voronoi delaunay analysis of voids in systems of nonspherical particles the voronoi network is known to be a useful tool for the structural description of voids in the packings of spheres produced by computer simulations in this article we extend the voronoi delaunay analysis to packings of nonspherical convex objects main properties of the voronoi network which are known for systems of spheres are valid for systems of any convex objects a general numerical algorithm for calculation of the voronoi network in three dimensions is proposed it is based on the calculation of the trajectory of the imaginary empty sphere of variable size moving inside a system the delaunay empty sphere method analysis of voids is presented for an ensemble of random straight lines and for a molecular dynamics model of liquid crystal the spatial distribution of voids and a simple percolation analysis are obtained the distributions of the bottleneck radii and the radii of spheres inscribed in the voids are calculated

tetrahedral mesh generation based on node insertion in crystal lattice arrangements and advancing front delaunay triangulation a method of unstructured technical mesh generation for general three dimensional domains is presented a conventional boundary representation is adopted as the basis for the description of solids with evolving geometry and topology the geometry of the surfaces is represented either analytically or by piecewise polynominal interpolation a preliminary surface mesh is generated by an advancing front method with the nodes inserted by hard sphere packing in physical space in accordance with a prescribed mesh density interior nodes are inserted in a face centered cubic fcc crystal lattice arrangements coupled to octree spatial subdivision with the local lattice parameter determined by a prespecified nodal density function prior to triangulation of the volume the preliminary surface mesh is preprocessed by a combination of local transformations and subdivisions in order to guarantee that the surface triangulation be a subcomplex of the volume delaunay triangulation a joint delaunay triangulation of the interior and boundary nodes which preserves the modified surface mesh is then constructed via an advancing front approach the resulting mesh is finally improved upon by the application of local transformations the overall time complexity of the mesher is o n log n the robustness and versatility of the approach as well as the good quality of the resulting meshes is demostrated with the aid of selected examples

a multigrid method with unstructured adaptive grids for steady euler equations the multigrid method based on multi stage jacobi relaxation earlier developed by the authors for structured grid calculations with euler equations is extended to unstructured grid applications the meshes are generated with delaunay triangulation algorithms and are adapted to the flow solution

a sinh transformation for evaluating two dimensional nearly singular boundary element integrals a new transformation technique is introduced for evaluating the two dimensional nearly singular integrals which arise in the solution of laplace s equation in three dimensions using the boundary element method when the source point is very close to the element of integration the integrals are evaluated using in a product fashion a transformation which has recently been used to evaluate one dimensional near singular integrals this sinh transformation method automatically takes into account the position of the projection of the source point onto the element and also the distance b between the source point and the element the method is straightforward to implement and when it is compared with a number of existing techniques for evaluating two dimensional near singular integrals it is found that the sinh method is superior to the existing methods considered both for potential integrals across the full range of b values considered 0 0 01 for smaller values of b the use of the lmethod is recommended for flux integrals copyright 2006 john wiley sons ltd

group actions on the complex projective plane 0 introduction in this paper we consider homologically trivial locally smooth actions of finite and compact lie groups on topological 4 manifolds having the same homology as cp2 the complex projective plane our main result states that all such actions are lowally complex linear and that the acting groups can also act linearly on cp2 in particular we obtain the classification of groups acting on both manifolds homotopy equivalent to cp2 g acts on the complex projective plane resp on the chern manifold if and only if g is isomorphic to a subgroup resp a pseudofree subgroup of pu 3 the paper is organized as follows in 1 a precise statement of the main result is given 2 examines tangent representations of some cyclic groups the main tool being the g signature theorem the relevant number theoretic computations are carried out in appendix 1 3 and 4 discuss abelian and nonabelian groups in terms of the numbers of fixed points euler characteristic of the fixed point set 5 proves that all tangent representations are complex 6 9 discuss several classes of groups acting without fixed points the terminology that we use to distinguish among different classes of fixed point free groups is adopted from the linear case 10 contains the proof of our main theorem and finally 11 is devoted to group actions on the chern manifold for completeness a list of fixed point free linear groups is included in appendix 2 acknowledgment i would like to thank professor john ewing for suggesting this problem to me and also for his support and encouragement i have been informed that i hambleton and r lee have independently obtained results similar to those presented here

evaluation of hypersingular and nearly singular integrals in the isogeometric boundary element method for acoustics abstract special integration routines for hypersingular and nearly singular integrals that arise in an isogeometric boundary element method igabem are presented the igabem is applied to an acoustic problem in the frequency domain that corresponds to the helmholtz equation in three dimensions a major topic of such an acoustical bem is the non uniqueness of the conventional boundary integral equation which can be corrected by the burton miller formulation at the cost of a hypersingular integral that requires the presented routines the combination of the bem with an isogeometric analysis iga is advantageous since the exact cad description is directly incorporated to the numerical simulation as the geometry is represented exactly the integration routines become even more relevant and cannot be masked by the geometrical approximation for the conventional approximation with plane lagrange elements a variety of different integration routines exist but are not investigated in the scope of an iga therefore the hypersingular integration scheme of guiggiani is adapted to be applicable to this new methodology a direct evaluation of the hypersingular integral is possible instead of an integration over the complete surface as in the often used regularized boundary integral equation additionally the sinh transformation for nearly singular integrals is introduced to achieve more accurate results with the same number of integration points as the conventional gaussian integration the new formulation is analyzed under the special case of the oscillatory kernels in acoustics

the voronoi tessellation generated from eigenvalues of complex random matrices the voronoi froth generated from eigenvalues of asymmetric complex random matrices is studied by numerical simulation it is more regular than the random voronoi froth irvf generated from a poisson process the existence of a unique tessellation called here random matrix voronoi froth rmvf follows from the universality of the distribution of eigenvalues which is also briefly commented on the geometrical and topological properties of the rmvf have been characterised an empirical and accurate distribution function is also proposed for the cell side length of a rvf deviations from the aboav weaire law are discussed their magnitude may be interpreted as a measure ofthe departure from an equilibrium structure in the frame of the statistical crystallography theory of rivier

structure of simple liquids as a percolation problem on the voronoi network analysis of the mutual arrangements of delaunay simplices is formulated as a site correlated percolation problem on the voronoi network typical cluster configurations of simplices of tetrahedral and octahedral form the main structural elements of simple liquids are found and percolation thresholds for them are obtained

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

the atomic strain tensor abstract a definition of the local atomic strain increments in three dimensions and an algorithm for computing them is presented an arbitrary arrangement of atoms is tessellated into delaunay tetrahedra identifying interstices and voronoi polyhedra identifying atomic domains the deformation gradient increment tensor for interstitial space is obtained from the displacement increments of the corner atoms of delaunay tetrahedra the atomic site strain increment tensor is then obtained by finding the intersection of the delaunay tetrahedra with the voronoi polyhedra accumulating the individual deformation gradient contributions of the intersected delaunay tetrahedra into the voronoi polyhedra an example application is discussed showing how the atomic strain clarifies the relative local atomic movement for a polymeric glass treated at the atomic level

adaptive h p and hp versions for boundary integral element methods recently very promising results in a so called hp version of the finite element method have been obtained the basic idea is a balanced combination of mesh refinement and increase of the polynomial degree of the shape functions this idea is applied to a boundary collocation method in this paper the new method is compared with adaptive h and p versions and it is shown in numerical examples that even in the presence of singularities in the exact solution exponential convergence is obtained

rapid and accurate computation of the distance function using grids we present two fast and simple algorithms for approximating the distance function for given isolated points on uniform grids the algorithms are then generalized to compute the distance to piecewise linear objects by incorporating the geometry of huygens principle in the reverse order with the classical viscosity solution theory for the eikonal equation u 1 the algorithms become almost purely algebraic and yield very accurate approximations the generalized closest point formulation used in the second algorithm provides a framework for further extension to compute the distance accurately to smooth geometric objects on different grid geometries without the construction of the voronoi diagrams this method provides a fast and simple translator of data commonly given in computational geometry to the volumetric representation used in level set methods

hp fast multipole boundary element method for 3d acoustics summary a fast multipole boundary element method fmbem extended by an adaptive mesh refinement algorithm for solving acoustic problems in three dimensional space is presented in this paper the collocation method is used and the burton miller formulation is employed to overcome the fictitious eigenfrequencies arising for exterior domain problems due to the application of the combined integral equation chbie the developed fmbem is feasible for all positive wave numbers even up to high frequencies in order to evaluate the hypersingular integral resulting from the burton miller formulation of the bie an integration technique for arbitrary element order is applied the fast multipole method combined with an arbitrary order h p mesh refinement strategy enables accurate computation of large scale systems numerical examples substantiate the high accuracy attainable by the developed fmbem while requiring only moderate computational effort at the same time this article is protected by copyright all rights reserved

the algebraic characterization of the exteriors of certain 2 knots we study 2 knots with virtually solvable group by applying recent work of freedman to the 4 manifolds obtained by surgery on such knots in particular we show that gluck reconstruction is the only ambiguity in recovering the cappell shaneson knots as top locally flat knots from their groups alone

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

mesh duality and legendre duality we describe a sense in which mesh duality is equivalent to legendre duality that is a general pair of meshes which satisfy a definition of duality for meshes are shown to be the projection of a pair of piecewise linear functions that are dual to each other in the sense of a legendre dual transformation in applications the latter functions can be a tangent plane approximation to a smoother function and a chordal plane approximation to its legendre dual convex examples include one from meteorology and also the relation between the delaunay mesh and the voronoi tessellation the latter are shown to be the projections of tangent plane and chordal approximations to the same paraboloid

detecting structure in two dimensions combining voronoi tessellation and percolation conventional source detection algorithms in high energy astrophysics and other fields mostly use spherical or quadratic sliding windows of varying size on two dimensionally binned representations of spatial event distributions in order to detect statistically significant event enhancements sources within a given field while this is a reasonably reliable technique for nearly pointlike sources with good statistics poor and extended sources are likely to be incorrectly assessed or even missed at all as the calculations are governed by nonphysical parameters like the bin size and the window geometry rather than by the actual data the approach presented here does not introduce any artificial bias but makes full use of the unbinned two dimensional event distribution a voronoi tessellation on a finite plane surface yields individual densities or fluxes for every single event the distribution of which allows the determination of the contribution from a random poissonian background field noise the application of a nonparametric percolation to the tessellation cells exceeding this noise level leads directly to a source list which is free of any assumptions about the source geometry high density fluctuations from the random background field will still be included in this tentative source list but can be easily eliminated in most cases by setting a lower threshold to the required number of events per source since no finite size detection windows or the like have been used this analysis yields automatically straightforward fluxes for every source finally accepted the main disadvantage of this approach is the considerable cpu time required for the construction of the voronoi tessellation it is thus applicable only to either small fields or low event density regions

about the delaunay voronoi tesselation abstract the topological description of the neighborhood of the constituing atoms in a dense random material liquids glasses is of a major importance for their characterization we discuss an algorithm for the delaunay construction which is global and independent of the dimension of the space and derive from it the full dual voronoi tesselation by an original realization of duality

variational delaunay approach to the generation of tetrahedral finite element meshes we describe an algorithm which generates tetrahedral decomposition of a general solid body whose surface is given as a collection of triangular facets the principal idea is to modify the constraints in such a way as to make them appear in an unconstrained triangulation of the vertex set apriori the vertex set positions are randomized to guarantee existence of a unique triangulation which satisfies the delaunay empty sphere property algorithms for robust parallelized construction of such triangulations are available in order to make the boundary of the solid appear as a collection of tetrahedral faces we iterate two operations edge flip and edge split with the insertion of additional vertex until all of the boundary facets are present in the tetrahedral mesh the outcome of the vertex insertion is another triangulation of the input surfaces but one which is represented as a subset of the tetrahedral faces to determine if a constraining facet is present in the unconstrained delaunay triangulation of the current vertex set we use the results of rajan which re formulate delaunay triangulation as a linear programming problem

digital topology and geometry in medical imaging a survey digital topology and geometry refers to the use of topologic and geometric properties and features for images defined in digital grids such methods have been widely used in many medical imaging applications including image segmentation visualization manipulation interpolation registration surface tracking object representation correction quantitative morphometry etc digital topology and geometry play important roles in medical imaging research by enriching the scope of target outcomes and by adding strong theoretical foundations with enhanced stability fidelity and efficiency this paper presents a comprehensive yet compact survey on results principles and insights of methods related to digital topology and geometry with strong emphasis on understanding their roles in various medical imaging applications specifically this paper reviews methods related to distance analysis and path propagation connectivity surface tracking image segmentation boundary and centerline detection topology preservation and local topological properties skeletonization and object representation correction and quantitative morphometry a common thread among the topics reviewed in this paper is that their theory and algorithms use the principle of digital path connectivity path propagation and neighborhood analysis

triangle distribution and equation of state for classical rigid disks the triangle distribution function f 3 for three mutual near neighbors in the plane describes basic aspects of short range order and statistical thermodynamics in two dimensional many particle systems this paper examines prospects for constructing a self consistent calculation for the rigid disk system f 3 we present several identities obeyed by f 3 a rudimentary closure suggested by scaled particle theory is introduced in conjunction with three of the basic identities this closure leads to an unique f 3 over the entire density range the pressure equation of state exhibits qualitatively correct behaviors in both the low density and the close packed limits but no intervening phase transition appears we discuss extensions to improved disk closures and to the three dimensional rigid sphere system

grid optimization for the boundary element method the quality of solution obtained using the boundary element method bem is dependent on how the boundary is discretized this is particularly true in domains of complex geometry a rule for grid optimization for the bem is derived on the bases of an asymptotic measure of the boundary element error that preserves the number of elements degrees of freedom three example problems are provided to show the advantages of grid optimization in terms of accuracy and cost

spin models on random lattices under certain conditions phase transitions in systems with quenched disorder are expected to exhibit a dierent behaviour than in the corresponding pure system here we discuss a series of monte carlo studies of a special type of such disordered systems namely spin models dened on quenched random lattices exhibiting geometrical disorder in the connectivity of the lattice sites in two dimensions we present results for the q state potts model on random tri valent 3 planar graphs which appear quite naturally in the dynamically triangulated random surface dtrs approach to quantum gravity as well as on poissonian random lattices of voronoi delaunay type both cases q64 and 4 are discussed which in the pure model without disorder give rise to second and rst order phase transitions respectively in three dimensions results for the ising model on poissonian random lattices are briey described we conclude with a comparison of the two types of connectivity disorder with the more standard case of bond disorder and a discussion of the distinguishing dierences c 2000 elsevier science b v all rights reserved

voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids voronoi cells and the notion of natural neighbours are used to develop a finite difference method for the diffusion operator on arbitrary unstructured grids natural neighbours are based on the voronoi diagram which partitions space into closest point regions the sibson and the laplace non sibsonian interpolants which are based on natural neighbours have shown promise within a galerkin framework for the solution of partial differential equations in this paper we focus on the laplace interpolant with a two fold objective first to unify the previous developments related to the laplace interpolant and to indicate its ties to some well known numerical methods and secondly to propose a voronoi cell finite difference scheme for the diffusion operator on arbitrary unstructured grids a conservation law in integral form is discretized on voronoi cells to derive a finite difference scheme for the diffusion operator on irregular grids the proposed scheme can also be viewed as a point collocation technique a detailed study on consistency is conducted and the satisfaction of the discrete maximum principle stability is established owing to symmetry of the laplace weight a symmetric positive definite stiffness matrix is realized which permits the use of efficient linear solvers on a regular rectangular or hexagonal grid the difference scheme reduces to the classical finite difference method numerical examples for the poisson equation with dirichlet boundary conditions are presented to demonstrate the accuracy and convergence of the finite difference scheme copyright 2003 john wiley sons ltd

on the statistics of area size in two dimensional thick voronoi diagrams cells of voronoi diagrams in two dimensions are usually considered as having edges of zero width however this is not the case in several experimental situations in which the thickness of the edges of the cells is relatively large in this paper the concept of a thick voronoi tessellation that is with edges of non zero width is introduced and the statistics of cell areas as thickness changes are analyzed

error indicators for adaptive mesh refinement in the boundary element method a new approach in this paper a new approach to the development of a posteriori error indicators for the direct boundary element method bem is presented it takes advantage of a unique opportunity offered by the collocation bem that is the possibility of obtaining more than one numerical solution from a given boundary element mesh unlike other approaches the error indicator introduced in the present paper does not require any explicit evaluation of the residual of the approximate solution in order to test the capability of the proposed error indicators to actually control a selective mesh refinement process an h adaptive refinement strategy is also described together with the criteria for refinement and for the termination of the adaptive process full details on the whole numerical implementation for two dimensional problems are provided in particular there is described a direct method for the numerical evaluation of cauchy principal value integrals numerical examples show that pointwise convergence is always obtained in a few steps and the final meshes look nearly as if they were designed by an expert analyst

circle actions and fundamental groups for homology 4 spheres we generalize work of fintushel and pao to give a topological classification of smooth circle actions on oriented 4 manifolds 2 satisfying h e2 0 we then use these ideas to construct infinite families of homology 4 spheres that do not admit effective circle actions and whose fundamental groups cannot be 3 manifold groups 0 introduction this paper studies several questions related to circle actions on homology 4 spheres and their fundamental groups the main result is a construction which yields infinite families of homology 4 spheres 24 more generally 24 satisfying h1 2 z 0 which do not support effective circle actions and whose fundamental groups cannot be 3 manifold groups this construction makes it clear that the gap between fundamental groups of homology 3 spheres and 4 spheres is quite large as one might expect the paper is organized as follows 1 quickly reviews fintushel s classification of s actions on simply connected 4 manifolds f1 f3 and pao s work on the topological classification p and gives a simple generalization to the case where h1 e z 0 we prove theorem 1 2 assume that 2 is an oriented smooth 4 manifold with h1 e z 0 and that s acts effectively on e then 2 may be decomposed as a connected sum of copies of s2 x s2 cp2 and a spun homology 3 sphere m3 hence 7ti 24 7 t m3 2 describes a construction of homology 4 spheres which generalizes the equivariant construction of 1 if k is a smooth knot in some homology sphere m3 we produce 24 with 71 24 71 m3kx d32 1 1 mlb la x vx where m and 1 are meridian and preferred longitude b and a are arbitrary integers 3 analyzes the case of torus knots in s3 using the theory of seifert manifolds and we use these results in 4 to prove received by the editors october 8 1980 and in revised form march 26 1981 1980 mathematics subject classification primary 57s15 57m99 1982 american mathematical society 0002 9947 82 00001073 04 50

adaptive boundary element methods for some first kind integral equations in this paper we present an adaptive boundary element method for the boundary integral equations of the first kind concerning the dirichlet problem and the neumann problem for the laplacian in a two dimensional lipschitz domain for the h version of the finite element galerkin discretization of the single layer potential and the hypersingular operator we derive a posteriori error estimates which guarantee a given bound for the error in the energy norm up to a multiplicative constant following eriksson and johnson this yields adaptive algorithms steering the mesh refinement numerical examples confirm that our adaptive algorithms yield automatically the expected convergence rate

a new algorithm for three dimensional voronoi tessellation abstract a new efficient algorithm for computing the voronoi tessellation in three dimensions is presented the voronoi tessellation is obtained by constructing delaunay tetrahedra the complete set of these tetrahedra corresponds to that of a single voronoi polyhedron

on the construction of the voronoi mesh on a sphere abstract a new construction of the voronoi mesh on the sphere is presented the main feature is that the algorithm adds points one at a time until the final voronoi mesh is built up by adding one point to an existing voronoi mesh of k points only local changes are needed to construct a voronoi mesh of k 1 points this construction is particularly well suited to time dependent problems since using information from the voronoi mesh at the previous time step allows us to reduce the construction to o n operations when the two configurations are close while the algorithm does not break down when they are far apart numerical experiments are presented to substantiate the o n operation count for a typical case