a g space theory and a weakened weak w2 form for a unified formulation of compatible and incompatible methods part ii applications to solid mechanics problems in part i of this paper we have established the g space theory and fundamentals for w 2 formulation part ii focuses on the applications of the g space theory to formulate w 2 models for solid mechanics problems we first define a bilinear form prove some of the important properties and prove that the w 2 formulation will be spatially stable and convergent to exact solutions we then present examples of some of the possible w 2 models including the sfem ns fem es fem ns pim es pim and cs pim we show the major properties of these models 1 they are variationally consistent in a conventional sense if the solution is sought in a proper h space compatible cases 2 they pass the standard patch test when the solution is sought in a proper g space with discontinuous functions incompatible cases 3 the stiffness of the discretized model is reduced compared with the finite element method fem model and possibly to the exact model allowing us to obtain upper bound solutions with respect to both the fem and the exact solutions and 4 the w 2 models are less sensitive to the quality of the mesh and triangular meshes can be used without any accuracy problems these properties and theories have been confirmed numerically via examples solved using a number of w 2 models including compatible and incompatible cases we shall see that the g space theory and the w 2 forms can formulate a variety of stable and convergent numerical methods with the fem as one special case

a stabilized conforming nodal integration for galerkin mesh free methods domain integration by gauss quadrature in the galerkin mesh free methods adds considerable complexity to solution procedures direct nodal integration on the other hand leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes a strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration for convergence an integration constraint ic is introduced as a necessary condition for a linear exactness in the mesh free galerkin approximation the gradient matrix of strain smoothing is shown to satisfy ic using a divergence theorem no numerical control parameter is involved in the proposed strain smoothing stabilization the numerical results show that the accuracy and convergent rates in the mesh free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method it is also demonstrated that the gauss integration method fails to meet ic in mesh free discretization for this reason the proposed method provides even better accuracy than gauss integration for galerkin mesh free method as presented in several numerical examples copyright 2001 john wiley sons ltd

a theoretical study on the smoothed fem s fem models properties accuracy and convergence rates incorporating the strain smoothing technique of meshfree methods into the standard finite element method fem liu et al have recently proposed a series of smoothed finite element methods s fem for solid mechanics problems in these s fem models the compatible strain fields are smoothed based on smoothing domains associated with entities of elements such as elements nodes edges or faces and the smoothed galerkin weak form based on these smoothing domains is then applied to compute the system stiffness matrix we present in this paper a general and rigorous theoretical framework to show properties accuracy and convergence rates of the s fem models first an assumed strain field derived from the hellinger reissner variational principle is shown to be identical to the smoothed strain field used in the s fem models we then define a smoothing projection operator to modify the compatible strain field and show a set of properties we next establish a general error bound of the s fem models some numerical examples are given to verify the theoretical properties established copyright 2010 john wiley sons ltd

community detection in degree corrected block models community detection is a central problem of network data analysis given a network the goal of community detection is to partition the network nodes into a small number of clusters which could often help reveal interesting structures the present paper studies community detection in degree corrected block models dcbms we first derive asymptotic minimax risks of the problem for a misclassification proportion loss under appropriate conditions the minimax risks are shown to depend on degree correction parameters community sizes and average within and between community connectivities in an intuitive and interpretable way in addition we propose a polynomial time algorithm to adaptively perform consistent and even asymptotically optimal community detection in dcbms

exponential error rates of sdp for block models beyond grothendieck s inequality in this paper we consider the cluster estimation problem under the stochastic block model we show that the semidefinite programming sdp formulation for this problem achieves an error rate that decays exponentially in the signal to noise ratio the error bound implies weak recovery in the sparse graph regime with bounded expected degrees as well as exact recovery in the dense regime an immediate corollary of our results yields error bounds under the censored block model moreover these error bounds are robust continuing to hold under heterogeneous edge probabilities and a form of the so called monotone attack significantly this error rate is achieved by the sdp solution itself without any further pre or post processing and improves upon existing polynomially decaying error bounds proved using the grothendieck s inequality our analysis builds on two key ingredients 1 showing that the graph has a well behaved spectrum even in the sparse regime after discounting an exponentially small number of edges and 2 an order statistics argument that governs the final error rate both arguments highlight the implicit regularization effect of the sdp formulation

minimax rates of community detection in stochastic block models in statistics as well as in computer science probability and applied mathematics community detection for the stochastic block model sbm is probably the most studied topic in network analysis many methodologies have been proposed some beautiful and signicant phase transition results are obtained in various settings in this paper we provide a general minimax theory for community detection it gives minimax rates of the mis match ratio for a wide rage of settings including homogeneous and inhomogeneous sbms dense and sparse networks nite and growing number of communities the minimax rates are exponential dierent from polynomial rates we often see in statistical literature an immediate consequence of the result is to establish threshold phenomenon for strong consistency exact recovery as well as weak consistency partial recovery we obtain the upper bound by a range of penalized likelihood type approaches the lower bound is achieved by a novel reduction from a global mis match ratio to a local clustering problem for one node through an exchangeability property

irregular dynamics and homoclinic orbits to hamiltonian saddle centers we consider 4 dimensional real analytic hamiltonian systems with a saddle center equilibrium related to a pair of real and a pair of imaginary eigenvalues and a homoclinic orbit to it we find conditions for the existence of transversal homoclinic orbits to periodic orbits of long period in every energy level sufficiently close to the energy level of the saddle center equilibrium we also consider one parameter families of reversible 4 dimensional hamiltonian systems we prove that the set of parameter values where the system has homoclinic orbits to a saddle center equilibrium has no isolated points we also present similar results for systems with heteroclinic orbits to saddle center equilibria 1997 john wiley sons inc

a g space theory and a weakened weak w2 form for a unified formulation of compatible and incompatible methods part i theory this paper introduces a g space theory and a weakened weak form w2 using the generalized gradient smoothing technique for a unified formulation of a wide class of compatible and incompatible methods the w2 formulation works for both finite element method settings and mesh free settings and w2 models can have special properties including softened behavior upper bounds and ultra accuracy part i of this paper focuses on the theory and fundamentals for w2 formulations a normed g space is first defined to include both continuous and discontinuous functions allowing the use of much more types of methods techniques to create shape functions for numerical models important properties and a set of useful inequalities for g spaces are then proven in the theory and analyzed in detail these properties ensure that a numerical method developed based on the w2 formulation will be spatially stable and convergent to the exact solutions as long as the physical problem is well posed the theory is applicable to any problems to which the standard weak formulation is applicable and can offer numerical solutions with special properties including close to exact stiffness upper bounds and ultra accuracy copyright 2009 john wiley sons ltd

meshfree and finite element nodal integration methods nodal integration can be applied to the galerkin weak form to yield a particle type method where stress and material history are located exclusively at the nodes and can be employed when using meshless or finite element shape functions this particle feature of nodal integration is desirable for large deformation settings because it avoids the remapping or advection of the state variables required in other methods to a lesser degree nodal integration can be desirable because it relies on fewer stress point evaluations than most other methods in this work aspects regarding stability consistency efficiency and explicit time integration are explored within the context of nodal integration both small and large deformation numerical examples are provided copyright 2007 john wiley sons ltd

global surfaces of section in non regular convex energy levels of hamiltonian systems in this paper we prove the existence of global sections of disk type in non regular and strictly convex energy levels of integrable and near integrable hamiltonian systems with two degrees of freedom this extends a result of hofer et al in ann math 2 148 1 197 289 1998 where the same statement is true provided the energy level is regular

nonintegrability of some hamiltonian systems scattering and analytic continuation we consider canonical two degrees of freedom analytic hamiltonian systems with hamiltonian functionh 1 2 p12 p22 u q1 q2 where u q1 q2 1 2 v2q12 2q22 o q12 q22 3 2 and q2 u q1 0 0 under some additional not so restrictive hypothesis we present explicit conditions for the exisstence of transversal homoclinic orbits to some periodic orbits of these systems we use a theorem of lerman 1991 and an analogy between one of its conditions with the usual one dimensional quantum scattering problem the study of the scattering equation leads us to an analytic continuation problem for the solutions of a linear second order differential equation we apply our results to some classical problems

renyi divergence measures for commonly used univariate continuous distributions abstract probabilistic distances also called divergences which in some sense assess how close two probability distributions are from one another have been widely employed in probability statistics information theory and related fields of particular importance due to their generality and applicability are the renyi divergence measures this paper presents closed form expressions for the renyi and kullback leibler divergences for nineteen commonly used univariate continuous distributions as well as those for multivariate gaussian and dirichlet distributions in addition a table summarizing four of the most important information measure rates for zero mean stationary gaussian processes namely renyi entropy differential shannon entropy renyi divergence and kullback leibler divergence is presented lastly a connection between the renyi divergence and the variance of the log likelihood ratio of two distributions is established thereby extending a previous result by song j stat plan infer 93 2001 on the relation between renyi entropy and the log likelihood function a table with the corresponding variance expressions for the univariate distributions considered here is also included

gradient enhanced damage for quasi brittle materials summary conventional continuum damage descriptions of material degeneration suffer from loss of well posedness beyond a certain level of accumulated damage as a consequence numerical solutions are obtained which are unacceptable from a physical point of view the introduction of higher order deformation gradients in the constitutive model is demonstrated to be an adequate remedy to this deficiency of standard damage models a consistent numerical solution procedure of the governing partial differential equations is presented which is shown to be capable of properly simulating localization phenomena

convergence in the incompressible limit of finite element approximations based on the hu washizu formulation the classical hu washizu mixed formulation for plane problems in elasticity is examined afresh with the emphasis on behavior in the incompressible limit the classical continuous problem is embedded in a family of hu washizu problems parametrized by a scalar for which alpha lambda big mu corresponds to the classical formulation with and being the lame parameters uniform well posedness in the incompressible limit of the continuous problem is established for 1 finite element approximations are based on the choice of piecewise bilinear approximations for the displacements on quadrilateral meshes conditions for uniform convergence are made explicit these conditions are shown to be met by particular choices of bases for stresses and strains and include bases that are well known as well as newly constructed bases though a discrete version of the spherical part of the stress exhibits checkerboard modes it is shown that a independent a priori error estimate for the displacement can be established furthermore a independent estimate is established for the post processed stress the theoretical results are explored further through selected numerical examples

a fully locking free isogeometric approach for plane linear elasticity problems a stream function formulation in this paper we study plane incompressible elastic problems by means of a stream function formulation such that a divergence free displacement field can be computed from a scalar potential the numerical scheme is constructed within the framework of nurbs based isogeometric analysis and we take advantage of the high continuity guaranteed by nurbs basis functions in order to obtain the displacement field from the potential differentiation as a consequence the obtained numerical scheme is automatically locking free in the presence of the incompressibility constraint a discontinuous galerkin approach is proposed to deal with multiple mapped possibly multiply connected domains extensive numerical results are provided to show the method capabilities 2007 elsevier b v all rights reserved

on the stability of double homoclinic loops we consider 2 degrees of freedom hamiltonian systems with an involutive symmetry and a pair of orbits bi asymptotic homoclinic to a saddle center equilibrium related to pairs of pure real and pure imaginary eigenvalues i we show that the stability of this double homoclinic loop is determined by the reflection coefficient of a one dimensional scattering problem and we also show that the mechanism for losing stability is the creation of an infinite heteroclinic chain connecting a sequence of periodic orbits that accumulates at the double loop

f bar based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids part i formulation and benchmarking this paper proposes a new technique which allows the use of simplex finite elements linear triangles in 2d and linear tetrahedra in 3d in the large strain analysis of nearly incompressible solids the new technique extends the f bar method proposed by de souza neto et al int j solids and struct 1996 33 3277 3296 and is conceptually very simple it relies on the enforcement of near incompressibility over a patch of simplex elements rather than the point wise enforcement of conventional displacement based finite elements within the framework of the f bar method this is achieved by assuming for each element of a mesh a modified f bar deformation gradient whose volumetric component is defined as the volume change ratio of a pre defined patch of elements the resulting constraint relaxation effectively overcomes volumetric locking and allows the successful use of simplex elements under finite strain near incompressibility as the original f bar procedure the present methodology preserves the displacement based structure of the finite element equations as well as the strain driven format of standard algorithms for numerical integration of path dependent constitutive equations and can be used regardless of the constitutive model adopted the new elements are implemented within an implicit quasi static environment in this context a closed form expression for the exact tangent stiffness of the new elements is derived this allows the use of the full newton raphson scheme for equilibrium iterations the performance of the proposed elements is assessed by means of a comprehensive set of benchmarking two and three dimensional numerical examples

achieving the bayes error rate in synchronization and block models by sdp robustly we study the statistical performance of semidefinite programming sdp relaxations for clustering under random graph models under the mathbb z 2 synchronization model censored block model cbm and stochastic block model sbm we show that sdp achieves an error rate of the form exp 1 o 1 bar n i here bar n is an appropriate multiple of the number of nodes and i is an information theoretic measure of the signal to noise ratio we provide matching lower bounds on the bayes error for each model and therefore demonstrate that the sdp approach is bayes optimal as a corollary our results imply that sdp achieves the optimal exact recovery threshold under each model furthermore we show that sdp is robust the above bound remains valid under semirandom versions of the models in which the observed graph is modified by a monotone adversary our proof is based on a novel primal dual analysis of sdp under a unified framework for all three models and the analysis shows that sdp tightly approximates a joint majority voting procedure

conditions for equivalence between the hu washizu and related formulations and computational behavior in the incompressible limit the relationship of the hu washizu mixed formulation to other mixed and enhanced formulations is examined in detail in the context of linear elasticity with a view to presenting a unified framework for such formulations the hu washizu formulation is considered in both its classical form and in a modified form that is suited to establishing well posedness in the incompressible limit recently established theoretical results on uniform well posedness are studied computationally by considering a range of numerical examples which illustrate where appropriate the good behavior predicted by the theory as well as the locking behavior that is evident in those problems in which the conditions for stability and convergence are not satisfied 2005 elsevier b v all rights reserved

assumed deformation gradient finite elements with nodal integration for nearly incompressible large deformation analysis an assumed strain finite element technique for non linear finite deformation is presented the weighted residual method enforces weakly the balance equation with the natural boundary condition and also the kinematic equation that links the elementwise and the assumed deformation gradient assumed gradient operators are derived via nodal integration from the kinematic weighted residual a variety of finite element shapes fits the derived framework four node tetrahedra eight 27 and 64 node hexahedra are presented here since the assumed deformation gradients are expressed entirely in terms of the nodal displacements the degrees of freedom are only the primitive variables displacements at the nodes the formulation allows for general anisotropic materials and no volumetric deviatoric split is required the consistent tangent operator is inexpensive and symmetric furthermore the material update and the tangent moduli computation are carried out exactly as for classical displacement based models the only deviation is the consistent use of the assumed deformation gradient in place of the displacement derived deformation gradient examples illustrate the performance with respect to the ability of the present technique to resist volumetric locking a constraint count can partially explain the insensitivity of the resulting finite element models to locking in the incompressible limit copyright 2008 john wiley sons ltd

fast and accurate 4 node quadrilateral an accurate and variationally consistent 4 node quadrilateral element is introduced where high coarse mesh accuracy and low mesh distortion sensitivity are characteristic qualities even when incompressibility is approached for plane strain one point quadrature integration procedure is adopted and a new improved stabilization technique is developed orthogonality conditions are utilized so that the patch test is satisfied for arbitrary quadrilaterals several numerical examples including a convergence rate study are presented which confirm the excellent performance of this element copyright c 2004 john wiley sons ltd

conservative dynamics unstable sets for saddle center loops we consider two degree of freedom hamiltonian systems with a saddle center loop namely an orbit homoclinic to a saddle center equilibrium related to pairs of pure real and pure imaginary i eigenvalues we study the topology of the sets of orbits that have the saddle center loop as their and limit set a saddle center loop as a periodic orbit is a closed loop in phase space and the above sets are analogous to the unstable and stable manifolds respectively of a hyperbolic periodic orbit

an edge based smoothed finite element method es fem with stabilized discrete shear gap technique for analysis of reissner mindlin plates abstract an edge based smoothed finite element method es fem for static free vibration and buckling analyses of reissner mindlin plates using 3 node triangular elements is studied in this paper the calculation of the system stiffness matrix is performed by using the strain smoothing technique over the smoothing domains associated with edges of elements in order to avoid the transverse shear locking and to improve the accuracy of the present formulation the es fem is incorporated with the discrete shear gap dsg method together with a stabilization technique to give a so called edge based smoothed stabilized discrete shear gap method es dsg the numerical examples demonstrated that the present es dsg method is free of shear locking and achieves the high accuracy compared to the exact solutions and others existing elements in the literature

a smoothed finite element method for plate analysis a quadrilateral element with smoothed curvatures for mindlin reissner plates is proposed the curvature at each point is obtained by a non local approximation via a smoothing function the bending stiffness matrix is calculated by a boundary integral along the boundaries of the smoothing elements smoothing cells numerical results show that the proposed element is robust computational inexpensive and simultaneously very accurate and free of locking even for very thin plates the most promising feature of our elements is their insensitivity to mesh distortion

an edge based smoothed finite element method for 3d analysis of solid mechanics problems summary the edge based smoothed finite element method es fem was proposed recently in liu nguyen thoi and lam to improve the accuracy of the fem for 2d problems this method belongs to the wider family of the smoothed fem for which smoothing cells are defined to perform the numerical integration over the domain later the face based smoothed fem fs fem was proposed to generalize the es fem to 3d problems according to this method the smoothing cells are centered along the faces of the tetrahedrons of the mesh in the present paper an alternative method for the extension of the es fem to 3d is investigated this method is based on an underlying mesh composed of tetrahedrons and the approximation of the field variables is associated with the tetrahedral elements however in contrast to the fs fem the smoothing cells of the proposed es fem are centered along the edges of the tetrahedrons of the mesh from selected numerical benchmark problems it is observed that the es fem is characterized by a higher accuracy and improved computational efficiency as compared with linear tetrahedral elements and to the fs fem for a given number of degrees of freedom copyright 2013 john wiley sons ltd

theoretical aspects of the smoothed finite element method sfem this paper examines the theoretical bases for the smoothed finite element method sfem which was formulated by incorporating cell wise strain smoothing operation into standard compatible finite element method fem the weak form of sfem can be derived from the hu washizu three field variational principle for elastic problems it is proved that 1d linear element and 2d linear triangle element in sfem are identical to their counterparts in fem while 2d bilinear quadrilateral elements in sfem are different from that of fem when the number of smoothing cells scs of the elements equals 1 the sfem solution is proved to be variationally consistent and has the same properties with those of fem using reduced integration when sc approaches infinity the sfem solution will approach the solution of the standard displacement compatible fem model when sc is a finite number larger than 1 the sfem solutions are not variationally consistent but energy consistent and will change monotonously from the solution of sfem sc 1 to that of sfem sc it is suggested that there exists an optimal number of sc such that the sfem solution is closest to the exact solution the properties of sfem are confirmed by numerical examples copyright 2006 john wiley sons ltd

norms of random matrices local and global problems abstract can the behavior of a random matrix be improved by modifying a small fraction of its entries consider a random matrix a with i i d entries we show that the operator norm of a can be reduced to the optimal order o n by zeroing out a small submatrix of a if and only if the entries have zero mean and finite variance moreover we obtain an almost optimal dependence between the size of the removed submatrix and the resulting operator norm our approach utilizes the cut norm and grothendieck pietsch factorization for matrices and it combines the methods developed recently by c le and r vershynin and by e rebrova and k tikhomirov

a low order finite element method for three dimensional linear elasticity problems with general meshes abstract the paper is concerned with a low order finite element method namely the staggered cell centered finite element method which has been proposed and analyzed in ong et al 2015 for two dimensional compressible and nearly incompressible linear elasticity problems in this work we extend the results to the three dimensional case and focus on the creating of the meshes in particular from a general primal mesh m we construct a polygonal dual mesh m and its submesh m in a way such that each dual control volume of m corresponds to a primal vertex and is a union macro element of some fixed number of adjacent tetrahedral elements of m the displacement is approximated by piecewise trilinear functions on the subdual mesh m and the pressure by piecewise constant functions on the dual mesh m as for two dimensional case such construction of the meshes and approximation spaces satisfies the macroelement condition which implies stability and convergence of the scheme numerical experiments are carried out to investigate the performance of the proposed method on various mesh types

a face based smoothed finite element method fs fem for 3d linear and geometrically non linear solid mechanics problems using 4 node tetrahedral elements this paper presents a novel face based smoothed finite element method fs fem to improve the accuracy of the finite element method fem for three dimensional 3d problems the fs fem uses 4 node tetrahedral elements that can be generated automatically for complicated domains in the fs fem the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the faces of the tetrahedral elements the results demonstrated that the fs fem is significantly more accurate than the fem using tetrahedral elements for both linear and geometrically non linear solid mechanics problems in addition a novel domain based selective scheme is proposed leading to a combined fs ns fem model that is immune from volumetric locking and hence works well for nearly incompressible materials the implementation of the fs fem is straightforward and no penalty parameters or additional degrees of freedom are used the computational efficiency of the fs fem is found better than that of the fem copyright 2008 john wiley sons ltd

matrix completion from a few entries let m be an n n matrix of rank r and assume that a uniformly random subset e of its entries is observed we describe an efficient algorithm which we call optspace that reconstructs m from e o rn observed entries with relative root mean square error 1 2 rmse c nr e 1 2 with probability larger than 1 1 n3 further if r o 1 and m is sufficiently unstructured then optspace reconstructs it exactly from e o n log n entries with probability larger than 1 1 n3 this settles in the case of bounded rank a question left open by candes and recht and improves over the guarantees for their reconstruction algorithm the complexity of our algorithm is o e r log n which opens the way to its use for massive data sets in the process of proving these statements we obtain a generalization of a celebrated result by friedman kahn szemeredi and feige ofek on the spectrum of sparse random matrices

stability and comparison of different linear tetrahedral formulations for nearly incompressible explicit dynamic applications this papers summarizes two linear tetrahedral fe formulations that have been recently proposed to overcome volumetric locking in nearly incompressible explicit dynamic applications in particular the average nodal pressure anp technique described by bonet and burton communications in numerical methods in engineering 1998 14 437 449 is briefly reviewed in addition the split based formulation proposed by zienkiewicz et al international journal for numerical methods in engineering 1998 43 565 583 is described here in terms of a time integration of the nodal jacobian this will make it simple to compare both techniques and will enable a new combined method to be presented the paper will then discuss the stability constraints that each technique places on the timestep size a von neuman stability analysis on simple 1 d uniform meshes will show that the anp element permits the use of much larger timesteps than the split based formulations finally numerical examples corroborating in 3 d this analytical conclusions will be presented copyright 2001 john wiley sons ltd

an edge based smoothed finite element method es fem for static free and forced vibration analyses of solids abstract this paper presents an edge based smoothed finite element method es fem to significantly improve the accuracy of the finite element method fem without much changing to the standard fem settings the es fem can use different shape of elements but prefers triangular elements that can be much easily generated automatically for complicated domains in the es fem the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the edges of the triangles intensive numerical results demonstrated that the es fem possesses the following excellent properties 1 the es fem model possesses a close to exact stiffness it is much softer than the overly stiff fem and much stiffer than the overly soft ns fem model 2 the results are often found superconvergence and ultra accurate much more accurate than the linear triangular elements of fem and even more accurate than those of the fem using quadrilateral elements with the same sets of nodes 3 there are no spurious non zeros energy modes found and hence the method is also temporally stable and works well for vibration analysis and 4 the implementation of the method is straightforward and no penalty parameter is used and the computational efficiency is better than the fem using the same sets of nodes in addition a novel domain based selective scheme is proposed leading to a combined es ns fem model that is immune from volumetric locking and hence works very well for nearly incompressible materials these properties of the es fem are confirmed using examples of static free and forced vibration analyses of solids

nonbacktracking spectrum of random graphs community detection and nonregular ramanujan graphs a nonbacktracking walk on a graph is a directed path such that no edge is the inverse of its preceding edge the nonbacktracking matrix of a graph is indexed by its directed edges and can be used to count nonbacktracking walks of a given length it has been used recently in the context of community detection and has appeared previously in connection with the ihara zeta function and in some generalizations of ramanujan graphs in this work we study the largest eigenvalues of the nonbacktracking matrix of the erd s renyi random graph and of the stochastic block model in the regime where the number of edges is proportional to the number of vertices our results confirm the spectral redemption conjecture in the symmetric case and show that community detection can be made on the basis of the leading eigenvectors above the feasibility threshold

locking effects in the finite element approximation of elasticity problems we consider the finite element approximation of the 2d elasticity problem when the poisson ratiov is close to 0 5 it is well known that the performance of certain commonly used finite elements deteriorates asv 0 a phenomenon calledlocking we analyze this phenomenon and characterize the strength of the locking androbustness of varioush version schemes using triangular and rectangular elements we prove that thep andh p versions are free of locking with respect to the error in the energy norm a generalization of our theory to the 3d problem is also discussed

a mimetic discretization of the stokes problem with selected edge bubbles a new mimetic finite difference method for the stokes problem is proposed and analyzed the mimetic discretization methodology can be understood as a generalization of the finite element method to meshes with general polygons polyhedrons in this paper the mimetic generalization of the unstable p 1 p 0 and the conditionally stable q1 p0 finite element is shown to be fully stable when applied to a large range of polygonal meshes moreover we show how to stabilize the remaining cases by adding a small number of bubble functions to selected mesh edges a simple strategy for selecting such edges is proposed and verified with numerical experiments

reconstruction and estimation in the planted partition model the planted partition model also known as the stochastic blockmodel is a classical cluster exhibiting random graph model that has been extensively studied in statistics physics and computer science in its simplest form the planted partition model is a model for random graphs on n nodes with two equal sized clusters with an between class edge probability of q and a within class edge probability of p although most of the literature on this model has focused on the case of increasing degrees ie pn qn rightarrow infty as n rightarrow infty the sparse case p q o 1 n is interesting both from a mathematical and an applied point of view a striking conjecture of decelle krzkala moore and zdeborova based on deep non rigorous ideas from statistical physics gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model in particular if p a n and q b n then decelle et al conjectured that it is possible to cluster in a way correlated with the true partition if a b 2 2 a b and impossible if a b 2 2 a b by comparison the best known rigorous result is that of coja oghlan who showed that clustering is possible if a b 2 c a b for some sufficiently large c we prove half of their prediction showing that it is indeed impossible to cluster if a b 2 2 a b furthermore we show that it is impossible even to estimate the model parameters from the graph when a b 2 2 a b on the other hand we provide a simple and efficient algorithm for estimating a and b when a b 2 2 a b following decelle et al our work establishes a rigorous connection between the clustering problem spin glass models on the bethe lattice and the so called reconstruction problem this connection points to fascinating applications and open problems

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

improved graph clustering graph clustering involves the task of dividing nodes into clusters so that the edge density is higher within clusters as opposed to across clusters a natural classic and popular statistical setting for evaluating solutions to this problem is the stochastic block model also referred to as the planted partition model in this paper we present a new algorithm a convexified version of maximum likelihood for graph clustering we show that in the classic stochastic block model setting it outperforms existing methods by polynomial factors when the cluster size is allowed to have general scalings in fact it is within logarithmic factors of known lower bounds for spectral methods and there is evidence suggesting that no polynomial time algorithm would do significantly better we then show that this guarantee carries over to a more general extension of the stochastic block model our method can handle the settings of semirandom graphs heterogeneous degree distributions unequal cluster sizes unaffiliated nodes partially observed graphs planted clique coloring and so on in particular our results provide the best exact recovery guarantees to date for the planted partition planted k disjoint cliques and planted noisy coloring models with general cluster sizes in other settings we match the best existing results up to logarithmic factors

homoclinic bifurcations in reversible hamiltonian systems abstract we study the existence of homoclic solutions for reversible hamiltonian systems taking the family of differential equations u iv au u f u b 0 as a model where f is an analytic function and a b real parameters these equations are important in several physical situations such as solitons and in the existence of finite energy stationary states of partial differential equations but no assumptions of any kind of discrete symmetry is made and the analysis here developed can be extended to others hamiltonian systems and successfully employed in situations where standard methods fail we reduce the problem of computing these orbits to that of finding the intersection of the unstable manifold with a suitable set and then apply it to concrete situations we also plot the homoclinic values configuration in parameters space giving a picture of the structural distribution and a geometrical view of homoclinic bifurcations

assumed strain finite element methods for conserving temporal integrations in non linear solid dynamics this paper presents a new class of assumed strain finite elements to use in combination with general energy momentum conserving time stepping algorithms so that these conservation properties in time are preserved by the fully discretized system in space and time the case of interest corresponds to nearly incompressible material responses in the fully non linear finite strain elastic and elastoplastic ranges the new elements consider the classical scaling of the deformation gradient with an assumed jacobian its determinant defined locally through a weighted averaging procedure at the element level the key aspect of the newly proposed formulation is the definition of the associated linearized strain operator or b bar operator the developments presented here start by identifying the conditions that this discrete operator must satisfy for the fully discrete system in time and space to inherit exactly the conservation laws of linear and angular momenta and the conservation dissipation law of energy for elastic and inelastic problems respectively care is also taken of the preservation of the relative equilibria and the corresponding group motions associated with the momentum conservation laws and characterized by purely rotational and translational motions superimposed to the equilibrium deformed configuration with these developments at hand a new general b bar operator is introduced that satisfies these conditions the new operator not only accounts for the spatial interpolations e g bilinear displacements with piece wise constant volume but also depends on the discrete structure of the equations in time the aforementioned conservation dissipation properties of energy and momenta are then proven to hold rigorously for the final numerical schemes unconditionally of the time step size and the material model elastic or elastoplastic different finite elements are considered in this framework including quadrilateral and triangular elements for plane problems and brick elements for three dimensional problems several representative numerical simulations are presented involving in particular the use of energy dissipating momentum conserving time stepping schemes recently developed by the author and co workers for general finite strain elastoplasticity in order to illustrate the properties of the new finite elements including these conservation dissipation properties in time and their locking free response in the quasi incompressible case copyright 2007 john wiley sons ltd

a locking free finite difference method on staggered grids for linear elasticity problems abstract a finite difference method on staggered grids is constructed on general nonuniform rectangular partition for linear elasticity problems stability optimal order error estimates in discrete h 1 norms on general nonuniform grids and second order superconvergence on almost uniform grids have been obtained these theoretical results are uniform about the lame constant 0 so the finite difference method is locking free the method and theoretical results can be extended to three dimensional problems numerical experiments using the method show agreement of the numerical results with theoretical analysis

community detection in sparse networks via grothendieck s inequality we present a simple and flexible method to prove consistency of semidefinite optimization problems on random graphs the method is based on grothendieck s inequality unlike the previous uses of this inequality that lead to constant relative accuracy we achieve any given relative accuracy by leveraging randomness we illustrate the method with the problem of community detection in sparse networks those with bounded average degrees we demonstrate that even in this regime various simple and natural semidefinite programs can be used to recover the community structure up to an arbitrarily small fraction of misclassified vertices the method is general it can be applied to a variety of stochastic models of networks and semidefinite programs

an edge based smoothed finite element method es fem for analyzing three dimensional acoustic problems abstract it is well known that one key issue of solving the helmholtz equation using finite element method fem is the accuracy deterioration in the solution with increasing wave number due to the numerical dispersion error such a numerical dispersion error is essentially caused by the overly stiff nature of the fem model to overcome this problem this paper presents an edge based smoothed finite element method es fem for analyzing acoustic problems using linear triangular and tetrahedron elements that can be generated automatically respectively for complicated two dimensional and three dimensional domains the discretized linear system equations for es fem are established using the smoothed galerkin weak form with smoothing domains associated with the edges of the triangles or surfaces of the tetrahedrons the edge based gradient smoothing operation provides proper softening effect makes the es fem model much softer than the overly stiff fem model and hence significantly reduces the numerical dispersion error numerical examples including a 2d problem of acoustic pressure distribution in a vehicle passenger compartment and a 3d problem about the acoustic pressure distribution in an engine chamber have been studied using the present es fem the results demonstrate that the es fem possesses the following advantages compared with the standard fem using the same meshes first es fem achieves similar convergence rate but better accuracy especially at high frequency second es fem is less sensitive to the mesh distortion meaning that the quality of mesh has less effect on the solution of es fem third it works well for triangular types of meshes and thus for the problems with complicated geometry

upper bound solution to elasticity problems a unique property of the linearly conforming point interpolation method lc pim it is well known that the displacement based fully compatible finite element method fem provides a lower bound in energy norm for the exact solution to elasticity problems it is however much more difficult to bound the solution from above for general problems in elasticity and it has been a dream of many decades to find a systematical way to obtain such an upper bound this paper presents a very important and unique property of the linearly conforming point interpolation method lc pim it provides a general means to obtain an upper bound solution in energy norm for elasticity problems this paper conducts first a thorough theoretical study on the lc pim we derive its weak form based on variational principles study a number of properties of the lc pim and prove that lc pim is variationally consistent and that it produces upper bound solutions we then demonstrate these properties through intensive numerical studies with many examples of 1d 2d and 3d problems using the lc pim together with the fem we now have a systematical way to numerically obtain both upper and lower bounds of the exact solution to elasticity problems as shown in these example problems copyright 2007 john wiley sons ltd

achieving exact cluster recovery threshold via semidefinite programming extensions resolving a conjecture of abbe bandeira and hall the authors have recently shown that the semidefinite programming sdp relaxation of the maximum likelihood estimator achieves the sharp threshold for exactly recovering the community structure under the binary stochastic block model sbm of two equal sized clusters the same was shown for the case of a single cluster and outliers extending the proof techniques in this paper it is shown that sdp relaxations also achieve the sharp recovery threshold in the following cases 1 binary sbm with two clusters of sizes proportional to network size but not necessarily equal 2 sbm with a fixed number of equal sized clusters and 3 binary censored block model with the background graph being erd s renyi furthermore a sufficient condition is given for an sdp procedure to achieve exact recovery for the general case of a fixed number of clusters plus outliers these results demonstrate the versatility of sdp relaxation as a simple general purpose computationally feasible methodology for community detection

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

on stability convergence and accuracy of bes fem and bfs fem for nearly incompressible elasticity abstract we present in this paper a rigorous theoretical framework to show stability convergence and accuracy of improved edge based and face based smoothed finite element methods bes fem and bfs fem for nearly incompressible elasticity problems the crucial idea is that the space of piecewise linear polynomials used for the displacements is enriched with bubble functions on each element while the pressure is a piecewise constant function the meshes of triangular or tetrahedral elements required by these methods can be generated automatically the enrichment induces a softening in the bilinear form allowing the weakened weak w 2 procedure to produce a high quality solution free from locking and that does not oscillate we prove theoretically that both methods confirm the uniform inf sup and convergence conditions four numerical examples are given to validate the reliability of the bes fem and bfs fem

stabilized conforming nodal integration in the natural element method a stabilized conforming nodal integration scheme is implemented in the natural neighbour method in conjunction with non sibsonian interpolation in this approach both the shape functions and the integration scheme are defined through use of first order voronoi diagrams the method illustrates improved performance and significant advantages over previous natural neighbour formulations the method also shows substantial promise for problems with large deformations and for the computation of higher order gradients copyright 2004 john wiley sons ltd

achieving exact cluster recovery threshold via semidefinite programming the binary symmetric stochastic block model deals with a random graph of n vertices partitioned into two equal sized clusters such that each pair of vertices is independently connected with probability p within clusters and q across clusters in the asymptotic regime of p a log n n and q b log n n for fixed a b and n to infty we show that the semidefinite programming relaxation of the maximum likelihood estimator achieves the optimal threshold for exactly recovering the partition from the graph with probability tending to one resolving a conjecture of abbe et al furthermore we show that the semidefinite programming relaxation also achieves the optimal recovery threshold in the planted dense subgraph model containing a single cluster of size proportional to n

on locking and robustness in the finite element method a numerical scheme for the approximation of a parameter dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value a robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter precise mathematical definitions for these terms are developed their quantitative characterization is given and some general theorems involving locking and robustness are proven a model problem involving heat transfer is analyzed in detail using this mathematical framework and various related computational results are described applications to some different problems involving locking are presented

a class of mixed assumed strain methods and the method of incompatible modes a three field mixed formulation in terms of displacements stresses and an enhanced strain field is presented which encompasses as a particular case the classical method of incompatible modes within this frame work incompatible elements arise as particular compatible mixed approximations of the enhanced strain field the conditions that the stress interpolation contain piece wise constant functions and be l2 ortho gonal to the enhanced strain interpolation ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation the preceding conditions are formulated in a form particularly convenient for element design as an illustration of the methodology three new elements are developed and shown to exhibit good performance a plane 3d elastic plastic quad an axisymmetric element and a thick plate bending quad the formulation described herein is suitable for non linear analysis

mixed finite elements for elasticity there have been many efforts dating back four decades to develop stable mixed finite elements for the stress displacement formulation of the plane elasticity system this requires the development of a compatible pair of finite element spaces one to discretize the space of symmetric tensors in which the stress field is sought and one to discretize the space of vector fields in which the displacement is sought although there are number of well known mixed finite element pairs known for the analogous problem involving vector fields and scalar fields the symmetry of the stress field is a substantial additional difficulty and the elements presented here are the first ones using polynomial shape functions which are known to be stable we present a family of such pairs of finite element spaces one for each polynomial degree beginning with degree two for the stress and degree one for the displacement and show stability and optimal order approximation we also analyze some obstructions to the construction of such finite element spaces which account for the paucity of elements available

information theoretic thresholds from the cavity method abstract vindicating a sophisticated but non rigorous physics approach called the cavity method we establish a formula for the mutual information in statistical inference problems induced by random graphs and we show that the mutual information holds the key to understanding certain important phase transitions in random graph models we work out several concrete applications of these general results for instance we pinpoint the exact condensation phase transition in the potts antiferromagnet on the random graph thereby improving prior approximate results contucci et al 2013 34 further we prove the conjecture from krzakala et al 2007 55 about the condensation phase transition in the random graph coloring problem for any number q 3 of colors moreover we prove the conjecture on the information theoretic threshold in the disassortative stochastic block model decelle et al 2011 35 additionally our general result implies the conjectured formula for the mutual information in low density generator matrix codes montanari 2005 73

entrywise eigenvector analysis of random matrices with low expected rank recovering low rank structures via eigenvector perturbation analysis is a common problem in statistical machine learning such as in factor analysis community detection ranking matrix completion among others while a large variety of bounds are available for average errors between empirical and population statistics of eigenvectors few results are tight for entrywise analyses which are critical for a number of problems such as community detection this paper investigates entrywise behaviors of eigenvectors for a large class of random matrices whose expectations are low rank which helps settle the conjecture in abbe bandeira and hall 2014 that the spectral algorithm achieves exact recovery in the stochastic block model without any trimming or cleaning steps the key is a first order approximation of eigenvectors under the ell infty norm begin equation u k approx frac au k lambda k end equation where u k and u k are eigenvectors of a random matrix a and its expectation mathbb e a respectively the fact that the approximation is both tight and linear in a facilitates sharp comparisons between u k and u k in particular it allows for comparing the signs of u k and u k even if u k u k infty is large the results are further extended to perturbations of eigenspaces yielding new ell infty type bounds for synchronization mathbb z 2 spiked wigner model and noisy matrix completion