a theory of algebraic cocycles we introduce the notion of an algebraic cocycle as the algebraic analogue of a map to an eilenberg maclane space using these cocycles we develop a cohomology theory for complex algebraic varieties the theory is bigraded functorial and admits gysin maps it carries a natural cup product and a pairing to l homology chern classes of algebraic bundles are defined in the theory there is a natural transformation to singular integral cohomology theory that preserves cup products computations in special cases are carried out on a smooth variety it is proved that there are algebraic cocycles in each algebraic rational p p cohomology class in this announcement we present the outlines of a cohomology theory for algebraic varieties based on a new concept of an algebraic cocycle details will appear in fl our cohomology is a companion to the l homology theory recently studied in l f l f1 l f2 fm this homology is a bigraded theory based directly on the structure of the space of algebraic cycles it admits a natural transformation to integral homology that generalizes the usual map taking a cycle to its homology class our new cohomology theory is similarly bigraded and based on the structure of the space of algebraic cocycles it carries a ring structure coming from the complex join an elementary construction of projective geometry and it admits a natural transformation to integral cohomology chern classes are defined in the theory and transform under to the usual ones our definition of cohomology is very far from a duality construction on l homology nonetheless there is a natural and geometrically defined kronecker pairing between our morphic cohomology and l homology the foundation stone of our theory is the notion of an effective algebraic cocycle which is of some independent interest roughly speaking such a cocycle on a variety x with values in a projective variety y is a morphism from x to the space of cycles on y when x is normal this is equivalent by graphing to a cycle on x y with equidimensional fibres over x such cocycles abound in algebraic geometry and arise naturally in many circumstances the simplest perhaps is that of a flat morphism f x y whose corresponding cocycle associates to x x the pullback cycle f x many more arise naturally from synthetic constructions in projective geometry we show that every variety is rich in cocycles indeed if 1991 mathematics subject classification primary 14f99 14c05

grothendieck standard conjectures morphic cohomology and hodge index theorem using morphic cohomology we produce a sequence of conjectures called morphic conjectures which terminates at the grothendieck standard conjecture a a refinement of hodge structures is given and with the assumption of morphic conjectures we prove a hodge index theorem we answer a question of friedlander and lawson by assuming the grothendieck standard conjecture b and prove that the topological filtration from morphic cohomology is equal to the grothendieck arithmetic filtration for some cases

an alternative alpha finite element method aαfem for free and forced structural vibration using triangular meshes an alternative alpha finite element method email protected using triangular elements is proposed that significantly improves the accuracy of the standard triangular finite elements and provides a superconvergent solution in the energy norm for the static analysis of two dimensional solid mechanics problems in the email protected the piecewise constant strain field of linear triangular fem models is enhanced by additional strain terms with an adjustable parameter a which results in an effectively softer stiffness formulation compared to a linear triangular element the element is further extended to the free and forced vibration analyses of solids several numerical examples show that the email protected achieves high reliability compared to other existing elements in the literature

on nirenberg type conditions for higher order systems on time scales we study the existence of periodic solutions for a nonlinear system of nth order differential equations on time scales assuming a suitable nirenberg type condition we prove the existence of at least one solution of the problem using mawhin s coincidence degree

a hybrid extended finite element level set method for modeling phase transformations a hybrid numerical method for modelling the evolution of sharp phase interfaces on fixed grids is presented we focus attention on two dimensional solidification problems where the temperature field evolves according to classical heat conduction in two subdomains separated by a moving freezing front the enrichment strategies of the extended finite element method x fem are employed to represent the jump in the temperature gradient that governs the velocity of the phase boundary a new approach with the x fem is suggested for this class of problems whereby the partition of unity is constructed with c1 polynomials and enriched with a c0 function this approach leads to jumps in temperature gradient occurring only at the phase boundary and is shown to significantly improve estimates for the front velocity temporal derivatives of the temperature field in the vicinity of the phase front are obtained with a projection that employs discontinuous enrichment in conjunction with a finer finite difference grid the level set method is used to represent the evolution of the phase interface an iterative procedure is adopted to satisfy the constraints on the temperature field on the phase boundary the robustness and utility of the method is demonstrated with several benchmark problems of phase transformation copyright 2002 john wiley sons ltd

some improvements of the fast marching method the fast marching method published by sethian proc natl acad sci usa 93 1996 pp 1591 1595 is an optimally efficient algorithm for solving problems of front evolution where the front speed is monotonic it has been used in a wide variety of applications such as robotic path planning r kimmel and j sethian fast marching methods for computing distance maps and shortest paths tech report 669 cpam university of california berkeley 1996 crack propagation m stolarska et al modelling crack growth by level sets in the extended finite element method comput methods appl mech engrg to appear internat j numer methods engrg 51 2001 pp 943 960 n sukumar d l chopp and b moran extended finite element method and fast marching method for three dimensional fatigue crack propagation j comput phys submitted seismology j sethian and a popovici geophysics 64 1999 pp 516 523 photolithography j sethian fast marching level set methods for three dimensional photolithography development in proceedings of the spie 1996 international symposium on microlithography santa clara ca 1996 and medical imaging r malladi and j sethian proc natl acad sci usa 93 1996 pp 9389 9392 it has also been a valuable tool for the implementation of modern level set methods where it is used to efficiently compute the distance to the front and or an extended velocity function in this paper we improve upon the second order fast marching method of sethian siam rev 41 1999 pp 199 235 by constructing a second order approximation of the interface generated from local data on the mesh the data is interpolated on a single box of the mesh using a bicubic approximation the distance to the front is then calculated by using a variant of newton s method to solve both the level curve equation and the orthogonality condition for the nearest point to a given node the result is a second order approximation of the distance to the interface which can then be used to produce second order accurate initial conditions for the fast marching method and a third order fast marching method

a novel model for biofilm growth and its resolution by using the hybrid immersed interface level set method in this work we propose a new model to simulate biofilm structures finger like as well as compact structures as a result of microbial growth in different environmental conditions at the same time the numerical method that we use in order to carry out the computational simulations is new to the biological community as far as we know the use of our model sheds light on the biological process of biofilm formation since it simulates some central issues of biofilm growth the pattern formation of heterogeneous structures such as finger like structures in a substrate transport limited regime and the formation of more compact structures in a growth limited regime the main advantage of our approach is that we consider several of the most relevant aspects of biofilm modeling particularly the existence and evolution of a biofilm liquid interface at the same time in order to perform numerical simulations we have used sophisticated numerical techniques based on mixing the immersed interface method and the level set method which are well described in the present work

techniques computations and conjectures for semi topological k theory we establish the existence of an atiyah hirzebruch like spectral sequence relating the morphic cohomology groups of a smooth quasi projective complex variety to its semi topological k groups this spectral sequence is compatible with and indeed is built from the motivic spectral sequence that relates the motivic cohomology and algebraic k theory of varieties and it is also compatible with the classical atiyah hirzebruch spectral sequence in algebraic topology in the second part of this paper we use this spectral sequence in conjunction with another computational tool that we introduce namely a variation on the integral weight filtration of the borel moore singular homology of complex varieties introduced by h gillet and c soule to compute the semi topological k theory of a large class of varieties in particular we prove that for curves surfaces toric varieties projective rational three folds and related varieties the semi topological k groups and topological k groups are isomorphic in all degrees permitted by cohomological considerations we also formulate integral conjectures relating semi topological k theory to topological k theory analogous to more familiar conjectures namely the quillen lichtenbaum and beilinson lichtenbaum conjectures concerning mod n algebraic k theory and motivic cohomology in particular we prove a local vanishing result for morphic cohomology which enables us to formulate precisely a conjectural identification of morphic cohomology by a suslin our computations verify that these conjectures hold for the list of varieties above

representation of singular fields without asymptotic enrichment in the extended finite element method summary in this paper we replace the asymptotic enrichments around the crack tip in the extended finite element method xfem with the semi analytical solution obtained by the scaled boundary finite element method sbfem the proposed method does not require special numerical integration technique to compute the stiffness matrix and it improves the capability of the xfem to model cracks in homogeneous and or heterogeneous materials without a priori knowledge of the asymptotic solutions a heaviside enrichment is used to represent the jump across the discontinuity surface we call the method as the extended sbfem numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics show that the proposed method yields accurate results with improved condition number a simple code is annexed to compute the terms in the stiffness matrix which can easily be integrated in any existing fem xfem code copyright 2013 john wiley sons ltd

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

algebraic cycles and homotopy theory examine its homotopy groups which after a certain idealization of the space turn out to be astonishingly simple in fact for complex projective space pn the structure can be understood completely this yields new information about the topology of the classical chow varieties and establishes an explicit relationship with universal cohomology operations in the general case complete computations are difficult but we shall establish a complex suspension theorem and lay the foundations for a theory based on the homotopy groups of chow varieties to state the results we must give precise meaning to the space of subvarieties we begin with the fundamental case of subvarieties in complex projective n space pln for each pair of integers p and d with d 1 and 0 p n consider the set wp d pn of all finite formal sums

intersections via resolutions abstract we investigate the viability of defining an intersection product on algebraic cycles on a singular algebraic variety by pushing forward intersection products formed on a resolution of singularities for varieties with resolutions having a certain structure including all varieties over a field of characteristic zero we obtain a stratification which reflects the geometry of the centers and the exceptional divisors this stratification is sufficiently fine that divisors can be intersected with r cycles for r 1 and 2 cycles can be intersected on a fourfold provided their incidences with the strata are controlled similar pairings are defined on a variety with one dimensional singular locus

algebraic cycles on real varieties and z 2 equivariant homotopy theory in this paper the spaces of algebraic cycles on a real projective variety x are studied as z 2 spaces under the action of the galois group gal c r in particular the equivariant homotopy type of the group of algebraic p cycles zp pc is computed a version of lawson homology for real varieties is proposed the real lawson homology groups are computed for a class of real varieties

numerically determined enrichment functions for the extended finite element method and applications to bi material anisotropic fracture and polycrystals strain singularities appear in many linear elasticity problems a very fine mesh has to be used in the vicinity of the singularity in order to obtain acceptable numerical solutions with the finite element method fem special enrichment functions describing this singular behavior can be used in the extended finite element method x fem to circumvent this problem these functions have to be known in advance but their analytical form is unknown in many cases li et al described a method to calculate singular strain fields at the tip of a notch numerically a slight modification of this approach makes it possible to calculate singular fields also in the interior of the structural domain we will show in numerical experiments that convergence rates can be significantly enhanced by using these approximations in the x fem the convergence rates have been compared with the ones obtained by the fem this was done for a series of problems including a polycrystalline structure copyright 2010 john wiley sons ltd

moving mesh strategy based on a gradient flow equation for two dimensional problems in this paper we introduce a moving mesh method for solving pdes in two dimensions it can be viewed as a higher dimensional generalization of the moving mesh pde mmpde strategy developed in our previous work for one dimensional problems w huang y ren and r d russell siam j numer anal 31 1994 pp 709 730 the mmpde is derived from a gradient flow equation which arises using a mesh adaptation functional in turn motivated from the theory of harmonic maps geometrical interpretations are given for the gradient equation and functional and basic properties of this mmpde are discussed numerical examples are presented where the method is used both for mesh generation and for solving time dependent pdes the results demonstrate the potential of the mesh movement strategy to concentrate the mesh points so as to adapt to special problem features and to also preserve a suitable level of mesh orthogonality

an adaptive level set approach for incompressible two phase flows we present a numerical method using the level set approach for solving incompressible two phase flow with surface tension in the level set approach the free surface is represented as the zero level set of a smooth function this has the effect of replacing the advection of density which has steep gradients at the free surface with the advection of the level set function which is smooth in addition the free surface can merge or break up with no special treatment we maintain the level set function as the signed distance from the free surface in order to accurately compute flows with high density ratios and stiff surface tension effects in this work we couple the level set scheme to an adaptive projection method for the incompressible navier stokes equations in order to achieve higher resolution of the free surface with a minimum of additional expense we present two dimensional axisymmetric and fully three dimensional results of air bubble and water drop computations

some remarks concerning mod nk theory the spectral sequence predicted by a beilinson relating motivic cohomology to algebraic k theory has been established for smooth quasi projective varieties over a field cf fs l1 among other properties verified this spectral sequence has the expected multiplicative behavior involving cup product in motivic cohomology and product in algebraic k theory and a good multiplicative mod n reduction relating mod n motivic cohomology to mod n algebraic k theory the purpose of this short paper is to draw two simple conclusions concerning mod n algebraick theory of smooth quasi projective varieties over an algebraically closed field by comparing this spectral sequence to its localization which we show converges to mod n etale k theory in theorem 2 1 we verify that the map from mod n algebraic k theory to mod n etale k theory of a smooth quasi projective variety x over an algebraically closed field in which n is invertible is surjective in degrees greater than or equal to twice the dimension of x this refines the surjectivity result of dfst and extends an argument by a suslin in s1 then in example 3 4 we show that certain projective smooth 3 folds x constructed by s bloch and h esnault have non zero elements in k0 x z n which are killed by multiplication by a sufficiently high power of the bott element k2 x z n although the existence of such examples is perhaps not surprising there have been attempts to prove that multiplication by is always injective the authors gratefully acknowledge useful conversations with h esnault and a suslin

extended isogeometric analysis based on pht splines for crack propagation near inclusions summary adaptive local refinement is one of the main issues for isogeometric analysis iga in this paper an adaptive extended iga xiga approach based on polynomial splines over hierarchical t meshes pht splines for modeling crack propagation is presented the pht splines overcome certain limitations of non uniform rational b splines nurbs based formulations in particular they make local refinements feasible to drive the adaptive mesh refinement we present a recovery based error estimator for the proposed method the method is based on the xiga method in which discontinuous enrichment functions are added to the iga approximation and this method does not require remeshing as the cracks grow in addition crack propagation is modeled by successive linear extensions that are determined by the stress intensity factors sifs under linear elastic fracture mechanics the proposed method has been used to analyze numerical examples and the sifs results were compared with reference results the findings demonstrate the accuracy and efficiency of the proposed method this article is protected by copyright all rights reserved

space time meshfree collocation method methodology and application to initial boundary value problems a novel space time meshfree collocation method stmcm for solving systems of non linear ordinary and partial differential equations by a consistent discretization in both space and time is proposed as an alternative to established mesh based methods the stmcm belongs to the class of truly meshfree methods i e the methods that do not have any underlying mesh but work on a set of nodes only without any a priori node to node connectivity instead the neighbouring information is established on the fly the stmcm is constructed using the interpolating moving least squares technique which allows a simplified implementation of boundary conditions due to fulfillment of the kronecker delta property by the kernel functions which is not the case for the major part of other meshfree methods the method is validated by several examples ranging from interpolation problems to the solution of pdes whereas the stmcm solutions are compared with either analytical or reference ones

harnack thom theorem for higher cycle groups and picard varieties we generalize the harnack thom theorem to relate the ranks of the lawson homology groups with 22 coefficients of a real quasiprojective variety with the ranks of its reduced real lawson homology groups in the case of zero cycle group we recover the classical harnack thom theorem and generalize the classical version to include real quasiprojective varieties we use weil s construction of picard varieties to construct reduced real picard groups and milnor s construction of universal bundles to construct some weak models of classifying spaces of some cycle groups these weak models are used to produce long exact sequences of homotopy groups which are the main tool in computing the homotopy groups of some cycle groups of divisors we obtain some congruences involving the picard number of a nonsingular real projective variety and the rank of its reduced real lawson homology groups of divisors

extended finite element method with edge based strain smoothing esm xfem for linear elastic crack growth abstract this paper presents a strain smoothing procedure for the extended finite element method xfem the resulting edge based smoothed extended finite element method esm xfem is tailored to linear elastic fracture mechanics and in this context to outperform the standard xfem in the xfem the displacement based approximation is enriched by the heaviside and asymptotic crack tip functions using the framework of partition of unity this eliminates the need for the mesh alignment with the crack and re meshing as the crack evolves edge based smoothing es relies on a generalized smoothing operation over smoothing domains associated with edges of simplex meshes and produces a softening effect leading to a close to exact stiffness super convergence and ultra accurate solutions the present method takes advantage of both the es fem and the xfem thanks to the use of strain smoothing the subdivision of elements intersected by discontinuities and of integrating the singular derivatives of the approximation functions is suppressed via transforming interior integration into boundary integration numerical examples show that the proposed method improves significantly the accuracy of stress intensity factors and achieves a near optimal convergence rate in the energy norm even without geometrical enrichment or blending correction

algebraic cycles representing cohomology operations in this paper we will show that certain universal homology classes which are fundamental in topology are algebraic to be specific the products of eilenberg maclane spaces mathcal k 2q equiv k mathbf z 2 times k mathbf z 4 times cdots times k mathbf z 2q have models which are limits of complex projective varieties precisely we have mathcal k 2q varinjlim nolimits d rightarrow infty mathcal c q d mathbf p n where mathcal c q d mathbf p n denotes the chow variety of effective cycles of codimension q and degree d on mathbf p mathbf c n it is natural to ask which elements in the homology of mathcal k 2q are represented by algebraic cycles in these approximations in this paper we find such representations for the even dimensional classes which are known as steenrod squares as well as their pontrjagin and join products these classes are dual to the cohomology classes which correspond to the basic cohomology operations also known as the steenrod squares

fast marching methods fast marching methods are numerical schemes for computing solutions to the nonlinear eikonal equation and related static hamilton jacobi equations based on entropy satisfying upwind schemes and fast sorting techniques they yield consistent accurate and highly efficient algorithms they are optimal in the sense that the computational complexity of the algorithms is o n log n where n is the total number of points in the domain the schemes are of use in a variety of applications including problems in shape offsetting computing distances from complex curves and surfaces shape from shading photolithographic development computing first arrivals in seismic travel times construction of shortest geodesics on surfaces optimal path planning around obstacles and visibility and reflection calculations in this paper we review the development of these techniques including the theoretical and numerical underpinnings provide details of the computational schemes including higher order versions and demonstrate the techniques in a collection of different areas

vanishing theorems for real algebraic cycles we establish the analogue of the friedlander mazur conjecture for teh s reduced lawson homology groups of real varieties which says that the reduced lawson homology of a real quasi projective variety x vanishes in homological degrees larger than the dimension of x in all weights as an application we obtain a vanishing of homotopy groups of the mod 2 topological groups of averaged cycles and a characterization in a range of indices of dos santos real lawson homology as the homotopy groups of the topological group of averaged cycles we also establish an equivariant poincare duality between equivariant friedlander walker real morphic cohomology and dos santos real lawson homology we use this together with an equivariant extension of the mod 2 beilinson lichtenbaum conjecture to compute some real lawson homology groups in terms of bredon cohomology

a spline based enrichment function for arbitrary inclusions in extended finite element method with applications to finite deformations summary a novel enrichment function which can model arbitrarily shaped inclusions within the framework of the extended finite element method is proposed the internal boundary of an arbitrary shaped inclusion is first discretized and a numerical enrichment function is constructed on the fly using spline interpolation we consider a piecewise cubic spline which is constructed from seven localized discrete boundary points the enrichment function is then determined by solving numerically a nonlinear equation which determines the distance from any point to the spline curve parametric convergence studies are carried out to show the accuracy of this approach compared with pointwise and linear segmentation of points for the construction of the enrichment function in the case of simple inclusions and arbitrarily shaped inclusions in linear elasticity moreover the viability of this approach is illustrated on a neo hookean hyperelastic material with a hole undergoing large deformation in this case the enrichment is able to adapt to the deformation and effectively capture the correct response without remeshing copyright 2013 john wiley sons ltd

a moving mesh finite element method for the two dimensional stefan problems an r adaptive moving mesh method is developed for the numerical solution of an enthalpy formulation of two dimensional heat conduction problems with a phase change the grid is obtained from a global mapping of the physical to the computational domain which is designed to cluster mesh points around the interface between the two phases of the material the enthalpy equation is discretised using a semiimplicit galerkin finite element method using linear basis functions the moving finite element method is applied to problems where the phase front is cusp shaped and where the interface changes topology

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

efficient methods for implicit geometrical representation of complex material microstructures summary we present two methods for initializing distance functions on adaptively refined finite element meshes to represent complex material microstructures from segmented x ray tomographic data implicit microstructure representation combined with the extended fem allows modelers to represent complex material microstructures with consistent mesh quality and accuracy in the first method a level set evolution equation is formulated and solved by the galerkin method on an adaptively refined mesh we show that the convergence and stability of this method is optimal for the order of elements used in the second approach we initialize the distance field by the fast marching method on a uniform grid and then project the solution onto the finite element mesh by least squares we show that this latter approach is superior in speed and accuracy as an example problem both methods are demonstrated in the initialization of distance fields for two inclusion phases within a al 7075 alloy copyright 2014 john wiley sons ltd

on the performance of strain smoothing for quadratic and enriched finite element approximations xfem gfem pufem by using the strain smoothing technique proposed by chen et al comput mech 2000 25 137 156 for meshless methods in the context of the finite element method fem liu et al comput mech 2007 39 6 859 877 developed the smoothed fem sfem although the sfem is not yet well understood mathematically numerical experiments point to potentially useful features of this particularly simple modification of the fem to date the sfem has only been investigated for bilinear and wachspress approximations and is limited to linear reproducing conditions the goal of this paper is to extend the strain smoothing to higher order elements and to investigate numerically in which condition strain smoothing is beneficial to accuracy and convergence of enriched finite element approximations we focus on three widely used enrichment schemes namely a weak discontinuities b strong discontinuities c near tip linear elastic fracture mechanics functions the main conclusion is that strain smoothing in enriched approximation is only beneficial when the enrichment functions are polynomial cases a and b but that non polynomial enrichment of type c lead to inferior methods compared to the standard enriched fem e g xfem copyright c 2011 john wiley sons ltd

a combined extended finite element and level set method for biofilm growth this paper presents a computational technique based on the extended finite element method xfem and the level set method for the growth of biofilms the discontinuous derivative enrichment of the standard finite element approximation eliminates the need for the finite element mesh to coincide with the biofilm fluid interface and also permits the introduction of the discontinuity in the normal derivative of the substrate concentration field at the biofilm fluid interface the xfem is coupled with a comprehensive level set update scheme with velocity extensions which makes updating the biofilm interface fast and accurate without need for remeshing the kinetics of biofilms are briefly given and the non linear strong and weak forms are presented the non linear system of equations is solved using a newton raphson scheme example problems including 1d and 2d biofilm growth are presented to illustrate the accuracy and utility of the method the 1d results we obtain are in excellent agreement with previous 1d results obtained using finite difference methods our 2d results that simulate finger formation and finger tip splitting in biofilms illustrate the robustness of the present computational technique copyright 2007 john wiley sons ltd

quaternionic algebraic cycles and reality in this paper we compute the equivariant homotopy type of spaces of algebraic cycles on real brauer severi varieties under the action of the galois group gal c r appropriate stabilizations of these spaces yield two equivariant spectra the first one classifies dupont seymour s quaternionic k theory and the other one classifies an equivariant cohomology theory 3 which is a natural recipient of characteristic classes kh x 3 x for quaternionic bundles over real spaces x

non planar 3d crack growth by the extended finite element and level sets part i mechanical model a methodology for solving three dimensional crack problems with geometries that are independent of the mesh is described the method is based on the extended finite element method in which the crack discontinuity is introduced as a heaviside step function via a partition of unity in addition branch functions are introduced for all elements containing the crack front the branch functions include asymptotic near tip fields that improve the accuracy of the method the crack geometry is described by two signed distance functions which in turn can be defined by nodal values consequently no explicit representation of the crack is needed examples for three dimensional elastostatic problems are given and compared to analytic and benchmark solutions the method is readily extendable to inelastic fracture problems copyright 2002 john wiley sons ltd

moving algebraic cycles of bounded degree the chow moving lemma is a theorem which asserts that a given algebraic s cycle on a smooth algebraic variety x can be moved within its rational equivalence class to intersect properly a given r cycle on x provided that r s dim x cf chow s2 in the past few years there has been considerable interest in studying spaces of algebraic cycles rather than simply cycles modulo an equivalence relation with this in mind it is natural to ask whether one can move a given bounded family of s cycles on the smooth variety x to intersect properly a given bounded family of r cycles the main point of this paper is to formulate and prove just such a result in theorem 3 1 we demonstrate that for any integer e and any smooth projective variety x one can simultaniously and algebraically move all effective s cycles of degree e on x so that each such cycle meets every effective r cycle of degree e on x in proper dimension the primary motivation for this moving lemma for cycles of bounded degree was the possibility of a duality theorem between cohomology and homology theories defined in terms of homotopy groups of cycle spaces using theorem 3 1 we have proved such a duality theorem for complex quasi projective varieties in f l2 we prove our moving lemma for varieties over an arbitary infinite field permitting a proof in f v of a duality theorem for motivic cohomology and homology the reader will find that our moving lemma has numerous good properties first of all the move is given as an algebraic move parametrized by a punctured projective line on chow varietes although this move is good only for s cycles of bounded degree it is defined on all effective s cycles moreover the move starts at time 0 by expressing an effective s cycle z as a difference of effective s cycles both of which have intersection properties no worse than z finally our moving lemma is applicable to smooth quasiprojective varieties for it is stated for a possibly singular projective variety x resulting in a conclusion of proper intersection off the singular locus of x the classical motivation for the moving lemma was to define an intersection product on algebraic cycles modulo rational equivalence thereby establishing the chow ring a x some of the classical literature overlooked the question of whether or not intersection of cycles defined via a moving lemma is independent of the move e g chow r s2 on the other hand cf chev s3 one direct consequence of our moving lemma is a proof for smooth quasi projective varieties that the intersection product is indeed well defined independent of the choice of move theorem 3 4 of course the intersection product now has an intrinsic formulation for all smooth algebraic varieties due to fulton

on the construction of blending elements for local partition of unity enriched finite elements for computational efficiency partition of unity enrichments are preferably localized to the sub domains where they are needed it is shown that an appropriate construction of the elements in the blending area the regionwhere the enriched elements blend to unenriched elements is often crucial for good performance of local partition of unity enrichments an enhanced strain formulation is developed which leads to good performance the optimal rate of convergence is achieved for polynomial enrichments it is shown that a proper choice of the finite element shape functions and partition of unity shape functions also improves the accuracy and convergence the methods are illustrated by several examples the examples deal primarily with the signed distance function enrichment for treating discontinuous derivatives inside an element but other enrichments are also considered results show that both methods provide optimal rates of convergence

a fully smoothed xfem for analysis of axisymmetric problems with weak discontinuities in this paper we propose a fully smoothed extended finite element method for axisymmetric problems with weak discontinuities the salient feature of the proposed approach is that all the terms in the stiffness and mass matrices can be computed by smoothing technique this is accomplished by combining the gaussian divergence theorem with the evaluation of indefinite integral based on smoothing technique which is used to transform the domain integral into boundary integral the proposed technique completely eliminates the need for isoparametric mapping and the computing of jacobian matrix even for the mass matrix when employed over the enriched elements the proposed technique does not require sub triangulation for the purpose of numerical integration the accuracy and convergence properties of the proposed technique are demonstrated with a few problems in elastostatics and elastodynamics with weak discontinuities it can be seen that the proposed technique yields stable and accurate solutions and is less sensitive to mesh distortion

a stable xfem formulation for multi phase problems enforcing the accuracy of the fluxes through lagrange multipliers summary when applied to diffusion problems in a multi phase setup the popular extended finite element method xfem strategy suffers from an inaccurate representation of the local fluxes in the vicinity of the interface the xfem enrichment improves the global quality of the solution but it is not enforcing any local feature to the fluxes thus the resulting numerical fluxes in the vicinity of the interface are not realistic in particular when conductivity ratios between the different phases are very high this paper introduces an additional restriction to the xfem formulation aiming at properly reproducing the features of the local fluxes in the transition zone this restriction is implemented through lagrange multipliers and the stability of the resulting mixed formulation is tested satisfactorily through the chapelle bathe numerical procedure several examples are presented and the solutions obtained show a spectacular improvement with respect to the standard xfem copyright 2013 john wiley sons ltd

graph mappings and poincare duality we give a new geometric description for a compact oriented pseduo manifold x of the poincare duality map from the integral cohomology of x to the integral homology of x our construction takes a multi valued lipschitz map on x with values in a sphere sn to its geometric measure theoretic graph in x sn and then to the slice of this graph as an integral cycle on x this construction is compatible with analogous constructions on algebraic cocycles on projective varieties employed by the authors and others

an efficient interface preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow in sussman smereka and osher j comp phys 94 1994 pp 146 159 a numerical scheme was presented for computing incompressible air water flows using the level set method crucial to the above method was a new iteration method for maintaining the level set function as the signed distance from the zero level set in this paper we implement a constraint along with higher order difference schemes in order to make the iteration method more accurate and efficient accuracy is measured in terms of the new computed signed distance function and the original level set function having the same zero level set we apply our redistancing scheme to incompressible flows with noticeably better resolved results at reduced cost we validate our results with experiment and theory we show that our distance level set scheme with the added constraint competes well with available interface tracking schemes for basic advection of an interface we perform basic accuracy checks and more stringent tests involving complicated interfacial structures as with all level set schemes our method is easy to implement

an explicit application of partition of unity approach to xfem approximation for precise reproduction of a priori knowledge of solution summary the application of the xfem to fracture mechanics is effective because a crack can be modeled independently from the meshes and a complex remeshing procedure can be avoided however the classical xfem has an essential problem in the approximation of partially enriched elements that is blending elements which causes a lack of accuracy for the weighted xfem although the numerical results show the effective improvements it was found that the issue of blending elements still remains upon detailed examination in the present paper the pu xfem is formulated as an explicit application of the partition of unity pu approach to the xfem in order to precisely reproduce a priori knowledge of the solution by enrichment the pu xfem is applied to two dimensional linear fracture mechanics and its effectiveness is verified it is consequently found out that the pu xfem precisely reproduces a priori knowledge of the solution and is therefore effective to completely solve the problem of the blending elements copyright 2013 john wiley sons ltd

integrating strong and weak discontinuities without integration subcells and example applications in an xfem gfem framework partition of unity methods such as the extended finite element method allows discontinuities to be simulated independently of the mesh int j numer meth engng 1999 45 601 620 this eliminates the need for the mesh to be aligned with the discontinuity or cumbersome re meshing as the discontinuity evolves however to compute the stiffness matrix of the elements intersected by the discontinuity a subdivision of the elements into quadrature subcells aligned with the discontinuity is commonly adopted in this paper we use a simple integration technique proposed for polygonal domains int j nuttier meth engng 2009 80 1 103 134 doi 10 1002 nme 2589 to suppress the need for element subdivision numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics and a multi material problem show that the proposed method yields accurate results owing to its simplicity the proposed integration technique can be easily integrated in any existing code copyright c 2010 john wiley sons ltd

non planar 3d crack growth by the extended finite element and level sets part ii level set update we present a level set method for treating the growth of non planar three dimensional cracks the crack is defined by two almost orthogonal level sets signed distance functions one of them describes the crack as a two dimensional surface in a three dimensional space and the second is used to describe the one dimensional crack front which is the intersection of the two level sets a hamilton jacobi equation is used to update the level sets a velocity extension is developed that preserves the old crack surface and can accurately generate the growing surface the technique is coupled with the extended finite element method which approximates the displacement field with a discontinuous partition of unity this displacement field is constructed directly in terms of the level sets so the discretization by finite elements requires no explicit representation of the crack surface numerical experiments show the robustness of the method both in accuracy and in treating cracks with significant changes in topology copyright 2002 john wiley sons ltd

the extended finite element method xfem for solidification problems an enriched finite element method for the multi dimensional stefan problems is presented in this method the standard finite element basis is enriched with a discontinuity in the derivative of the temperature normal to the interface the approximation can then represent the phase interface and the associated discontinuity in the temperature gradient within an element the phase interface can be evolved without re meshing or the use of artificial heat capacity techniques the interface is described by a level set function that is updated by a stabilized finite element scheme several examples are solved by the proposed method to demonstrate the accuracy and robustness of the method copyright 2001 john wiley sons ltd

numerical integration over arbitrary polygonal domains based on schwarz christoffel conformal mapping this paper presents a new numerical integration technique oil arbitrary polygonal domains the polygonal domain is mapped conformally to the unit disk using schwarz christoffel mapping and a midpoint quadrature rule defined oil this unit disk is used this method eliminates the need for a two level isoparametric mapping usually required moreover the positivity of the jacobian is guaranteed numerical results presented for a few benchmark problems in the context of polygonal finite elements show that the proposed method yields accurate results

rational isomorphisms between k theories and cohomology theories the well known isomorphism relating the rational algebraic k theory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map the segre map of infinite loop spaces moreover the associated chern character map on rational homotopy groups is shown to be a ring isomorphism a technique is introduced that establishes a useful general criterion for a natural transformation of functors on quasi projective complex varieties to induce a homotopy equivalence of semi topological singular complexes since semi topological k theory and morphic cohomology can be formulated as the semi topological singular complexes associated to algebraic k theory and motivic cohomology this criterion provides a rational isomorphism between the semi topological k theory groups and the morphic cohomology groups of a smooth complex variety consequences include a riemann roch theorem for the chern character on semi topological k theory and an interpretation of the topological filtration on singular cohomology groups in k theoretic terms

derivative recovery and a posteriori error estimate for extended finite elements this paper is the first attempt at error estimation for extended finite elements the goal of this work is to devise a simple and effective local a posteriori error estimate for partition of unity enriched finite element methods such as the extended finite element method xfem in each element the local estimator is the l2 norm of the difference between the raw xfem strain field and an enhanced strain field computed by extended moving least squares xmls derivative recovery obtained from the raw nodal xfem displacements the xmls construction is tailored to the nature of the solution the technique is applied to linear elastic fracture mechanics in which near tip asymptotic functions are added to the mls basis the xmls shape functions are constructed from weight functions following the diffraction criterion to represent the discontinuity the result is a very smooth enhanced strain solution including the singularity at the crack tip results are shown for two and three dimensional linear elastic fracture mechanics problems in mode i and mixed mode the effectivity index of the estimator is close to 1 and improves upon mesh refinement for the studied near tip problem it is also shown that for the linear elastic fracture mechanics problems treated the proposed estimator outperforms one of the superconvergent patch recovery technique of zienkiewicz and zhu which is only c0 parametric studies of the general performance of the estimator are also carried out

a partition of unity based finite element method for level sets level set methods have recently gained much popularity to capture discontinuities including their possible propagation typically the partial differential equations that arise in level set methods in particular the hamilton jacobi equation are solved by finite difference methods however finite difference methods are less suited for irregular domains moreover it seems slightly awkward to use finite differences for the capturing of a discontinuity while in a subsequent stress analysis finite elements are normally used for this reason we here present a finite element approach to solving the governing equations of level set methods after a review of the governing equations the initialization of the level sets the discretization on a finite domain and the stabilization of the resulting finite element method will be discussed special attention will be given to the proper treatment of the internal boundary condition which is achieved by exploiting the partition of unity property of finite element shape functions finally a quantitative analysis including accuracy analysis is given for a one dimensional example and a qualitative example is given for a two dimensional case with a curved discontinuity

numerical schemes for the hamilton jacobi and level set equations on triangulated domains borrowing from techniques developed for conservation law equations numerical schemes which discretize the hamilton jacobi h j level set and eikonal equations on triangulated domains are presented the first scheme is a provably monotone discretization for the h j equations unfortunately the basic scheme lacks lipschitz continuity of the numerical hamiltonian by employing a virtual edge flipping technique local lipschitz continuity of the numerical flux is restored on acute triangulations next schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous hamiltonians these schemes possess a discrete maximum principle on arbitrary triangulations and exhibit local lipschitz continuity of the numerical hamiltonian finally a class of petrov galerkin approximations is considered these schemes are stabilized via a least squares bilinear form the petrov galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy discretization of the level set equation also requires the numerical approximation of a mean curvature term a simple mass lumped galerkin approximation is presented in section 6 and analyzed using maximum principle analysis the use of unstructured meshes permits several forms of mesh adaptation which have been incorporated into numerical examples these numerical examples include discretizations of convex and nonconvex forms of the h j equation the eikonal equation and the level set equation

generalized abel jacobi map on lawson homology we construct an abel jacobi type map on the homologically trivial part of lawson homology groups it generalizes the abel jacobi map constructed by griffiths by using a result of h clemens we answer affirmatively the question whether there exists a smooth projective complex variety with infinitely generated lawson homology groups l ph 2p k x bbb q when k0 as a corollary we find for any nonnegative integer j a smooth complex projective variety x carrying infinitely generated semi topological k groups k sst j x bbb q

diffusional evolution of precipitates in elastic media using the extended finite element and the level set methods a sharp interface numerical formulation using an eulerian description aimed at modeling diffusional evolution of precipitates produced by phase transformations in elastic media is presented the extended finite element method xfem is used to solve the field equations and the level set method is used to evolve the precipitate matrix interface this new formulation is capable of handling microstructures with arbitrarily shaped particles and capturing their topological transitions without needing the mesh to conform with the precipitate matrix interface the xfem makes it possible to model the precipitate and the matrix to be both elastically anisotropic and inhomogeneous with ease the interface evolution velocity is evaluated using a domain integral scheme 1 that is consistent with the sharp interface numerical examples modeling two distinct phases of particle evolution growth dendritic evolution and equilibration ostwald ripening are presented to overcome the issue of grid anisotropy in growth simulations a random grid rotation scheme is implemented in conjunction with a bicubic spline interpolation scheme growing shapes are dendritic while equilibrium shapes are squarish and in this respect our simulation results are in agreement with those presented in the literature 2 4