generalized multiple scale reproducing kernel particle methods abstract an approach to unify reproducing kernel methods under one large umbrella and an extension to include time and spatial shifting are proposed the study is divided into three major topics the groundwork is set by revisiting the fourier analysis of discrete systems the multiresolution concept and its significance in devising the reproducing kernel methods and its discrete counterpart reproducing kernel particle methods are explained an edge detection technique based on multiresolution analysis is developed this wavelet approach together with particle methods gives rise to a straightforward h adaptivity algorithm by using this framework a hermite reproducing kernel method is also proposed and its relation to wavelet methods is presented it is also shown that the new approach generalizes existing kernel methods and it can easily be degenerated into other widely used methods such as partition of unity moving least square interpolants smooth particle hydrodynamics scaling functions and wavelets and multiple scale analysis furthermore the hermite reproducing kernel particle method a particle based discrete version of the hermite reproducing kernel method is developed finally multiple scale methods based on frequency and wave number shifting techniques are presented a stability analysis is also presented for newmark time integration schemes for the low frequency equation numerical examples are presented throughout the paper to illustrate the flexibility and accuracy of this class of multiple scale methods

analytical and belief propagation studies of random constraint satisfaction problems with growing domains we study solution space structure and solution finding algorithms of a representative hard random constraint satisfaction problem with growing domains known as model rb using rigorous methods we show that solutions are grouped into disconnected clusters before the theoretical satisfiability phase transition point using the cavity method it is shown that the entropy density obtained by belief propagation bp on random model rb instances which corresponds well to the analytical results vanishes as the control parameter constraint tightness approaches the satisfiability threshold from an algorithmic point of view we find that reinforced bp which performs much better than all existing algorithms allows us to find solutions efficiently for instances in the regime that is very close to the satisfiability transition these results also can shed light on the effectiveness of bp reinforcement on problems with a large number of states

random constraint satisfaction easy generation of hard satisfiable instances in this paper we show that the models of random csp instances proposed by xu and li k xu w li exact phase transitions in random constraint satisfaction problems journal of artificial intelligence research 12 2000 93 103 k xu w li many hard examples in exact phase transitions with application to generating hard satisfiable instances technical report corr report cs cc 0302001 revised version in theoretical computer science 355 2006 291 302 are of theoretical and practical interest indeed these models called rb and rd present several nice features first it is quite easy to generate random instances of any arity since no particular structure has to be integrated or property enforced in such instances then the existence of an asymptotic phase transition can be guaranteed while applying a limited restriction on domain size and on constraint tightness in that case a threshold point can be precisely located and all instances have the guarantee to be hard at the threshold i e to have an exponential tree resolution complexity next a formal analysis shows that it is possible to generate forced satisfiable instances whose hardness is similar to unforced satisfiable ones this analysis is supported by some representative results taken from an intensive experimentation that we have carried out using complete and incomplete search methods

on the application of the method of fundamental solutions for the study of the stress state of a plate subjected to elastic plastic deformation abstract the plane elastoplastic problem for the stress state of a plate with a narrowing subjected to uniaxial extension is considered the behaviour of the material that hardens with plastic deformation is characterised by the ramberg osgood stress strain relation since this relation provides a smooth continuous curve over the whole elastic and plastic deformation range the same governing equation can be used for both deformation regions the paper provides a method for solving the resulting nonlinear boundary value problem the algorithm is based on meshless methods i e the method of fundamental solutions and the method of particular solutions together with a picard iteration process the approximate solution i e the stress function obtained in each iteration step is a linear combination of fundamental and particular solutions it can thus be further used to compute the values of stresses and some effective material parameters i e the young modulus and the poisson ratio at any point of the domain

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

further results for enhanced strain methods with isoparametric elements abstract in this contribution the enhanced strain finite element method is investigated further with particular attention given to the analysis of the method for isoparametric elements it is shown that the results established earlier b d reddy and j c simo stability and convergence of a class of enhanced strain methods siam j numer anal 32 1995 in press for affine equivalent meshes carry over to the case of isoparametric elements that is the method is stable and convergent provided that a set of three criteria is satisfied and for plane elements at least convergence is at the same rate as in the standard method a procedure for recovering the stress is shown to lead to an approximate stress which converges at the optimal rate to the actual stress with regard to computations a new basis for enhanced strains is introduced and numerical results are presented for this basis as well as for existing bases the results for enhanced strain methods are distinguished by the fact that the constant appearing in asymptotic error estimates is much smaller for these methods than for standard approaches

some novel developments in finite element procedures for gradient dependent plasticity summary improved algorithms are proposed for a gradient plasticity theory in which the laplacian of an invariant plastic strain measure enters the yield function particular attention is given to the type of finite elements that can be used within the format of gradient dependent plasticity assuming a weak satisfaction of the yield function mixed finite elements are developed in which the invariant plastic strain measure and the displacements are discretized two families of finite elements are developed one in which the invariant plastic strain measure is interpolated using c continuous polynomials and one in which penalty enhanced c continuous interpolants are used the performance of both families of finite elements is assessed numerically in one dimensional and two dimensional boundary value problems the regularizing effect of the used gradient enhancement in computations of elastoplastic solids is demonstrated both for mesh refinement and for the directional bias of the grid lines

localisation in a cosserat continuum under static and dynamic loading conditions abstract localisation studies have been carried out for an unconstrained elasto plastic strain softening cosserat continuum because of the presence of an internal length scale in this continuum model a perfect convergence is found upon mesh refinement a finite constant width of the localisation zone and a finite energy dissipation are computed under static as well as under transient loading conditions because of the existence of rotational degrees of freedom in a cosserat continuum additional wave types arise and wave propagation becomes dispersive this has been investigated analytically and numerically for an elastic cosserat continuum and an excellent agreement has been found between both solutions

strain driven homogenization of inelastic microstructures and composites based on an incremental variational formulation the paper investigates computational procedures for the treatment of a homogenized macro continuum with locally attached micro structures of inelastic constituents undergoing small strains the point of departure is a general internal variable formulation that determines the inelastic response of the constituents of a typical micro structure as a generalized standard medium in terms of an energy storage and a dissipation function consistent with this type of inelasticity we develop a new incremental variational formulation of the local constitutive response where a quasi hyperelastic micro stress potential is obtained from a local minimization problem with respect to the internal variables it is shown that this local minimization problem determines the internal state of the material for finite increments of time we specify the local variational formulation for a setting of smooth single surface inelasticity and discuss its numerical solution based on a time discretization of the internal variables the existence of the quasi hyperelastic stress potential allows the extension of homogenization approaches of elasticity to the incremental setting of inelasticity focusing on macro strain driven micro structures we develop a new incremental variational formulation of the global homogenization problem where a quasi hyperelastic macro stress potential is obtained from a global minimization problem with respect to the fine scale displacement fluctuation field it is shown that this global minimization problem determines the state of the micro structure for finite increments of time we consider three different settings of the global variational problem for prescribed linear displacements periodic fluctuations and constant stresses on the boundary of the micro structure and discuss their numerical solutions based on a spatial discretization of the fine scale displacement fluctuation field the performance of the proposed methods is demonstrated for the model problem of von mises type elasto visco plasticity of the constituents and applied to a comparative study of micro to macro transitions of inelastic composites copyright 2002 john wiley sons ltd

variational multiscale element free galerkin method for 2d burgers equation a new numerical method which is based on the coupling between variational multiscale method and meshfree methods is developed for 2d burgers equation with various values of re the proposed method takes full advantage of meshfree methods therefore no mesh generation and mesh recreation are involved meanwhile compared with the variational multiscale finite element method a strong assumption that is the fine scale vanishes identically over the element boundaries although non zero within the elements is not needed subsequently two problems which have an available analytical solution of their own are solved to analyze the convergence behaviour of the proposed method finally a 2d burgers equation having large re is solved and the results have also been compared with the ones computed by two other methods the numerical results show that the proposed method can indeed obtain accurate numerical results for 2d burgers equation having large re which does not refer to the choice of a proper stabilization parameter

an analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids abstract this paper presents an analysis of strong discontinuities in inelastic solids at finite strains solutions exhibiting this type of discontinuities characterized by a discontinuous displacement field are shown to make physical and mathematical sense in a classical multiplicative plasticity continuum model if the softening modulus is reinterpreted as a singular distribution physically the strainsoftening is localized along the discontinuity conditions for the appearance of strong discontinuities in the geometrically nonlinear range are characterized as it is the response of the material during localization in addition these analytical results are exploited in the design of a new class of finite element methods the proposed methods fall within the class of enhanced strain methods and lead to solutions independent of the mesh size and insensitive to mesh alignment without requiring any regularization of the solutions by numerical parameters like a characteristic length

on the phase transitions of random k constraint satisfaction problems constraint satisfaction has received increasing attention over the years intense research has focused on solving all kinds of constraint satisfaction problems csps in this paper first we propose a random csp model named k csp that guarantees the existence of phase transitions under certain circumstances the exact location of the phase transition is quantified and experimental results are provided to illustrate the performance of the proposed model second we revise the model k csp to a random linear csp by incorporating certain linear structure to constraint relations we also prove the existence of the phase transition and exhibit its exact location for this random linear csp model

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

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

the schwarz alternating method in solid mechanics abstract we advance the schwarz alternating method as a means for concurrent multiscale coupling in finite deformation solid mechanics we prove that the schwarz alternating method converges to the solution of the problem on the entire domain and that the convergence rate is geometric provided that each of the subdomain problems is well posed i e their corresponding energy density functions are quasi convex it is shown that the use of a newton type method for the solution of the resultant nonlinear system leads to two kinds of block linearized systems depending on the treatment of the dirichlet boundary conditions the first kind is a symmetric block diagonal linear system in which each diagonal block is the tangent stiffness of each subdomain i e the off diagonal blocks are all zero and the coupling terms appear only on the right hand side the second kind is a nonsymmetric block system with off diagonal coupling terms several variants of the schwarz alternating method are proposed for the first kind of linear system including one in which the schwarz alternating iterations and the newton iterations are combined into a single scheme this version of the method is particularly attractive as it lends itself to a minimally intrusive implementation into existing finite element codes finally we demonstrate the performance of the proposed variants of the schwarz alternating method on several one dimensional and three dimensional examples

analysis and adaptive modeling of highly heterogeneous elastic structures in this paper we develop a theory and methodology for obtaining approximate solutions to boundary value problems describing the deformation of highly heterogeneous linearly elastic structures the method which represents a significant departure from traditional homogenization methods provides a systematic and rigorous approach towards resolving the effects of microstructure of different scales on the macroscopic response of complex heterogeneous structures an early variant of this method was first introduced in io the method referred to as hdpm homogenized dirichlet projection method proceeds by first solving an auxiliary homogenized boundary value problem describing the deformation of a structure with the same exterior geometry but with a selected set of uniform material properties the adequacy of this homogenized solution is then determined using an a posteriori error estimate that provides a measure of the error of the homogenized solution in subdomains of the structure compared to the solution of the fine scale heterogeneous problem without specific knowledge of this fine scale solution in those subdomains where the homogenized solution is deemed acceptable it is retained however in subdomains where the homogenized solution is inadequate as is determined when the estimated error exceeds a preset tolerance a local boundary value problem is constructed by projecting the homogenized displacements onto a partition designed to isolate these subdomains these local boundary value problems are then solved in those subdomains where the homogenized solution is inaccurate using the exact microstructure with the approximate local boundary conditions a posteriori error estimates are made to ascertain the quality of the resulting solution if the quality of the solution remains inadequate above a preset error tolerance a two stage adaptive procedure is implemented stage i material adaptivity corresponds to modifying the homogenized structure s material properties if after stage i the solution quality is still inadequate the subdomains of local solution are enlarged thereby modeling in greater detail the actual microstructure and the local solution process is repeated stage ii subdomain unrefinement the main feature of this method is that only in subdomains where the error in the usual homogenized solution is above a preset tolerance is the microstructure taken into account the cost of this method is shown to be orders of magnitude cheaper than direct huge scale computational simulations of micromechanical events the results of several numerical experiments are provided to demonstrate the method and verify theoretical estimates in this work we present a method for the solution of boundary value problems modeling the deformation of structures composed of highly heterogeneous linearly elastic materials we concentrate on materials formed by a homogeneous matrix embedded with particulate matter of different properties fig 1 however the results presented are not limited to materials of this type and can in theory be used for materials with virtually any microstructure a straightforward calculation reveals that an accurate numerical approximation of fine scale solutions of

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

hierarchical modeling of heterogeneous solids in this paper a methodology is introduced for the development of adaptive methods for hierarchical modeling of elastic heterogeneous bodies the approach is based on the idea of computing an estimate of the modeling error introduced by replacing the actual fine scale material tensor with that of a homogenized material and to adaptively refine the material description until a prespecified error tolerance is met this process generates a family of coarse scale solutions in which the solution corresponding to the fine scale model of the body which embodies the exact microstructure loading and boundary conditions represents the highest level of sophistication in a family of continuum models the adaptive strategy developed can lead to a new non uniform description of material properties which reflects the loading and boundary conditions a post processing technique is also introduced which endows the coarse scale solutions with fine scale information through a local solution process convergence of the adaptive algorithm is proven and modeling error estimates as a function of scale of material description are presented preliminary results of several numerical experiments are given to confirm estimates and to illustrate the promise of the approach in practical applications

on a multi scale element free galerkin method for the stokes problem abstract in this paper a multi scale element free galerkin method is presented for the stokes problem the new method is based on the hughes variational multi scale formulation and arises from a decomposition of the velocity field into coarse resolved scales and fine unresolved scales in this method an unresolved model is obtained in which unresolved scales are incorporated analytically through the bubble functions modeling of the unresolved scales corrects the lack of the stability of the standard element free galerkin formulation and the resulting stabilized formulation possesses superior properties like that of the streamline upwind petrov galerkin supg method and the galerkin least squares gls method the method allows equal order basis for pressure and velocity because it violates the celebrated babuska brezzi condition a significant feature of the present method is that the structure of the stabilization tensor appears naturally via the solution of the fine scale problem numerical results for example problems confirm that this method has some excellent properties such as better stability and accuracy

the finite element method for the mechanically based model of non local continuum summary in this paper the finite element method fem for the mechanically based non local elastic continuum model is proposed in such a model each volume element of the domain is considered mutually interacting with the others beside classical interactions involved by the cauchy stress field by means of central body forces that are monotonically decreasing with their inter distance and proportional to the product of the interacting volume elements the constitutive relations of the long range interactions involve the product of the relative displacement of the centroids of volume elements by a proper distance decaying function which accounts for the decrement of the long range interactions as long as distance increases in this study the elastic problem involving long range central interactions for isotropic elastic continuum will be solved with the aid of the fem the accuracy of the solution obtained with the proposed fem code is compared with other solutions obtained with galerkins approximation as well as with finite difference method moreover a parametric study regarding the effect of the material length scale in the mechanically based model and in the kroner eringen non local elasticity has been investigated for a plane elasticity problem copyright 2011 john wiley sons ltd

locating the phase transition in binary constraint satisfaction problems abstract the phase transition in binary constraint satisfaction problems i e the transition from a region in which almost all problems have many solutions to a region in which almost all problems have no solutions as the constraints become tighter is investigated by examining the behaviour of samples of randomly generated problems in contrast to theoretical work which is concerned with the asymptotic behaviour of problems as the number of variables becomes larger this paper is concerned with the location of the phase transition in finite problems the accuracy of a prediction based on the expected number of solutions is discussed it is shown that the variance of the number of solutions can be used to set bounds on the phase transition and to indicate the accuracy of the prediction a class of sparse problems for which the prediction is known to be inaccurate is considered in detail it is shown that for these problems the phase transition depends on the topology of the constraint graph as well as on the tightness of the constraints

a space time formulation for multiscale phenomena abstract we develop subgrid scale models for a class of nonsymmetric linear evolution operators by applying the variational multiscale method in space time the results generalize those of hughes 14 which were confined to the steady case the subgrid scale models are shown to be a paradigm for bubble function finite element methods and provide a theoretical and practical framework for the development of so called stabilized methods

two solutions to diluted p spin models and xorsat problems we derive analytical solutions for p spin models with finite connectivity at zero temperature these models are the statistical mechanics equivalent of p xorsat problems in theoretical computer science we give a full characterization of the phase diagram location of the phase transitions static and dynamic together with a description of the clustering phenomenon taking place in configurational space we use two alternative methods the cavity approach and a rigorous derivation

a variational multiscale approach to strain localization formulation for multidimensional problems abstract the multiscale approach to strain localization problems developed earlier is generalized to higher dimensions the key idea is the identification of the fine scale field with a component of the displacement that has a large gradient the difference between the total displacement and the fine scale is the coarse scale field the formulation is variationally based with the weak form as the point of departure with suitable assumptions on the space of fine scale fields the weak form is separable into coarse and fine scale parts the weak form of the fine scale problem is used to eliminate the fine scales from the formulation this is achieved by way of a projection onto the fine scale space and allows a reformulation of the full problem in terms of coarse scale components alone in this framework fine scale interpolations are identified that ensure sharp resolution of localized displacement irrespective of the underlying finite element mesh it is demonstrated that by correctly accounting for the microstructural fine scale behavior the formulation results in solutions devoid of pathological mesh sensitivity for problems wherein the detailed structure of the localized displacement is of interest recovery of the fine scale components can be performed to reconstruct the entire displacement field several numerical examples are presented that demonstrate the robustness of the formulation

ideals associated to deformations of singular plane curves we consider in this paper the geometry of certain loci in deformation spaces of plane curve singularities these loci are the equisingular locus es which parametrizes equisingular or topologically trivial deformations the equigeneric locus eg which parametrizes deformations of constant geometric genus and the equiclassical locus ec which parametrizes deformations of constant geometric genus and class the class of a reduced plane curve is the degree of its dual it was previously known that the tangent space to es corresponds to an ideal called the equisingular ideal and that the support of the tangent cone to eg corresponds to the conductor ideal we show that the support of the tangent cone to ec corresponds to an ideal which we call the equiclassical ideal by studying these ideals we are able to obtain information about the geometry and dimensions of es ec and eg this allows us to prove some theorems about the dimensions of families of plane curves with certain specified singularities

constructing an asymptotic phase transition in random binary constraint satisfaction problems the standard models used to generate random binary constraint satisfaction problems are described at the problem sizes studied experimentally a phase transition is seen as the constraint tightness is varied however achlioptas et al showed that if the problem size number of variables increases while the remaining parameters are kept constant asymptotically almost all instances are unsatisfiable in this paper an alternative scheme for one of the standard models is proposed in which both the number of values in each variable s domain and the average degree of the constraint graph are increased with problem size it is shown that with this scheme there is asymptotically a range of values of the constraint tightness in which instances are trivially satis able with probability at least 0 5 and a range in which instances are almost all unsatisfiable hence there is a crossover point at some value of the constraint tightness between these two ranges this scheme is compared to a similar scheme due to xu and li

reproducing kernel hierarchical partition of unity part ii applications in this part of the work the meshless hierarchical partition of unity proposed in 1 referred here as part i is used as a multiple scale basis in numerical computations to solve practical problems the applications discussed in the present work fall into two categories 1 a wavelet adaptivity refinement procedure and 2 a so called wavelet petrov galerkin procedure in the applications of wavelet adaptivity the hierarchical reproducing kernels are used as a multiple scale basis to compute the numerical solutions of the helmholtz equation a model equation of wave propagation problems and to simulate shear band formation in an elasto viscoplastic material a problem dictated by the presence of the high gradient deformation in both numerical experiments numerical solutions with high resolution are obtained by inserting the wavelet like basis into the primary interpolation function basis a process that may be viewed as a spectral p type refinement by using the interpolant that has synchronized convergence property as a weighting function a wavelet petrov galerkin procedure is proposed to stabilize computations of some pathological problems in numerical computations such as advection diffusion problems and stokes flow problem it offers an alternative procedure in stablized methods and also provides some insight or new interpretation of the method detailed analysis has been carried out on the stability and convergence of the wavelet petrov galerkin method copyright 1999 john wiley sons ltd

multiscale stabilized solutions to one dimensional systems of conservation laws abstract we present a variational multiscale formulation for the numerical solution of one dimensional systems of conservation laws the key idea of the proposed formulation originally presented by hughes comput methods appl mech engrg 127 1995 387 401 is a multiple scale decomposition into resolved grid scales and unresolved subgrid scales incorporating the effect of the subgrid scales onto the coarse scale problem results in a finite element method with enhanced stability properties capable of accurately representing the sharp features of the solution in the formulation developed herein the multiscale split is invoked prior to any linearization of the equations special attention is given to the choice of the matrix of stabilizing coefficients and the discontinuity capturing diffusion the methodology is applied to the one dimensional simulation of three phase flow in porous media and the shallow water equations these numerical simulations clearly show the potential and applicability of the formulation for solving highly nonlinear nearly hyperbolic systems on very coarse grids application of the numerical formulation to multidimensional problems is presented in a forthcoming paper

spectral analysis of the dissipation of the residual based variational multiscale method abstract we analyze the application of the residual based variational multiscale vms method to the solution of a generic partial differential equation we prove that the model terms introduced by this method are dissipative under certain restrictions that are likely to be satisfied by spectral approximations of conservation laws in particular we demonstrate that the fourier spectral approximations of burgers equation and the incompressible navier stokes equations satisfy these conditions hence we show that the residual based vms model for these equations is dissipative even though the form of the model term does not resemble an eddy viscosity we analyze the scale dependence of this dissipation using fully resolved numerical solutions and conclude that the vms model accurately captures the scale dependence of the cross stress terms u u and under predicts the contribution from the reynolds stress terms u u these observations have implications on the use of the residual based vms method in spectral approximations of nonlinear conservation laws

clustering of solutions in the random satisfiability problem using elementary rigorous methods we prove the existence of a clustered phase in the random k sat problem for k geq 8 in this phase the solutions are grouped into clusters which are far away from each other the results are in agreement with previous predictions of the cavity method and give a rigorous confirmation to one of its main building blocks it can be generalized to other systems of both physical and computational interest

plane curves of minimal degree with prescribed singularities we prove that there exists a positive such that for any integer d 3 and any topological types s1 sn of plane curve singularities satisfying

geometry of families of nodal curves on the blown up projective plane let p2r be the projective plane blown up at r generic points denote by e0 e1 er the strict transform of a generic straight line on p2 and the exceptional divisors of the blown up points on p2r respectively we consider the variety virr of all irreducible curves c with k nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness irreducibility and non emptiness moreover we extend our conditions for the smoothness and the irreducibility on families of reducible curves for r 9 we give the complete answer concerning the existence of nodal curves in virr

non local yield limit degradation presented is a new type of a non local continuum model which avoids problems of convergence at mesh refinement and spurious mesh sensitivity in a softening continuum characterized by degradation of the yield limit the key idea which has recently been proposed in a general context and has already been applied to softening damage due to stiffness degradation is to apply the non local concept only to those parameters which cause the degradation while keeping the definition of the strains local compared to the previously advanced fully non local continuum formulation the new approach has the advantage that the stresses are subjected to the standard differential equations of equilibrium and standard boundary or interface conditions the new formulation exhibits no zero energy periodic modes imbrication of finite elements is unnecessary and finite elements with standard continuity requirements are sufficient two dimensional finite element solutions with up to 3248 degrees of freedom are presented to document convergence and efficacy the formulation is applied to tunnel excavation in a soil stabilized by cement grouting with the objective of preventing cave in burst of the tunnel sides due to compression softening

on the survey propagation equations in random constraint satisfiability problems in this paper we study the existence of a solution of the survey propagation equations for a given instance of a random problem in the framework of constraint satisfiability we consider the concrete examples of k satisfiability and coloring we conjecture that when the number of variables goes to infinity the solution of the survey propagation equations for a given instance can be obtained by finding the supposed unique solution of the corresponding equations on an infinite tree we conjecture that the survey propagation equations on a random infinite tree have a unique solution in the suitable range of parameters we also present analytic arguments that indicate that the survey propagation equations do have solutions in the satisfiable phase for simplicity of notation the argument is presented in the case of the coloring problem the same results extend to other optimization problems where exist configurations that have cost zero i e in the satisfiable phase on a random graph the solutions of th

meshfree analysis of softening elastoplastic solids using variational multiscale method a meshfree multiscale method is presented for efficient analysis of elastoplastic solids in the analysis of softening elastoplastic solids standard finite element methods or meshfree methods typically yield mesh dependent results the reason for this well known effect is the loss of ellipticity of the boundary value problem in this work the scale decomposition is carried out based on a variational form of the problem a coarse scale is designed to represent global behavior and a fine scale to represent local behavior a fine scale region is detected from the local failure analysis of an acoustic tensor to indicate a region where deformation changes abruptly each scale variable is approximated using a meshfree method meshfree approximation is well suited for adaptivity as a method of increasing the resolution a partition of unity based extrinsic enrichment is used in particular fine scale approximations are designed to appropriately represent local behavior by using a localization angle moreover the regularization effect through the convexification of non convex potential is embedded to represent fine scale behavior each scale problem is solved iteratively the proposed method is applied to shear band problems in the results of analysis about pure shear and compression problems straight shear bands can be captured and mesh insensitive results are obtained curved shear bands can also be captured without mesh dependency in the analysis of indentation problem

clustering phase of a general constraint satisfaction problem model d k csp abstract relation between problem hardness and solution space structure is an important research aspect model d k csp is a random model of constraint satisfaction problem which generates very hard instances when r 1 or r is near 1 where r represents normalized constraint density we study the clustering phase of the model and find that if r is below and close to 1 the solution space contains many widely distributed and well separated small frozen clusters which should be the reason why the generated instances are hard to solve

a study of strain localization in a multiple scale framework the one dimensional problem abstract this work approaches strain localization by recognizing the multiple scales inherent in the problem a component associated with the region of localized strain typically one with high gradient is referred to as the fine scale it represents the microstructure the field obtained by removing the fine scale from the total solution is referred to as the coarse scale the aim of the multiple scale method advanced here is to derive a model for the coarse scale field that accounts for the finite scale this process eliminates the fine scale from the problem yet retains its effect in applying this framework to the nonlinear problems with which localized strains are associated a crucial step is a first order approximation of the relevant relations the fully nonlinear problem is solved by an iterative scheme by accounting directly for the microstructure the multiple scale model recovers the regularizing effects of various alternative formulations for softening strain localization it thus presents itself as a unifying framework for such models numerical solutions are shown to be invariant with respect to the discretization furthermore for cases in which the displacements assume a distinct profile within the localization band the multiple scale model provides a resolution that can be made as accurate as desired even with the coarsest mesh possible the model is applied to strain localization problems that arise in inviscid and viscoplastic solids numerical simulations are presented that demonstrate the efficacy of the approach

fe2 multiscale in linear elasticity based on parametrized microscale models using proper generalized decomposition abstract in recent years a tremendous growth of activity in multiscale modelling has been produced to get the mechanical response of highly heterogenous materials where the complexity of solving numerically all microscale details is not feasible in this paper we present a multiscale framework based on the decomposition of the displacement field into coarse macro and fine micro scales earlier proposed in the variational multiscale approach or the hp d method the novelty of this work lies in solving the microscale step where a multidimensional parametrized model of a generic rve is solved by means of the multidimensional model reduction technique named as proper generalized decomposition pgd as result of this previous and off line step the displacement field over the rve is obtained by simple algebraic operations for any combination of parameters boundary condition material properties loads etc allowing a significant reduction in computational cost when solving the macro scale problem for problems where the detailed structure of the localized displacement strain and stress fields are of interest recovery of the fine scale components can be performed immediately the basic concepts and several 1 d and 2 d linear elastic problems are presented to demonstrate the robustness of the formulation and computer time saving

phase transitions in the coloring of random graphs we consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color using the cavity method we present a detailed and systematic analytical study of the space of proper colorings solutions we show that for a fixed number of colors and as the average vertex degree number of constraints increases the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses first at the clustering transition the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard afterward the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state eventually above the coloring threshold no more solutions are available

a variational principle for gradient plasticity abstract we elaborate on a generalized plasticity model which belongs to the class of gradient models suggested earlier by aifantis and co workers the generalization of the conventional theory of plasticity has been accomplished by the inclusion of higher order spatial gradients of the equivalent plastic strain in the yield condition first it is shown how these gradients affect the critical condition for the onset of localization and allow for a wavelength selection analysis leading to estimates for the width and or spacing of shear bands due to the presence of higher order gradients additional boundary conditions for the equivalent plastic strain are required this question and also the associated problem of the formulation and solution of general boundary value problems were left open in the previous work we demonstrate here that upon assuming a certain type of additional boundary conditions the structural symmetries of the gradient dependent constitutive model are such that there exists a variational principle for the displacement rates and the rate of the equivalent plastic strain the variational principle can serve as a basis for the numerical solution of boundary value problems in the sense of the finite element method explicit expressions for the tangent stillness matrix and the generalized nodal point forces are given

irreducibility of equisingular families of curves in 1985 joe harris proved the long standing claim of severi that equisingular families of plane nodal curves are irreducible whenever they are nonempty for families with more complicated singularities this is no longer true given a divisor d on a smooth projective surface it thus makes sense to look for conditions which ensure that the family v irr d s 1 s r of irreducible curves in the linear system d l with precisely r singular points of types s 1 s r is irreducible considering different surfaces including general surfaces in p 3 c and products of curves we produce a sufficient condition of the type r i 1 deg x s i 2 d k 2 where is some constant and x s i some zero dimensional scheme associated to the singularity type our results carry the same asymptotics as the best known results in this direction in the plane case even though the coefficient is worse for most of the surfaces considered these are the only known results in that direction

random k satisfiability problem from an analytic solution to an efficient algorithm we study the problem of satisfiability of randomly chosen clauses each with k boolean variables using the cavity method at zero temperature we find the phase diagram for the k 3 case we show the existence of an intermediate phase in the satisfiable region where the proliferation of metastable states is at the origin of the slowdown of search algorithms the fundamental order parameter introduced in the cavity method which consists of surveys of local magnetic fields in the various possible states of the system can be computed for one given sample these surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems one such algorithm is shown here for the k 3 satisfiability problem with very good performances

existence of curves with prescribed topological singularities throughout this paper we study the existence of irreducible curves c on smooth projective surfaces with singular points of prescribed topological types s 1 s r there are necessary conditions for the existence of the type r i 1 s i c 2 c k for some fixed divisor k on and suitable coefficients and and the main sufficient condition that we find is of the same type saying it is asymptotically proper ten years ago general results of this quality were not known even for the case e p 2 c an important ingredient for the proof is a vanishing theorem for invertible sheaves on the blown up e of the form o d r i 1 m i e i deduced from the kawamata vieweg vanishing theorem its proof covers the first part of the paper while the middle part is devoted to the existence theorems in the last part we investigate our conditions on ruled surfaces products of elliptic curves surfaces in p 3 c and k3 surfaces

on the variety of plane curves of degree with nodes and cusps let pn be the projective space which parametrizes all plane curves of degree d and v d k the subvariety of pn consisting of all reduced and irreducible plane curves of degree d with 3 nodes and k cusps as their only singularities in this paper we prove that v d k is irreducible if k 3 except possibly when k 3 and d 5 or 6

a two scale finite element relaxation analysis of shear bands in non convex inelastic solids small strain theory for standard dissipative materials abstract we propose a fundamentally new approach to the treatment of shear band localizations in strain softening elastic plastic solids at small strains based on energy minimization principles associated with micro structure developments the point of departure is a general internal variable formulation that determines the inelastic response as a standard dissipative medium consistent with this type of inelasticity we consider an incremental variational formulation of the local constitutive response where a quasihyperelastic stress potential is obtained from a local constitutive minimization problem with respect to the internal variables the existence of this variational formulation allows the definition of the material stability of an inelastic solid based on weak convexity conditions of the incremental stress potential in analogy to treatments in finite elasticity furthermore localization phenomena are interpreted as micro structure developments on multiple scales associated with non convex incremental stress potentials in analogy to elastic phase decomposition problems these micro structures can be resolved by the relaxation of non convex energy functionals based on a convexification of the stress potential the relaxed problem provides a well posed formulation for a mesh objective analysis of localizations as close as possible to the non convex original problem we develop based on an approximated rank one convexification of the incremental stress potential a computational two scale procedure for a mesh objective treatment of localization problems it constitutes a local minimization problem for a relaxed incremental stress potential with just one scalar variable representing the intensity of the micro shearing of a rank one laminate aligned to the shear band this problem is sufficiently robust with regard to applications to large scale inhomogeneous deformation processes of elastic plastic solids the performance of the proposed energy relaxation method is demonstrated for a representative set of numerical simulations of straight and curved shear bands which illustrate the mesh independence of the results

properties of discontinuous bifurcation solutions in elasto plasticity explicit expressions for the spectral properties of the bifurcation problem involving discontinuities for general elastic plastic materials are presented it then follows that the classical value of the critical hardening modulus derived by rice 1976 proc 14th iutam congr delft the netherlands pp 207 220 north holland amsterdam is in fact the only possible one furthermore from the spectral analysis it follows in a very straightforward fashion that bifurcation displaying elastic unloading on one side of the singular surface can never precede bifurcation with plastic loading on both sides of this surface explicit analytical results for the critical bifurcation directions and the corresponding hardening modulus are derived for non associated volumetric flow rules while the devialoric portion is associated the considered yield and potential functions may depend on all three stress invariants and may involve mixed isotropic and kinematic hardening the result obtained by rudnicki and rice 1975 j mech phys solids 23 371 394 for a drucker pruger material appear as a special case other criteria that are investigated arc those of coulomb and rankine less

a multiscale finite element method for the helmholtz equation it is well known that when the standard galerkin method is applied to the helmholtz equation it exhibits an error in the wavenumber and the solution does not therefore preserve the phase characteristics of the exact solution improvements on the galerkin method including galerkin least squares gls methods have been proposed however these approaches rely on a dispersion analysis of the underlying difference stencils in order to reduce error in the solution in this paper we propose a multiscale finite element for the helmholtz equation the method employs a multiscale variational formulation which leads to a subgrid model in which subgrid scales are incorporated analytically through appropriate green s functions it is shown that entirely new and accurate methods emerge and that gls methods can be obtained as special cases of the more general subgrid model

fe formulation of a nonlocal plasticity theory a nonlocal continuum plasticity theory is presented the nonlocal field introduced here is defined as a certain weighted average of the corresponding local field taken over all the material points in the body hereby a quantity with the dimension of length occurs as a material parameter when this so called internal length is equal to zero the local classical plasticity theory is regained in the present model the yield function will depend on a nonlocal field the consistency condition and the integration algorithm result in integral equations for determination of the field of plastic multipliers the integral equations are classified as fredholm equations of the second kind and the existence of a solution will be commented upon after discretization a matrix equation is obtained and an algorithm for finding the solution is proposed for a generalized von mises material a plane boundary value problem is solved with a fe method since the nonlocal quantities are integrals c0 continuous elements are sufficient the solution strategy is split into a displacement estimate for equilibrium and the integration of constitutive equations in the numerical simulations shear band formation is analysed and the results display mesh insensitivity

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

a variational multiscale method to model crack propagation at finite strains this contribution presents a hierarchical variational multiscale framework to model propagating discontinuities at finite strains thereby the deformation map is decomposed into coarse scale and fine scale displacements which results in a decoupled system of coarse scale and fine scale equations both are solved numerically by means of the finite element method whereby crack propagation is taken into account at the fine scale growing cracks are numerically handled by the introduction of discontinuous elements a locality assumption on the fine scale solution and an adaptive scheme to resize the fine scale domain are introduced and demonstrated to increase the efficiency of the method copyright 2009 john wiley sons ltd

reproducing kernel hierarchical partition of unity part i formulation and theory this work is concerned with developing the hierarchical basis for meshless methods a reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart to form such hierarchical partition a class of basic wavelet functions are introduced based upon the built in consistency conditions the differential consistency conditions for the hierarchical kernel functions are derived it serves as an indispensable instrument in establishing the interpolation error estimate which is theoretically proven and numerically validated for a special interpolant with different combinations of the hierarchical kernels a synchronized convergence effect may be observed being different from the conventional legendre function based p type hierarchical basis the new hierarchical basis is an intrinsic pseudo spectral basis which can remain as a partition of unity in a local region because the discrete wavelet kernels form a partition of nullity these newly developed kernels can be used as the multi scale basis to solve partial differential equations in numerical computation as a p type refinement copyright 1999 john wiley sons ltd