a priori and a posteriori analysis of the quasinonlocal quasicontinuum method in 1d for a next nearest neighbour pair interaction model in a periodic domain a priori and a posteriori analyses of the quasinonlocal quasicontinuum method qnl qc are presented the results are valid for large deformations and essentially guarantee a one to one correspondence between atomistic solutions and qnl qc solutions the analysis is based on consistency error estimates in negative norms novel a priori and a posteriori stability estimates and a quantitative inverse function theorem

multiresolution molecular mechanics convergence and error structure analysis abstract we present a convergence and error structure analysis of the newly proposed energy based concurrent atomistic continuum method multiresolution molecular statics mms yang et al 2013 the ability to perform this study lies in the property that mms is fully consistent with full atomistics this work investigates the effects of 1 energy sampling schemes 2 linearity of interatomic interaction and 3 smoothness of the solution on the convergence characteristics and error structure of mms for this purpose the nonlocal harmonic and lennard jones lj potentials together with quadratic deformation uniformly distributed and random force loadings are employed to test the convergence for a one dimensional tensile loading model and a two dimensional cantilever beam bending model the key finding is that under certain conditions mms shows monotonic convergence and convergence rates that are similar to those in finite element method fem also the condition under which either discretization or sampling error dominates is determined for each specific sampling scheme based on the convergence and error structure analysis suggestions on how to effectively choose an mms model are given

a model of hysteresis with two inputs we formulate and analyze a new model of vector hysteresis for the case of two input signals we prove the essential mathematical properties of this model and we present the solutions to two identification problems connected with our model

approximately conserved quantity in the henon heiles problem a possible relation between an approximately conserved quantity in the h enon heiles problem and the integrals of the toda lattice is investigated a relation good to fifth order exactly and seventh order when averaged is proposed a form of averaging appears to be significant in the successful construction of approximately conserved quantities

numerical solution of a virtual internal bond model for material fracture a virtual internal bond vib model is proposed recently in mechanical engineering literatures for simulating dynamic fracture the model is a nonlinear wave equation of mixed type hyperbolic or elliptic there is instability in the elliptic region and usual numerical methods might not work we examine the artificial viscosity method for the model and apply central type schemes directly to the corresponding viscous system to ensure appropriate numerical viscous term for such a mixed type problem we provide a formal justification of indicating convergence of the scheme despite the difficulty of the type change the exact solution of a riemann problem is used to demonstrate the numerical method for one dimensional case we then generalize the method to a two dimensional material with a triangular or hexagonal lattice structure computational results for a two dimensional example are given

finite element analysis of cauchy born approximations to atomistic models this paper is devoted to a new finite element consistency analysis of cauchy born approximations to atomistic models of crystalline materials in two and three space dimensions through this approach new atomistic cauchy born models are introduced and analyzed these intermediate models can be seen as first level atomistic quasicontinuum approximations in the sense that they involve only short range interactions the analysis and the models developed herein are expected to be useful in the design of coupled atomistic continuum methods in more than one dimension taking full advantage of the symmetries of the atomistic lattice we show that the consistency error of the models considered both in energies and in dual w1 p type norms is mathcal o varepsilon 2 where varepsilon denotes the interatomic distance in the lattice

stability instability and error of the force based quasicontinuum approximation due to their algorithmic simplicity and high accuracy force based model coupling techniques are popular tools in computational physics for example the force based quasicontinuum qcf approximation is the only known pointwise consistent quasicontinuum approximation for coupling a general atomistic model with a finite element continuum model in this paper we present a detailed stability and error analysis of this method our optimal order error estimates provide a theoretical justification for the high accuracy of the qcf approximation they clearly demonstrate that the computational efficiency of continuum modeling can be utilized without a significant loss of accuracy if defects are captured in the atomistic region the main challenge we need to overcome is the fact that the linearized qcf operator is typically not positive definite moreover we prove that no uniform inf sup stability condition holds for discrete versions of the w1 p w1 q duality pairing with 1 p 1 q 1 if 1 p however we were able to establish an inf sup stability condition for a discrete version of the w1 w1 1 duality pairing which leads to optimal order error estimates in a discrete w1 norm

one dimensional mapping modulated phases and lyapunov exponent for the antiferromagnetic q state potts and multi site exchange interaction ising models abstract we describe the bifurcation structure period doubling and chaos for the antiferromagnetic q state potts model on the bethe lattice and three site interaction ising model on husimi one in a magnetic field by using the recursion relation technique a chaotic behavior of the magnetic susceptibility for the models is observed at low temperatures the resulting one dimensional rational mapping has a positive lyapunov exponent in the region of the chaotic regime for the antiferromagnetic q state potts q 2 and three site interaction ising models we discuss modulated phases for the antiferromagnetic q state potts q 2 and q 2 and three site interaction ising model at low temperatures the q state potts model q 2 has only one modulated phase with 1 2 pinching corresponding to the 2 cycle the q state potts q 2 and three site interaction ising models have an infinite number of modulated phases with different pinching numbers we construct the first modulated phase after the first bifurcation point

magnetic properties lyapunov exponent and superstability of the spin 12 ising heisenberg model on a diamond chain abstract the exactly solvable spin 1 2 ising heisenberg model on a diamond chain has been considered we have found the exact results for the magnetization using the recursion relation method the existence of the magnetization plateau has been observed at one third of the saturation magnetization in the antiferromagnetic case some ground state properties of the model are examined at low temperatures the system has two ferrimagnetic fri1 and fri2 phases and one paramagnetic prm phase lyapunov exponents for the various values of the exchange parameters and temperatures have been analyzed it has also been shown that the maximal lyapunov exponent exhibits plateau lyapunov exponents exhibit different behavior for two ferrimagnetic phases we have found the existence of the supercritical point for the multi dimensional rational mapping of the spin 1 2 ising heisenberg model on a diamond chain for the first time in the absence of the external magnetic field and t 0 in the antiferromagnetic case

from molecular models to continuum mechanics we present here a limiting process allowing us to write some continuum mechanics models as a natural asymptotic of molecular models the approach is based on the hypothesis that the macroscopic displacement is equal to the microscopic one we carry out the corresponding calculations in the case of two body energies including higher order terms and also in the case of thomas fermi type models

nonlinear resonance in systems with hysteresis a general approach is suggested for the analysis of nonlinear resonance in control systems including an integral linear term and a nonlinear term with hysteresis this approach uses a special property of nonlinear terms which is valid for many important classes of hysteresis operators for systems with such hysteresis nonlinearities we introduce a method to derive conditions for the existence of forced oscillations and for systems with a parameter of nonlinear resonance at infinity

signs and approximate magnitudes of lyapunov exponents in continuous time dynamical systems methods for algebraically determining the signs and the magnitudes of lyapunov exponents of a given dynamical system are studied the existence of zero lyapunov exponents for the toda henon heiles and rossler systems are shown the approximate lyapunov spectra of lorenz and rossler systems are computed

goal oriented adaptive mesh refinement for the quasicontinuum approximation of a frenkel kontorova model abstract the quasicontinuum approximation e b tadmor m ortiz r phillips quasicontinuum analysis of defects in solids philos mag a 73 6 1996 1529 1563 is a method to reduce the atomistic degrees of freedom of a crystalline solid by piecewise linear interpolation from representative atoms that are nodes for a finite element triangulation in regions of the crystal with a highly nonuniform deformation such as around defects every atom must be a representative atom to obtain sufficient accuracy but the mesh can be coarsened away from such regions to remove atomistic degrees of freedom while retaining sufficient accuracy we present an error estimator and a related adaptive mesh refinement algorithm for the quasicontinuum approximation of a generalized frenkel kontorova model that enables a quantity of interest to be efficiently computed to a predetermined accuracy

theory based benchmarking of the blended force based quasicontinuum method we formulate an atomistic to continuum coupling method based on blending atomistic and continuum forces our precise choice of blending mechanism is informed by theoretical predictions we present a range of numerical experiments studying the accuracy of the scheme focusing in particular on its stability these experiments confirm and extend the theoretical predictions and demonstrate a superior accuracy of b qcf over energy based blending schemes

convergence analysis of a quasi continuum approximation for a two dimensional material without defects in many applications materials are modeled by a large number of particles or atoms where any particle can interact with any other the computational cost is very high since the number of atoms is huge recently much attention has been paid to a so called quasi continuum qc method which is a mixed atomistic continuum model the qc method uses an adaptive finite element framework to effectively integrate the majority of the atomistic degrees of freedom in regions where there is no serious defect however numerical analysis of this method is still in its infancy in this paper we will conduct a convergence analysis of the qc method in the case when there is no defect we will also remark on the case when the defect region is small the difference between our analysis and conventional analysis is that our exact atomistic solution is not a solution of a continuous partial differential equation but a discrete lattice scale solution which is not approximately related to any conventional partial differential equation

a priori error analysis of two force based atomistic continuum models of a periodic chain the force based quasicontinuum qcf approximation is a non conservative atomistic continuum hybrid model for the simulation of defects in crystals we present an a priori error analysis of the qcf method applied to a one dimensional periodic chain that is valid for an arbitrary interaction range large deformations and takes coarse graining into account our main tool in this analysis is a new concept of atomistic stress moreover we formulate a new atomistic continuum coupling mechanism based on coupling stresses instead of forces and extend the a priori analysis to this new method we show that the new method has several theoretical advantages over the original qcf method

an optimal order error analysis of the one dimensional quasicontinuum approximation we derive a model problem for quasicontinuum approximations that allows a simple yet insightful analysis of the optimal order convergence rate in the continuum limit for both the energy based quasicontinuum approximation and the quasi nonlocal quasicontinuum approximation for simplicity the analysis is restricted to the case of second neighbor interactions and is linearized about a uniformly stretched reference lattice the optimal order error estimates for the quasi nonlocal quasicontinuum approximation are given for all strains up to the continuum limit strain for fracture the analysis is based on an explicit treatment of the coupling error at the atomistic to continuum interface combined with an analysis of the error due to the atomistic and continuum schemes using the stability of the quasicontinuum approximation

theoretical and numerical analysis for the quasi continuum approximation of a material particle model in many applications materials are modeled by a large number of particles or atoms where any one of particles interacts with all others near or nearest neighbor interaction is expected to be a good simplification of the full interaction in the engineering community in this paper we shall analyze the approximate error between the solution of the simplified problem and that of the full interaction problem so as to answer the question mathematically for a one dimensional model a few numerical methods have been designed in the engineering literature for the simplified model recently much attention has been paid to a finite element like quasicontinuum qc method which utilizes a mixed atomistic continuum approximation model no numerical analysis has been done yet in the paper we shall estimate the error of the qc method for this one dimensional model possible ill posedness of the method and its modification are discussed as well

mathematical models for hysteresis the various existing classical models for hysteresis preisach ishlinskii duhem madelung are surveyed as well a more modern treatments by contemporary workers the emphasis is on a clear mathematical description of the formulation and properties of each model in addition the authors try to make the reader aware of the many open questions in the study of hysteresis

an analysis of node based cluster summation rules in the quasicontinuum method we investigate two examples of node based cluster summation rules that have been proposed for the quasicontinuum qc method a force based approach and an energy based approach which is a generalization of the nonlocal qc method we show that even for the case of nearest neighbor interaction in a one dimensional periodic chain both of these approaches create large errors that cannot be removed by increasing the cluster size when used with graded and more generally nonsmooth meshes we offer some suggestions for how the accuracy of cluster summation rules may be improved

analysis of energy based blended quasi continuum approximations the development of patch test consistent quasi continuum energies for multidimensional crystalline solids modeled by many body potentials remains a challenge the original quasi continuum energy qce r miller and e tadmor model simul mater sci eng 17 2009 053001 has been implemented for many body potentials in two and three space dimensions but it is not patch test consistent we propose that by blending the atomistic and corresponding cauchy born continuum models of qce in an interfacial region with thickness of a small number k of blended atoms a general blended quasi continuum energy bqce can be developed with the potential to significantly improve the accuracy of qce near lattice instabilities such as dislocation formation and motion in this paper we give an error analysis of the blended quasi continuum energy bqce for a periodic one dimensional chain of atoms with next nearest neighbor interactions our analysis includes the optimization of the blending function for an improved convergence rate we show that the ell 2 strain error for the nonblended qce energy which has low order o varepsilon 1 2 where varepsilon is the atomistic length scale m dobson and m luskin siam j numer anal 47 2009 pp 2455 2475 p ming and j z yang multiscale model simul 7 2009 pp 1838 1875 can be reduced by a factor of k 3 2 for an optimized blending function where k is the number of atoms in the blending region the qce energy has been further shown to suffer from a o 1 error in the critical strain at which the lattice loses stability m dobson m luskin and c ortner j mech phys solids 58 2010 pp 1741 1757 we prove that the error in the critical strain of bqce can be reduced by a factor of k 2 for an optimized blending function thus demonstrating that the bqce energy for an optimized blending function has the potential to give an accurate approximation of the deformation near lattice instabilities such as crack growth

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

quadrature rule type approximations to the quasicontinuum method for long range interatomic interactions a quadrature rule type method is presented to approximate the quasicontinuum method for atomistic mechanics for both the short range and long range interaction cases the complexity of this method depends on the number of representative particles but not on the total number of particles simple analysis and numerical experiments are provided to illustrate the accuracy and performance of the method it is shown that for the same accuracy the quadrature rule type method is much less costly than the quasicontinuum method

properties of the preisach model for hysteresis as shown by krasnosel skii the classical preisach model allows to construct a hysteresis operator wbetween spaces of real functions of time this construction via the definition of a measure mu in the so called preisach plane is recalled characterizations in terms of mu are given for several mapping and continuity properties of w in various function spaces for the invertibility of w and for the corresponding mapping and continuity properties of the inverse

a posteriori error estimation and adaptive algorithm for atomistic continuum coupling in two dimensions atomistic continuum coupling methods aim to achieve optimal balance between accuracy and efficiency adaptivity is the key for the efficient implementation of such methods in this paper we carry

iterative methods for the force based quasicontinuum approximation analysis of a 1d model problem force based atomistic continuum hybrid methods are the only known pointwise consistent methods for coupling a general atomistic model to a finite element continuum model for this reason and due to their algorithmic simplicity force based coupling methods have become a popular class of atomistic continuum hybrid models as well as other types of multiphysics models however the recently discovered unusual stability properties of the linearized force based quasicontinuum qcf approximation especially its indefiniteness present a challenge to the development of efficient and reliable iterative methods we present analytic and computational results for the generalized minimal residual gmres solution of the linearized qcf equilibrium equations we show that the gmres method accurately reproduces the stability of the force based approximation and conclude that an appropriately preconditioned gmres method results in a reliable and efficient solution method

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

semilinear duhem model for rate independent and rate dependent hysteresis the classical duhem model provides a finite dimensional differential model of hysteresis in this paper we consider rate independent and rate dependent semilinear duhem models with provable properties the vector field is given by the product of a function of the input rate and linear dynamics if the input rate function is positively homogeneous then the resulting input output map of the model is rate independent yielding persistent nontrivial input output closed curve that is hysteresis at arbitrarily low input frequency if the input rate function is not positively homogeneous the input output map is rate dependent and can be approximated by a rate independent model for low frequency inputs sufficient conditions for convergence to a limiting input output map are developed for rate independent and rate dependent models finally the reversal behavior and orientation of the rate independent model are discussed

a priori and a posteriori w 1 infty error analysis of a qc method for complex lattices in this paper we derive regularity results for equilibria of multilattices under an external force and prove a priori and a posteriori error estimates for a multiscale numerical method for computing such equilibria the estimates are derived in a w 1 infty norm in one space dimension one of the features of our analysis is that we establish an equivalent way of formulating the coarse grained problem which greatly simplifies derivation of the error bounds both a priori and a posteriori we illustrate our error estimates with numerical experiments

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