on a class of hemivariational inequalities at resonance abstract we consider a class of noncoercive hemivariational inequalities involving the p laplacian at resonance we use the unilateral growth condition so the energy functional is nonsmooth nonconvex and its effective domain does not coincide with the whole space w 1 p 0 to avoid this difficulty we study the problem in finite dimensional spaces using the mountain pass theorem for locally lipschitz functionals and then we pass to the limit to obtain the existence of solutions

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

the complex lorentzian leech lattice and the bimonster ii let d be the incidence graph of the projective plane over f3 the artin group of the graph d maps onto the bimonster and a complex hyperbolic reflection group acting on 13 dimensional complex hyperbolic space y the generators of the artin group are mapped to elements of order 2 resp 3 in the bimonster resp let y y be the complement of the union of the fixed points of reflections in daniel allcock has conjectured that the orbifold fundamental group of y surjects onto bimonster in this article we study the reflection group we show that th e artin group of d maps to the orbifold fundamental group of y thus answering a question in allcock s article a monstrous proposal and taking one step towards the proof of allcock s conjecture the finite group aut d acts on y we make a detailed study of the complex hyperbolic line fixed by the subgroup l3 3 aut d we define meromorphic automorphic forms on y invariant under with poles along mirrors we show that the restriction of these forms to the complex hyperbolic line fixed by l3 3 gives meromorphic modular forms of level 13

anatomy of the monster ii abstract we summarise much of what is currently known about the subgroup structure of the fischer griess monster introduction the publication of the atlas of finite groups 1 henceforth referred to as the atlas marked the end of a chapter in the history of group theory in more ways than one the classification of finite simple groups had been completed and using new computational techniques the last of the sporadic groups had been constructed and enough progress had been made in classifying their maximal subgroups to make the rest seem like a tidying up job since the publication of the atlas these computational methods have been pushed further the classification of the maximal subgroups for fi 22 th fi 23 j 4 and fi 24 was completed by kleidman linton parker and wilson in 7 8 9 10 12 13 14 this means that appendix 2 of the atlas of brauer characters 11 when collated with the atlas itself completes the listing for all sporadic groups except the baby monster and monster for the baby monster four new maximal subgroups s 6 l 3 4 2 2 s 6 s 6 4 l 2 31 and l 2 49 2 3 were given in appendix 2 of 11 and wilson has since found four more m 11 l 3 3 l 2 17 2 and l 2 ll 2 for references see 23 24 25 with work on the 2 locals by meierfrankenfeld and shpektorov 15 an end to the baby monster problem is in sight with a lot of work also having been done on maximal subgroups of generic groups this leaves only the monster unfortunately the degree of its smallest representation over any field 196882 is too big for present day computational techniques to make any headway

an explicit dynamics extended finite element method part 2 element by element stable explicit explicit dynamic scheme abstract this paper presents a general explicit time integration scheme for dynamics simulations using the extended finite element method with standard critical time step we use the generalized mass lumping technique proposed in part i of this paper this technique allows us to consider arbitrary enrichment functions in the x fem for explicit dynamics simulations in this second part the proposed approach allows the use of standard finite elements critical time step estimates for that purpose we develop a classical element by element strategy that couples the standard central difference scheme with the unconditionally stable explicit scheme proposed by chang s y chang an explicit method with improved stability property int j numer methods engrg 77 8 2008 1100 1120 this scheme coupled with x fem allows us to recover fe critical time steps independently of the enrichment functions considered furthermore a study of the stability property of this new explicit scheme is proposed in the x fem framework both with fixed enrichments and evolving enrichments with time some examples illustrate the good properties of the new explicit numerical time scheme and some applications to dynamics crack growth are given

critical point theory for nonsmooth functionals abstract in this paper we develop critical point theory for nonsmooth functional f h 0 1 r defined by f u 1 2 i j 1 a i j x u d i u d j u d x g x u d x the corresponding deformation lemmas are proved with the application to the bifurcation for quasilinear schrodinger equations in mind we extend the obtained results to that for a functional defined on a product space and prove a generalized saddle point theorem

a rigorous and unified mass lumping scheme for higher order elements abstract in dynamic analysis with explicit time integration schemes a lumped mass matrix lmm is preferable because lmm can avoid solving the large scale simultaneous algebraic equations mathematically rigorous mass lumping schemes such as the mass lumping by nodal quadrature and the row sum technique are applicable to only linear or bilinear elements for higher order elements such as 8 node serendipity elements the diagonal scaling procedure is the only lumping method that can be recommended to generate positive definite diagonal element mass matrices unfortunately there is no mathematical theory in support of this approach this study proposes a general mass lumping scheme applicable to higher order elements where the virtual work of initial force is integrated over the problem domain that is viewed as the manifold covered by the finite element patches by a series of numerical experiments both free and forced vibration problems it is suggested that even in the implicit time integration scheme the consistent mass matrix cmm can be superseded by the proposed lmm furthermore the proposed lmm has much stronger adaptability to distorted meshes

cracking node method for dynamic fracture with finite elements a new method for modeling discrete cracks based on the extended finite element method is described in the method the growth of the actual crack is tracked and approximated with contiguous discrete crack segments that lie on finite element nodes and span only two adjacent elements the method can deal with complicated fracture patterns because it needs no explicit representation of the topology of the actual crack path a set of effective rules for injection of crack segments is presented so that fracture behavior beginning from arbitrary crack nucleations to macroscopic crack propagation is seamlessly modeled the effectiveness of the method is demonstrated with several dynamic fracture problems that involve complicated crack patterns such as fragmentation and crack branching copyright 2008 john wiley sons ltd

locally toroidal polytopes of rank 6 and sporadic groups abstract we augment the list of finite universal locally toroidal regular polytopes of type 3 3 4 3 3 due to p mcmullen and e schulte adding as well as removing entries this disproves a related long standing conjecture our new universal polytope is related to a well known y shaped presentation for the sporadic simple group fi 22 and admits s 4 o 8 2 s 3 as the automorphism group we also discuss further extensions of its quotients in the context of y shaped presentations as well we note that two known examples of finite universal polytopes of type 3 3 4 3 3 are related to y shaped presentations of orthogonal groups over f 2 mixing construction is used in a number of places to describe covers and 2 covers

critical point theory for nondifferentiable functionals and applications abstract we consider minimization results for locally lipschitzian functionals on a banach space by introducing a compactness condition of the palais smale type which is suggested by the variational principle of ekeland applications are given to some semilinear elliptic problems with discontinuous nonlinearities

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

stable imposition of stiff constraints in explicit dynamics for embedded finite element methods summary we investigate various strategies to enforce the kinematics at an embedded interface for transient problems within the extended finite element method in particular we focus on explicit time integration of the semi discrete equations of motion and extend both dual and primal variational frameworks for constraint enforcement to a transient regime we reiterate the incompatibility of the dual formulation with purely explicit time integration and the severe restrictions placed by the courant friedrichs levy condition on primal formulations we propose an alternate consistent formulation for the primal method and derive an estimate for the stabilization parameter which is more amenable in an explicit dynamics framework importantly the use of the new estimate circumvents the need for any tolerances as an interface approaches an element boundary we also show that with interfacial constraints existing mass lumping schemes can lead to prohibitively small critical time steps accordingly we propose a mass lumping procedure which provides a more favorable estimate these techniques are then demonstrated on several benchmark numerical examples where we compare and contrast the accuracy of the primal methods against the dual methods in enforcing the constraints copyright 2012 john wiley sons ltd

variational methods for non differentiable functionals and their applications to partial differential equations the interest in extending the theory of nonlinear partial differential equations pde to pde with discontinuous nonlinearities dnde has been increasing daily the reasons are twofold 1 many free boundary problems may be reduced to boundary value problems of dnde cf chang 16 71 chang and jiang 9 and fleishman and mahar 121 2 sometimes in dealing with a pde problem it is beneficial to put the original pde into a large category for instance dnde cf chang 7 simply speaking a dnde is as follows

on generation of lumped mass matrices in partition of unity based methods summary the partition of unity based methods such as the extended finite element method and the numerical manifold method are able to construct global functions that accurately reflect local behaviors through introducing locally defined basis functions beyond polynomials in the dynamic analysis of cracked bodies using an explicit time integration algorithm as a result huge difficulties arise in deriving lumped mass matrices because of the presence of those physically meaningless degrees of freedom associated with those locally defined functions observing no spatial derivatives of trial or test functions exist in the virtual work of inertia force we approximate the virtual work of inertia force in a coarser manner than the virtual work of stresses where we inversely utilize the from local to global skill the proposed lumped mass matrix is strictly diagonal and can yield the results in agreement with the consistent mass matrix but has more excellent dynamic property than the latter meanwhile the critical time step of the numerical manifold method equipped with an explicit time integration scheme and the proposed mass lumping scheme does not decrease even if the crack in study approaches the mesh nodes a very excellent dynamic property copyright 2017 john wiley sons ltd

modeling holes and inclusions by level sets in the extended finite element method a methodology to model arbitrary holes and material interfaces inclusions without meshing the internal boundaries is proposed the numerical method couples the level set method s osher j a sethian j comput phys 79 1 1988 12 to the extended finite element method x fem n moes j dolbow t belytschko int j numer methods engrg 46 1 1999 131 in the x fem the finite element approximation is enriched by additional functions through the notion of partition of unity the level set method is used for representing the location of holes and material interfaces and in addition the level set function is used to develop the local enrichment for material interfaces numerical examples in two dimensional linear elastostatics are presented to demonstrate the accuracy and potential of the new technique

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

on existence of solutions for parabolic hemivariational inequalities abstract a class of parabolic hemivariational inequalities is considered in which the energy function is locally lipschitz not necessarily convex or differentiable and the nonlinear evolution operator is multivalued and generalized pseudomonotone using techniques of multivalued analysis and a surjectivity result for l generalized pseudomonotone operators we prove the existence for a cauchy problem and a periodic problem some applications to nonsmooth mechanics are also indicated

stability of multi time step partitioned integrators for first order finite element systems abstract mixed time integration procedures are those in which different time steps and different integrators are used in different parts of the mesh a rigorous stability analysis is provided for the first time for this type of time integration for first order semidiscretizations such as arise in diffusion problems the stability analysis is verified by some numerical examples

arbitrary branched and intersecting cracks with the extended finite element method summary extensions of a new technique for the nite element modelling of cracks with multiple branches multiple holes and cracks emanating from holes are presented this extended nite element method x fem allows the representation of crack discontinuities and voids independently of the mesh a standard displacementbased approximation is enriched by incorporating discontinuous elds through a partition of unity method a methodology that constructs the enriched approximation based on the interaction of the discontinuous geometric features with the mesh is developed computation of the stress intensity factors sif in dierent examples involving branched and intersecting cracks as well as cracks emanating from holes are presented to demonstrate the accuracy and the robustness of the proposed technique copyright 2000 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

x fem explicit dynamics for constant strain elements to alleviate mesh constraints on internal or external boundaries this paper deals with the use of the extended finite element method x fem for rapid dynamic problems to solve the equations of motion a common technique is the explicit direct integration with a newmark scheme since this temporal scheme is only conditionally stable the critical time step must be determined it is generally induced by mesh constraints the idea of the paper is to weaken constraints on mesh generation algorithms so that the critical time step is as large as possible using the x fem one allows a non conformity between mesh and discontinuities such as cracks holes or interfaces in a first part we present a summary about direct integration schemes and about the extended finite element method then we focus on the theoretical description of a 1d x fem finite element and its generalization to 2d and 3d finite elements then dynamic numerical simulations are shown they concern structures under impact with holes or external boundaries not exactly matched by the mesh comparisons are made with numerical results coming from the abaqus software it shows that developments are satisfactory we conclude with some outlooks concerning this work

an explicit dynamics extended finite element method part 1 mass lumping for arbitrary enrichment functions this paper presents a general mass lumping technique for explicit dynamics simulations using the extended finite element method with arbitrary enrichment functions the proposed mass lumping technique is a generalization of previously published results for cracks and holes time step estimates are studied for crack singular enrichment functions and for hole enrichment in both cases we show that the critical time step does not tend to zero and is of the same order as that of the same unenriched element the performance of the method is demonstrated on several numerical examples that compare well with classical finite elements and previously published x fem results

element local level set method for three dimensional dynamic crack growth summary an approximate level set method for three dimensional crack propagation is presented in this method the discontinuity surface in each cracked element is defined by element local level sets ellss the local level sets are generated by a fitting procedure that meets the fracture directionality and its continuity with the adjacent element crack surfaces in a least square sense a simple iterative procedure is introduced to improve the consistency of the generated element crack surface with those of the adjacent cracked elements the discrete discontinuity is treated by the phantom node method which is a simplified version of the extended finite element method xfem the ells method and the phantom node technology are combined for the solution of dynamic fracture problems numerical examples for three dimensional dynamic crack propagation are provided to demonstrate the effectiveness and robustness of the proposed method copyright q 2009 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

mass lumping strategies for x fem explicit dynamics application to crack propagation this paper deals with the numerical modelling of cracks in the dynamic case using the extended finite element method more precisely we are interested in explicit algorithms we prove that by using a specific lumping technique the critical time step is exactly the same as if no crack were present this somewhat improves a previous result for which the critical time step was reduced by a factor of square root of 2 from the case with no crack the new lumping technique is obtained by using a lumping strategy initially developed to handle elements containing voids to be precise the results obtained are valid only when the crack is modelled by the heaviside enrichment note also that the resulting lumped matrix is block diagonal blocks of size 2 2 for constant strain elements linear simplex elements the critical time step is not modified when the element is cut thanks to the lumped mass matrix the critical time step never tends to zero moreover the lumping techniques conserve kinetic energy for rigid motions in addition tensile stress waves do not propagate through the discontinuity hence the lumping techniques create neither error on kinetic energy conservation for rigid motions nor wave propagation through the crack both these techniques will be used in a numerical experiment copyright 2007 john wiley sons ltd

presenting the baby monster abstract we prove that y 433 is isomorphic to 2 2 f 2 where f 2 is fischer s baby monster group this answers the last open problem concerning the description of y groups the result implies in particular that y 553 and y 555 are fabulous systems of generators for the monster and bimonster respectively in a sense introduced by j conway we also give a new independent proof that y 555 is the bimonster

a critical point theorem via the ekeland variational principle abstract the aim of this paper is to establish the existence of a local minimum for a continuously g teaux differentiable function possibly unbounded from below without requiring any weak continuity assumption several special cases are also emphasized moreover a novel definition of palais smale condition which is more general than the usual one is presented and a mountain pass theorem is pointed out as a consequence multiple critical points theorems are then established finally as an example of applications an elliptic dirichlet problem with critical exponent is investigated

y groups via transitive extension abstract motivated by an earlier observation by j h fischer around 1980 b conway conjectured that the coxeter diagram y555 on fig 1 together with a single additional so called spider relation ab1c1ab2c2ab3c3 10 1 form a presentation for the wreath product of the monster group m and a group of order 2 this conjecture was proved by s p norton and the author in 1990 the original proof was rather involved relying on simple connectedness results for certain diagram geometries on numerous data obtained by coset enumeration on a computer and on some delicate calculations with subgroups coming from the 26 node theorem in the present work we follow an inductive approach to the identification of y groups by considering larger y groups as transitive extensions of smaller ones along these lines we obtain an alternative identification proof for the y groups which is almost computer free we refer to only one result of double coset enumeration our approach provides a uniform understanding of the y groups particularly of features such as centres and redundant generators

non abelian representations of some sporadic geometries for a point line incidence system s p l with three points per line we define the universal representation group of s as formula we prove that if g is the 2 local parabolic geometry of the sporadic simple groupf1 the monster orf2 the baby monster thenr g f1or 2 f2 respectively

26 implies the bimonster the bimonster or wreathed square 2 of the monster group is presented by the coxeter relations of the 26 node incidence graph of the projective plane of order 3 along with the additional relations that all free 12 gons of this diagram generate symmetric groups s we prove this 12 using purely geometric arguments as well as the celebrated ivanov norton theorem 6 7 9 this elementary presentation provides a natural connection from the projective plane of order 3 to the bimonster previously this connection was known in the other direction 2 3 we also consider other groups that can be obtained in a similar fashion a useful reference on coxeter and reflection groups is 5 we use the atlas 1 notation for finite groups

nonlinear elliptic eigenvalue problems with discontinuities abstract in this paper we study the existence of solution for two different eigenvalue problems the first is nonlinear and the second is semilinear our approach is based on results from the nonsmooth critical point theory in the first theorem we prove the existence of at least two nontrivial solutions when is in a half axis in the second theorem based on a nonsmooth variant of the generalized mountain pass theorem we prove the existence of at least one nontrivial solution for every r

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

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

elliptic problems with discontinuities in this paper we prove two existence theorems for elliptic problems with discontinuities the first one is a noncoercive dirichlet problem and the second one is a neumann problem we do not use the method of upper and lower solutions for neumann problems we assume that f is nondecreasing we use the critical point theory for locally lipschitz functionals