blow up and global existence for a general class of nonlocal nonlinear coupled wave equations abstract we study the initial value problem for a general class of nonlinear nonlocal coupled wave equations the problem involves convolution operators with kernel functions whose fourier transforms are nonnegative some well known examples of nonlinear wave equations such as coupled boussinesq type equations arising in elasticity and in quasi continuum approximation of dense lattices follow from the present model for suitable choices of the kernel functions we establish local existence and sufficient conditions for finite time blow up and as well as global existence of solutions of the problem

the rigorous approximation of long wavelength capillary gravity waves in a previous paper we proved that long wavelength solutions of the water wave problem in the case of zero surface tension split up into two wave packets one moving to the right and one to the left where each of these wave packets evolves independently as a solution of a korteweg de vries kdv equation in this paper we examine the effect of surface tension on this scenario we find that we obtain three different physical regimes depending on the strength of the surface tension for weak surface tension the propagation of the wave packets is very similar to that in the zero surface tension case for strong surface tension the evolution is again governed by a pair of kdv equations but the coefficients in these equations have changed in such a way that the kdv soliton now represents a wave of depression on the fluid surface finally at a special intermediate value of the surface tension where the bond number equals the kdv description breaks down and it is necessary to introduce a new approximating equation the kawahara equation which is a fifth order nonlinear partial differential equation in each of these regimes we give rigorous estimates of the difference between the solution of the appropriate modulation equation and the solution of the true water wave problem

global existence and blow up of solutions for a general class of doubly dispersive nonlocal nonlinear wave equations this study deals with the analysis of the cauchy problem of a general class of nonlocal nonlinear equations modeling the bi directional propagation of dispersive waves in various contexts the nonlocal nature of the problem is reflected by two different elliptic pseudodifferential operators acting on linear and nonlinear functions of the dependent variable respectively the well known doubly dispersive nonlinear wave equation that incorporates two types of dispersive effects originated from two different dispersion operators falls into the category studied here the class of nonlocal nonlinear wave equations also covers a variety of well known wave equations such as various forms of the boussinesq equation local existence of solutions of the cauchy problem with initial data in suitable sobolev spaces is proven and the conditions for global existence and finite time blow up of solutions are established

global existence and asymptotic behaviour of solutions for the generalized boussinesq equation abstract in this paper we prove the existence and the uniqueness of global solution for the cauchy problem for the generalized boussinesq equation under some assumptions we also show that the l norm of small solution of the cauchy problem for the generalized boussinesq equation decays to zero as t tends to the infinity

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

the logic of nonmonotonicity research note abstract shoham 14 presents a general semantic framework in which many nonmonotonic logics the so called preference logics can be represented we present another means of giving general semantics for preference logics using models of the conditional logic c this has the advantage that we also provide an axiomatization for preference logics insofar as all nonmonotonic logics are preference logics c is the logic of nonmonotonicity

global existence and asymptotic behavior of solution for the rosenau equation with hydrodynamical damped term abstract we consider the cauchy problem for rosenau equation with hydrodynamical damped term in n dimensional space under smallness condition on the initial data we prove the global existence and asymptotic behavior of the solution in the time weighted sobolev spaces by the contraction mapping principle

differentiability of the blow up curve for one dimensional nonlinear wave equations consider the cauchy problem rac artial 2 u artial t 2 umimits i 1 n rac artial 2 u artial x i 2 f u x n athbb r n t 0 ueft x 0 ight feft x ight eft x n athbb r n ight u t eft x 0 ight geft x ight eft x n athbb r n ight with n 3 it is well known that if f u is superlinear i e feft u ight a eft u ight p ifeft u ight o nfty eft a 0 p 1 ight 0 1 then solutions generally blow up in finite time see 2 6

generalized bridging domain method for coupling finite elements with discrete elements abstract the concurrent coupling of finite elements and discrete elements is an effective dof reduction methodology for reproducing the granular flow phenomenon as discrete element method does in this paper we present a novel coupling strategy named the generalized bridging domain method this method introduces independent functions to weight the material properties of the continuum and those of the discrete element model and then the equilibrium of each new model is guaranteed by imposing compensation forces the compensation force of continuum is determined by force and displacement compatibility requirements of continuum with respect to discrete elements and vice versa utilizing different weighting functions four typical coupling methods the bridging domain method edge to edge coupling separate domain coupling and separate edge coupling are obtained additionally a new integration algorithm with multiple time steps is developed for the separate edge coupling the numerical performance of the separate domain coupling where displacement compatibility condition of continuum and that of discrete elements are individually enforced by the lagrange multiplier method has been investigated in detail results of several numerical examples show that the separate domain coupling outperforms other methods in avoiding spurious reflection and the separate edge coupling is effective enough for coupling finite elements with discrete elements due to the truncation of high frequency waves there exists energy loss but at an acceptable level if the waves can be resolved by the macroscopic finite element model

zero curvature formulations of dual hierarchies zero curvature formulations are given for the dual hierarchies of standard soliton equation hierarchies recently introduced by olver and rosenau including the physically interesting fuchssteiner fokas camassa holm hierarchy

a bridging domain and strain computation method for coupled atomistic continuum modelling of solids we present a multiscale method that couples atomistic models with continuum mechanics the method is based on an overlapping domain decomposition scheme constraints are imposed by a lagrange multiplier method to enforce displacement compatibility in the overlapping subdomain in which atomistic and continuum representations overlap an efficient version of the method is developed for cases where the continuum can be modelled as a linear elastic material an iterative scheme is utilized to optimize the coupled configuration conditions for the regularity of the constrained matrices are determined a method for computing strain in atomistic models and handshake domains is formulated based on a moving least square approximation which includes both extensional and angle bending terms it is shown that this method exactly computes the linear strain field applications to the fracture of defected single layer atomic sheets and nanotubes are given copyright 2006 john wiley sons ltd

blow up and scattering of solution for a generalized boussinesq equation in this paper we consider the cauchy problem for a class of boussinesq equation we obtain the existence and uniqueness of the local solutions for a class of nonlinearity of the perturbation blow up solutions are obtained furthermore the global existence and nonlinear scattering for small amplitude solutions are established

global existence and asymptotic behavior of solution for the sixth order boussinesq equation with damped term abstract we investigate the small data cauchy problem for the sixth order boussinesq equation with damped term the decay estimates of solutions to the corresponding linear equation are given by using the dyadic decomposition and some properties of bessel functions then we prove the global existence and asymptotic of the small amplitude solution in the time weighted sobolev space by the contraction mapping principle

existence and stability of traveling waves for a class of nonlocal nonlinear equations in this article we are concerned with the existence and orbital stability of traveling wave solutions of a general class of nonlocal wave equations utt luxx b u p 1 u xx p 1 the main characteristic of this class of equations is the existence of two sources of dispersion characterized by two coercive pseudo differential operatorsl and b members of the class arise as mathematical models for the propagation of dispersive waves in a wide variety of situations for instance all boussinesq type equations and the so called double dispersion equation are members of the class we first establish the existence of traveling wave solutions to the nonlocal wave equations considered we then obtain results on the orbital stability or instability of traveling waves for the case l i corresponding to a class of klein gordon type equations we give an almost complete characterization of the values of the wave velocity for which the traveling waves are orbitally stable or unstable by blow up

a note on the camassa holm equation abstract we consider a generalized camassa holm equation we improve the compactness argument of 5 6 7 17 establishing the existence of global weak solutions in h 1 without the use of an ole nik type estimate the potential interest of this improvement is linked to convergence analysis of numerical schemes for which it may not be easy to verify an ole nik type estimate

existence and scattering of small solutions to a boussinesq type equation of sixth order abstract in this paper we consider the existence and uniqueness of the global small solution as well as the small data scattering result to the cauchy problem for a boussinesq type equation of sixth order with the nonlinear term f u behaving as u p p 1 as u 0 in r n n 1 the main method and techniques used in our paper are the littlewood paley dyadic decomposition the stationary phase estimate and some properties of bessel function

semantical considerations on nonmonotonic logic abstract commonsense reasoning is nonmonotonic in the sense that we often draw on the basis of partial information conclusions that we later retract when we are given more complete information some of the most interesting products of recent attempts to formalize nonmonotonic reasoning are the nonmonotonic logics of mcdermott and doyle 1 2 these logics however all have peculiarities that suggest they do not quite succeed in capturing the intuitions that prompted their development in this paper we reconstruct nonmonotonic logic as a model of an ideally rational agent s reasoning about his own beliefs for the resulting system called autoepistemic logic we define an intuitively based semantics for which we can show autoepistemic logic to be both sound and complete we then compare autoepistemic logic with the approach of mcdermott and doyle showing how it avoids the peculiarities of their nonmonotonic logic

peakons and coshoidal waves traveling wave solutions of the camassa holm equation abstract camassa and holm 1 have recently derived a new integrable wave equation for the special case 0 they showed it has solitary waves of the form c exp x ct which they named peakons in this work we derive a perturbation series for general which converges even at the peakon limit we also give three analytical representations for the spatially periodic generalization of the peakon the coshoidal wave the three representations are i a closed form analytical solution ii a fourier series with coefficients that are explicit rational functions and iii an imbricate fourier series which is the superposition of an infinite number of peakons each separated from its neighbors by distance p where p is the spatial period lastly we have numerically tested the soliton superposition principle although the camassa holm equation is integrable for general it appears that imbricating solitary waves generates an exact spatially periodic solution only for the special cases 0 c 1 2 however the imbricate soliton series is a very good approximate solution for general even when the spatial period is small and the solution resembles a sine wave more than a solitary wave

a coupled molecular dynamics and extended finite element method for dynamic crack propagation a multiscale method is presented which couples a molecular dynamics approach for describing fracture at the crack tip with an extended finite element method for discretizing the remainder of the domain after recalling the basic equations of molecular dynamics and continuum mechanics the discretization is discussed for the continuum subdomain where the partition of unity property of finite element shape functions is used since in this fashion the crack in the wake of its tip is naturally modelled as a traction free discontinuity next the zonal coupling method between the atomistic and continuum models is recapitulated finally it is discussed how the stress has been computed in the atomic subdomain and a two dimensional computation is presented of dynamic fracture using the coupled model the result shows multiple branching which is reminiscent of recent results from simulations on dynamic fracture using cohesive zone models

stress strain relationship for granular materials based on the hypothesis of best fit abstract stress strain relationships of a granular system are derived based on the premise that the mean field of displacement is the best fit of actual particle displacements based on the best fit hypothesis two fundamental relationships are derived 1 the average strain as a function of contact displacements and 2 the mean field of contact force as a function of stress these two relationships lead to a stress strain law without the kinematic constraint of uniform strain for an assembly with isotropic packing structure closed form stress strain relationships are explicitly derived in which the modulus can be determined from the inter particle stiffness the difference between the derived stress strain relationship and that derived from voigt hypothesis is discussed the effect of fabric variation is illustrated in the theory discrete element analyses are also performed and the results are used to compare with the estimated moduli from the proposed theory in the comparison the solution bounds for granular assembly system are illustrated

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

kawahara dynamics in dispersive media abstract in a previous paper it was shown that in the long wave limit the water wave problem without surface tension can be described approximately by two decoupled kdv equations here we study a model problem and prove that in a degenerate case which occurs for the water wave problem with surface tension and near other codimension two points at which the coefficient in front of the leading order dispersive term in the equation of motion vanishes the long wave limit can be rigorously approximated by two decoupled kawahara equations

solitonic solutions for a variable coefficient variant boussinesq system in the long gravity waves variable coefficient variant boussinesq vcvb system is able to describe the nonlinear and dispersive long gravity waves traveling in two horizontal directions with varying depth in this paper with symbolic computation a lax pair associated with the vcvb system under some constraints for variable coefficients is derived and based on the lax pair two sorts of basic darboux transformations are presented by applying the darboux transformations some solitonic solutions are obtained with the relevant constraints given in the text in addition the vcvb system is transformed to a variable coefficient broer kaup system solitonic solutions and procedure of getting them could be helpful to solve the nonlinear and dispersive problems in fluid dynamics

a 3 d discrete element method for dry granular flows of ellipsoidal particles abstract we use 3 d discrete element method dem to simulate dry granular flows of non spherical particles in our simulation code non spherical particles are represented by clusters of spheres for contact detection and for contact force calculation for particles in contact the normal force displacement nfd model is simulated by a partially latching spring system whereas the tangential force displacement tfd model is simulated by an incremental relation that corresponds to a simplification of the mindlin and deresiewicz r d mindlin h deresiewicz asme j appl mech 20 1953 327 344 contact mechanics theory to establish the validity of the simulation procedure benchmark tests such as superball and superbean simulations were carried out we present various results obtained from the simulation of ellipsoidal particles flowing down a chute having a bumpy bottom these ellipsoidal particles had their physical properties obtained from experiments

a logic for default reasoning the need to make default assumptions is frequently encountered in reasoning about incompletely specified worlds inferences sanctioned by default are best viewed as beliefs which may well be modified or rejected by subsequent observations it is this property which leads to the non monotonicity of any logic of defaults in this paper we propose a logic for default reasoning we then specialize our treatment to a very large class of commonly occuring defaults for this class we develop a complete proof theory and show how to interface it with a top down resolution theorem prover finally we provide criteria under which the revision of derived beliefs must be effected

propositional knowledge base revision and minimal change abstract the semantics of revising knowledge bases represented by sets of propositional sentences is analyzed from a model theoretic point of view a characterization of all revision schemes that satisfy the gardenfors rationality postulates is given in terms of minimal change with respect to an ordering among interpretations revision methods proposed by various authors are surveyed and analyzed in this framework the correspondences between gardenfors like rationality postulates and minimal changes with respect to other orderings are also investigated

a completely integrable hamiltonian system the dynamical system characterized by the hamiltonian h q p j k 1 n p jp kf q dj q k with f x cos x sin x is completely integrable here n is an arbitrary positive integer and are 4 arbitrary constants and real and both real or both imaginary

decay and scattering of solutions for a generalized boussinesq equation abstract in this paper we consider the long time behavior of small solutions of the cauchy problem for a generalized boussinesq equation a scattering operator and the nonlinear scattering for small amplitude solutions of the boussinesq equation are established under certain hypotheses

formation of singularities in solutions to nonlinear hyperbolic equations on etudie des systemes de lois de conservation hyperboliques et d equations d ondes completement non lineaires

the cauchy problem for a one dimensional nonlinear elastic peridynamic model this paper studies the cauchy problem for a one dimensional nonlinear peridynamic model describing the dynamic response of an infinitely long elastic bar the issues of local well posedness and smoothness of the solutions are discussed the existence of a global solution is proved first in the sublinear case and then for nonlinearities of degree at most three the conditions for finite time blow up of solutions are established

existence of perturbed solitary wave solutions to a model equation for water waves abstract we prove the existence of travelling wave solutions to a fifth order partial differential equation which is a formal asymptotic approximation for water waves with surface tension these travelling waves are arbitrarily small perturbations of solitary waves but are not solitary waves themselves because they approach small amplitude oscillations for large values of the independent variable this result suggests that for bond numbers less than one third there are branches of travelling wave solutions to the water wave equations which are perturbations of supercritical elevation solitary waves and which bifurcate from froude number one and bond number one third

conditioning analysis of nonlocal integral operators in fractional sobolev spaces we study the conditioning of nonlocal integral operators with singular and integrable kernels in fractional sobolev spaces these operators are used for instance in peridynamics formulation and nonlocal diffusion in one dimension 1d we present sharp quantification of the extremal eigenvalues in all three parameters size of nonlocality mesh size and regularity of the fractional sobolev space we accomplish sharpness both rigorously and numerically for the minimal eigenvalue we obtain sharpness analytically by using a nonlocal characterization of sobolev spaces we verify this estimate by exploiting the toeplitz property of the stiffness matrix however the analytical approach fails to give sharp quantification of the maximal eigenvalue hence in 1d we take an algebraic approach by directly working with the stiffness matrix entries which have complicated expressions due to all three parameters we systematically characterize the nonzero entries and dramatically simplify their expressions by u

large time existence for 3d water waves and asymptotics we rigorously justify in 3d the main asymptotic models used in coastal oceanography including shallow water equations boussinesq systems kadomtsev petviashvili kp approximation green naghdi equations serre approximation and full dispersion model we first introduce a variable nondimensionalized version of the water waves equations which vary from shallow to deep water and which involves four dimensionless parameters using a nonlocal energy adapted to the equations we can prove a well posedness theorem uniformly with respect to all the parameters its validity ranges therefore from shallow to deep water from small to large surface and bottom variations and from fully to weakly transverse waves

the time dependent amplitude equation for the swift hohenberg problem precise estimates for the validity of the amplitude approximation for the swift hohenberg equation are given in a fully time dependent framework it is shown that small solutions of ordero which are modulated like stationary solutions have an evolution which is well described in the amplitude approximation for a time of ordero 2 for the proofs we use techniques for nonlinear semigroups and oscillatory integrals

on the l2 and the h1 couplings for an overlapping domain decomposition method using lagrange multipliers in this paper a comparison of the l 2 and the h couplings is made for an overlapping domain decomposition method using lagrange multipliers the analysis of the local equations arising from the formulation of the coupling of two mechanical models shows that continuous weight functions are required for the l 2 coupling term whereas both discontinuous and continuous weight functions can be used for the h 1 coupling the choice of the lagrange multiplier space is discussed and numerically studied the paper ends with some numerical examples of an end loaded cantilever beam and a cracked plate under tension and shear it is shown that the continuity enforced with the h 1 coupling leads to a link with a flexibility that can be beneficial for coupling a very coarse mesh with a very fine one to limit the effect of the volume coupling on the global response a narrow coupling zone is recommended in this case volume coupling tends to a surface coupling especially with a l 2 coupling

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

a two scale model of granular materials a concurrent two scale and two method approach for modeling the mechanical behavior of dry frictional non cohesive granular materials is presented in domains of large eventually discontinuous deformation the material is modeled on the grain scale using a three dimensional discrete element method in the remaining domain the material is considered continuous and modeled by the finite element method using an elasto plastic constitutive equation whose parameters are fit to the particle model via a homogenization scheme the discrete and finite element models are coupled at the interface by the arlequin method an overlapping domain is introduced where the overall virtual work is interpolated between both models and compatibility is assured by kinematic constraints for this purpose the discrete particle displacements are split into a fine and coarse scale part and equality of the coarse scale part and the continuum solution is enforced through the penalty method

finite propagation speed for the camassa holm equation we prove that any classical solution of the camassa holm equation will have compact support if its initial data has this property

on a class of nonlocal wave equations from applications we study equations from the area of peridynamics which is a nonlocal extension of elasticity the governing equations form a system of nonlocal wave equations we take a novel approach by applying operator theory methods in a systematic way on the unbounded domain n we present three main results as main result 1 we find that the governing operator is a bounded function of the governing operator of classical elasticity as main result 2 a consequence of main result 1 we prove that the peridynamic solutions strongly converge to the classical solutions by utilizing for the first time strong resolvent convergence in addition main result 1 allows us to incorporate local boundary conditions in particular into peridynamics this avenue of research is developed in companion papers providing a remedy for boundary effects as main result 3 employing spherical bessel functions we give a new practical series representation of the solution which allows straightforward numerical treatment with symbolic computation

the long wave limit for the water wave problem i the case of zero surface tension the korteweg de vries equation boussinesq equation and many other equations can be formally derived as approximate equations for the two dimensional water wave problem in the limit of long waves here we consider the classical problem concerning the validity of these equations for the water wave problem in an infinitely long canal without surface tension we prove that the solutions of the water wave problem in the long wave limit split up into two wave packets one moving to the right and one to the left where each of these wave packets evolves independently as a solution of a korteweg de vries equation our result allows us to describe the nonlinear interaction of solitary waves 2000 john wiley sons inc

global solution for a generalized boussinesq equation this paper considers the existence and uniqueness of the solution to the cauchy problem for a class of boussinesq equation proves the global existence and finite time blowup of the solution to the problem by the potential well method

saturation nonmonotonic reasoning and the closed world assumption abstract to deal with the problem of implicit negative information in databases it is necessary to use a nonmonotonic form of reasoning the form of reasoning described in this paper uses an ordering relation on interpretations of the type i is inferior to j if every elementary fact true in i is true in j in simple cases this rule may be defined by a set p of formulas implies a formula q if every minimal model of p satisfies q in more general cases this definition leads to paradox if p is consistent and has no minimal model p must not imply q and its negation we have 1 1 carefully formulated the definition of a special type of implication called sub implication to avoid this type of paradox 2 2 proved that if p is a set of clauses then p has a minimal model and more generally that every model of a set of clauses is minored by one of its minimal models 3 3 developed in this last case subject to certain conditions such as the use of a particular set of clauses called groundable clauses or g clauses a decidable algorithm to compute whether p implies q q being practically any formula

some tricks from the symmetry toolbox for nonlinear equations generalizations of the camassa holm equation abstract the main subject of the paper is to give a survey and to present new methods on how integrability results i e results for symmetry groups inverse scattering formulations action angle transformations and the like can be transferred from one equation to others in case the equations are not related by backlund transformations as a main example the so called camassa holm equation is chosen for which the relevant results are obtained by having a look on the korteweg de vries kdv equation the camassa holm equation turns out to be a different factorization equation of the kdv it describes shallow water waves and reconciles the properties which were known for different orders of shallow water wave approximations we follow here an old method already marginally mentioned in fuchssteiner and fokas 1981 and fuchssteiner 1983 and recently applied by others olver and rosenau 1995 the method allows an immediate recovery of the recursion operator for the camassa holm equation from the invariance structure of the kdv although both equations are not related by backlund transformations however in addition and different from other approaches from there by use of the squared eigenfunction relation for the kdv equation the lax pair formulation for the different factorization equation is derived for the example under consideration it is of course the one obtained in camassa and holm 1993 since the methods proposed here can be transferred to any compatible factorization of recursion operators its application even in the special case which was chosen for illustration leads to a large class of integrable equations among which the camassa holm equation can be found as well as a three parameter family of generalizations of the equation the advantage of the general approach to the lax pair presentation is that direct transformations between action and angle variables are obtained so using this lax pair formulation as a novel result a direct transformation between action and angle variables for the camassa holm equation is derived further novel results in the paper are a hodograph link back from a backlund transformation of the camassa holm equation to a particular member of the kdv hierarchy additional symmetries and the construction of the conformal algebra for the hierarchy of the camassa holm equation the methods involved include hereditary operators bi hamiltonian formulations nilpotent flows and scaling symmetries

the arlequin method as a flexible engineering design tool by superposing and gluing models the arlequin method offers an extended modelling framework for the design of engineering structures this paper aims at developing the numerical aspects of the approach and at showing how it can be used with great flexibility and in a consistent manner to change locally a global mechanical model the capabilities of the arlequin method and the effectiveness of the implemented numerical tools are exemplified by 1 d 2 d and 3 d numerical applications

ghost forces and spurious effects in atomic to continuum coupling methods by the arlequin approach the main objective of this work is to identify spurious effects and propose a corrective method when coupling a particle model involving long range interaction potentials with a first gradient homogenized model using the arlequin framework the most significant spurious effects are generally created by the so called ghost forces that arise in coupling methods based on the minimization of a global energy functional they depend as well on the coupling formulation itself on the notion of representative volume element and on the discretization of the continuum model the proposed corrective technique is based on post processing of the approximate solution by introducing dead forces that can be systematically evaluated and consistently inserted within the arlequin formulation efficiency of the corrective procedure is demonstrated on 1d and 2d examples copyright 2010 john wiley sons ltd

the long wave limit for a boussinesq equation we consider the long wave limit for a boussinesq equation it is shown that for spatially localized initial conditions the dynamics are under a korteweg de vries kdv regime on a large time interval i e in this limit the solutions split up into two counterpropagating wave packets where each of the wave packets evolve independently and approximately as a solution of a kdv equation for the proof exact estimates between the long wave solutions of the boussinesq equation and the approximations obtained via the decoupled set of kdv equations are demonstrated we expect that this result holds for the water wave problem too

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