validity of cauchy born hypothesis in multi scale modeling of plastic deformations abstract the cauchy born cb hypothesis has been widely used in multi scale modeling of crystalline nano structures the violation of cb hypothesis in stress space and the transition to plasticity which is equivalent to the violation of cb hypothesis in strain space are generally confused and it becomes crucial to differentiate between the two distinct phenomena the violation of the former usually occurs at high values of stress and at regions where the surface effects are manifest while the violation of the latter occurs at low stresses when the material loses its strength to tolerate the applied loading in this paper a novel technique is developed to investigate the validity of cb hypothesis in stress space in the presence of plastic deformations the method is based on the definition of a validity surface in the space of stress and the identification of those violated regions of the domain that do not satisfy the validity criterion at each step of the analysis the stress of violated regions is corrected so that the stress over the whole domain remains within the validity surface the procedure is similar to the return mapping algorithm extensively used in the theory of plasticity in this manner a quantitative measure is provided that indicates how improperly the cb hypothesis is violated numerical simulations for gold nano structures illustrate that for certain configurations the violation of cb hypothesis is limited to those parts of the domain with severe surface effects however for general configurations the violation of cb hypothesis depends on the type of loading and the complexity of the domain in such cases the implementation of stress correction procedure is important for the accuracy of solution finally the results of numerical simulations are compared with those of molecular statics ms analyses to illustrate the efficiency of proposed computational model

invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

protected gates for topological quantum field theories we study restrictions on locality preserving unitary logical gates for topological quantum codes in two spatial dimensions a locality preserving operation is one which maps local operators to local operators for example a constant depth quantum circuit of geometrically local gates or evolution for a constant time governed by a geometrically local bounded strength hamiltonian locality preserving logical gates of topological codes are intrinsically fault tolerant because spatially localized errors remain localized and hence sufficiently dilute errors remain correctable by invoking general properties of two dimensional topological field theories we find that the locality preserving logical gates are severely limited for codes which admit non abelian anyons in particular there are no locality preserving logical gates on the torus or the sphere with m punctures if the braiding of anyons is computationally universal furthermore for ising anyons on the m punctured sphere locality preserving gates must be elements of the logical pauli group we derive these results by relating logical gates of a topological code to automorphisms of the verlinde algebra of the corresponding anyon model and by requiring the logical gates to be compatible with basis changes in the logical hilbert space arising from local f moves and the mapping class group

stability and size dependency of cauchy born hypothesis in three dimensional applications abstract the cauchy born hypothesis cb provides a hierarchical approach in the molecular theory of crystal elasticity to relate the continuum and atomic deformations this kinematic theory has been extensively used as the constitutive law of continuum regions in multi scale models in these models the fine scale is proposed to describe the real behavior of crystalline structure wherever the continuum description fails the main objective of this article is to investigate the stability and size dependency of cb hypothesis in three dimensional applications by direct comparison of information between atomistic and continuous description of a medium the sutton chen many body potential is used for the gold metal to consider the real metallic behavior in numerical simulations two failure criteria are introduced in the strain and stress domains the validity surfaces are derived for the cauchy born hypothesis and the size effect of specimens is investigated on the convergency of results it is shown that the gold crystal deforms homogeneously inside the validity surface in which the material is elastic and the cb has remained valid it is observed that although the deformation is inhomogeneous and the cb is invalid outside the validity surface the crystalline structure may exhibit elastic or plastic behavior in this region moreover it is numerically shown that the size dependency of validity surface decreases with the increase of the size of specimens these observations are meticulously investigated by loading and unloading several cubic specimens using molecular dynamics simulation

combinatorial quantization of the hamiltonian chern simons theory ii motivated by a recent paper of fock and rosly 6 we describe a mathematically precise quantization of the hamiltonian chern simons theory we introduce the chern simons theory on the lattice which is expected to reproduce the results of the continuous theory exactly the lattice model enjoys the symmetry with respect to a quantum gauge group using this fact we construct the algebra of observables of the hamiltonian chern simons theory equipped with a operation and a positive inner product

existence results for semilinear differential equations with nonlocal and impulsive conditions abstract this paper is concerned with the existence for impulsive semilinear differential equations with nonlocal conditions using the techniques of approximate solutions and fixed point existence results are obtained for mild solutions when the impulsive functions are only continuous and the nonlocal item is lipschitz in the space of piecewise continuous functions is not lipschitz and not compact is continuous in the space of bochner integrable functions respectively

finite elements for materials with strain gradient effects a finite element implementation is reported of the fleck hutchinson phenomenological strain gradient theory this theory fits within the toupin mindlin framework and deals with first order strain gradients and the associated work conjugate higher order stresses in conventional displacement based approaches the interpolation of displacement requires c1 continuity in order to ensure convergence of the finite element procedure for higher order theories mixed type finite elements are developed herein for the fleck hutchinson theory these elements use standard c0 continuous shape functions and can achieve the same convergence as c1 elements these c0 elements use displacements and displacement gradients as nodal degrees of freedom kinematic constraints between displacement gradients are enforced via the lagrange multiplier method the elements developed all pass a patch test the resulting finite element scheme is used to solve some representative linear elastic boundary value problems and the comparative accuracy of various types of element is evaluated copyright 1999 john wiley sons ltd

the postulations a la d alembert and a la cauchy for higher gradient continuum theories are equivalent a review of existing results in order to found continuum mechanics two different postulations have been used the first introduced by lagrange and piola starts by postulating how the work expended by internal interactions in a body depends on the virtual velocity field and its gradients then by using the divergence theorem a representation theorem is found for the volume and contact interactions which can be exerted at the boundary of the considered body this method assumes an a priori notion of internal work regards stress tensors as dual of virtual displacements and their gradients deduces the concept of contact interactions and produces their representation in terms of stresses using integration by parts the second method conceived by cauchy and based on the celebrated tetrahedron argument starts by postulating the type of contact interactions which can be exerted on the boundary of every suitably regular part of a body then it proceeds by proving the existence of stress tensors from a balance type postulate in this paper we review some relevant literature on the subject discussing how the two postulations can be reconciled in the case of higher gradient theories finally we underline the importance of the concept of contact surface edge and wedge s order forces

on general isotropic tensor functions of one tensor this paper discusses general isotropic real symmetric tensor valued functions of one real symmetric tensor by exploiting the eigenprojection based spectral representation of such functions eigenprojection based closed formulae for the function derivatives are derived the derived formulae provide an alternative to the eigenvector based derivative expressions proposed by chadwick and ogden arch rat mech anal 1971 44 54 68 for the same class of functions and are an extension of the eigenprojection based derivative expressions obtained by carlson and hoger quart appl math 1986 44 3 409 423 for a subset of the present class of functions the material presented here is restricted to two and three dimensional spaces which are of particular relevance to continuum mechanics for completeness algorithms for closed form computation of the isotropic tensor functions as well as their derivatives based on the eigenprojection representation are also presented these should be of interest to computational mechanics researchers dealing with functions of the present type copyright 2004 john wiley sons ltd

hierarchical modeling in the mechanics of materials many problems in the mechanics of materials involve the operation of either multiple spatial or temporal scales simultaneously as a result an important thrust of recent work in this area has been the development of methods allowing for the consideration of multiple scales simultaneously in this paper we examine hierarchical approaches to modeling problems of this kind with special reference to the way in which information can be fed from one scale to the next in models of plasticity

the asymptotic behavior of the solutions of the cauchy problem generated by ϕ accretive operators abstract the purpose of this paper is to study the asymptotic behavior of the solutions of certain type of differential inclusions posed in banach spaces in particular we obtain the abstract result on the asymptotic behavior of the solution of the boundary value problem u t p u u 1 u f on 0 u u on 0 u 0 x u 0 x in where is a bounded open domain in r n with smooth boundary f t x is a given l 1 function on 0 1 and 1 p p represents the p laplacian operator is the associated neumann boundary operator and a maximal monotone graph in r r with 0 0

a bridging domain method for coupling continua with molecular dynamics abstract a bridging domain method for coupling continuum models with molecular models is described in this method the continuum and molecular domains are overlapped in a bridging subdomain where the hamiltonian is taken to be a linear combination of the continuum and molecular hamiltonians we enforce the compatibility in the bridging domain by lagrange multipliers or by the augmented lagrangian method an explicit algorithm for dynamic solutions is developed results show that this multiscale method can avoid spurious wave reflections at the molecular continuum interface without any additional filtering procedures even for problems with significant nonlinearities the method is also shown to naturally handle the coupling of the continuum energy equation with the molecular subdomain a multiple time step algorithm is also developed within this framework

on a nonlocal problem for semilinear differential equations with upper semicontinuous nonlinearities in general banach spaces abstract sufficient conditions on the existence of mild solutions for the following semilinear nonlocal evolution inclusion with upper semicontinuous nonlinearity u t a t u t f t u t 0 t d u 0 g u are given when g is completely continuous and lipschitz continuous in general banach spaces respectively an example concerning the partial differential equation is also presented

multiresolution molecular mechanics atomistic continuum coupling methods combine accurate atomistic methods and efficient continuum methods to simulate the behavior of highly ordered crystalline systems coupled methods utilize the advantages of both approaches to simulate systems at a lower computational cost while retaining the accuracy associated with atomistic methods many concurrent atomistic continuum coupling methods have been proposed in the past however their true computational efficiency has not been demonstrated the present work presents an efficient implementation of a concurrent coupling method called the multiresolution molecular mechanics mmm for serial parallel and adaptive analysis first we present the features of the software implemented along with the associated technologies the scalability of the software implementation is demonstrated and the competing effects of multiscale modeling and parallelization are discussed then the algorithms contributing to the efficiency of the software are presented these include algorithms for eliminating latent ghost atoms from calculations and measurement based dynamic balancing of parallel workload the efficiency improvements made by these algorithms are demonstrated by benchmark tests the efficiency of the software is found to be on par with lammps a state of the art molecular dynamics md simulation code when performing full atomistic simulations speed up of the mmm method is shown to be directly proportional to the reduction of the number of the atoms visited in force computation finally an adaptive mmm analysis on a nanoindentation problem containing over a million atoms is performed yielding an improvement of 6 3 8 5 times in efficiency over the full atomistic md method for the first time the efficiency of a concurrent atomistic continuum coupling method is comprehensively investigated and demonstrated

multiresolution molecular mechanics surface effects in nanoscale materials abstract surface effects have been observed to contribute significantly to the mechanical response of nanoscale structures the newly proposed energy based coarse grained atomistic method multiresolution molecular mechanics mmm yang to 2015 57 is applied to capture surface effect for nanosized structures by designing a surface summation rule sr s within the framework of mmm combined with previously proposed bulk summation rule sr b the mmm summation rule sr mmm is completed sr s and sr b are consistently formed within sr mmm for general finite element shape functions analogous to quadrature rules in finite element method fem the key idea to the good performance of sr mmm lies in that the order or distribution of energy for coarse grained atomistic model is mathematically derived such that the number position and weight of quadrature type sampling atoms can be determined mathematically the derived energy distribution of surface area is different from that of bulk region physically the difference is due to the fact that surface atoms lack neighboring bonding as such sr s and sr b are employed for surface and bulk domains respectively two and three dimensional numerical examples using the respective 4 node bilinear quadrilateral 8 node quadratic quadrilateral and 8 node hexahedral meshes are employed to verify and validate the proposed approach it is shown that mmm with sr mmm accurately captures corner edge and surface effects with less 0 3 degrees of freedom of the original atomistic system compared against full atomistic simulation the effectiveness of sr mmm with respect to high order element is also demonstrated by employing the 8 node quadratic quadrilateral to solve a beam bending problem considering surface effect in addition the introduced sampling error with sr mmm that is analogous to numerical integration error with quadrature rule in fem is very small

existence of nondensely defined evolution equations with nonlocal conditions abstract this paper is concerned with the existence for nondensely defined evolution equations with nonlocal conditions using the techniques of fixed point theory and approximate solutions existence results are obtained for integral solutions when the nonlocal item is lipschitz continuous or continuous respectively examples are also given to illustrate our results

local and nonlocal continuum modeling of inelastic periodic networks applied to stretching dominated trusses we present a nonlocal continuum model and its numerical implementation to describe the macroscale response of periodic discrete networks via second order homogenization the scale bridging technique is applied to the specific example of stretching dominated elastic and inelastic periodic truss networks experiments on small scale truss structures have highlighted the importance of nodal connections on the effective stiffness and strength therefore we describe the mechanics of trusses by accounting for the stretching of truss members and the deformation of nodes for the representative 2d examples of lattices having square and triangular architectures and for example bar and nodal constitutive laws we show that a simple continuum model based on affinely deforming a representative unit cell is sufficient to reproduce the nonlinear elastic behavior of discrete trusses by contrast localization that arises e g from inelastic deformation requires a refined model this is where the presented nonlocal continuum model is capable of accurately capturing details of localized deformation we illustrate the performance of the model by comparing the results of example finite element simulations using the continuum constitutive model to discrete lattice calculations with elastic plastic bars optimal performance is achieved when the representative unit cell of the continuum model agrees with the actual size of the discrete truss unit cell which accounts for size effects even in regimes where a separation of scales between finite element size and unit cell size does not strictly apply

existence results and asymptotic behavior for nonlocal abstract cauchy problems the purpose of this paper is to study the existence and asymptotic behavior of solutions for cauchy problems with nonlocal initial datum generated by accretive operators in banach spaces

multi scale modeling of surface effect via the boundary cauchy born method in this paper a novel multi scale approach is developed for modeling of the surface effect in crystalline nano structures the technique is based on the cauchy born hypothesis in which the strain energy density of the equivalent continua is calculated by means of inter atomic potentials the notion of introducing the surface effect in the finite element method is based on the intrinsic function of quadratures called as an indicator of material behavior the information of quadratures is derived by interpolating the data from probable representative atoms in their proximity the technique is implemented by the definition of reference boundary cb elements which enable to capture not only the surface but also the edge and corner effects as the surface effect is important in small scale simulation the relative number of boundary cb elements increases which leads to predomination of boundary effects in the model in order to implement the equivalent continua in boundary value problems the updated lagrangian formulation of nonlinear finite element is derived the numerical simulation of the proposed model together with the direct comparison with fully atomistic model indicates that the technique provides promising results for facile modeling of boundary effects and investigating its effect on the mechanical response of metallic nano scale devices copyright 2010 john wiley sons ltd

observables in the turaev viro and crane yetter models we define an invariant of graphs embedded in a 3 manifold and a partition function for 2 complexes embedded in a triangulated 4 manifold by specifying the values of variables in the turaev viro and crane yetter state sum models in the case of the three dimensional invariant we prove a duality formula relating its fourier transform to another invariant defined via the colored jones polynomial in the case of the four dimensional partition function we give a formula for it in terms of a regular neighborhood of the 2 complex and the signature of its complement some examples are computed which show that the partition function determines an invariant which can detect non locally flat surfaces in a 4 manifold

dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment summary a methodology is developed for switching from a continuum to a discrete discontinuity where the governing partial dierential equation loses hyperbolicity the approach is limited to rate independent materials so that the transition occurs on a set of measure zero the discrete discontinuity is treated by the extendednite element method xfem whereby arbitrary discontinuities can be incorporated in the model without remeshing loss of hyperbolicity is tracked by a hyperbolicity indicator that enables both the crack speed and crack direction to be determined for a given material model a new method was developed for the case when the discontinuity ends within an element it facilitates the modelling of crack tips that occur within an element in a dynamic setting the method is applied to several dynamic crack growth problems including the branching of cracks copyright 2003 john wiley sons ltd

on higher gradients in continuum atomistic modelling abstract continuum atomistic modelling denotes a mixed approach combining the usual framework of continuum mechanics with atomistic features like e g interaction or rather pair potentials thereby the kinematics are typically characterized by the so called cauchy born rule representing atomic distance vectors in the spatial configuration as an affine mapping of the atomic distance vectors in the material configuration in terms of the local deformation gradient the application of the cauchy born rule requires sufficiently homogeneous deformations of the underlying crystal the model is no more valid if the deformation becomes inhomogeneous nevertheless the development of microstructures with inhomogeneous deformation is inevitable in the present work the cauchy born rule is thus extended to capture inhomogeneous deformations by the incorporation of the second order deformation gradient the higher order equilibrium equation as well as the appropriate boundary conditions are presented for the case of finite deformations the constitutive law for the piola kirchhoff stress and the additional higher order stress are represented for the simplified case of pair potential based energy density functions finally a deformation inhomogeneity measure is introduced and studied for a particular non homogeneous simple shear like deformation

a general approximation of the exponential cauchy born hypothesis to model arbitrarily shaped shell like nanostructures within continuum mechanics summary in this paper a continuum membrane theory and its subsequent finite element approximation for the description of arbitrary shell like nanostructures such as graphene based nanostructures is presented this is carried out by applying a multiscale approach where the continuum membrane is linked to the underlying atomistic lattice this linkage is performed by the exponential generalization of the cauchy born hypothesis because the classical cauchy born hypothesis is restricted to three dimensional bulk structures and is thus not applicable to shell like structures however the approximations of the exponential cauchy born hypothesis published so far are limited to structures with a planar reference configuration in this paper we present an extended approximation which does not require the reference configuration to be planar and is thus applicable to arbitrarily shaped shell like nanostructures a detailed elaboration of the related finite element implementation with important computational aspects is presented finally the accuracy of the proposed method and its implementation is verified with several numerical examples copyright 2015 john wiley sons ltd

a second order homogenization procedure for multi scale analysis based on micropolar kinematics the paper presents a higher order homogenization scheme based on non linear micropolar kinematics representing the macroscopic variation within a representative volume element rve of the material on the microstructural level the micro macro kinematical coupling is introduced as a second order taylor series expansion of the macro displacement field and the microstructural displacement variation is gathered in a fluctuation term this approach relates strongly to second gradient continuum formulations presented by e g kouznetsova et al int j numer meth engng 2002 54 1235 1260 thus establishing a link between second gradient and micropolar theories the major difference of the present approach as compared to second gradient formulations is that an additional constraint is placed on the higher order deformation gradient in terms of the micropolar stretch the driving vehicle for the derivation of the homogenized macroscopic stress measures is the hill mandel condition postulating the equivalence of microscopic and macroscopic homogenized virtual work thereby the resulting homogenization procedure yields not only a stress tensor conjugated to the micropolar stretch tensor but also the couple stress tensor conjugated to the micropolar curvature tensor the paper is concluded by a couple of numerical examples demonstrating the size effects imposed by the homogenization of stresses based on the micropolar kinematics copyright 2006 john wiley sons ltd

symplectic structures associated to lie poisson groups the lie poisson analogues of the cotangent bundle and coadjoint orbits of a lie group are considered for the natural poisson brackets the symplectic leaves in these manifolds are classified and the corresponding symplectic forms are described thus the construction of the kirillov symplectic form is generalized for lie poisson groups

on existence and local attractivity of solutions of a quadratic volterra integral equation of fractional order in the paper we study the existence of solutions of a nonlinear quadratic volterra integral equation of fractional order this equation is considered in the banach space of real functions defined continuous and bounded on an unbounded interval moreover we show that solutions of this integral equation are locally attractive

nonlocal nonlinear differential equations with a measure of noncompactness in banach spaces abstract in this paper we give the existence of integral solutions for nonlinear differential equations with nonlocal initial conditions under the assumptions of the hausdorff measure of noncompactness in separable and uniformly smooth banach spaces

on the existence and asymptotic behaviour of solutions of an evolution equation and an application to the feynman kac theorem abstract in this paper we study the existence of mild solutions of a semilinear evolution equation on an unbounded interval the rapidity of the growth of those solutions is characterized moreover we investigate the local and global attractivity of solutions of an equation in question and we describe their asymptotic behaviour our investigations are conducted in a real separable banach space

two dimensional lattice gauge theory based on a quantum group in this article we analyse a two dimensional lattice gauge theory based on a quantum group the algebra generated by gauge fields is the lattice algebra introduced recently by a yu alekseev h grosse and v schomerus in 1 we define and study wilson loops this theory is quasi topological as in the classical case which allows us to compute the correlation functions of this theory on an arbitrary surface

existence of classical solutions to nonautonomous nonlocal parabolic problems abstract we obtain a new existence theorem for classical solutions to nonautonomous equations with nonlocal initial conditions u t a t u t f t u t t s t u s g u u 0 in a banach space x where t s 0 f g are given x valued functions and a t is a sectorial operator not necessarily densely defined in x for each t 0 t both banach s contraction principle and schauder s fixed point theorem as well as the theory of evolution families and interpolation spaces are employed in our approach a concrete example is shown to illustrate the existence theorem

an η approximation method in nonlinear vector optimization abstract in this paper a new approach for a solution of a nonlinear multiobjective programming problem is introduced an equivalent approximated vector optimization problem is constructed by a modification of the objective and the constraint functions in the original multiobjective programming problem the connection between weak efficient points in the original multiobjective programming problem and its equivalent approximated vector optimization problem is proved in this way optimality conditions for nonlinear constrained multiobjective programming problems having invex and or generalized invex objective and constraint functions with respect to the same functions are obtained

asymptotic behavior of the solutions of non autonomous systems in banach spaces abstract in this paper we study the weak and strong convergence of the integral solution of the following nonlinear evolution inclusion u t a t u t f t t 0 u 0 x 0 where a t t 0 is a family of m accretive operators and f is a multi function which extends some results in this area

variational formulation of softening phenomena in fracture mechanics the one dimensional case we show that discrete models of atoms subject to nearest neighbour non linear interactions approximate continua allowing for softening and fracture a detailed study of local minima and stationary points is carried out scale effects are discussed

finite size effects in molecular dynamics simulations static structure factor and compressibility i theoretical method a method is presented for determining the static structure factor s q of a uniform bulk fluid for all wave vector magnitudes q from computer simulation data for a finite n particle system the method is based on the usual fourier transform relation between s q and the radial distribution function g r and on the theory of explicit finite size effects in particular a taylor series expansion in powers of 1 n for mathit g mathit n r the n particle counterpart of g r a time dependent generalization of the method to the intermediate scattering function i q t is also presented implicit finite size effects are also considered but are found to be negligible for systems to which we have applied the method textcopyright 1996 the american physical society

multi time step explicit implicit method for non linear structural dynamics we present a method with domain decomposition to solve time dependent non linear problems this method enables arbitrary numeric schemes of the newmark family to be coupled with different time steps in each subdomain this coupling is achieved by prescribing continuity of velocities at the interface we are more specifically interested in the coupling of implicit explicit numeric schemes taking into account material and geometric non linearities the interfaces are modelled using a dual schur formulation where the lagrange multipliers represent the interfacial forces unlike the continuous formulation the discretized formulation of the dynamic problem is unable to verify simultaneously the continuity of displacements velocities and accelerations at the interfaces we show that within the framework of the newmark family of numeric schemes continuity of velocities at the interfaces enables the definition of an algorithm which is stable for all cases envisaged to prove this stability we use an energy method i e a global method over the whole time interval in order to verify the algorithms properties then we propose to extend this to non linear situations in the following cases implicit linear explicit non linear explicit non linear explicit non linear and implicit non linear explicit non linear finally we present some examples showing the feasibility of the method copyright 2001 john wiley sons ltd

multiresolution molecular mechanics a unified and consistent framework for general finite element shape functions we present a general mathematical framework for the newly proposed energy based concurrent atomistic continuum method multiresolution molecular mechanics mmm yang et al 2013 the main features of the generalized framework are 1 consistency with the atomistic framework by directly employing the interatomic potential to calculate force and energy 2 simple procedure for analytically deriving the optimal summation rule for any given finite element shape function employed in the coarse grained region the procedure for obtaining the optimal summation rule is developed based on deriving and then fitting the atomic energy distribution within an element under the constraint of a given shape function to validate the generalized framework test problems including non local harmonic and anharmonic models undergoing tensile shear and bending deformations will be solved using linear bilinear and quadratic elements respectively results obtained using the proposed optimal summation rules for the different element types will be compared with gauss quadrature for accuracy through error structure analyses it is found that the proposed summation rule always outperforms gauss quadrature even when the latter employs more quadrature points than the former it is argued that widely used numerical quadrature techniques such as gauss quadrature are not optimal for coarse grained atomic energy approximation because they do not account for the discrete nature of the atoms in contrast the present summation rule is derived consistently from the underlying atomic energy distribution and thus has better accuracy and smaller computational cost

impulsive problems for semilinear differential equations with nonlocal conditions abstract this paper is concerned with the existence of impulsive semilinear differential equations with nonlocal conditions using the technique of fixed point theory existence results are obtained for mild solutions when the nonlocal item is lipschitz is not lipschitz and not compact respectively

existence of mild solutions for semilinear evolution equations with non local initial conditions abstract in this paper we study the existence of mild solutions for a class of semilinear evolution equations with non local initial conditions and extend some related results in this direction

the braided heisenberg group the braided groups and braided matrices b r for the solution r of the yang baxter equation associated to the quantum heisenberg group are computed it is also shown that a particular extension of the quantum heisenberg group is dual to the heisenberg universal enveloping algebra uq h and this result is used to derive an action of uq h on the braided groups the various covariance properties are then demonstrated using the braided heisenberg group as an explicit example in addition the braided heisenberg group is found to be self dual finally a physical application to a system of n braided harmonic oscillators is discussed an isomorphism is found between the n fold braided and unbraided tensor products and the usual free time evolution is shown to be equivalent to an action of a primitive generator of uq h on the braided tensor product

validity of the cauchy born rule applied to discrete cellular scale models of biological tissues the development of new models of biological tissues that consider cells in a discrete manner is becoming increasingly popular as an alternative to continuum methods based on partial differential equations although formal relationships between the discrete and continuum frameworks remain to be established for crystal mechanics the discrete to continuum bridge is often made by assuming that local atom displacements can be mapped homogeneously from the mesoscale deformation gradient an assumption known as the cauchy born rule cbr although the cbr does not hold exactly for noncrystalline materials it may still be used as a first order approximation for analytic calculations of effective stresses or strain energies in this work our goal is to investigate numerically the applicability of the cbr to two dimensional cellular scale models by assessing the mechanical behavior of model biological tissues including crystalline honeycomb and noncrystalline reference states the numerical procedure involves applying an affine deformation to the boundary cells and computing the quasistatic position of internal cells the position of internal cells is then compared with the prediction of the cbr and an average deviation is calculated in the strain domain for center based cell models we show that the cbr holds exactly when the deformation gradient is relatively small and the reference stress free configuration is defined by a honeycomb lattice we show further that the cbr may be used approximately when the reference state is perturbed from the honeycomb configuration by contrast for vertex based cell models a similar analysis reveals that the cbr does not provide a good representation of the tissue mechanics even when the reference configuration is defined by a honeycomb lattice the paper concludes with a discussion of the implications of these results for concurrent discrete and continuous modeling adaptation of atom to continuum techniques to biological tissues and model classification

multi scale constitutive modelling of heterogeneous materials with a gradient enhanced computational homogenization scheme a gradient enhanced computational homogenization procedure that allows for the modelling of microstructural size effects is proposed within a general non linear framework in this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element rve this enables us to incorporate the microstructural size and to account for non uniform macroscopic deformation fields within the microstructural cell every microstructural constituent is modelled as a classical continuum and the rve problem is formulated in terms of standard equilibrium and boundary conditions from the solution of the microstructural boundary value problem the macroscopic stress tensor and the higher order stress tensor are derived based on an extension of the hill mandel condition this automatically delivers the microstructurally based constitutive response of the higher order macro continuum and deals with the microstructural size in a natural way several examples illustrate the approach particularly the microstructural size effects

analysis of blended atomistic continuum hybrid methods we present a comprehensive error analysis of two prototypical atomistic to continuum coupling methods of blending type the energy based and the force based quasicontinuum methods our results are valid in two and three dimensions for finite range many body interactions e g eam type and in the presence of lattice defects we consider point defects and dislocations the two key ingredients in the analysis are 1 new force and energy consistency error estimates and 2 a new technique for proving energy norm stability of a c couplings that requires only the assumption that the exact atomistic solution is a stable equilibrium

link invariants and combinatorial quantization of hamiltonian chern simons theory we define and study the properties of observables associated to any link in r where is a compact surface using the combinatorial quantization of hamiltonian chern simons theory these observables are traces of holonomies in a non commutative yang mills theory where the gauge symmetry is ensured by a quantum group we show that these observables are link invariants taking values in a non commutative algebra the so called moduli algebra when s2 these link invariants are pure numbers and are equal to reshetikhin turaev link invariants

a continuum model for size dependent deformation of elastic films of nano scale thickness abstract ultra thin elastic films of nano scale thickness with an arbitrary geometry and edge boundary conditions are analyzed an analytical model is proposed to study the size dependent mechanical response of the film based on continuum surface elasticity by using the transfer matrix method along with an asymptotic expansion technique of small parameter closed form solutions for the mechanical field in the film is presented in terms of the displacements on the mid plane the asymptotic expansion terminates after a few terms and exact solutions are obtained the mid plane displacements are governed by three two dimensional equations and the associated edge boundary conditions can be prescribed on average solving the two dimensional boundary value problem yields the three dimensional response of the film the solution is exact throughout the interior of the film with the exception of a thin boundary layer having an order of thickness as the film in accordance with the saint venant s principle

on the generalized virial theorem and eshelby tensors the formal relationships between the scalar and tensorial virials and eshelby tensors have been presently investigated the key idea is to evaluate the eshelby stress from discrete or atomistic simulations for a structured body conceived as a numerical homogenization method to reconstitute the macroscopic continuum behavior in multiscale modelling approaches extending first the writing of the scalar virial to a material format it is shown that the average of the elaborated scalar material virial is the trace of the material eshelby stress the spatial and material virials are further related to eachother in the framework of hyperelasticity and a tensorial extension of the material virial is provided interpretation of those results from the microscopic point of view shows that eshelby stress may be identified and calculated at the discrete level from the average of the virial tensor consideration of the material version of the virial theorem further leads to express eshelby stress versus the average of the internal tensorial material virial and of the kinetic energy the average scalar virial is further identified to the grand potential in a thermodynamic context a definition of the material scalar virial for a second order continuum is lastly proposed based on the identification of a second order eshelby stress and in line with the second order cauchy born rule

nonlinear differential equations with nonlocal conditions in banach spaces abstract in this paper we study the existence of integral solutions for nonlinear differential equations with nonlocal initial conditions in banach spaces we derive conditions under which the integral solutions exist

kitaev lattice models as a hopf algebra gauge theory we prove that kitaev s lattice model for a finite dimensional semisimple hopf algebra h is equivalent to the combinatorial quantisation of chern simons theory for the drinfeld double d h this shows that kitaev models are a special case of the older and more general combinatorial models this equivalence is an analogue of the relation between turaev viro and reshetikhin turaev tqfts and relates them to the quantisation of moduli spaces of flat connections we show that the topological invariants of the two models the algebra of operators acting on the protected space of the kitaev model and the quantum moduli algebra from the combinatorial quantisation formalism are isomorphic this is established in a gauge theoretical picture in which both models appear as hopf algebra valued lattice gauge theories we first prove that the triangle operators of a kitaev model form a module algebra over a hopf algebra of gauge transformations and that this module algebra is isomorphic to the lattice algebra in the combinatorial formalism both algebras can be viewed as the algebra of functions on gauge fields in a hopf algebra gauge theory the isomorphism between them induces an algebra isomorphism between their subalgebras of invariants which are interpreted as gauge invariant functions or observables it also relates the curvatures in the two models which are given as holonomies around the faces of the lattice this yields an isomorphism between the subalgebras obtained by projecting out curvatures which can be viewed as the algebras of functions on flat gauge fields and are the topological invariants of the two models

a quasicontinuum methodology for multiscale analyses of discrete microstructural models summary many studies in different research fields use lattice models to investigate the mechanical behavior of materials full lattice calculations are often performed to determine the influence of localized microscale phenomena on large scale responses but they are usually computationally expensive in this study the quasicontinuum qc method phil mag a 1996 73 1529 1563 is extended towards lattice models that employ discrete elements such as trusses and beams the qc method is a multiscale approach that uses a triangulation to interpolate the lattice model in regions with small fluctuations in the deformation field while in regions of high interest the exact lattice model is obtained by refining the triangulation to the internal spacing of the lattice interpolation ensures that the number of unknowns is reduced while summation ensures that only a selective part of the underlying lattice model must be visited to construct the governing equations as the qc method has so far only been applied to atomic lattice models the existing summation procedures have been revisited for structural lattice models containing discrete elements this has led to a new qc method that makes use of the characteristic structure of the considered truss network the proposed qc method is to the best of the authors knowledge the only qc method that does not need any correction at the interface between the interpolated and the fully resolved region and at the same time gives exact results unlike the cluster qc methods in its present formulation the proposed qc method can only be used for lattice models containing nearest neighbor interactions but with some minor adaptations it can also be used for lattices with next nearest neighbor interactions such as atomic lattices copyright 2011 john wiley sons ltd