a finite element method for the simulation of strong and weak discontinuities in solid mechanics in this paper we introduce and analyze a finite element method for elasticity problems with interfaces the method allows for discontinuities internal to the elements in the approximation across the interface we propose a general approach that can handle both perfectly and imperfectly bonded interfaces without modifications of the code for the case of linear elasticity we show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh we present numerical examples for the linear case as well as for contact and crack propagation model problems

pre and post gel size distributions in ir reversible polymerisation a class of irreversible coagulation processes can be modelled by smoluchowski s coagulation equation with rate constants kij a b i j cij non negative a b and c for c not 0 a gelation transition occurs the authors obtain explicit solutions for the size distribution ck t with ck 0 delta k1 next they construct and solve the equations for reversible polymerisation by incorporating break up processes in the kinetic equation with a unimolecular fragmentation rate fij lambda ninjkij ni j the degeneracy factors nk obey k 1 nk 1 2 sigma kijninj with i j k and n1 1 and the strength parameter lambda exp g kbt where the binding energy g to infinity for irreversible coagulation explicit results are only given for flory s polymerisation models raf and braf 1 in the vicinity of the gel point the authors verify the scaling hypothesis and calculate critical exponents

exact solutions for the coagulation fragmentation equation the authors present new exact equilibrium and time dependent solutions to the two main versions of the coagulation fragmentation equation equilibria are first found for the pure coagulation and pure fragmentation equations equilibrium solutions for the full coagulation fragmentation equation are then constructed when the kernels governing the reaction rates are chosen to be in a particular form time dependent solutions are also derived for the pure fragmentation equation

an extended finite element method with analytical enrichment for cohesive crack modeling a recent approach to fracture modeling has combined the extended finite element method xfem with cohesive zone models most studies have used simplified enrichment functions to represent the strong discontinuity but have lacked an analytical basis to represent the displacement gradients in the vicinity of the cohesive crack in this study enrichment functions based upon an existing analytical investigation of the cohesive crack problem are proposed these functions have the potential of representing displacement gradients in the vicinity of the cohesive crack and allow the crack to incrementally advance across each element key aspects of the corresponding numerical formulation and enrichment functions are discussed a parameter study for a simple mode i model problem is presented to evaluate if quasi static crack propagation can be accurately followed with the proposed formulation the effects of mesh refinement and mesh orientation are considered propagation of the cohesive zone tip and crack tip time variation of the cohesive zone length and crack profiles are examined the analysis results indicate that the analytically based enrichment functions can accurately track the cohesive crack propagation of a mode i crack independent of mesh orientation a mixed mode example further demonstrates the potential of the formulation copyright 2008 john wiley sons ltd

a triangle of interconnected coagulation models a number of new coagulation models depending on a parameter is derived the dependence is considered in two different ways if the parameter takes its maximal value then in the first case we obtain a new discrete kinetic equation we demonstrate that its continuous version is simply the oort hulst s coagulation model in the second case the maximum of the corresponding parameter yields the smoluchowski coagulation equation at the minimal values of both parameters we arrive at another new kinetic equation these three models form a triangle connecting the two known coagulation equations situated in its vertexes smoluchowski and oort hulst equations via an alteration of the parameters also a comparative analysis of these three models is presented as an advantage of the oort hulst approach we compute the coagulation front and establish a connection between the infringement of the mass conservation law and convergence of the coagulation front to infinity

cohesive and non cohesive fracture by higher order enrichment of xfem summary a comprehensive study is performed on the use of higher order terms of the crack tip asymptotic fields as enriching functions for the extended fem xfem for both cohesive and traction free cracks for traction free cracks the williams asymptotic field is used to obtain highly accurate stress intensity factors sifs directly from the enriched degrees of freedom without any post processing the low accuracy of the results of the original research on this subject by liu et al int j numer meth engng 2004 59 1103 1118 is remedied here by appropriate modifications of the enrichment scheme the modifications are simple and can be easily included into an xfem computer code for cohesive cracks the relevant asymptotic field is used and two widely used criteria including the sifs criterion and the stress criterion are examined for the crack growth simulation both linear and nonlinear cohesive laws are used for the stress criterion averaging is avoided due to the highly accurate crack tip approximation because of the higher order enrichment then a modified stress criterion is proposed which is shown to be applicable to a wider class of problems several numerical examples including straight and curved cracks stationary and growing cracks single and multiple cracks and traction free and cohesive cracks are studied to investigate in detail the robustness and efficiency of the proposed enrichment scheme copyright 2012 john wiley sons ltd

the extended generalized finite element method an overview of the method and its applications an overview of the extended generalized finite element method gefm xfem with emphasis on methodological issues is presented this method enables the accurate approximation of solutions that involve jumps kinks singularities and other locally non smooth features within elements this is achieved by enriching the polynomial approximation space of the classical finite element method the gefm xfem has shown its potential in a variety of applications that involve non smooth solutions near interfaces among them are the simulation of cracks shear bands dislocations solidification and multi field problems copyright 2010 john wiley sons ltd

cohesive zone models higher order continuum theories and reliability methods for computational failure analysis a concise overview is given of various numerical methods that can be used to analyse localization and failure in engineering materials the importance of the cohesive zone approach is emphasized and various ways to incorporate the cohesive zone methodology in discretization methods are discussed numerical representations of cohesive zone models suffer from a certain mesh bias for discrete representations this is caused by the initial mesh design while for smeared representations it is rooted in the ill posedness of the rate boundary value problem that arises upon the introduction of decohesion a proper representation of the discrete character of cohesive zone formulations which avoids any mesh bias can be obtained elegantly when exploiting the partition of unity property of finite element shape functions the effectiveness of the approach is demonstrated for some examples at different scales moreover examples are shown how this concept can be used to obtain a proper transition from a plastifying or damaging continuum to a shear band with gross sliding or to a fully open crack true discontinuum when adhering to a continuum description of failure higher order continuum models must be used meshless methods are ideally suited to assess the importance of the higher order gradient terms as will be shown finally regularized strain softening models are used in finite element reliability analyses to quantify the probability of the emergence of various possible failure modes

on exact solutions for the coagulation fragmentation equation it is pointed out that the equilibrium solutions for pure coagulation and pure fragmentation which dubovskii et al ibid vol 25 p 4737 1992 purport to obtain are in error and that such solutions do not in fact exist

a simple proof of a theorem by uhlenbeck and yau a subbundle of a hermitian holomorphic vector bundle e h can be metrically and differentially defined by the orthogonal projection onto itself a weakly holomorphic subbundle of e h is by definition an orthogonal projection lying in the sobolev space l21 of l2 sections of end e with l2 first order derivatives in the sense of distributions which satisfies furthermore id d 0 a weakly holomorphic subbundle of e h is shown to define a coherent subsheaf of open image in new window e and implicitly a holomorphic subbundle of e in the complement of an analytic subset of codimension 2 this result provided the key technical argument to the proof given by uhlenbeck and yau for the kobayashi hitchin correspondence on compact kahler manifolds we give here a much simpler proof based on current theory the idea is to construct local meromorphic sections of im which locally span the fibers we first make this construction on one dimensional submanifolds of x and subsequently extend it by means of a hartogs type theorem of shiffman s

comment on singular solutions to the stationary coagulation equation we comment on our paper exact solutions for the coagulation fragmentation equation ibid vol 25 p 4737 1992

a comment on the article a finite element method for simulation of strong and weak discontinuities in solid mechanics by a hansbo and p hansbo comput methods appl mech engrg 193 2004 3523 3540

the generalized finite element method this paper describes a pilot design and implementation of the generalized finite element method gfem as a direct extension of the standard finite element method sfem or fem which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the fem the development of the gfem is illustrated for the laplacian in two space dimensions in domains which may include several hundreds of voids and or cracks for which the construction of meshes used by the fem is practically impossible the two main capabilities are 1 it can construct the approximation using meshes which may overlap part or all of the domain boundary 2 it can incorporate into the approximation handbook functions which are known analytically or are generated numerically and approximate well the solution of the boundary value problem in the neighborhood of corner points voids cracks etc the main tool is a special integration algorithm which we call the fast remeshing approach which is robust and works for any domain with arbitrary complexity the incorporation of the handbook functions into the gfem is done by employing the partition of unity method pum the presented formulations and implementations can be easily extended to the multi material medium where the voids are replaced by inclusions of various shapes and sizes and to the case of the elasticity problem this work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the gfem which is needed before analogous implementations are attempted in the three dmensional and nonlinear cases which are the cases of main interest for future work

the yang mills flow and the atiyah bott formula on compact kahler manifolds we study the yang mills flow on a holomorphic vector bundle e over a compact k ahler manifold x along a solution of the flow we show that the curvature endomorphism i lambda f a t approaches in l 2 an endomorphism with constant eigenvalues given by the slopes of the quotients from the harder narasimhan filtration of e this proves a sharp lower bound for the hermitian yang mills functional and thus the yang mills functional generalizing to arbitrary dimension a formula of atiyah and bott first proven on riemann surfaces furthermore we show any reflexive extension to all of x of the limiting bundle e infty is isomorphic to rm gr rm hns e verifying a conjecture of bando and siu

improving the accuracy of xfem crack tip fields using higher order quadrature and statically admissible stress recovery this study is concerned with improving the accuracy of crack tip fields obtained using the extended generalized finite element method xfem first the numerical integration of the element stiffness matrices which guarantees convergence with quadrature of not only the regular nodal displacements but also additional degrees of freedom corresponding to the enrichment functions is studied as the accuracy of the stresses obtained by direct differentiation of the converged with quadrature regular nodal displacements and of the coefficients corresponding to enrichment functions is still not satisfactory a statically admissible stress recovery sar scheme is introduced sar uses basis functions which meet the equilibrium equations within the domain and the local traction conditions on the boundary and moving least squares mls to fit the stresses at sampling points e g quadrature points obtained by the xfem important parameters controlling the accuracy of crack tip fields using the xfem and sar namely the order of quadrature the number of retained terms in the crack tip asymptotic field the number of enriched layers and use of arbitrary branch functions a proper choice of the sampling points in the enriched element and the size of the domain of influence doi of mls are investigated copyright 2005 john wiley sons ltd

extended finite element method for three dimensional crack modelling an extended finite element method x fem for three dimensional crack modelling is described a discontinuous function and the two dimensional asymptotic crack tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity this enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed stress intensity factors sifs for planar three dimensional cracks are presented which are found to be in good agreement with benchmark solutions copyright 2000 john wiley sons ltd

improved implementation and robustness study of the x fem for stress analysis around cracks numerical crack propagation schemes were augmented in an elegant manner by the x fem method the use of special tip enrichment functions as well as a discontinuous function along the sides of the crack allows one to do a complete crack analysis virtually without modifying the underlying mesh which is of industrial interest especially when a numerical model for crack propagation is desired this paper improves the implementation of the x fem method for stress analysis around cracks in three ways first the enrichment strategy is revisited the conventional approach uses a topological enrichment only the elements touching the front are enriched we suggest a geometrical enrichment in which a given domain size is enriched the improvements obtained with this enrichment are discussed second the conditioning of the x fem both for topological and geometrical enrichments is studied a preconditioner is introduced so that off the shelf iterative solver packages can be used and perform as well on x fem matrices as on standard fem matrices the preconditioner uses a local nodal cholesky based decomposition third the numerical integration scheme to build the x fem stiffness matrix is dramatically improved for tip enrichment functions by the use of an ad hoc integration scheme a 2d benchmark problem is designed to show the improvements and the robustness

numerical simulation and convergence analysis of a finite volume scheme for solving general breakage population balance equations this paper presents a finite volume scheme fvs for solving general breakage population balance equations bpbes in particular the number density based bpbe is transformed to the form of a mass conservation law then it becomes easy to apply the well known fvss that have an important property of mass conservation following kumar and warnecke for fixed pivot fp method 16 and cell average technique cat 15 the stability and the convergence analysis of the semi discretized fvs are studied unlike the cat and the fp method the fvs is second order consistent independently of the type of meshes we also observe that fvs gives second order convergence rate on four different types of uniform and non uniform meshes with non decreasing behavior of mesh width nevertheless one order better accuracy than the fp method is achieved on locally uniform meshes it is also noticed that on non uniform random meshes the fvs shows one order and two orders higher accuracy than the cat and the fp method respectively the mathematical results of convergence analysis are validated numerically by taking three test problems the numerical simulations are also compared with the results obtained by the cat and the fp method

existence of gelling solutions for coagulation fragmentation equations we study the smoluchowski coagulation fragmentation equation which is an infinite set of non linear ordinary differential equations describing the evolution of a mono disperse system of particles in a well stirred solution approximating the solutions of the smoluchowski equations by a sequence of finite markov chains we investigate the qualitative behavior of the solutions we determine a device on the finite chains which can detect the gelation phenomena the density dropping phenomena it shows how the gelation phenomena are reflected on the sequence of finite markov chains using this device we determine various types of gelation kernels and get the bounds of gelation times

maximal representations of uniform complex hyperbolic lattices let rho be a maximal representation of a uniform lattice gamma subset rm su n 1 n geq 2 in a classical lie group of hermitian type h we prove that necessarily h rm su p q with p geq qn and there exists a holomorphic or antiholomorphic rho equivariant map from complex hyperbolic space to the symmetric space associated to rm su p q this map is moreover a totally geodesic homothetic embedding in particular up to a representation in a compact subgroup of rm su p q the representation rho extends to a representation of rm su n 1 in rm su p q

mathematical analysis of finite volume preserving scheme for nonlinear smoluchowski equation abstract this study presents the convergence analysis of the recently developed finite volume preserving scheme forestier coste and mancini 2012 for approximating a coalescence or smoluchowski equation the idea of the finite volume scheme is to preserve the total volume in the system by modifying the coalescence kernel using the notion of overlapping bins cells the consistency of the finite volume scheme is examined thoroughly in order to prove second order convergence on uniform non uniform smooth and locally uniform grids independently of the aggregation kernel the theoretical observations of order of convergence is verified using the experimental order of convergence for analytically tractable kernels

incremental secant modulus iteration scheme and stress recovery for simulating cracking process in quasi brittle materials using xfem in this paper an incremental secant modulus iteration scheme using the extended generalized finite element method xfem is proposed for the simulation of cracking process in quasi brittle materials described by cohesive crack models whose softening law is composed of linear segments the leading term of the displacement asymptotic field at the tip of a cohesive crack which ensures a displacement discontinuity normal to the cohesive crack face is used as the enrichment function in the xfem the opening component of the same field is also used as the initial guess opening profile of a newly extended cohesive segment in the simulation of cohesive crack propagation a statically admissible stress recovery sar technique is extended to cohesive cracks with special treatment of non homogeneous boundary tractions the application of locally normalized co ordinates to eliminate possible ill conditioning of sar and the influence of different weight functions on sar are also studied several mode i cracking problems in quasi brittle materials with linear and bilinear softening laws are analysed to demonstrate the usefulness of the proposed scheme as well as the characteristics of global responses and local fields obtained numerically by the xfem

convergence analysis of sectional methods for solving breakage population balance equations ii the cell average technique this work presents the convergence of the cell average technique kumar et al in powder technol 179 205 228 2007 for solving breakage population balance equation similarly to our paper kumar and warnecke numerische math 2008 of this series we study convergence on four different types of meshes a second order convergence is proved for uniform locally uniform and non uniform smooth meshes finally the scheme is analyzed on random mesh and it is found that the scheme is only first order accurate nevertheless we obtain for locally uniform as well as for random mesh one order higher accuracy than the fixed pivot technique discussed by the authors in the first paper all mathematical observations of convergence analysis are also validated numerically and numerical results are compared with the results of the first part

multiscale aggregating discontinuities a method for circumventing loss of material stability new methods for the analysis of failure by multiscale methods that invoke unit cells to obtain the subscale response are described these methods called multiscale aggregating discontinuities are based on the concept of perforated unit cells which exclude subdomains that are unstable i e exhibit loss of material stability using this concept it is possible to compute an equivalent discontinuity at the coarser scale including both the direction of the discontinuity and the magnitude of the jump these variables are then passed to the coarse scale model along with the stress in the unit cell the discontinuity is injected at the coarser scale by the extended finite element method analysis of the procedure shows that the method is consistent in power and yields a bulk stress strain response that is stable applications of this procedure to crack growth in heterogeneous materials are given copyright 2007 john wiley sons ltd

a regularized xfem framework for embedded cohesive interfaces this work focuses on xfem modeling of elastodamaging interfaces characterized by regularized kinematics after regularization of the displacement discontinuity a process zone of finite width with continuous strain fields emerges the stress fields are independently energy deduced modeled via specific constitutive laws and subsequently introduced within the virtual work principle the methodology can address cohesive interfaces with vanishing and finite thickness in a unified way two dimensional results for a strain localization problem and several delamination tests are discussed in detail and compared with previous numerical experimental results