numerical modelling of crack propagation automatic remeshing and comparison of different criteria modelling of a crack propagating through a finite element mesh under mixed mode conditions is of prime importance in fracture mechanics three different crack growth criteria and the respective crack paths prediction for several test cases are compared the maximal circumferential stress criterion the strain energy density fracture criterion and the criterion of the maximal strain energy release rate are implemented using advanced finite element techniques a fully automatic remesher enables to deal with multiple boundaries and multiple materials the propagation of the crack is calculated with both remeshing and nodal relaxation several examples are presented to check for the robustness of the numerical techniques and to study specific features of each criterion

the cauchy and characteristic boundary value problems of random rigid perfectly plastic media abstract effects of spatial random fluctuations in the yield condition of rigid perfectly plastic continuous media are analysed in cases of cauchy and characteristic boundary value problems a weakly random plastic microstructure is modeled on a continuum mesoscale by an isotropic yield condition with the yield limit taken as a locally averaged random field the solution method is based on a stochastic generalization of the method of sliplines whose significant feature is that the deterministic characteristics are replaced by the forward evolution cones containing random characteristics comparisons of response of this random medium and of a deterministic homogeneous medium with a plastic limit equal to the average of the random one are carried out numerically in several specific examples of the two boundary value problems under study an application of the method is given to the limit analysis of a cylindrical tube under internal traction the major conclusion is that weak material randomness always leads to a relatively stronger scatter in the position and field variables as well as to a larger size of the domain of dependence effects which are amplified by both presence of shear traction and inhomogeneity in the boundary data additionally it is found that there is hardly any difference between stochastic slipline fields due to either gaussian or uniform noise in the plastic limit

a higher order scheme for two dimensional quasi static crack growth simulations an efficient scheme for the simulation of quasi static crack growth in two dimensional linearly elastic isotropic specimens is presented the crack growth is simulated in a stepwise manner where an extension to the already existing crack is added in each step in a local coordinate system each such extension is represented as a polynomial of some user specified degree n the coefficients of the polynomial describing an extension are found by requiring that the mode ii stress intensity factor is equal to zero at certain points of the extension if a crack grows from a pre existing crack so that a kink develops the leading term describing the crack shape close to the kink will in a local coordinate system be proportional to x 3 2 we therefore allow the crack extensions to contain such a term in addition to the monomial terms the discontinuity in the crack growth direction at a kink the kink angle is determined by requiring that the mode ii stress intensity factor should be equal to zero for an infinitesimal extension of the existing crack to implement the scheme accurate values of the stress intensity factors and t stress are needed in each step of the simulation these fracture parameters are computed using a previously developed integral equation of the second kind c 2007 elsevier b v all rights reserved less

structure of metastable states in the hopfield model an upper bound for the number of metastable states in the hopfield model is calculated as a function of the hamming fraction from an input pattern for all finite values of alpha the ratio of number of patterns to nodes the hamming fraction from the input pattern to the nearest metastable state is infinite when alpha 0 113 the bound also implies that there is a gap between a set of states close to the input pattern and another set centred around the hamming fraction 0 5 from it

on the numerical modelling of quasi static crack growth in linear elastic fracture mechanics in this paper we present a strategy for the simulation of a propagating crack under mixed mode linear elastic conditions using a discontinuous finite element method a key issue is to accurately compute the incremental change in the kink angle of the propagating crack during subsequent steps we have chosen to work with a domain formulation of the material force vector as a criteria for the propagation direction we describe the theoretical background together with the numerical implementation in detail and show some results for different loading conditions copyright 2005 john wiley sons ltd

a quasi static discontinuous galerkin configurational force crack propagation method for brittle materials summary this paper presents a framework for r adaptive quasi static configurational force cf brittle crack propagation cast within a discontinuous galerkin dg symmetric interior penalty sipg finite element scheme cracks are propagated in discrete steps with a staggered algorithm along element interfaces which align themselves with the predicted crack propagation direction the key novelty of the work is the exploitation of the dg face stiffness terms existing along element interfaces to propagate a crack in a mesh independent r adaptive quasi static fashion driven by the cf at the crack tip this adds no new degrees of freedom to the data structure additionally as dg methods have element specific degrees of freedom a geometry driven p adaptive algorithm is also easily included allowing for more accurate solutions of the cf on a moving crack front further for non determinant systems we introduce an average boundary condition that restrains rigid body motion leading to a determinant system to the authors knowledge this is first time such a boundary condition has been described the proposed formulation is validated against single and multiple crack problems with single and mixed mode cracks demonstrating the predictive capabilities of the method this article is protected by copyright all rights reserved

mls based variable node elements compatible with quadratic interpolation part ii application for finite crack element two dimensional finite crack elements for simulation of propagating cracks are developed using the moving least square mls approximation the mapping from the parental domain to the physical element domain is implicitly obtained from mls approximation with the shape functions and their derivatives calculated and saved only at the numerical integration points the mls based variable node elements are extended to construct the crack elements which allow the discontinuity of crack faces and the crack tip singularity the accuracy of the crack elements is checked by calculating the stress intensity factor under mode i loading the crack elements turn out to be very efficient and accurate for simulating crack propagations only with the minimal amount of element adjustment and node addition as the crack tip moves numerical results and comparison to the results from other works demonstrate the effectiveness and accuracy of the present scheme for the crack elements copyright 2005 john wiley sons ltd

geometry update driven by material forces for simulation of brittle crack growth in functionally graded materials functionally graded materials fgms are advanced materials that possess continuously graded properties such that the growth of cracks is strongly dependent on the gradation of the material in this work a thermodynamic consistent framework for crack propagation in fgms is presented by applying a dissipation inequality to a time dependent migrating control volume the direction of crack growth is obtained in terms of material forces as a result of the principle of maximum dissipation in the numerical implementation a staggered algorithm deformation update for fixed geometry followed by geometry update for fixed deformation is employed within each time increment the geometry update is a result of the incremental crack propagation which is driven by material forces the corresponding mesh is generated by combining delaunay triangulation with local mesh refinement furthermore a newton algorithm is proposed taking into account mesh transfer of displacements for crack propagation in incremental elasticity in two numerical examples brittle crack propagation in fgms is investigated for various directions of strength gradation within the structures copyright 2008 john wiley sons ltd

fast and robust level set update for 3d non planar x fem crack propagation modelling abstract in the x fem framework the need to represent a discontinuity independently of the structural mesh relies on the level set technique hence crack propagation can be simulated by an update of two distinct level sets the evolution of which is described by differential equations the aim of this paper is to analyse the resolution of these equations in order to formulate a robust and fast numerical process allowing 3d crack propagation simulations even in presence of high kink angles occurring in mixed mode propagation the numerical integration is accomplished by means of a robust finite difference upwind scheme applied to an auxiliary regular grid an alternative level set update equation and a fast localisation of the integration domain specifically developed for crack propagation problems are formulated and proposed in the paper in order to gain in stability robustness and performance

an implicit geometrical approach to level sets update for 3d non planar x fem crack propagation abstract in the x fem framework the crack geometry is described independently of the structural mesh usually by means of two level set functions thanks to the versatility and wide applicability of the level set methodology the main difficulty in the adoption of the level set approach is the formulation of an efficient and robust update scheme allowing to calculate the new level set values associated to the new crack configuration after a propagation step in this paper an implicit geometrical methodology is formulated in order to overcome the stability and performance issues affecting the resolution of the hamilton jacobi partial differential equations governing the level set update the calculation of the new level sets values is based on geometrical considerations only which enormously reduces the computational time and increases the robustness of the update calculations in contrast to other geometrical approaches this methodology can be applied both to 2d and 3d problems and the crack front and surface are always implicitly described by means of the two level sets this avoids both the need to describe somehow the crack surface and front and the creation of algorithms to track their evolution which improves further the performance of the methodology the numerical tests reported in the paper show that the proposed methodology is 30 times faster in 2d and 3 6 times faster in 3d than classic approaches based on the resolution of the hamilton jacobi equations and 2 7 times faster than explicit geometrical approaches the proposed methodology can also manage the separation of a crack front in several fronts during the propagation and the merging of several fronts into one single front

higher order xfem for curved strong and weak discontinuities the extended finite element method xfem enables the accurate approximation of solutions with jumps or kinks within elements optimal convergence rates have frequently been achieved for linear elements and piecewise planar interfaces higher order convergence for arbitrary curved interfaces relies on two major issues i an accurate quadrature of the galerkin weak form for the cut elements and ii a careful formulation of the enrichment which should preclude any problems in the blending elements for i we employ a strategy of subdividing the elements into subcells with only one curved side reference elements that are higher order on only one side are then used to map the integration points to the real element for ii we find that enrichments for strong discontinuities are easily extended to higher order accuracy in contrast problems in blending elements may hinder optimal convergence for weak discontinuities different formulations are investigated including the corrected xfem numerical results for several test cases involving strong or weak curved discontinuities are presented quadratic and cubic approximations are investigated optimal convergence rates are achieved using the standard xfem for the case of a strong discontinuity close to optimal convergence rates for the case of a weak discontinuity are achieved using the corrected xfem copyright 2009 john wiley sons ltd

enriched element free galerkin methods for crack tip fields summary the element free galerkin efg method is a meshless method for solving partial di erential equations which uses only a set of nodal points and a cad like description of the body to formulate the discrete model it has been used extensively for fracture problems and has yielded good results when adequate refinement is used near the crack tip but stresses tend to be oscillatatory near the crack tip unless substantial refinement is used an enriched efg formulation for fracture problems is proposed two methods are used 1 adding the asymptotic fields to the trial function and 2 augmenting the basis by the asymptotic fields a local mapping of the enriched fields for curved cracks is also described results show that both methods greatly reduce stress oscillations and allow the calculation of accurate stress intensity factors with far fewer degrees of freedom 1997 by john wiley sons ltd

michell trusses in the presence of microscale material randomness limitation of optimality the classical problem of a michell optimal truss concerns a minimum weight design of a planar truss that transmits a given load to a given rigid foundation with the requirement that the axial stresses in the bars of the truss stay within an allowable range the present study considers this problem when the truss is made of a material with random microstructure that is when is a random field the trusses tending to the optimal state can be determined through a net of characteristics generalized to a stochastic setting while in the classical case of a homogeneous material this net gives the minimum weight as its spacing tends to zero the presence of a random microstructure prevents the attainment of this state basically the finer the net the stronger the scatter of characteristics which forces one to use more structural material to compensate for these fluctuations in effect there is a limitation to the attainment of the optimality of the michell truss made of a hypothetical perfectly homogeneous material

complementarity problem arising from static growth of multiple cracks and mls based numerical manifold method abstract the full response of a brittle structure containing multiple cracks to loading under the servo control is of vital importance in the evaluation of properties of the structure during crack growth the fracture toughness condition at all the crack tips as well as the equilibrium condition should be obeyed leading to a nonlinear complementarity problem ncp the vector valued function in the ncp depends implicitly on the cracking increments which in turn determine the stress field the stress field can be obtained through solving a mixed variational problem since the degrees of freedom in the discrete variational problem vary with cracking not only in magnitude but also in number the jacobian matrix of the ncp is hard to compute and it cannot be expected to be solved by the newton methods that involve the calculation of jacobian matrices therefore a well scaled projection contraction algorithm is designed the proposed procedure is able to simulate growth of multiple cracks in a natural way allowing the crack tips to stop anywhere with no sensitivity to node configuration or cracking increments through the analysis of some examples that have been widely tested many interesting and profound phenomena are found which have never been revealed

polygon scaled boundary finite elements for crack propagation modelling summary an automatic crack propagation modelling technique using polygon elements is presented a simple algorithm to generate a polygon mesh from a delaunay triangulated mesh is implemented the polygon element formulation is constructed from the scaled boundary finite element method sbfem treating each polygon as a sbfem subdomain and is very efficient in modelling singular stress fields in the vicinity of cracks stress intensity factors are computed directly from their definitions without any nodal enrichment functions an automatic remeshing algorithm capable of handling any n sided polygon is developed to accommodate crack propagation the algorithm is simple yet flexible because remeshing involves minimal changes to the global mesh and is limited to only polygons on the crack paths the efficiency of the polygon sbfem in computing accurate stress intensity factors is first demonstrated for a problem with a stationary crack four crack propagation benchmarks are then modelled to validate the developed technique and demonstrate its salient features the predicted crack paths show good agreement with experimental observations and numerical simulations reported in the literature copyright 2012 john wiley sons ltd

optimal storage properties of neural network models the authors calculate the number p alpha n of random n bit patterns that an optimal neural network can store allowing a given fraction f of bit errors and with the condition that each right bit is stabilised by a local field at least equal to a parameter k for each value of alpha and k there is a minimum fraction fmin of wrong bits they find a critical line alpha c k with alpha c 0 2 the minimum fraction of wrong bits vanishes for alpha alpha c k the calculations are done using a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is locally stable in a finite region of the k alpha plane including the line alpha c k but there is a line above which the solution becomes unstable and replica symmetry must be broken

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

a simple method for crack growth in mixed mode with x fem abstract in this paper a new method for level set update is proposed in the context of crack propagation modeling with the extended finite element method x fem and level sets compared with the existing methods such as the resolution of the hamilton jacobi equations this new method is much simpler because it does not required complex manipulations of the level sets this method called the projection method uses both a classical discretization of the surface of the crack segments for 2d cracks and triangles for 3d cracks and a level set representation of the crack this discretization is updated with respect to the position of the new crack front then the level sets are re computed using the true distance to the new crack by an orthogonal projection of each node of the structure onto the new crack surface then numerical illustrations are given on 2d and 3d academic examples finally three illustrations are given on 3d industrial applications

singular functions for heterogeneous composites with cracks and notches the use of equilibrated singular basis functions abstract in this paper a novel method is presented to solve problems with weak singularities in two dimensional heterogeneous media using equilibrated singular basis functions this especially includes crack problems in composites of functionally graded material types the method is mainly presented in a boundary formulation although it may be found quite useful in other mesh based or mesh less approaches as the extended finite element method xfem the present paper considers harmonic and elasticity problems the most distinguished advantage of the present method is that the solution progress advances without absolutely any knowledge of the analytical singularity order of the problem to this end the partial differential equation of the problem is approximately satisfied in a weighted residual approach after developing the formulation in a mapped polar co ordination some primary basis functions generated from chebyshev polynomials and trigonometric functions along with corresponding weight functions are employed the numerical examples either selected from the well known literature or solved by well established techniques will demonstrate the capability of the method in problems related to composite materials

a marching cubes based failure surface propagation concept for three dimensional finite elements with non planar embedded strong discontinuities of higher order kinematics summary this paper presents an advanced failure surface propagation concept based on the marching cubes algorithm initially proposed in the field of computer graphics and applies it to the embedded finite element method when modeling three dimensional 3d solids at failure the propagation of the failure surface representing a crack or shear band should not exhibit a strong sensitivity to the details of the finite element discretization this results in the need for a propagation of the discrete failure zone through the individual finite elements which is possible for finite elements with embedded strong discontinuities whereas for two dimensional calculations the failure zone propagation location is easily predicted by the maximal principal stress direction more advanced strategies are needed to achieve a smooth failure surface in 3d simulations an example for such method is the global tracking algorithm which predicts the crack path by a scalar level set function computed on the basis of the solution of a simplified heat conduction like problem its prediction may though lead to various scenarios on how the failure surface may propagate through the individual finite elements in particular for a hexahedral eight node finite element 256 such cases exist to capture all those possibilities the marching cubes algorithm is combined with the global tracking algorithm and the finite elements with embedded strong discontinuities in this work in addition because many of the possible cases result in non planar failure surfaces within a single finite element and because the local quantities used to describe the kinematics of the embedded strong discontinuities are physically meaningful in a strict sense only for planar failure surfaces a remedy for such scenarios is proposed various 3d failure propagation simulations outline the performance of the proposed concept copyright 2013 john wiley sons ltd

multiconnected neural network models a generalisation of the hopfield model which includes interactions between p 2 ising spins is considered the exact storage capacity behaves as np 1 2 p 1 ln n when the number of nodes n is large in the limit p to infinity the thermodynamics of the model can be solved exactly without using the replica method at zero temperature a solution which is completely correlated with the input pattern exists for alpha alpha c where alpha c to infinity as p to infinity and this solution has lower energy than the spin glass solution if alpha alpha 1 1 4 ln 2 where the number of patterns n 2 alpha p np 1 for finite values of p the correlation with the input pattern is not complete for p 3 and 4 approximate values of alpha c and alpha 1 are obtained and for p to infinity the replica symmetric approximation gives alpha c approximately p 4 ln p

on the growth and decay of acceleration waves in random media we study the effects of material spatial randomness on the growth to shock or decay of acceleration waves in the deterministic formulation such waves are governed by a bernoulli equation d d x x x 2 in which the material coefficients and represent the dissipation and elastic nonlinearity respectively in the case of a random microstructure the wavefront sees the local details it is a mesoscale window travelling through a random continuum upon a stochastic generalization of the bernoulli equation both coefficients become stationary random processes and the critical amplitude c as well as the distance to form a shock x become random variables we study the character of these variables especially as compared to the deterministic setting for various cases of the random process i one white noise ii two independent white noises iii two correlated gaussian noises and iv an ornstein uhlenbeck process situations of fully positively negatively or zero correlated noises in and are investigated in detail particular attention is given to the determination of the average critical amplitude c equations for the evolution of the moments of the probability of formation of a shock wave within a given distance x and the average distance to form a shock wave specific comparisons of these quantities are made with reference to a homogeneous medium defined by the mean values of the x process

three unfinished works on the optimal storage capacity of networks the optimal storage properties of three different neural network models are studied for two of these models the architecture of the network is a perceptron with or j interactions whereas for the third model the output can be an arbitrary function of the inputs analytic bounds and numerical estimates of the optimal capacities and of the minimal fraction of errors are obtained for the first two models the third model can be solved exactly and the exact solution is compared to the bounds and to the results of numerical simulations used for the two other models