an error based approach to complementary formulations of static field solutions the constitutive relationship between fields in electromagnetic media is cast in error form for well posed static problems the error splits systematically into complementary functionals error minimization provides a universal variational principle that generates complementary and dual solution formulations the error which is wholly attributed to the numerical constraints in approximate solutions is always defined and computable locally and globally

a tensor artificial viscosity using a mimetic finite difference algorithm abstract we have developed a two dimensional tensor artificial viscosity for finite difference shock wave computations the discrete viscosity tensor is formed by multiplying the gradient of velocity tensor by a scalar term the scalar term is based on the form of viscosity first presented by kurapatanko and also contains a limiter designed to switch off the viscosity for shockless compression and rigid body rotation mimetic discretizations are used to derive the form of the momentum and energy equations for a nonorthogonal grid where the viscosity tensor is evaluated at the zone edges the advantage of the tensor viscosity is a reduction of the dependence of the solution on the relation of the grid to the flow structure

volume inequalities for isotropic measures a direct approach to ball s simplex inequality is presented this approach which does not use the brascamp lieb inequality also gives barthe s characterization of the simplex for ball s inequality and extends it from discrete to arbitrary measures it also yields the dual inequality along with equality conditions and it does both for arbitrary measures

discrete geometric formulation of admittance boundary conditions for frequency domain problems over tetrahedral dual grids in this work it is shown how admittance boundary conditions for electromagnetic boundary value problems in the frequency domain can be formulated for the discrete geometric approach the details are presented for primal grids composed of tetrahedra and barycentric dual grids

a discrete geometric approach to solving time independent schrodinger equation the time independent schrodinger equation stems from quantum theory axioms as a partial differential equation this work aims at providing a novel discrete geometric formulation of this equation in terms of integral variables associated with precise geometric elements of a pair of three dimensional interlocked grids one of them based on tetrahedra we will deduce in a purely geometric way a computationally efficient discrete counterpart of the time independent schrodinger equation in terms of a standard symmetric eigenvalue problem moreover boundary and interface conditions together with non homogeneity and anisotropy of the media involved are accounted for in a straightforward manner this approach yields to a sensible computational advantage with respect to the finite element method where a generalized eigenvalue problem has to be solved instead such a modeling tool can be used for analyzing a number of quantum phenomena in modern nano structured devices where the accounting of the real 3d geometry is a crucial issue

general affine surface areas abstract two families of general affine surface areas are introduced basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for l affine surface areas are established

inequalities for mixed p affine surface area we prove new alexandrov fenchel type inequalities and new affine isoperimetric inequalities for mixed p affine surface areas we introduce a new class of bodies the illumination surface bodies and establish some of their properties we show for instance that they are not necessarily convex we give geometric interpretations of l p affine surface areas mixed p affine surface areas and other functionals via these bodies the surprising new element is that not necessarily convex bodies provide the tool for these interpretations

formulation of the eddy current problem in multiply connected regions in terms of h in this paper various formulations for the eddy current problem are presented the formulations are based on solving directly for the magnetic field h and they differ from each other mainly by how the field on the boundary is treated the electromagnetic problem is studied in connection with the fivefold decomposition of the space of square integrable vector fields within a bounded region this provides us with numerical approaches with clear signposts about how to solve the eddy current problem in multiply connected domains besides the fivefold decomposition another essential tool in our approach is whitney elements as they provide the structure needed to retain consistency between the continuous and discrete problems the paper demonstrates the usefulness of these mathematical tools in solving electromagnetic field problems 1998 john wiley sons ltd

geometric interpretation of finite dimensional eddy current formulations this paper gives an explicit geometric interpretation of finite element formulations of the eddy current problem the paper shows step by step how the eddy current problem can be implemented in a finite element kind of software system exploiting familiar geometric ideas such as lengths of edges areas of faces volumes of tetrahedra and the mutual interconnections between the simplices of a mesh the approach is a specific case of the so called geometric techniques copyright 2006 john wiley sons ltd

on geometric discretization of elasticity this paper presents a geometric discretization of elasticity when the ambient space is euclidean this theory is built on ideas from algebraic topology exterior calculus and the recent developments of discrete exterior calculus we first review some geometric ideas in continuum mechanics and show how constitutive equations of linearized elasticity similar to those of electromagnetism can be written in terms of a material hodge star operator in the discrete theory presented in this paper instead of referring to continuum quantities we postulate the existence of some discrete scalar valued and vector valued primal and dual differential forms on a discretized solid which is assumed to be a triangulated domain we find the discrete governing equations by requiring energy balance invariance under time dependent rigid translations and rotations of the ambient space there are several subtle differences between the discrete and continuous theories for example power of tractions in the discrete theory is written on a layer of cells with a nonzero volume we obtain the compatibility equations of this discrete theory using tools from algebraic topology we study a discrete cosserat medium and obtain its governing equations finally we study the geometric structure of linearized elasticity and write its governing equations in a matrix form we show that in addition to constitutive equations balance of angular momentum is also metric dependent all the other governing equations are topological

the even orlicz minkowski problem the classical minkowski problem leads to the lp minkowski problem and now to the orlicz minkowski problem existence is demonstrated for the even orlicz minkowski problem a byproduct is a new approach to the solution of the classical minkowski problem

new affine measures of symmetry for convex bodies abstract grunbaum introduced measures of symmetry for convex bodies that measure how far a given convex body is from a centrally symmetric one here we introduce new measures of symmetry that measure how far a given convex body is from one with enough symmetries to define these new measures of symmetry we use affine covariant points we give examples of convex bodies whose affine covariant points are far apart in particular we give an example of a convex body whose centroid and santalo point are far apart

a characterization of affine surface area abstract we show that every upper semicontinuous and equi affine invariant valuation on the space of d dimensional convex bodies is a linear combination of affine surface area volume and the euler characteristic

a classification of sl n invariant valuations a classification of upper semicontinuous and sl n invariant valuations on the space of n dimensional convex bodies is established as a consequence complete characterizations of cen t ro affine and l p affine surface areas are obtained the proofs make use of a new sl n shaping process for convex bodies

new orlicz affine isoperimetric inequalities abstract the orlicz brunn minkowski theory received considerable attention recently and many results in the l p brunn minkowski theory have been extended to their orlicz counterparts the aim of this paper is to develop orlicz l affine and geominimal surface areas for a single convex body as well as for multiple convex bodies which generalize the l p mixed affine and geominimal surface areas fundamental concepts in the l p brunn minkowski theory our extensions are different from the general affine surface areas by ludwig in 21 moreover our definitions for orlicz l affine and geominimal surface areas reveal that these affine invariants are essentially the infimum supremum of v k l the orlicz mixed volume of k and the polar body of l where l runs over all star bodies and all convex bodies respectively with volume of l equal to the volume of the unit euclidean ball b 2 n properties for the orlicz l affine and geominimal surface areas such as affine invariance and monotonicity are proved related orlicz affine isoperimetric inequalities are also established

lp dual affine surface area abstract according to the notion of l p affine surface area by lutwak in this paper we introduce the concept of l p dual affine surface area further we establish the affine isoperimetric inequality and the blaschke santalo inequality for l p dual affine surface area besides the dual brunn minkowski inequality for l p dual affine surface area is presented

computation of stationary 3d halo currents in fusion devices with accuracy control this paper addresses the calculation of the resistive distribution of halo currents in three dimensional structures of large magnetic confinement fusion machines a neumann electrokinetic problem is solved on a geometry so complicated that complementarity is used to monitor the discretization error an irrotational electric field is obtained by a geometric formulation based on the electric scalar potential whereas three geometric formulations are compared to obtain a solenoidal current density a formulation based on the electric vector potential and two geometric formulations inspired from mixed and mixed hybrid finite elements the electric vector potential formulation is usually considered impractical since an enormous computing power is wasted by the topological pre processing it requires to solve this challenging problem we present novel algorithms based on lazy cohomology generators that enable to save orders of magnitude computational time with respect to all other state of the art solutions proposed in literature believing that our results are useful in other fields of scientific computing the proposed algorithm is presented as a detailed pseudocode in such a way that it can be easily implemented

convergence analysis of the high order mimetic finite difference method we prove second order convergence of the conservative variable and its flux in the high order mfd method the convergence results are proved for unstructured polyhedral meshes and full tensor diffusion coefficients for the case of non constant coefficients we also develop a new family of high order mfd methods theoretical result are confirmed through numerical experiments

lp dual curvature measures abstract a new family of geometric borel measures on the unit sphere is introduced special cases include the l p surface area measures which extend the classical surface area measure of aleksandrov and fenchel jessen and l p integral curvature which extends alkesandrov s integral curvature in the l p brunn minkowski theory it also includes the dual curvature measures which are the duals of federer s curvature measures in the dual brunn minkowski theory this partially unifies the classical theory of mixed volumes and the newer theory of dual mixed volumes

constitutive equations for discrete electromagnetic problems over polyhedral grids in this paper a novel approach is proposed for constructing discrete counterparts of constitutive equations over polyhedral grids which ensure both consistency and stability of the algebraic equations discretizing an electromagnetic field problem the idea is to construct discrete constitutive equations preserving the thermodynamic relations for constitutive equations in this way consistency and stability of the discrete equations are ensured at the base a purely geometric condition between the primal and the dual grids has to be satisfied for a given primal polyhedral grid by properly choosing the dual grid numerical experiments demonstrate that the proposed discrete constitutive equations lead to accurate approximations of the electromagnetic field

the planar orlicz minkowski problem in the l1 sense abstract in this paper we solve l p minkowski problem for l 1 data and all p 0 and orlicz minkowski problem with two nonlinear terms in l 1 sense a byproduct is the blaschke santalo inequality which was previously established for only constant data and now is shown to hold for l 1 data

constrained convex bodies with extremal affine surface areas abstract given a convex body k r n and p r we introduce and study the extremal inner and outer affine surface areas i s p k sup k k as p k and o s p k inf k k as p k where as p k denotes the l p affine surface area of k and the supremum is taken over all convex subsets of k and the infimum over all convex compact subsets containing k the convex body that realizes i s 1 k in dimension 2 was determined in 3 where it was also shown that this body is the limit shape of lattice polytopes in k in higher dimensions no results are known about the extremal bodies we use a thin shell estimate of 23 and the lowner ellipsoid to give asymptotic estimates on the size of i s p k and o s p k surprisingly it turns out that both quantities are proportional to a power of volume

new lp affine isoperimetric inequalities we prove new lp affine isoperimetric inequalities for all p 8 1 we establish for all p n a duality formula which shows that lp affine surface area of a convex body k equals affine surface area of the polar body k

dual orlicz brunn minkowski theory dual orlicz lϕ affine and geominimal surface areas abstract this paper aims to develop the basic theory for the dual orlicz l affine and geominimal surface areas of star bodies basic properties of these new affine invariants are established moreover related orlicz affine isoperimetric cyclic santalo type and alexandrov fenchel type inequalities are obtained

an a posteriori driven adaptive mixed high order method with application to electrostatics in this work we propose an adaptive version of the recently introduced mixed high order method and showcase its performance on a comprehensive set of academic and industrial problems in computational electromagnetism the latter include in particular the numerical modeling of comb drive and mems devices mesh adaptation is driven by newly derived residual based error estimators the resulting method has several advantageous features it supports fairly general meshes it enables arbitrary approximation orders and has a moderate computational cost thanks to hybridization and static condensation the a posteriori driven mesh refinement is shown to significantly enhance the performance on problems featuring singular solutions allowing to fully exploit the high order of approximation

an analysis of finite volume finite element and finite difference methods using some concepts from algebraic topology in this paper we apply the ideas of algebraic topology to the analysis of the finite volume and finite element methods illuminating the similarity between the discretization strategies adopted by the two methods in the light of a geometric interpretation proposed for the role played by the weighting functions in finite elements we discuss the intrinsic discrete nature of some of the factors appearing in the field equations underlining the exception represented by the constitutive term the discretization of which is maintained as the key issue for numerical methods devoted to field problems we propose a systematic technique to perform this task present a rationale for the adoption of two dual discretization grids and point out some optimization opportunities in the combined selection of interpolation functions and cell geometry for the finite volume method finally we suggest an explanation for the intrinsic limitations of the classical finite difference method in the construction of accurate high order formulas for field problems

valuations and surface area measures we consider valuations defined on polytopes containing the origin which have measures on the sphere as values we show that the classical surface area measure is essentially the only such valuation which is sl n contravariant of degree one moreover for all real p an lp version of the above result is established for gl n contravariant valuations of degree p this provides a characterization of the lp surface area measures from the lp brunn minkowski theory mathematics subject classification 52b45 28a10

the santalo regions of a convex body motivated by the blaschke santalo inequality we define for a convex body k in r and for t r the santalo regions s k t of k we investigate properties of these sets and relate them to a concept of affine differential geometry the affine surface area of k let k be a convex body in r for x int k the interior of k let k be the polar body of k with respect to x it is well known that there exists a unique x0 int k such that the product of the volumes k k0 is minimal see for instance sch this unique x0 is called the santalo point of k moreover the blaschke santalo inequality says that k k0 v n where vn denotes the volume of the n dimensional euclidean unit ball b 0 1 with equality if and only if k is an ellipsoid for t r we consider here the sets s k t x k k k v2 n t following e lutwak we call s k t a santalo region of k observe that it follows from the blaschke santalo inequality that the santalopoint x0 s k 1 and that s k 1 x0 if and only if k is an ellipsoid thus s k t has non empty interior for some t 1 if and only if k is not an ellipsoid in the first part of this paper we show some properties of s k t and give estimates on the size of s k t this question was asked by e lutwak the paper was written while both authors stayed at msri supported by a grant from the national science foundation msc classification 52

critical analysis of the spanning tree techniques two algorithms based upon a tree cotree decomposition called in this paper spanning tree technique stt and generalized spanning tree technique gstt have been shown to be useful in computational electromagnetics the aim of this paper is to give a rigorous description of the gstt in terms of homology and cohomology theories together with an analysis of its termination in particular the authors aim to show by concrete counterexamples that various problems related with both stt and gstt algorithms exist the counterexamples clearly demonstrate that the failure of stt and gstt is not an exceptional event but something that routinely occurs in practical applications

lp mixed geominimal surface area abstract this paper deals with l p geominimal surface area and its extension to l p mixed geominimal surface area we give an integral formula of l p geominimal surface area by the p petty body and introduce the concept of l p mixed geominimal surface area which is a natural extension of l p geominimal surface area some inequalities such as analogues of alexandrov fenchel inequalities blaschke santalo inequalities and affine isoperimetric inequalities for l p mixed geominimal surface areas are obtained

convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes the stability and convergence properties of the mimetic finite difference method for diffusion type problems on polyhedral meshes are analyzed the optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved

p q mixed geominimal surface area and p q mixed affine surface area abstract in this paper the p q mixed geominimal surface area and the p q mixed affine surface area are introduced they are extensions of the l p geominimal surface area and the l p affine surface area respectively affine invariance under unimodular affine transformations and continuity for the p q mixed geominimal surface area and the p q mixed affine surface area are discussed the existence the uniqueness and the uniform boundness of the p q mixed petty bodies are obtained affine isoperimetric inequalities for the p q mixed geominimal surface area and the p q mixed affine surface area are established

dual orlicz brunn minkowski theory abstract in this paper a dual orlicz brunn minkowski theory is presented an orlicz radial sum and dual orlicz mixed volumes are introduced the dual orlicz minkowski inequality and the dual orlicz brunn minkowski inequality are established the variational formula for the volume with respect to the orlicz radial sum is proved the equivalence between the dual orlicz minkowski inequality and the dual orlicz brunn minkowski inequality is demonstrated orlicz intersection bodies are defined and the orlicz busemann petty problem is posed

construction of a finite element basis of the first de rham cohomology group and numerical solution of 3d magnetostatic problems we devise an efficient algorithm for the finite element approximation of harmonic fields and the numerical solution of three dimensional magnetostatic problems in particular we construct a finite element basis of the first de rham cohomology group of the computational domain the proposed method works for general topological configurations and does not need the determination of cutting surfaces

generators of h_1 gamma_ h mathbbz for triangulated surfaces construction and classification we consider a bounded lipschitz polyhedron omega subset mathbb r 3 of general topology equipped with a tetrahedral triangulation that induces a mesh gamma h of the surface partial omega we seek a maximal set of surface edge cycles that are independent in h 1 gamma h mathbb z and bounding with respect to the exterior of omega we present an algorithm for constructing suitable 1 cycles in gamma h first representatives of a basis of the homology group h 1 gamma h mathbb z are constructed merely using the combinatorial description of the surface mesh gamma h then a duality pairing based on linking numbers is used to determine those combinations that are bounding with respect to mathbb r 3 setminus omega this is the key to circumventing a triangulation of the exterior region mathbb r 3 setminus omega in the computations for shape regular quasi uniform families of meshes the asymptotic complexity of the algorithm is shown to be o n2 where n is the number of edges of gamma h the scheme provides an essential preprocessing step for all boundary element methods for eddy current simulation which rely on discrete divergence free vectorfields and their description through stream functions

lp mixed affine surface area abstract according to the three notions of mixed affine surface area l p affine surface area and l p mixed affine surface area proposed by lutwak in this article we give the concept of ith l p mixed affine surface area such that the first and second notions of lutwak are its special cases further some lutwak s results are extended associated with this concept besides applying this concept we establish an inequality for the volumes and dual quermassintegrals of a class of star bodies

orlicz centroid bodies the sharp affine isoperimetric inequality that bounds the volume of the centroid body of a star body from below by the volume of the star body itself is the busemann petty centroid inequality a decade ago the lp analogue of the classical busemannpetty centroid inequality was proved here the definition of the centroid body is extended to an orlicz centroid body of a star body and the corresponding analogue of the busemann petty centroid inequality is established for convex bodies the centroids of the intersections of an origin symmetric body with half spaces form the surface of a convex body this centroid body is a concept that dates back at least to dupin the classical affine isoperimetric inequality that relates the volume of a convex body with that of its centroid body was conjectured by blaschke and established in a landmark work by petty 53 petty s inequality became known as the busemann petty centroid inequality because in establishing his inequality petty not only made critical use of busemann s random simplex inequality but as petty stated he reinterpreted it see e g the books by gardner 12 leichtweiss 25 schneider 55 and thompson 58 for reference the concept of a centroid body had a natural extension in what became known as the lp brunn minkowski theory and its dual this theory had its origins in the early 1960s when firey see e g schneider 55 introduced his concept of lp compositions of convex bodies three decades later in 34 and 35 these firey minkowski lp combinations were shown to lead to an embryonic lp brunn minkowski theory this theory and its dual has witnessed a rapid growth see e g 1 9 17 23 26 32 34 44 46 48 54 56 57 59 62 the lp analogues of centroid bodies became a central focus within the lp brunn minkowski theory and its dual and establishing the lpanalogue of the busemann petty centroid inequality became a major goal this was accomplished by the authors of the present paper in 37 with an independent approach presented by campi and gronchi 3 the lp centroid bodies quickly became objects of interest in asymptotic mathematics subject classification mr 52a40 research supported in part by nsf grant dms 0706859 proof copy 1 not for distribution 2 erwin lutwak deane yang and gaoyong zhang geometric analysis see e g 10 11 24 49 50 51 52 and even the theory of stable distributions see e g 48 using concepts introduced by ludwig 29 haberl and schuster 21 were led to establish asymmetric versions of the lp busemann petty centroid inequality that for bodies that are not origin symmetric are stronger than the lp busemann petty centroid inequality presented in 37 and 3 the asymmetric inequalities obtained by haberl and schuster are ideally suited for non symmetric bodies this can be seen by looking at the haberl schuster version of the lp analogue of the classical blaschke santalo inequality that was presented in 46 while for origin symmetric bodies the lp extension of 46 does recover the original blaschke santalo inequality as p for arbitrary bodies only the haberl schuster version does so the works of haberl and schuster 21 see also 22 ludwig and reitzner 32 and ludwig 31 have demonstrated the clear need to move beyond the lp brunn minkowski theory to what we are calling an orlicz brunn minkowski theory this need is not only motivated by compelling geometric considerations such as those presented in ludwig and reitzner 32 but also by the desire to obtain sobolev bounds see 22 of a far more general nature this paper is the second in a series intended to develop a few of the elements of an orlicz brunn minkowski theory and its dual here we define the orlicz centroid body establish some of its basic properties and most importantly establish what we call the orlicz busemannpetty centroid inequality for orlicz centroid bodies we consider convex r 0 such that 0 0 this means that must be decreasing on 0 and increasing on 0 we require that one of these is happening strictly so i e is either strictly decreasing on 0 or strictly increasing on 0 the class of such will be denoted by c if k is a star body see section 1 for precise definitions with respect to the origin in rn with volume k and c then we define the orlicz centroid body k of k as the convex body whose support function at x rn is given by h k x inf 0 1 k k x y dy 1 where x y denotes the standard inner product of x and y in rn and the integration is with respect to lebesgue measure in rn when p t t p with p 1 then

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

the orlicz brunn minkowski inequality abstract the orlicz brunn minkowski theory originated with the work of lutwak yang and zhang in 2010 in this paper we first introduce the orlicz addition of convex bodies containing the origin in their interiors and then extend the l p brunn minkowski inequality to the orlicz brunn minkowski inequality furthermore we extend the l p minkowski mixed volume inequality to the orlicz mixed volume inequality by using the orlicz brunn minkowski inequality

a geometric integral formulation for eddy currents summary in the computational electromagnetism community it is known how the differential formulation of an eddy current problem can be translated into a finite dimensional system of equations involving circulations and fluxes by means of the so called discrete geometric approach this is done by exploiting the geometric structure behind maxwell s equations in this paper we will show how the same discrete geometric approach can be profitably used also to discretize an eddy current problem formulated in an integral way we rely on a purely geometric definition of a novel set of face vector basis functions that we use to construct the discrete counterparts matrices of both the ohm s constitutive relation and of the integral relation between the magnetic vector potential and the eddy current density vector the symmetry and positive definiteness of such matrices will be demonstrated and their geometric structure will be apparent copyright q 2010 john wiley sons ltd

the spherical convex floating body abstract for a convex body on the euclidean unit sphere the spherical convex floating body is introduced the asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated this gives rise to a new spherical area measure the floating area remarkably this floating area turns out to be a spherical analogue to the classical affine surface area from affine differential geometry several properties of the floating area are established

the log brunn minkowski inequality abstract for origin symmetric convex bodies i e the unit balls of finite dimensional banach spaces it is conjectured that there exist a family of inequalities each of which is stronger than the classical brunn minkowski inequality and a family of inequalities each of which is stronger than the classical minkowski mixed volume inequality it is shown that these two families of inequalities are equivalent in that once either of these inequalities is established the other must follow as a consequence all of the conjectured inequalities are established for plane convex bodies

lattice electromagnetic theory from a topological viewpoint the language of differential forms and topological concepts are applied to study classical electromagnetic theory on a lattice it is shown that differential forms and their discrete counterparts cochains provide a natural bridge between the continuum and the lattice versions of the theory allowing for a natural factorization of the field equations into topological field equations i e invariant under homeomorphisms and metric field equations the various potential sources of inconsistency in the discretization process are identified distinguished and discussed a rationale for a consistent extension of the lattice theory to more general situations such as to irregular lattices is considered

a steiner formula in the lp brunn minkowski theory abstract we prove an analogue of the classical steiner formula for the l p affine surface area of a minkowski outer parallel body for any real parameter p we show that the classical steiner formula and the steiner formula of lutwak s dual brunn minkowski theory are special cases of this new steiner formula this new steiner formula and its localized versions lead to new curvature measures that have not appeared before in the literature they have the intrinsic volumes of the classical brunn minkowski theory as well as the dual quermassintegrals of the dual brunn minkowski theory as special cases properties of these new quantities are investigated a connection to information theory among them a steiner formula for the s th mixed l p affine surface area of a minkowski outer parallel body for any real parameters p and s is also given

complementary geometric formulations for electrostatics summary the simultaneous use of a pair of complementary discrete formulations for electrostatic boundary value problems bvps allows to accurately compute electromagnetic quantities such as capacitance or electrostatic force with a minimum computational effort in fact the two formulations provide the upper and lower bounds for these quantities and their averages result quite close to the exact solution even for extremely coarse meshes despite the potential benefit to the many three dimensional large scale applications taking advantage of this feature is not exploited in practice due to theoretical difficulties in the potential design the aim of this paper is to fill this gap by rigorously introducing a pair of three dimensional complementary geometric formulations to solve electrostatic bvps on domains covered by conformal polyhedral meshes in particular an original formulation based on a vector potential is introduced by using cohomology theory with integer coefficients it is shown how the so called thick links are needed which are representatives of the second cohomology group generators of the dielectric region two easy to implement graph theoretic algorithms to automatically find such a basis with optimal computational complexity are described some benchmark problems are presented to show how the simultaneous use of both formulations yields to a sensible computational advantage therefore solvers based on complementary formulations should be embedded in the next generation of electromagnetic computer aided engineering cae softwares copyright 2010 john wiley sons ltd

orlicz john ellipsoids abstract the orlicz john ellipsoids which are in the framework of the booming orlicz brunn minkowski theory are introduced for the first time it turns out that they are generalizations of the classical john ellipsoid and the evolved l p john ellipsoids the analog of ball s volume ratio inequality is established for the new orlicz john ellipsoids the connection between the isotropy of measures and the characterization of orlicz john ellipsoids is demonstrated

renyi divergence and lp affine surface area for convex bodies abstract we show that the fundamental objects of the l p brunn minkowski theory namely the l p affine surface areas for a convex body are closely related to information theory they are exponentials of renyi divergences of the cone measures of a convex body and its polar we give geometric interpretations for all renyi divergences d not just for the previously treated special case of relative entropy which is the case 1 now no symmetry assumptions are needed and if at all only very weak regularity assumptions are required previously the relative entropies appeared only after performing second order expansions of certain expressions now already first order expansions make them appear thus in the new approach we detect faster details about the boundary of a convex body

lp minkowski problem with not necessarily positive data abstract the l p minkowski problem in the plane has been the object of a number of recent investigations here we present various existence results for 0 p 2 in the case 0 p 2 a necessary and sufficient condition for the solvability of the problem is given

automatic generation of cuts on large sized meshes for the t ω geometric eddy current formulation in this paper a geometric t x formulation to solve eddy current problems on a tetrahedral mesh is presented when non simply connected conducting regions are considered the formulation requires the socalled thick cuts while in the literature more attention is usually given to the so called thin cuts while the automatic construction of thin cuts has been theoretically solved many years ago no implementation of an algorithm to compute the thick cuts which can be used in practice exists so far in this paper we propose how to fill this gap by introducing an algorithm to automatically compute the thick cuts on real sized meshes based on a belted tree and a tree cotree decomposition the belted tree is constructed by means of a homology computation by exploiting efficient reduction methods a number of benchmarks are presented to demonstrate the generality and the robustness of the algorithm a rigorous definition of thick cuts which necessarily has to rely on cohomology is presented in addition

asynchronous space time domain decomposition method with localized uncertainty quantification abstract the computational cost associated with uncertainty quantification of engineering problems featuring localized phenomenon can be reduced by confining the random variability of the model parameters within a region of interest in this case a localized treatment of mesh and time resolutions is required to capture the effect of the confined material uncertainty on the global response we present a computational approach for localized uncertainty quantification with the capability of asynchronous treatment of mesh and time resolutions in particular we allow each subdomain to have its local uncertainty representation and the corresponding mesh and time resolutions as a result computing resources can be directed toward a small region of interest where a model with high spatial and temporal resolutions is required to verify the numerical implementation we consider elastic wave propagation in an axially loaded beam moreover we perform convergence studies with respect to the spatial and temporal discretizations as well as the size of an uncertain subdomain a projectile impacting a composite sandwich plate is considered as an engineering application for the proposed method

lyz ellipsoid and petty projection body for log concave functions abstract the aims of this paper are to develop the lyz ellipsoid and petty projection body for log concave functions which correspond to the lyz ellipsoid and petty projection body for convex bodies when restricted to the subclass of characteristic functions moreover the continuous sl n contravariant valuation on a subclass of log concave functions is classified

aggregation based algebraic multigrid for convection diffusion equations we consider the iterative solution of large sparse linear systems arising from the upwind finite difference discretization of convection diffusion equations the system matrix is then an m matrix with nonnegative row sum and further when the convective flow has zero divergence the column sum is also nonnegative possibly up to a small correction term we investigate aggregation based algebraic multigrid methods for this class of matrices a theoretical analysis is developed for a simplified two grid scheme with one damped jacobi postsmoothing step an uncommon feature of this analysis is that it applies directly to problems with variable coefficients e g to problems with recirculating convective flow on the basis of this theory we develop an approach in which a guarantee is given on the convergence rate thanks to an aggregation algorithm that allows an explicit control of the location of the eigenvalues of the preconditioned matrix some issues that remain beyond the analysis are discussed in the

extended affine surface area during the twentieth century many of the notions of classical differential geometry have been extended so as to be defined on the boundaries of arbitrary convex bodies without smoothness requirements for example the projection measures quermassintegrals are the natural extensions of the integrals of mean curvature see santa16 27 pp 215 2321 for a discussion the surface area measures of aleksandrov fenchel jessen are extensions of the indefinite integrals of the elementary symmetric functions of the principal radii of curvature while federer s curvature measures when restricted to convex hypersurfaces are extensions of the indefinite integrals of the elementary symmetric functions of the principal curvatures see schneider 30 pp 27 331 one of the important concepts of classical differential geometry which has been difficult to extend is afline surface area this has often led to difficulties which when circumvented have been done so only by considerable effort and ingenuity for example suppose we are given a measure on the unit sphere which satisfies the hypothesis of the minkowski problem the solution of the minkowski problem guarantees the existence of a convex body whose surface area measure is the given measure from the given measure we can find the surface area of the body the minkowski mixed volume inequality can be used to obtain various upper bounds for the volume of the body however we cannot even inquire about the affine surface area of the body unless we can somehow convince ourselves that the convex body is sufficiently smooth results regarding aftine surface area have a surprising number of applications for example a number of geometric inequalities which appear to have no connection with affine surface area are in fact consequences of inequalities involving afline surface area see for example petty 25 and also 151 since polytopes have zero afftne surface area it seems quite surprising that the notion of afline surface area arises frequently in the area of polytope approximation see the survey of gruber 9 recently leichtweiss 11 121 presented a definition of extended afftne

mimetic finite difference method for the stokes problem on polygonal meshes various approaches to extend finite element methods to non traditional elements general polygons pyramids polyhedra etc have been developed over the last decade the construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis the mimetic finite difference mfd method works on general polygonal meshes and has many similarities with low order finite element methods both schemes try to preserve the fundamental properties of the underlying physical and mathematical models the essential difference between the two schemes is that the mfd method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices since no extension of basis functions inside the mesh elements is required practical implementation of the mfd method is simple for polygonal meshes that may include degenerate and non convex elements in this article we present a new mfd method for the stokes problem on arbitrary polygonal meshes and analyze its stability the method is developed for the general case of tensor coefficients which allows us to apply it to a linear elasticity problem as well numerical experiments show for the velocity variable second order convergence in a discrete l 2 norm and first order convergence in a discrete h 1 norm for the pressure variable first order convergence is shown in the l 2 norm

divergence for s concave and log concave functions abstract we prove new entropy inequalities for log concave and s concave functions that strengthen and generalize recently established reverse log sobolev and poincare inequalities for such functions this leads naturally to the concept of f divergence and in particular relative entropy for s concave and log concave functions we establish their basic properties among them the affine invariant valuation property applications are given in the theory of convex bodies

a new set of basis functions for the discrete geometric approach by exploiting the geometric structure behind maxwell s equations the so called discrete geometric approach allows to translate the physical laws of electromagnetism into discrete relations involving circulations and fluxes associated with the geometric elements of a pair of interlocked grids the primal grid and the dual grid to form a finite dimensional system of equations discrete counterparts of the constitutive relations must be introduced in addition they are referred to as constitutive matrices which must comply with precise properties symmetry positive definiteness consistency in order to guarantee the stability and consistency of the overall finite dimensional system of equations the aim of this work is to introduce a general and efficient set of vector functions associated with the edges and faces of a polyhedral primal grids or of a dual grid obtained from the barycentric subdivision of the boundary of the primal grid these vector functions comply with precise specifications which allow to construct stable and consistent discrete constitutive equations for the discrete geometric approach in the framework of an energetic method

sharp convex lorentz sobolev inequalities new sharp lorentz sobolev inequalities are obtained by convexifying level sets in lorentz integrals via the lp minkowski problem new lp isocapacitary and isoperimetric inequalities are proved for lipschitz star bodies it is shown that the sharp convex lorentz sobolev inequalities are analytic analogues of isocapacitary and isoperimetric inequalities

matching experimental and three dimensional numerical models for structural vibration problems with uncertainties abstract the simulation model which examines the dynamic behavior of real structures needs to address the impact of uncertainty in both geometry and material parameters this article investigates three dimensional finite element models for structural dynamics problems with respect to both model and parameter uncertainties the parameter uncertainties are determined via laboratory measurements on several beam like samples the parameters are then considered as random variables to the finite element model for exploring the uncertainty effects on the quality of the model outputs i e natural frequencies the accuracy of the output predictions from the model is compared with the experimental results to this end the non contact experimental modal analysis is conducted to identify the natural frequency of the samples the results show a good agreement compared with experimental data furthermore it is demonstrated that geometrical uncertainties have more influence on the natural frequencies compared to material parameters and material uncertainties are about two times higher than geometrical uncertainties this gives valuable insights for improving the finite element model due to various parameter ranges required in a modeling process involving uncertainty

orlicz projection bodies abstract minkowski s projection bodies have evolved into l p projection bodies and their asymmetric analogs these all turn out to be part of a far larger class of orlicz projection bodies the analog of the classical petty projection inequality is established for the new orlicz projection bodies

affine perimeter and limit shape it is proved here that as n infinity almost all convex 1 n z 2 lattice polygons lying in a given convex compact set k subset of r 2 are very close to a fixed convex set k 0 the distinguishing property of k 0 is that its affine perimeter is the largest among all convex sets contained in k

piecewise uniform bases and energetic approach for discrete constitutive matrices in electromagnetic problems in the paper we introduce piecewise uniform edge and face vector functions on a simplicial primal cell complex having geometric structure common to whitney s vector functions we also introduce piecewise uniform bases functions in the barycentric dual complex where the analogous of whitney s functions does not exist by using these piecewise uniform bases functions and by exploiting an energetic approach we construct symmetric positive definite constitutive matrices for discrete maxwell s equations we also prove that these constitutive matrices are deeply related to those of the finite elements with whitney s bases functions copyright 2005 john wiley sons ltd

base functions and discrete constitutive relations for staggered polyhedral grids an electromagnetic problem can be discretized on a pair of interlocked primal dual grids according to discrete geometric approaches like the finite integration technique fit or the cell method cm the critical aspect is however the construction of the discrete counterparts of the constitutive relations assuring stability and consistency of the overall discrete system of algebraic equations initially only orthogonal cartesian grids where considered more recently primal grids of tetrahedra and oblique prisms with triangular base can be handled with this paper a novel set of edge and face vector functions for general polyhedral primal grids is presented complying with precise specifications which allow to construct stable and consistent discrete constitutive equations in the framework of an energetic approach

relative entropy of cone measures and lp centroid bodies let k be a convex body in r n we introduce a new affine invariant which we call k t hat can be found in three different ways a as a limit of normalized lp affine surface areas b as the relative entropy of the cone measure of k and the cone measure of k c as the limit of the volume difference of k and lp centroid bodies we investigate properties of k and of related new invariant quantities in particular we show new affine isoperimetric inequalities and we show an information inequality for convex bodies

efficient solution of the schroedinger poisson equations in layered semiconductor devices in this paper we present several mathematical models that can be used to create approximate solutions of the three dimensional schroedinger poisson equation in layered semiconductor devices a general algorithmic strategy that can be used to create efficient solution procedures for each of these models is described computational results demonstrating the accuracy and efficiency that can be obtained with the use of these models is presented

homology and cohomology computation in finite element modeling a homology and cohomology solver for finite element meshes is represented it is an integrated part of the finite element mesh generator gmsh we demonstrate the exploitation of the cohomology computation results in a finite element solver and use an induction heating problem as a working example the homology and cohomology solver makes the use of a vector scalar potential formulation straightforward this gives better overall performance than a vector potential formulation cohomology computation also clarifies the lumped parameter coupling of the problem and enables the user to obtain useful postprocessing data as a part of the finite element solution

mimetic discretizations for maxwell s equations we have constructed reliable finite difference methods for approximating the solution to maxwell s equations using accurate discrete analogs of differential operators that satisfy the identities and theorems of vector and tensor calculus in discrete form the numerical approximation does not have spurious modes and mimics many fundamental properties of the underlying physical problem including conservation laws symmetries in the solution and the nondivergence of particular vector fields numerical examples demonstrate the high quality of the method when the medium is strongly discontinuous and for nonorthogonal nonsmooth computational grids

relative entropies for convex bodies we introduce a new class of not necessarily convex bodies and show among other things that these bodies provide yet another link between convex geometric analysis and information theory namely they give geometric interpretations of the relative entropy of the cone measures of a convex body and its polar and related quantities such interpretations were first given by paouris and werner for symmetric convex bodies in the context of the lp centroid bodies there the relative entropies appear after performing second order expansions of certain expressions now no symmetry assumptions are needed moreover using the new bodies already first order expansions make the relative entropies appear thus these bodies detect faster details of the boundary of a convex body than the lp centroid bodies

flag numbers and floating bodies abstract we investigate weighted floating bodies of polytopes we show that the weighted volume depends on the complete flags of the polytope this connection is obtained by introducing flag simplices which translate between the metric and combinatorial structure our results are applied in spherical and hyperbolic space this leads to new asymptotic results for polytopes in these spaces we also provide explicit examples of spherical and hyperbolic convex bodies whose floating bodies behave completely different from any convex body in euclidean space