construction of invariant whiskered tori by a parameterization method part ii quasi periodic and almost periodic breathers in coupled map lattices we construct quasi periodic and almost periodic solutions for coupled hamiltonian systems on an infinite lattice which is translation invariant the couplings can be long range provided that they decay moderately fast with respect to the distance for the solutions we construct most of the sites are moving in a neighborhood of a hyperbolic fixed point but there are oscillating sites clustered around a sequence of nodes the amplitude of these oscillations does not need to tend to zero in particular the almost periodic solutions do not decay at infinity we formulate an invariance equation solutions of this equation are embeddings of an invariant torus on which the motion is conjugate to a rotation we show that if there is an approximate solution of the invariance equation that satisfies some non degeneracy conditions there is a true solution close by the proof of this emph a posteriori theorem is based on a nash moser iteration which does not use transformation theory simpler versions of the scheme were developed in e fontich r de la llave y sire emph j differential equations bf 246 3136 2009 one technical tool important for our purposes is the use of weighted spaces that capture the idea that the maps under consideration are local interactions using these weighted spaces the estimates of iterative steps are similar to those in finite dimensional spaces in particular the estimates are independent of the number of nodes that get excited using these techniques given two breathers we can place them apart and obtain an approximate solution which leads to a true solution nearby by repeating the process infinitely often we can get solutions with infinitely many frequencies which do not tend to zero at infinity

stabilized finite element formulation for elastic plastic finite deformations abstract this paper presents a stabilized finite element formulation for nearly incompressible finite deformations in hyperelastic plastic solids such as metals an updated lagrangian finite element formulation is developed where mesh dependent terms are added to enhance the stability of the mixed finite element formulation this formulation circumvents the restriction on the displacement and pressure fields due to the babuska brezzi condition and provides freedom in choosing interpolation functions in the incompressible or nearly incompressible limit typical in metal forming applications moreover it facilitates the use of low order simplex elements i e p1 p1 reducing the degrees of freedom required for the solution in the incompressible limit when stable elements are necessary linearization of the weak form is derived for implementation into a finite element code numerical experiments with p1 p1 elements show that the method is effective in incompressible conditions and can be advantageous in metal forming analysis

stress analysis around crack tips in finite strain problems using the extended finite element method fracture of rubber like materials is still an open problem indeed it deals with modelling issues crack growth law bulk behaviour and computational issues robust crack growth in 2d and 3d incompressibility the present study focuses on the application of the extended finite element method x fem to large strain fracture mechanics for plane stress problems two important issues are investigated the choice of the formulation used to solve the problem and the determination of suitable enrichment functions it is demonstrated that the results obtained with the method are in good agreement with previously published works

variational and projection methods for the volume constraint in finite deformation elasto plasticity this paper focuses on the treatment of volume constraints which in the context of elasto plasticity typically arise as a result of assuming volume preserving plastic flow projection methods based on the modification of the discrete gradient operator b often proposed on an ad hoc basis are systematically obtained in the variational context furnished by a three field hu washizu principle the fully nonlinear formulation proposed here is based on a local multiplicative split of the deformation gradient into volume preserving and dilatational parts without relying on rate forms of the weak form of momentum balance this approach fits naturally in a formulation of plasticity based on the multiplicative decomposition of the deformation gradient and enables one to exactly enforce the condition of volume preserving plastic flow within the framework proposed in this paper rate forms and incrementally objective algorithms are entirely bypassed

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

a two dimensional version of the goldschmidt sims conjecture abstract the goldschmidt sims conjecture asserts that there is a finite number of conjugacy classes of edge transitive lattices in the automorphism group of a regular tree with prime valence we prove a similar theorem for irreducible lattices transitive on the 2 cells of the product of two regular trees of prime valences

nekhoroshev theorem for small amplitude solutions in nonlinear schrodinger equations we prove a nekhoroshev type result 26 27 for the nonlinear schrodinger equation with vanishing or periodic boundary conditions on 0 pi here m in mathbb r is a parameter and varphi mathbb r to mathbb r is a function analytic in a neighborhood of the origin and such that varphi 0 0 varphi 0 not 0 more precisely we consider the cauchy problem for 0 1 with initial data which extend to analytic entire functions of finite order and prove that all the actions of the linearized system are approximate constants of motion up to times growing faster than any negative power of the size of the initial datum the proof is obtained by a method which applies to hamiltonian perturbations of linear pde s with the following properties i the linear dynamics is periodic ii there exists a finite order birkhoff normal form which is integrable and quasi convex as a function of the action variables eq 0 1 satisfies i and ii when restricted to a level surface of left vert u l 2 right vert which is an integral of motion the main technical tool used in the proof is a normal form lemma for systems with symmetry which is also proved here

on the order of borel subgroups of group amalgams and an application to locally transitive graphs abstract a permutation group is called semiprimitive if each of its normal subgroups is either transitive or semiregular given nontrivial finite transitive permutation groups l 1 and l 2 with l 1 not semiprimitive we construct an infinite family of rank two amalgams of permutation type l 1 l 2 and borel subgroups of strictly increasing order as an application we show that there is no bound on the order of edge stabilisers in locally l 1 l 2 graphs we also consider the corresponding question for amalgams of rank k 3 we completely resolve this by showing that the order of the borel subgroup is bounded by the permutation type l 1 l k only in the trivial case where each of l 1 l k is regular

almost periodic solutions for a class of linear schrodinger equations with almost periodic forcing in this paper the existence of almost periodic solutions corresponding to full dimensional invariant tori is shown for a class of linear schrodinger equations with almost periodic forcing subject to periodic boundary conditions it is proved via an improved kolmogorov arnold moser reducibility

on locally semiprimitive graphs and a theorem of weiss abstract in this paper we investigate graphs that admit a group acting arc transitively such that the local action is semiprimitive with a regular normal nilpotent subgroup this type of semiprimitive group is a generalisation of an affine group we show that if the graph has valency coprime to six then there is a bound on the order of the vertex stabilisers depending on the valency alone we also prove a detailed structure theorem for the vertex stabilisers in the remaining case this is a contribution to an ongoing project to investigate the validity of the potocnik spiga verret conjecture

second order hamiltonian equations on t and almost periodic solutions abstract motivated by problems arising in nonlinear pde s with a hamiltonian structure and in high dimensional dynamical systems we study a suitable generalization to infinite dimensions of second order hamiltonian equations of the type xv x t n x x1 xn extending methods from quantitative perturbation theory kolmogorov arnold moser theory nash moser implicit function theorem etc we construct uncountably many almost periodic solutions for the infinite dimensional system i i x i z d x t z d endowed with the compact topology the hamiltonian structure is reflected by being a generalized gradient such a result is derived under suitable analyticity assumptions on i but without requiring any smallness conditions

a finite element method for the computational modelling of cohesive cracks the present contribution is concerned with the computational modelling of cohesive cracks in quasi brittle materials whereby the discontinuity is not limited to interelement boundaries but is allowed to propagate freely through the elements in the elements which are intersected by the discontinuity additional displacement degrees of freedom are introduced at the existing nodes therefore two independent copies of the standard basis functions are used one set is put to zero on one side of the discontinuity while it takes its usual values on the opposite side and vice versa for the other set to model inelastic material behaviour a discrete damage type constitutive model is applied formulated in terms of displacements and tractions at the surface some details on the numerical implementation are given concerning the failure criterion the determination of the direction of the discontinuity and the integration scheme finally numerical examples show the performance of the method

compatible strain mixed finite element methods for 2d compressible nonlinear elasticity abstract in this paper using the hilbert complexes of nonlinear elasticity the approximation theory for hilbert complexes and the finite element exterior calculus we introduce a new class of mixed finite element methods for 2d nonlinear elasticity compatible strain mixed finite element methods csfem we consider a hu washizu type mixed formulation and choose the displacement the displacement gradient and the first piola kirchhoff stress tensor as independent unknowns we use the underlying spaces of the hilbert complexes as the solution and test spaces we discretize the hilbert complexes and introduce a new class of mixed finite element methods for nonlinear elasticity by using the underlying finite element spaces of the discrete hilbert complexes this automatically enforces the trial spaces of the displacement gradient to satisfy the classical hadamard jump condition which is a necessary condition for the compatibility of non smooth displacement gradients the underlying finite element spaces of csfems are the tensorial analogues of the standard nedelec and raviart thomas elements of vector fields these spaces respect the global topologies of the domains in the sense that they can reproduce certain topological properties of the bodies regardless of the refinement level of meshes by solving several numerical examples we demonstrate that csfems have a good performance for bending problems in the near incompressible regime and for bodies with complex geometries csfems are capable of accurately approximating stresses and perform well in problems that standard enhanced strain methods suffer from the hourglass instability moreover csfems provide a convenient framework for modeling inhomogeneities

existence theorem for solitary waves on lattices in this article we give an existence theorem for localized travelling wave solutions on one dimensional lattices with hamiltonian h sum limits n in mathbb z left tfrac 1 2 p n 2 v q n 1 q n right wherev is the potential energy due to nearest neighbour interactions until now apart from rare integrable lattices like the toda latticev ab 1 e b b 1 the only evidence for existence of such solutions has been numerical our result in particular recovers existence of solitary waves in the toda lattice establishes for the first time existence of solitary waves in the nonintegrable cubic and quartic latticesv 1 2 2 1 3a 3 v 1 2 2 1 4b 4 thereby confirming the numerical findings in 1 and shedding new light on the recurrence phenomena in these systems observed first by fermi pasta and ulam 2 and shows that contrary to widespread belief the presence of exact solitary waves is not a peculiarity of integrable systems but generic in this class of nonlinear lattices the approach presented here is new and quite general and should also be applicable to other forms of lattice equations the travelling waves are sought as minimisers of a naturally associated variational problem obtained via hamilton s principle and existence of minimisers is then established using modern methods in the calculus of variations the concentration compactness principle of p l lions 3

localized breather like solution in a discrete klein gordon model and application to dna abstract non propagating localized oscillating solutions are studied in the case of a discrete one dimensional system with the 2 3 power onsite potential a multiple scale asymptotic perturbation expansion is not adequate for describing the large amplitude oscillations we consider starting from an exact pulse like solution in the continuum we derive an approximate oscillating solution our numerical simulations reveal that whereas the excitations are strongly damped in the strong intersite coupling case they are long lived in the discrete cases an additional modulation phenomenon of the oscillation seen in the simulation is explained by combination of the breather oscillations with the zero group velocity phonons modes then applying the green s functions formalism in analogy with a previous work s takeno et al prog theor phys suppl 94 1988 we derive a very accurate expression for the discrete breather with a quasi infinite lifetime lastly we demonstrate how such excitations can explain why the breathers make the dominant contributions in the dynamics of a simple model of dna

periodic and quasi periodic solutions of nonlinear wave equations via kam theory in this paper the nonlinear wave equation u u u xx v x u x t varepsilon u 3 x t 0 is studied it is shown that for a large class of potentials v x one can use kam methods to construct periodic and quasi periodic solutions in time for this equation

a nurbs based interface enriched generalized finite element method for problems with complex discontinuous gradient fields summary a non uniform rational b splines nurbs based interface enriched generalized finite element method is introduced to solve problems with complex discontinuous gradient fields observed in the structural and thermal analysis of the heterogeneous materials the presented method utilizes generalized degrees of freedom and enrichment functions based on nurbs to capture the solution with non conforming meshes a consistent method for the generation and application of the nurbs based enrichment functions is introduced these enrichment functions offer various advantages including simplicity of the integration possibility of different modes of local solution refinement and ease of implementation in addition we show that these functions well capture weak discontinuities associated with highly curved material interfaces the convergence accuracy and stability of the method in the solution of two dimensional elasto static problems are compared with the standard finite element scheme showing improved accuracy finally the performance of the method for solving problems with complex internal geometry is highlighted through a numerical example copyright 2015 john wiley sons ltd

small divisors with spatial structure in infinite dimensional hamiltonian systems a general perturbation theory of the kolmogorov arnold moser type is described concerning the existence of infinite dimensional invariant tori in nearly integrable hamiltonian systems the key idea is to consider hamiltonians with aspatial structure and to express all quantitative aspects of the theory in terms of rather general weight functions on such structures this approach combines great flexibility with an effective control of the vrious interactions in infinite dimensional systems

on the elimination of non resonance harmonics given a weakly coupled hamiltonian system with short range one dimensional interactions andany initial conditions a canonical change of variables is constructed which yields a new hamiltonian consisting of three parts an integrable term a resonant term whose effects are localized in those regions of the system which give small denominators in the kolmogorov arnol d moser iteration scheme and a non resonant interaction term which is very small in particular much much smaller than our original interactions the conditions which allow such a transformation to be constructed are independent of the number of degrees of freedom in the system as are the estimates on the size of the various terms thus if the resonances are sparsely distributed through the system most of the sites in the transformed hamiltonian behave essentially like an integrable system at least for as long a time as the trajectory of the system lies within the region where the canonical transformation is defined in subsequent work it is shown that this time is long and once again independent of the number of degrees of freedom in the system

bounds on the trajectories of a system of weakly coupled rotators in this paper we study a classical mechanical system of weakly coupled rotators on a one dimensional lattice such systems are of interest in statistical mechanics we prove that for any site in the system there is a large set of initial conditions for which there exists a canonical change of variables such that the trajectory of that site in the transformed system is essentially indistinguishable from that of an integrable system for a long but finite time alternatively the trajectory of this site lies very close to a torus in the phase space of the system for times very long in comparison with the typical period of the unperturbed rotators all the estimates in this theory areindependent of the number of degrees of freedom in the system we propose this mechanism as an explanation of certain numerical experiments

a partition of unity finite element method for obtaining elastic properties of continua with embedded thin fibres the numerical analysis of large numbers of arbitrarily distributed discrete thin fibres embedded in a continuum is a computationally demanding process in this contribution we propose an approach based on the partition of unity property of finite element shape functions that can handle discrete thin fibres in a continuum matrix without meshing them this is made possible by a special enrichment function that represents the action of each individual fibre on the matrix our approach allows to model fibre reinforced materials considering matrix fibres and interfaces between matrix and fibres individually each with its own elastic constitutive law copyright 2010 john wiley sons ltd

construction of invariant whiskered tori by a parameterization method part i maps and flows in finite dimensions abstract we present theorems which provide the existence of invariant whiskered tori in finite dimensional exact symplectic maps and flows the method is based on the study of a functional equation expressing that there is an invariant torus we show that given an approximate solution of the invariance equation which satisfies some non degeneracy conditions there is a true solution nearby we call this an a posteriori approach the proof of the main theorems is based on an iterative method to solve the functional equation the theorems do not assume that the system is close to integrable nor that it is written in action angle variables hence we can deal in a unified way with primary and secondary tori it also does not assume that the hyperbolic bundles are trivial and much less that the hyperbolic motion can be reduced to constant linear map the a posteriori formulation allows us to justify approximate solutions produced by many non rigorous methods e g formal series expansions numerical methods the iterative method is not based on transformation theory but rather on successive corrections this makes it possible to adapt the method almost verbatim to several infinite dimensional situations which we will discuss in a forthcoming paper we also note that the method leads to fast and efficient algorithms we plan to develop these improvements in forthcoming papers

existence and stability of quasi periodic breathers in networks of ginzburg landau oscillators abstract it is shown that there is an attracting quasi periodic breather in networks of ginzburg landau oscillators

construction of quasi periodic breathers via kam technique by developing a kam theorem which involves an infinitely multiple normal frequency it is shown that there are plenty of breathers quasi periodic in time and super exponentially localized in space for the networks of weakly coupled oscillators this answers an open problem by aubry a2 in case the linearized system has no continuous spectrum

a computational approach to handle complex microstructure geometries in multiscale analysis of components there is usually a need to solve microstructures with complex geometries in this paper we use the extended finite element method x fem to solve scales involving complex geometries the x fem allows one to use meshes not necessarily matching the physical surface of the problem while retaining the accuracy of the classical finite element approach for material interfaces this is achieved by introducing a new enrichment strategy although the mesh does not need to conform to the physical surfaces it needs to be fine enough to capture the geometry of these surfaces a simple algorithm is described to adaptively refine the mesh to meet this geometrical requirement numerical experiments on the periodic homogenization of two phase complex cells demonstrate the accuracy and simplicity of the x fem c 2003 elsevier b v all rights reserved

enrichment of enhanced assumed strain approximations for representing strong discontinuities addressing volumetric incompressibility and the discontinuous patch test summary we present a geometrically nonlinear assumed strain method that allows for the presence of arbitrary intra finite element discontinuities in the deformation map special attention is placed on the coarsemesh accuracy of these methods and their ability to avoid mesh locking in the incompressible limit given an underlying mesh and an arbitrary failure surface we first construct an enriched approximation for the deformation map with the nonlinear analogue of the extended finite element method x fem with regard to the richer space of functions spanned by the gradient of the enriched approximation we then adopt a broader interpretation of variational consistency for the construction of the enhanced strain in particular in those elements intersected by the failure surface we construct enhanced strain approximations which are orthogonal to piecewise constant stress fields contrast is drawn with existing strong discontinuity approaches where the enhanced strain variations in localized elements were constructed to be orthogonal to constant nominal stress fields importantly the present formulation gives rise to a symmetric tangent stiffness matrix even in localized elements the present modification also allows for the satisfaction of a discontinuous patch test wherein two different constant stress fields on each side of the failure surface lie in the solution space we demonstrate how the proposed modifications eliminate spurious stress oscillations along the failure surface particularly for nearly incompressible material response additional numerical examples are provided to illustrate the efficacy of the modified method for problems in hyperelastic fracture mechanics copyright c 2000 john wiley sons ltd

perturbations of lower dimensional tori for hamiltonian systems abstract this paper provides a generalized kolmogorov arnold moser theorem for lower dimensional tori in hamiltonian systems which applies to multiple normal frequency case the proof is based on newton s iteration method and a generalized version of small divisor conditions

a generalized finite element method for polycrystals with discontinuous grain boundaries summary we present a generalized finite element method for the analysis of polycrystals with explicit treatment of grain boundaries grain boundaries and junctions understood as loci of possible displacement discontinuity are inserted into finite elements by exploiting the partition of unity property of finite element shape functions consequently the finite element mesh does not need to conform to the polycrystal topology the formulation is outlined and a numerical example is presented to demonstrate the potential and accuracy of the approach the proposed methodology can also be used for branched and intersecting cohesive cracks and comparisons are made to a related approach int j numer meth engng 2000 48 1741 copyright 2006 john wiley sons ltd

quasi periodic breathers in hamiltonian networks of long range coupling this work is concerned with hamiltonian networks of weakly and long range coupled oscillators with either variable or constant on site frequencies we derive an infinite dimensional kam like theorem by which we establish that given any n sites of the lattice there is a positive measure set of small amplitude quasi periodic breathers solutions of the hamiltonian network that are quasi periodic in time and exponentially localized in space having n frequencies which are only slightly deformed from the on site frequencies

an unfitted hp adaptive finite element method based on hierarchical b splines for interface problems of complex geometry generating finite element discretizations with direct interface parameterizations constitutes a considerable computational expense in case of complex interface geometries the paper at hand introduces a b spline finite element method which circumvents parameterization of interfaces and offers fast and easy meshing irrespective of the geometric complexity involved its core idea is the adaptive approximation of discontinuities by hierarchical grid refinement which adds several levels of local basis functions in the close vicinity of interfaces but unfitted to their exact location so that a simple regular grid of knot span elements can be maintained numerical experiments show that an hp refinement strategy which simultaneously increases the polynomial degree of the b spline basis and the levels of refinement around interfaces achieves exponential rates of convergence despite the presence of discontinuities it is also demonstrated that the hierarchical b spline fem can be used to transfer the recently introduced finite cell concept to geometrically nonlinear problems its computational performance imposition of unfitted boundary conditions and fast hierarchical grid generation are illustrated for a set of benchmark problems in one two and three dimensions and the advantages of the regular grid approach for complex geometries are demonstrated by the geometrically nonlinear simulation of a voxel based foam composite

travelling breathers with exponentially small tails in a chain of nonlinear oscillators we study the existence of travelling breathers in klein gordon chains which consist of one dimensional networks of nonlinear oscillators in an anharmonic on site potential linearly coupled to their nearest neighbors travelling breathers are spatially localized solutions which appear time periodic in a referential in translation at constant velocity approximate solutions of this type have been constructed in the form of modulated plane waves whose envelopes satisfy the nonlinear schrodinger equation m remoissenet phys rev b33 n 4 2386 1986 j giannoulis and a mielke nonlinearity17 p 551 565 2004 in the case of travelling waves where the phase velocity of the plane wave equals the group velocity of the wave packet the existence of nearby exact solutions has been proved by iooss and kirchgassner who have obtained exact solitary wave solutions superposed on an exponentially small oscillatory tail g iooss k kirchgassner commun math phys 211 439 464 2000 however a rigorous existence result has been lacking in the more general case when phase and group velocities are different this situation is examined in the present paper in a case when the breather period and the inverse of its velocity are commensurate we show that the center manifold reduction method introduced by iooss and kirchgassner is still applicable when the problem is formulated in an appropriate way this allows us to reduce the problem locally to a finite dimensional reversible system of ordinary differential equations whose principal part admits homoclinic solutions to quasi periodic orbits under general conditions on the potential for an even potential using the additional symmetry of the system we obtain homoclinic orbits to small periodic ones for the full reduced system for the oscillator chain these orbits correspond to exact small amplitude travelling breather solutions superposed on an exponentially small oscillatory tail their principal part excluding the tail coincides at leading order with the nonlinear schrodinger approximation

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

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

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

dynamical systems on lattices with decaying interaction ii hyperbolic sets and their invariant manifolds abstract this is the second part of the work devoted to the study of maps with decay in lattices here we apply the general theory developed in fontich et al 2011 3 to the study of hyperbolic sets in particular we establish that any close enough perturbation with decay of an uncoupled lattice map with a hyperbolic set has also a hyperbolic set with dynamics on the hyperbolic set conjugated to the corresponding of the uncoupled map we also describe how the decay properties of the maps are inherited by the corresponding invariant manifolds

a theory of semiprimitive groups abstract a transitive permutation group is semiprimitive if each of its normal subgroups is transitive or semiregular interest in this class of groups is motivated by two sources problems arising in universal algebra related to collapsing monoids and the graph restrictive problem for permutation groups here we develop a theory of semiprimitive groups which encompasses their structure their quotient actions and a method by which all finite semiprimitive groups are constructed we also extend some results from the theory of primitive groups to semiprimitive groups and conclude with open problems of a similar nature

amalgams of prime index this serves as a motivation for our main theorem from the theory of the classification of finite simple groups indeed our result classifies all such configurations and in particular it says that in all the interesting cases we must have ip n p i d 2 and ip p n p i 3 or 5 the basic definitions we use are the same as in gs which we quickly repeat for the convenience of the reader an umalgatn consists of two monomorphisms 4 and dz of groups such that

a generalized finite element method for the simulation of three dimensional dynamic crack propagation this paper is aimed at presenting a partition of unity method for the simulation of three dimensional dynamic crack propagation the method is a variation of the partition of unity finite element method and hp cloud method in the context of crack simulation this method allows for modeling of arbitrary dynamic crack propagation without any remeshing of the domain in the proposed method the approximation spaces are constructed using a partition of unity pu and local enrichment functions the pu is provided by a combination of shepard and finite element partitions of unity this combination of pus allows the inclusion of arbitrary crack geometry in a model without any modification of the initial discretization it also avoids the problems associated with the integration of moving least squares or conventional shepard partitions of unity used in several meshless methods the local enrichment functions can be polynomials or customized functions these functions can efficiently approximate the singular fields around crack fronts the crack propagation is modeled by modifying the partition of unity along the crack surface and does not require continuous remeshings or mappings of solutions between consecutive meshes as the crack propagates in contrast with the boundary element method the proposed method can be applied to any class of problems solvable by the classical finite element method in addition the proposed method can be implemented into most finite element data bases several numerical examples demonstrating the main features and computational efficiency of the proposed method for dynamic crack propagation are presented

invariant manifolds of parabolic fixed points i existence and dependence on parameters abstract in this paper we study the existence and regularity of stable manifolds associated to fixed points of parabolic type in the differentiable and analytic cases using the parametrization method the parametrization method relies on a suitable approximate solution of a functional equation in the case of parabolic points if the manifolds have dimension two or higher in general this approximation cannot be obtained in the ring of polynomials but as a sum of homogeneous functions and it is given in 4 assuming a sufficiently good approximation is found here we provide an a posteriori result which gives a true invariant manifold close to the approximate one in the differentiable case in some cases there is a loss of regularity we also consider the case of parabolic periodic orbits of periodic vector fields and the dependence of the manifolds on parameters examples are provided we apply our method to prove that in several situations namely related to the parabolic infinity in the elliptic spatial three body problem these invariant manifolds exist and do have polynomial approximations

solution of nearly incompressible field problems using a generalized finite element approach abstract it is well known that for problems approaching incompressible limits standard finite element approaches suffer from suboptimal convergence due to a phenomenon known as locking we analyze this phenomenon and present a robust generalized finite element formulation that enables optimal solution convergence this approach is explored for the two dimensional incompressible elasticity equations with a variable poisson s ratio as well as the two dimensional lid driven cavity stokes flow problem using a penalty pressure formulation

uncertainty quantification in homogenization of heterogeneous microstructures modeled by xfem an extended finite element method xfem coupled with a monte carlo approach is proposed to quantify the uncertainty in the homogenized effective elastic properties of multiphase materials the methodology allows for an arbitrary number aspect ratio location and orientation of elliptic inclusions within a matrix without the need for fine meshes in the vicinity of tightly packed inclusions and especially without the need to remesh for every different generated realization of the microstructure moreover the number of degrees of freedom in the enriched elements is dynamically reallocated for each monte carlo sample run based on the given volume fraction the main advantage of the proposed xfem based methodology is a major reduction in the computational effort in extensive monte carlo simulations compared with the standard fem approach monte carlo and xfem appear to work extremely efficiently together the monte carlo approach allows for the modeling of the size aspect ratios orientations and spatial distribution of the elliptical inclusions as random variables with any prescribed probability distributions numerical results are presented and the uncertainty of the homogenized elastic properties is discussed copyright 2011 john wiley sons ltd

dynamical systems on lattices with decaying interaction i a functional analysis framework abstract we consider weakly coupled map lattices with a decaying interaction that is we consider systems which consist of a phase space at every site such that the dynamics at a site is little affected by the dynamics at far away sites we develop a functional analysis framework which formulates quantitatively the decay of the interaction and is able to deal with lattices such that the sites are manifolds this framework is very well suited to study systematically invariant objects one obtains that the invariant objects are essentially local we use this framework to prove a stable manifold theorem and show that the manifolds are as smooth as the maps and have decay properties i e the derivatives of one of the coordinates of the manifold with respect to the coordinates at far away sites are small other applications of the framework are the study of the structural stability of maps with decay close to uncoupled possessing hyperbolic sets and the decay properties of the invariant manifolds of their hyperbolic sets in the companion paper by fontich et al 2011 10

thermomechanical modeling of regressing heterogeneous solid propellants a numerical framework based on the generalized finite element method gfem is developed to capture the coupled effects of thermomechanical deformations and thermal gradients on the regression rate of a heterogeneous solid propellant the thermomechanical formulation is based on a multiplicative split of the deformation gradient and regression of the heterogeneous solid propellant is simulated using the level set method a spatial mesh convergence study is performed on a non regressing solid heterogeneous propellant system to examine the consistency of the coupled thermomechanical gfem solver the overall accuracy spatial and temporal of the coupled thermomechanical solver for regressing solid propellants is obtained from a periodic sandwich propellant configuration where the effects of thermomechanical deformations on its regression rate is investigated finally the effects of thermomechanical deformations in a regressing two dimensional heterogeneous propellant pack are studied and time average regression rates are reported

a high order generalized fem for through the thickness branched cracks this paper presents high order implementations of a generalized finite element method for through the thickness three dimensional branched cracks this approach can accurately represent discontinuities such as triple joints in polycrystalline materials and branched cracks independently of the background finite element mesh representative problems are investigated to illustrate the accuracy of the method in combination with various discretizations and refinement strategies the combination of local refinement at crack fronts and high order continuous and discontinuous enrichments proves to be an excellent combination which can deliver convergence rates close to that of problems with smooth solutions copyright 2007 john wiley sons ltd

interface material failure modeled by the extended finite element method and level sets multi phase materials are characterized by the fact that they possess a specific heterogeneous microstructure this feature often necessitates complex formulations in order to obtain realistic mechanical behavior with macroscopic material models furthermore such complex models cause substantial difficulties when the corresponding material parameters need to be identified and their physical meaning interpreted in multiscale analyses usually the microstructure of a material point is resolved directly thus the material behavior at the material point can be modeled in an elegant natural and simple way the present contribution aims at a detailed geometric modeling of multi phase materials as well as at a local mechanical modeling of material interfaces and interfacial failure the geometry of the material distribution within the microstructure is described through level set functions the mechanical modeling of material interfaces and interfacial cracks is accomplished by the extended finite element method x fem the combination of both the level set method and the x fem allows one to model such internal features of the microstructure without the adaptation of the mesh thus expensive meshing procedures can be avoided and considerable flexibility is obtained with respect to the variation of the material design in the x fem the finite element approximation is enriched by appropriate functions through the concept of partition of unity remarkably the level set method cannot only be applied to describe the geometry of the material interfaces and cracks respectively but also to support and even to develop the corresponding enrichment function

generalized finite element enrichment functions for discontinuous gradient fields a general gfem xfem formulation is presented to solve two dimensional problems characterized by c0 continuity with gradient jumps along discrete lines such as those found in the thermal and structural analysis of heterogeneous materials or in line load problems in homogeneous media the new enrichment functions presented in this paper allow solving problems with multiple intersecting discontinuity lines such as those found at triple junctions in polycrystalline materials and in actively cooled microvascular materials with complex embedded networks we show how the introduction of enrichment functions yields accurate finite element solutions with meshes that do not conform to the geometry of the discontinuity lines the use of the proposed enrichments in both linear and quadratic approximations is investigated as well as their combination with interface enrichment functions available in the literature through a detailed convergence study we demonstrate that quadratic approximations do not require any correction to the method to recover optimal convergence rates and that they perform better than linear approximations for the same number of degrees of freedom in the solution of these types of problems in the linear case the effectiveness of correction functions proposed in the literature is also investigated copyright 2009 john wiley sons ltd

generalized finite element method for modeling nearly incompressible bimaterial hyperelastic solids an extension of the generalized finite element method to the class of mixed finite element methods is presented to tackle heterogeneous systems with nearly incompressible non linear hyperelastic material behavior in particular heterogeneous systems with large modulus mismatch across the material interface undergoing large strains are investigated using two formulations one based on a continuous deformation map the other on a discontinuous one a bimaterial patch test is formulated to assess the ability of the two formulations to reproduce constant stress fields while a mesh convergence study is used to examine the consistency of the formulations finally compression of a model heterogeneous propellant pack is simulated to demonstrate the robustness of the proposed discontinuous deformation map formulation

an interface enriched generalized fem for problems with discontinuous gradient fields summary a new generalized fem is introduced for solving problems with discontinuous gradient fields the method relies on enrichment functions associated with generalized degrees of freedom at the nodes generated from the intersection of the phase interface with element edges the proposed approach has several advantages over conventional generalized fem formulations such as a lower computational cost easier implementation and straightforward handling of dirichlet boundary conditions a detailed convergence study of the proposed method and a comparison with the standard fem are presented for heat transfer problems the method achieves the optimal rate of convergence using meshes that do not conform to the interfaces present in the domain while achieving a level of accuracy comparable to that of the standard fem with conforming meshes various application problems are presented including the conjugate heat transfer problem encountered in microvascular materials copyright 2011 john wiley sons ltd

resonances radiation damping and instabilitym in hamiltonian nonlinear wave equations we consider a class of nonlinear klein gordon equations which are hamiltonian and are perturbations of linear dispersive equations the unperturbed dynamical system has a bound state a spatially localized and time periodic solution we show that for ge neric nonlinear hamiltonian perturbations all small amplitude so lutions decay to zero as time tends to infinity at an anomalously slow rate in particular spatially localized and time periodic solutions of the linear problem are destroyed by generic nonlinear hamiltonian perturbations via slow radiation of energy to infinity these solutions can therefore be thought of as metastable states the main mecha nism is a nonlinear resonant interaction of bound states eigenfunc tions and radiation continuous spectral modes leading to energy transfer from the discrete to continuum modes this is in contrast to the kam theory in which appropriate nonresonance conditions im ply the persistence of invariant tori a hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the fermi golden rule arising in the theory of resonances in quantum mechanics the techniques used involve i a time dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues ii a generalization of the hamiltonian normal form appropriate for infinite dimensional dis persive systems and iii ideas from scattering theory the arguments are quite general and we expect them to apply to a large class of

the kam theory of systems with short range interactions i the existence of quasiperiodic trajectories for hamiltonian systems consisting of long chains of nearly identical subsystems with interactions which decay rapidly with increasing distance between the interacting components is studied such models are of interest in statistical mechanics it is shown that nonergodic motions persist for much larger perturbations than prior work indicated if the number of degrees of freedom of the system isn the allowed perturbation decreases only as an inverse power ofn as the number of degrees of freedom increases rather than the inverse power ofn which previous estimates yielded

asymptotic localization of energy in nondisordered oscillator chains we study two popular one dimensional chains of classical anharmonic oscillators the rotor chain and a version of the discrete nonlinear schrodinger chain we assume that the interaction between neighboring oscillators controlled by the parameter 0 is small we rigorously establish that the thermal conductivity of the chains has a nonperturbative origin with respect to the coupling constant and we provide strong evidence that it decays faster than any power law in as 0 the weak coupling regime also translates into a high temperature regime suggesting that the conductivity vanishes faster than any power of the inverse temperature to our knowledge it is the first time that a clear connection has been established between kam like phenomena and thermal conductivity 2015 wiley periodicals inc

an a posteriori kam theorem for whiskered tori in hamiltonian partial differential equations with applications to some ill posed equations the goal of this paper is to develop a kam theory for tori with hyperbolic directions which applies to hamiltonian partial differential equations even to some ill posed ones the main result has an a posteriori format that is we show that if there is an approximate solution of an invariance equation which also satisfies some non degeneracy conditions then there is a true solution nearby this allows besides dealing with the quasi integrable case for the validation of numerical computations or formal perturbative expansions as well as for obtaining quasi periodic solutions in degenerate situations the a posteriori format also has other automatic consequences smooth dependence on parameters bootstrap of regularity etc we emphasize that the non degeneracy conditions required are just quantities evaluated on the approximate solution no global assumptions on the system such as twist hence they are readily verifiable in perturbation expansions we will pay attention to the quantitative relations between the sizes of the approximation and the non degeneracy conditions this will allow us to prove what experts call small twist theorems the non degeneracy conditions vanishes as the perturbation goes to zero but much slower than the error of the approximation the method of proof is based on an iterative method for solving a functional equation for the parameterization of the torus expressing that the range of the parameterization admits an evolution and is invariant we also solve functional equations for bundles which imply that are invariant under the linearization the iterative method does not use transformation theory nor action angle variables the main result does not assume that the system is close to integrable more surprisingly we do not need that the equations we study define an evolution for all initial conditions and are well posed even if the systems we study do not admit solutions for all initial conditions we show that there is a systematic way to choose initial conditions on which one can define an evolution which is quasi periodic we first develop an abstract theorem then we show how this abstract result applies to some concrete examples the examples considered in this paper are the scalar boussinesq equation and the boussinesq system both are pde models that aim to describe water waves in the long wave limit for these equations we construct small amplitude time quasi periodic solutions which are even in the spatial variable the strategy for the abstract theorem is inspired by that in fontich et al electron res announc math sci 16 9 22 2009 j differ equ 246 8 3136 3213 2009 the main part of the paper is to study infinite dimensional analogues of dichotomies which applies even to ill posed equations and which is stable under addition of unbounded perturbations this requires that we assume smoothing properties we also present very detailed bounds on the change of the splittings under perturbations

stabilized four node tetrahedron with nonlocal pressure for modeling hyperelastic materials summary non linear hyperelastic response of reinforced elastomers is modeled using a novel three dimensional mixed finite element method with a nonlocal pressure field the element is unconditionally convergent and free of spurious pressure modes nonlocal pressure is obtained by an implicit gradient technique and obeys the helmholtz equation physical motivation for this nonlocality is shown an implicit finite element scheme with consistent linearization is presented finally several hyperelastic examples are solved to demonstrate the computational algorithm including the inf sup and verifications tests copyright q 2008 john wiley sons ltd

a nonlinear manifold based reduced order model for multiscale analysis of heterogeneous hyperelastic materials a new manifold based reduced order model for nonlinear problems in multiscale modeling of heterogeneous hyperelastic materials is presented the model relies on a global geometric framework for nonlinear dimensionality reduction isomap and the macroscopic loading parameters are linked to the reduced space using a neural network the proposed model provides both homogenization and localization of the multiscale solution in the context of computational homogenization to construct the manifold we perform a number of large three dimensional simulations of a statistically representative unit cell using a parallel finite strain finite element solver the manifold based reduced order model is verified using common principles from the machine learning community both homogenization and localization of the multiscale solution are demonstrated on a large three dimensional example and the local microscopic fields as well as the homogenized macroscopic potential are obtained with acceptable engineering accuracy

smooth dependence of thermodynamic limits of srb measures the thermodynamic limits of srb measures for uniformly hyperbolic sets are smoothly dependent on the map in an appropriate functional space