a set of s of n data points in real d dimensional space r d where distances are measured using any minkowski metric in nearest neighbor searching we preprocess s into a data structure so that given any query point q r d is the closest point of s to q can be reported quickly given any positive real e data point p is a 1 e approximate nearest neighbor of q if its distance from q is within a factor of 1 e of the distance to the true nearest neighbor we show that it is possible to preprocess a set of n points in r d in o dn log n time and o dn space so that given a query point q r d and e 0 a 1 e approximate nearest neighbor of q can be computed in o c d e log n time where c d e d 1 6d e d is a factor depending only on dimension and e in general we show that given an integer k 1 1 e approximations to the k nearest neighbors of q can be computed in additional o kd log n time

a particle partition of unity method for the solution of elliptic parabolic and hyperbolic pdes in this paper we present a meshless discretization technique for instationary convection diffusion problems it is based on operator splitting the method of characteristics and a generalized partition of unity method we focus on the discretization process and its quality the method may be used as an h version or a p version even for general particle distributions the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid we discuss the implementational aspects of the proposed method furthermore we present the results of numerical examples where we considered instationary convection diffusion instationary diffusion linear advection and elliptic problems

anisotropic centroidal voronoi tessellations and their applications in this paper we introduce a novel definition of the anisotropic centroidal voronoi tessellation acvt corresponding to a given riemann metric tensor a directional distance function is used in the definition to simplify the computation we provide algorithms to approximate the acvt using the lloyd iteration and the construction of anisotropic delaunay triangulation under the given riemannian metric the acvt is applied to the optimization of two dimensional anisotropic delaunay triangulation to the generation of surface cvt and high quality triangular mesh on general surfaces various numerical examples demonstrating the effectiveness of the proposed method are presented

a natural neighbour based moving least squares approach for the element free galerkin method the element free galerkin method efg and the natural element method nem are two well known and widely used meshless methods whereas the efg method can represent moving boundaries like cracks only by modifying the weighting functions the nem requires an adaptation of the nodal set up but on the other hand the nem is computationally more efficient than efg in this paper a new concept for the automatic adjustment of nodal influence domains in the efg method is presented in order to obtain an efficiency similar to the nem this concept is based on the definition of natural neighbours for each meshless node which can be determined from a voronoi diagram of the nodal set up in this approach adapted nodal influence domains are obtained by interpolating the distances to the natural neighbours depending on the direction in the paper we show that this concept leads especially for problems with grading node density to a reduced number of influencing nodes at the interpolation points and consequently a significant reduction of the numerical effort copyright 2006 john wiley sons ltd

characterization and semiadditivity of the c 1 harmonic capacity the c 1 harmonic capacity c plays a central role in problems of approximation by harmonic functions in the c 1 norm in n 1 in this paper we prove the comparability between the capacity c and its positive version c as a corollary we deduce the semiadditivity of c this capacity can be considered as a generalization in n 1 of the continuous analytic capacity in c moreover we also show that the so called inner boundary conjecture fails for dimensions n 1 unlike in the case n 1

centroidal voronoi tessellation based reduced order modeling of complex systems noindent a reduced order modeling methodology based on centroidal voronoi tessellations cvts is introduced cvts are special voronoi tessellations for which the generators of the voronoi diagram are also the centers of mass means of the corresponding voronoi cells for discrete data sets cvts are closely related to the h means and k means clustering techniques a discussion of reduced order modeling for complex systems such as fluid flows is given to provide a context for the application of reduced order bases then detailed descriptions of cvt based reduced order bases and how they can be constructed from snapshot sets and how they can be applied to the low cost simulation of complex systems are given subsequently some concrete incompressible flow examples are used to illustrate the construction and use of cvt based reduced order bases the cvt based reduced order modeling methodology is shown to be effective for these examples

grid generation and optimization based on centroidal voronoi tessellations centroidal voronoi tessellations cvts are voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding voronoi regions such tessellations are of use in very diverse applications including data compression clustering analysis cell biology territorial behavior of animals and optimal allocation of resources in this paper we explore the use of cvts in grid generation in connection with finite element approximations of partial differential equations we being by describing these tessellations and methods for their determination we then discuss their application to mesh generation and finish with some examples of their use for the solution of partial differential equations

c m approximation by solutions of elliptic equations and calderon zygmund operators soit xcr n compact et soit f une fonction deux fois continument differentiable dans un voisinage de x dont le laplacien s annule en chaque point de x alors il y a une suite f j telle que chaque f j est une fonction harmonique sur un voisinage dependant de j de x et f j f uniformement sur x pour o 2

convergence of the lloyd algorithm for computing centroidal voronoi tessellations centroidal voronoi tessellations cvts are voronoi tessellations of a bounded geometric domain such that the generating points of the tessellations are also the centroids mass centers of the corresponding voronoi regions with respect to a given density function centroidal voronoi tessellations may also be defined in more abstract and more general settings due to the natural optimization properties enjoyed by cvts they have many applications in diverse fields the lloyd algorithm is one of the most popular iterative schemes for computing the cvts but its theoretical analysis is far from complete in this paper some new analytical results on the local and global convergence of the lloyd algorithm are presented these results are derived through careful utilization of the optimization properties shared by cvts numerical experiments are also provided to substantiate the theoretical analysis

upper bounds for the distances between adjacent zeros of solutions of delay differential equations abstract this paper is concerned with upper bounds for the distances between adjacent zeros of solutions of first order linear delay differential equations with variable coefficients of the form x t p t x t 0 t t where 0 and p t c t r by introducing a sequence of testing functions we are able to derive upper bounds that are better than those in the literature

some calderon zygmund kernels and their relations to wolff capacities and rectifiability we consider the calderon zygmund kernels k alpha n x x i 2n 1 x 2n 1 alpha i 1 d in mathbb r d for 0 alpha le 1 and n in mathbb n we show that on the plane for 0 alpha 1 the capacity associated to the kernels k alpha n is comparable to the riesz capacity c frac 2 3 2 alpha frac 3 2 of non linear potential theory as consequences we deduce the semiadditivity and bilipschitz invariance of this capacity furthermore we show that for any borel set e subset mathbb r d with finite length the l 2 mathcal h 1 lfloor e boundedness of the singular integral associated to k 1 n implies the rectifiability of the set e we thus extend to any ambient dimension results previously known only in the plane

the semiadditivity of continuous analytic capacity and the inner boundary conjecture let a e be the continuous analytic capacity of a compact set e c c in this paper we obtain a characterization of a in terms of curvature of measures with zero linear density and we deduce that a is countably semiadditive this result has important consequences for the theory of uniform rational approximation on compact sets in particular it implies the so called inner boundary conjecture 1 introduction the continuous analytic capacity a was introduced by erokhin and vitushkin see vi in the 1950s in order to study problems of uniform rational approximation in compact subsets of the complex plane up to now the geometric properties of the capacity a have not been well understood it has been an open question if a is semiadditive as a set function that is to say if

riesz transforms and harmonic lip1 capacity in cantor sets we estimate the l norm of the s dimensional riesz transforms on some cantor sets in rd to this end we show that the riesz transforms truncated at different scales behave in quasiorthogonal way as an application we obtain some precise numerical estimates for the lipschitz harmonic capacity of these sets

meshfree probabilistic determination of point sets and support regions for meshless computing new algorithms are presented for the determination of point sets and associated support regions that can then be used in meshless computing methods the algorithms are probabilistic in nature so that they are totally meshfree i e they do not require at any stage the use of any coarse or fine boundary conforming or superimposed meshes computational examples are provided that show for both uniform and non uniform point distributions that the algorithms result in high quality point sets and high quality support regions furthermore the algorithms lend themselves well to parallelization

adaptive mixed hybrid and penalty discontinuous galerkin method for two phase flow in heterogeneous media in this paper we present a hybrid method which consists of a mixed hybrid finite element method and a penalty discontinuous galerkin method for the approximation of a fractional flow formulation of a two phase flow problem in heterogeneous media with discontinuous capillary pressure the fractional flow formulation is comprised of a wetting phase pressure equation and a wetting phase saturation equation which are coupled through a total velocity and the saturation affected coefficients for the wetting phase pressure equation the continuous mixed hybrid finite element method space can be utilized due to a fundamental property that the wetting phase pressure is continuous while it can reduce the computational cost by using less degrees of freedom and avoiding the post processing of velocity reconstruction this method can also keep several good properties of the discontinuous galerkin method which are important to the fractional flow formulation such as the local mass balance continuous normal flux and capability of handling the discontinuous capillary pressure for the wetting phase saturation equation the penalty discontinuous galerkin method is utilized due to its capability of handling the discontinuous jump of the wetting phase saturation furthermore an adaptive algorithm for the hybrid method together with the centroidal voronoi delaunay triangulation technique is proposed five numerical examples are presented to illustrate the features of proposed numerical method such as the optimal convergence order the accurate and efficient velocity approximation and the applicability to the simulation of water flooding in oil field and the oil trapping or barrier effect phenomena

centroidal voronoi tessellations applications and algorithms a centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions we give some applications of such tessellations to problems in image compression quadrature finite difference methods distribution of resources cellular biology statistics and the territorial behavior of animals we discuss methods for computing these tessellations provide some analyses concerning both the tessellations and the methods for their determination and finally present the results of some numerical experiments