an online method for interpolating linear parametric reduced order models a two step online method is proposed for interpolating projection based linear parametric reduced order models roms in order to construct a new rom for a new set of parameter values the first step of this method transforms each precomputed rom into a consistent set of generalized coordinates the second step interpolates the associated linear operators on their appropriate matrix manifold real time performance is achieved by precomputing inner products between the reduced order bases underlying the precomputed roms the proposed method is illustrated by applications in mechanical and aeronautical engineering in particular its robustness is demonstrated by its ability to handle the case where the sampled parameter set values exhibit a mode veering phenomenon

nonlinear manifold learning for meshfree finite deformation thin shell analysis calculations on general point set surfaces are attractive because of their flexibility and simplicity in the preprocessing but present important challenges the absence of a mesh makes it nontrivial to decide if two neighboring points in the three dimensional embedding are nearby or rather far apart on the manifold furthermore the topology of surfaces is generally not that of an open two dimensional set ruling out global parametrizations we propose a general and simple numerical method analogous to the mathematical theory of manifolds in which the point set surface is described by a set of overlapping charts forming a complete atlas we proceed in four steps 1 partitioning of the node set into subregions of trivial topology 2 automatic detection of the geometric structure of the surface patches by nonlinear dimensionality reduction methods 3 parametrization of the surface using smooth meshfree here maximum entropy approximants and 4 gluing together the patch representations by means of a partition of unity each patch may be viewed as a meshfree macro element we exemplify the generality flexibility and accuracy of the proposed approach by numerically approximating the geometrically nonlinear kirchhoff love theory of thin shells we analyze standard benchmark tests as well as point set surfaces of complex geometry and topology

information loss in coarse graining of stochastic particle dynamics recently a new class of approximating coarse grained stochastic processes and associated monte carlo algorithms were derived directly from microscopic stochastic lattice models for the adsorption desorption and diffusion of interacting particles 12 13 15 the resulting hierarchy of stochastic processes is ordered by the level of coarsening in the space time dimensions and describes mesoscopic scales while retaining a significant amount of microscopic detail on intermolecular forces and particle fluctuations here we rigorously compute in terms of specific relative entropy the information loss between non equilibrium exact and approximating coarse grained adsorption desorption lattice dynamics our result is an error estimate analogous to rigorous error estimates for finite element finite difference approximations of partial differential equations we prove this error to be small as long as the level of coarsening is small compared to the range of interaction of the microscopic model this result gives a first mathematical reasoning for the parameter regimes for which approximating coarse grained monte carlo algorithms are expected to give errors within a given tolerance

abelianisation of orthogonal groups and the fundamental group of modular varieties abstract we study the commutator subgroup of integral orthogonal groups belonging to indefinite quadratic forms we show that the index of this commutator is 2 for many groups that occur in the construction of moduli spaces in algebraic geometry in particular the moduli of k3 surfaces we give applications to modular forms and to computing the fundamental groups of some moduli spaces

analysis of in plane wave propagation in periodic structures with sierpinski carpet unit cells abstract in this study the dispersion relationships and characteristics of directional periodic structure waveguides with sierpinski carpet unit cells were investigated finite element method was used to analyze the sierpinski carpet periodic structures with square pb cylinders inserted in the rubber background the dispersion curves of unit cells with different stages were investigated first then group velocities of periodic structures at given frequencies were calculated and used to analyze the directional waveguide characteristics in the sierpinski carpet periodic structures the dynamic responses of finite periodic structures were investigated to observe the directional features and the results were used to support the fact that sierpinski carpet unit cells with different stages can control the propagation behavior of elastic waves in specific ways the dispersion curves grew compressed as the stage increased multiple band characteristics were observed in the case of the second stage fractal configuration the directional propagation characteristics in sierpinski carpet periodic structures can be utilized to further extend the range of vibration reduction zones the results of this study have significant potential in terms of the application of sierpinski carpet periodic structures for vibration isolation

variational time integrators for finite dimensional thermo elasto dynamics without heat conduction this paper focuses on the formulation of variational integrators vis for finite dimensional thermo elastic systems without heat conduction the dynamics of these systems happen to have a hamiltonian structure after thermal displacements are introduced it is then possible to formulate integrators by taking advantage of standard methods in vi the class of integrators we construct have some remarkable features a they are symplectic b they exactly conserve the entropy of the system or in other words they exactly satisfy the second law of the thermodynamics for reversible adiabatic processes c they nearly exactly conserve the value of the energy for very long times and d they exactly conserve linear and angular momentum we first describe how to adapt any vi for the classical mechanical systems to integrate adiabatic thermo elastic ones and then formulate three new types of integrators the first class based on a generalized trapezoidal rule gives rise to two first order explicit integrators and a second order implicit one by composing then the two first order integrators we construct a second order explicit one finally we formulate a fourth order implicit integrator which is a symplectic partitioned runge kutta method the performance of these new algorithms is showcased through numerical examples copyright 2011 john wiley sons ltd

a hierarchy of rational timoshenko dispersion relations abstract this paper shows that the conventional timoshenko dispersion relation for bending waves is the 0 1 member of a two parameter family m n of approximations to the exact dispersion relation obtained from linear elasticity higher members of the family are shown to converge rapidly to the exact dispersion relation in particular any number of branches can be captured accurately over their entire length i e up to arbitrarily high frequencies and wavenumbers the theory admits a rigorous accuracy analysis and demonstrates that the conventional timoshenko dispersion relation is a completely rational approximation thus answering some previous doubts about this matter lower members of the two parameter family include the euler bernoulli dispersion relation and a rayleigh type inertia correction especially useful is the timoshenko 1 2 dispersion relation which extends the conventional timoshenko 0 1 dispersion relation by capturing the first four branches of the exact dispersion relation rather than merely the first two and is accurate over a wide range an inertia corrected timoshenko dispersion relation is also derived and is shown to include many previous approximations as special cases

equation free modelling of evolving diseases coarse grained computations with individual based models we demonstrate how direct simulation of stochastic individualbased models can be combined with continuum numericalanalysis techniques to study the dynamics of evolving diseases sidestepping the necessity of obtaining explicit populationlevel models the approach analyses the unavailable in closed form coarse macroscopic equations estimating the necessary quantities through appropriately initialized short bursts of individualbased dynamic simulation we illustrate this approach by analysing a stochastic and discrete model for the evolution of disease agents caused by point mutations within individual hosts building up from classical susceptibleinfectedrecovered and susceptibleinfectedrecoveredsusceptible models our example uses a onedimensional lattice for variant space and assumes a finite number of individuals macroscopic computational tasks enabled through this approach include stationarystate computation coarse projective integration parametric continuation and stability analysis

periodicity effects of axial waves in elastic compound rods abstract floquet analysis is applied to the bernoulli euler model for axial waves in a periodic rod explicit asymptotic formulae for the stop band borders are given and the topology of the stop band pattern is explained eigenfrequencies of the symmetric unit cell are determined by the phase closure principle and their correspondence with stop band formation is shown steady state and transient dynamics of a periodic rod of finite length are analysed numerically and the difference in structural response when excitation is done in either stop or pass bands is demonstrated a physical interpretation of the underlying mechanisms of stop bands is proposed

dimensional model reduction in non linear finite element dynamics of solids and structures a general approach to the dimensional reduction of non linear finite element models of solid dynamics is presented for the newmark implicit time discretization the computationally most expensive phase is the repeated solution of the system of linear equations for displacement increments to deal with this it is shown how the problem can be formulated in an approximation ritz basis of much smaller dimension similarly the explicit newmark algorithm can be also written in a reduced dimension basis and the computation time savings in that case follow from an increase in the stable time step length in addition the empirical eigenvectors are proposed as the basis in which to expand the incremental problem this basis achieves approximation optimality by using computational data for the response of the full model in time to construct a reduced basis which reproduces the full system in a statistical sense because of this global time viewpoint the basis need not be updated as with reduced bases computed from a linearization of the full finite element model if the dynamics of a finite element model is expressed in terms of a small number of basis vectors the asymptotic cost of the solution with the reduced model is lowered and optimal scalability of the computational algorithm with the size of the model is achieved at the same time numerical experiments indicate that by using reduced models substantial savings can be achieved even in the pre asymptotic range furthermore the algorithm parallelizes very efficiently the method we present is expected to become a useful tool in applications requiring a large number of repeated non linear solid dynamics simulations such as convergence studies design optimization and design of controllers of mechanical systems

wave propagation in continuous periodic structures research contributions from southampton 1964 1995 abstract after brief reference to some early studies by other investigators this paper focuses mainly on methods developed at the university of southampton since 1964 to analyze and predict the free and forced wave motion in continuous periodic engineering structures beginning with receptance methods which have been applied to periodic beams and rib skin structures it continues with a method of direct solution of the wave equation this uses floquet s principle and has been applied to beams and quasi one dimensional periodic plates and cylindrical shells sample curves of the propagation and attenuation constants pertaining to these structures are presented a limited discussion of the transfer matrix then follows after which the method of space harmonics is introduced as the method best suited to the prediction of sound radiated from a vibrating periodic structure reviewed next are some theorems and variational principles relating to periodic structures which have been developed at southampton and which form a basis for finding natural frequencies of finite structures or for computing free and forced wave notion by energy methods this has led to the finite element method in its standard and hierarchical forms being used to study wave motion in genuine two dimensional and three dimensional structures examples of this work are shown the method of phased array receptance functions is then introduced as possibly the easiest way of setting up exact equations for the propagation constants of uniform quasi one dimensional periodic structures a summary is finally presented of the limited and early work performed at southampton on simple disordered periodic structures

fractal model for coarse grained nonlinear partial differential equations spatially coarse grained or effective versions of nonlinear partial differential equations must be closed with a model for the unresolved small scales for systems that are known to display fractal scaling we propose a model based on synthetically generating a scale invariant field at small scales using fractal interpolation and then analytically evaluating its effects on the large resolved scales the procedure is illustrated for the forced burgers equation solved numerically on a coarse grid detailed comparisons with direct simulation of the full burgers equation and with an effective viscosity model are presented s0031 9007 96 02215 6

optimal prediction of underresolved dynamics abstract a method is presented for computing the average solution of problems that are too complicated for adequate resolution but where information about the statistics of the solution is available the method involves computing average derivatives by interpolation based on linear regression and an updating of a measure constrained by the available crude information examples are given

the forced vibration of one dimensional multi coupled periodic structures an application to finite element analysis abstract a general theory for the forced vibration of multi coupled one dimensional periodic structures is presented as a sequel to a much earlier general theory for free vibration starting from the dynamic stiffness matrix of a single multi coupled periodic element it derives matrix equations for the magnitudes of the characteristic free waves excited in the whole structure by prescribed harmonic forces and or displacements acting at a single periodic junction the semi infinite periodic system excited at its end is first analysed to provide the basis for analysing doubly infinite and finite periodic systems in each case total responses are found by considering just one periodic element an already known method of reducing the size of the computational problem is reexamined expanded and extended in detail involving reduction of the dynamic stiffness matrix of the periodic element through a wave coordinate transformation use of the theory is illustrated in a combined periodic structure finite element analysis of the forced harmonic in plane motion of a uniform flat plate excellent agreement between the computed low frequency responses and those predicted by simple engineering theories validates the detailed formulations of the paper the primary purpose of the paper is not towards a specific application but to present a systematic and coherent forced vibration theory carefully linked with the existing free wave theory

singular boundary method for wave propagation analysis in periodic structures abstract a strong form boundary collocation method the singular boundary method sbm is developed in this paper for the wave propagation analysis at low and moderate wavenumbers in periodic structures the sbm is of several advantages including mathematically simple easy to program meshless with the application of the concept of origin intensity factors in order to eliminate the singularity of the fundamental solutions and avoid the numerical evaluation of the singular integrals in the boundary element method due to the periodic behaviors of the structures the sbm coefficient matrix can be represented as a block toeplitz matrix by employing three different fast toeplitz matrix solvers the computational time and storage requirements are significantly reduced in the proposed sbm analysis to demonstrate the effectiveness of the proposed sbm formulation for wave propagation analysis in periodic structures several benchmark examples are presented and discussed the proposed sbm results are compared with the analytical solutions the reference results and the comsol software

analysis of in plane wave propagation in hexagonal and re entrant lattices plane wave propagation in hexagonal and re entrant lattices is investigated through the application of bloch s theorem two dimensional dispersion relations are estimated and analyzed to illustrate peculiar properties of re entrant configurations and investigate the directional behavior of hexagonal lattices for varying construction angles special attention is devoted to the determination of phase and group wave velocities velocity plots are obtained for varying frequency and direction of wave propagation to analyze the dispersion characteristics of the lattices and further investigate their directional characteristics

unresolved computation and optimal predictions we present methods for predicting the solution of time dependent partial differential equations when that solution is so complex that it cannot be properly resolved numerically but when prior statistical information can be found the sparse numerical data are viewed as constraints on the solution and the gist of our proposal is a set of methods for advancing the constraints in time so that regression methods can be used to reconstruct the mean future for linear equations we offer general recipes for advancing the constraints the methods are generalized to certain classes of nonlinear problems and the conditions under which strongly nonlinear problems and partial statistical information can be handled are briefly discussed our methods are related to certain data acquisition schemes in oceanography and meteorology c 1999 john wiley sons inc

a manifold learning based reduced order model for springback shape characterization and optimization in sheet metal forming abstract the parameters of a stamping process include the geometry of the tools the shape of the initial sheet blank the material constitutive law and the process parameters when designing the overall process one has to also take into account the springback effect that appears when the tools are removed and additional surfaces are cut off the goal then is to obtain a final shape as close as possible to the desired shape while satisfying the admissibility constraints on the variable parameters as well as the feasibility constraints frequently expressed in the form of forming limit diagrams in the present paper we represent the post springback shape by a level set function then rather than rely on arbitrarily selected case dependent measurement locations as in the numisheet benchmark problems we build a reduced order shape space where this level set evolves by extending our recent shape manifold approach to the problem of springback assessment for 3d shapes next we propose an optimization algorithm designed to minimize the gap between the post springback and the desired final shapes the required level set functions are generated from a corresponding set of springback shapes predicted by finite element simulations using our approach we determine the minimal number of parameters needed in order to uniquely characterize the final formed shape regardless of complexity finally we demonstrate the approach using an industrial test case springback assessment of the deep drawing operation of an automotive strut tower

local maximum entropy approximation schemes a seamless bridge between finite elements and meshfree methods we present a one parameter family of approximation schemes which we refer to as local maximum entropy approximation schemes that bridges continuously two important limits delaunay triangulation and maximum entropy max ent statistical inference local max ent approximation schemes represent a compromise in the sense of pareto optimality between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width local max ent approximation schemes are entirely defined by the node set and the domain of analysis and the shape functions are positive interpolate affine functions exactly and have a weak kronecker delta property at the boundary local max ent approximation may be regarded as a regularization or thermalization of delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation local max ent approximation schemes can be taken as a convenient basis for the numerical solution of pdes in the style of meshfree galerkin methods in test cases characterized by smooth solutions we find that the accuracy of local max ent approximation schemes is vastly superior to that of finite elements

asymptotic and numerical analyses for mechanical models of heat baths a mechanical model of a particle immersed in a heat bath is studied in which a distinguished particle interacts via linear springs with a collection of n particles with variable masses and random initial conditions the jth particle oscillates with frequency jp where p is a parameter for p 1 2 the sequence of random processes that describe the trajectory of the distinguished particle tends almost surely as n to the solution of an integro differential equation with a random driving term the mean convergence rate is 1 np 1 2 we further investigate whether the motion of the distinguished particle can be well approximated by an integration scheme the symplectic euler scheme when the product of time step h and highest frequency np is of order 1 that is when high frequencies are underresolved for 1 2 p 1 the numerical solution is found to converge to the exact solution at a reduced rate of log h h2 1 p these results shed light on existing numerical data

the response of two dimensional periodic structures to point harmonic forcing abstract this analysis is concerned with the response of an infinite two dimensional periodic structure to point harmonic loading initially a damped finite system is considered and the system size is then allowed to tend to infinity it is shown that the response in the far field can be expressed in terms of the properties of the phase constant surfaces which arise in the analysis of plane wave motion through the system within a pass band it is found that the physical nature of the response is strongly affected by the presence or otherwise of a caustic in the absence of a caustic the response amplitude has a relatively smooth spatial distribution if a caustic is present then the response amplitude has a complex spatial pattern and a dead region of very low response occurs the theory is applied to a two dimensional mass spring periodic system and a comparison is made with an exact calculation for the response of a large finite system excellent agreement is obtained between the two sets of results the work has application to all types of two dimensional periodic structures including orthogonally stiffened plates and shells and it raises the possibility of designing a periodic structure to act as a spatial filter to isolate sensitive equipment from a localized excitation source

nonlinear manifold learning for model reduction in finite elastodynamics abstract model reduction in computational mechanics is generally addressed with linear dimensionality reduction methods such as principal components analysis pca hypothesizing that in many applications of interest the essential dynamics evolve on a nonlinear manifold we explore here reduced order modeling based on nonlinear dimensionality reduction methods such methods are gaining popularity in diverse fields of science and technology such as machine perception or molecular simulation we consider finite deformation elastodynamics as a model problem and identify the manifold where the dynamics essentially take place the slow manifold by nonlinear dimensionality reduction methods applied to a database of snapshots contrary to linear dimensionality reduction the smooth parametrization of the slow manifold needs special techniques and we use local maximum entropy approximants we then formulate the lagrangian mechanics on these data based generalized coordinates and develop variational time integrators our proof of concept example shows that a few nonlinear collective variables provide similar accuracy to tens of pca modes suggesting that the proposed method may be very attractive in control or optimization applications furthermore the reduced number of variables brings insight into the mechanics of the system under scrutiny our simulations also highlight the need of modeling the net effect of the disregarded degrees of freedom on the reduced dynamics at long times

on the optimum support size in meshfree methods a variational adaptivity approach with maximum entropy approximants we present a method for the automatic adaption of the support size of meshfree basis functions in the context of the numerical approximation of boundary value problems stemming from a minimum principle the method is based on a variational approach and the central idea is that the variational principle selects both the discretized physical fields and the discretization parameters here those defining the support size of each basis function we consider local maximum entropy approximation schemes which exhibit smooth basis functions with respect to both space and the discretization parameters the node location and the locality parameters we illustrate by the poisson linear and non linear elasticity problems the effectivity of the method which produces very accurate solutions with very coarse discretizations and finds unexpected patterns of the support size of the shape functions copyright 2009 john wiley sons ltd

equivalences of twisted k3 surfaces we prove that two derived equivalent twisted k3 surfaces have isomorphic periods the converse is shown for k3 surfaces with large picard number it is also shown that all possible twisted derived equivalences between arbitrary twisted k3 surfaces form a subgroup of the group of all orthogonal transformations of the cohomology of a k3 surface

nonlinear model reduction via discrete empirical interpolation a dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition pod method for constructing reduced order models for time dependent and or parametrized nonlinear partial differential equations pdes in the presence of a general nonlinearity the standard pod galerkin technique reduces dimension in the sense that far fewer variables are present but the complexity of evaluating the nonlinear term remains that of the original problem the original empirical interpolation method eim is a modification of pod that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by pod we propose a discrete empirical interpolation method deim a variant that is suitable for reducing the dimension of systems of ordinary differential equations odes of a certain type as presented here it is applicable to odes arising from finite difference discretization of time dependent pdes and or parametrically dependent steady state problems however the approach extends to arbitrary systems of nonlinear odes with minor modification our contribution is a greatly simplified description of the eim in a finite dimensional setting that possesses an error bound on the quality of approximation an application of deim to a finite difference discretization of the one dimensional fitzhugh nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long time integration that fully captures nonlinear limit cycle behavior we also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results

maximum entropy methods for time harmonic acoustics abstract this paper explores the application of maximum entropy methods max ent to time harmonic acoustic problems max ent basis functions are mesh free approximants that are constructed observing an equivalence between basis functions and discrete probability distributions and applying jaynes s maximum entropy principle they are c continuous and therefore they are particularly suited for the resolution of helmholtz problems where classical finite element methods show a poor accuracy in the high frequency region in addition it was recently shown that max ent approximants can be blended with isogeometric basis functions on the boundary of the domain this preserves the correct representation of the boundary like in isogeometric analysis with the advantage that the discretization of the interior of the domain is straightforward in this paper the max ent mathematical formulation is reviewed and then some numerical applications are studied including a 2d car cavity geometry defined by b spline curves in all cases if the same nodal discretization is used finite elements results are significantly improved

moduli spaces of twisted k3 surfaces and cubic fourfolds motivated by the relation between twisted k3 surfaces and special cubic fourfolds we construct moduli spaces of polarized twisted k3 surfaces of any fixed degree and order we do this by mimicking the construction of the moduli space of untwisted polarized k3 surfaces as a quotient of a bounded symmetric domain

a hierarchy of high order theories for symmetric modes in an elastic layer abstract a hierarchy of high order theories for symmetric modes in an infinitely long elastic layer of the constant thickness is derived by means of the inertia corrected polynomial approximations for each member of the hierarchy boundary conditions for layers of the finite length are formulated the forcing problems at several approximation levels are solved with the use of the bi orthogonality conditions accuracy of these approximations is assessed by comparison of results with the exact solution of the rayleigh lamb problem special attention is paid to the power flow analysis in alternative modal excitation cases and to the applicability of the saint venant s principle in stationary elasto dynamics

variational time integrators the purpose of this paper is to review and further develop the subject of variational integration algorithms as it applies to mechanical systems of engineering interest in particular the conservation properties of both synchronous and asynchronous variational integrators avis are discussed in detail we present selected numerical examples which demonstrate the excellent accuracy conservation and convergence characteristics of avis in these tests avis are found to result in substantial speed ups at equal accuracy relative to explicit newmark a mathematical proof of convergence of the avis is also presented in this paper finally we develop the subject of horizontal variations and configurational forces in discrete dynamics this theory leads to exact path independent characterizations of the configurational forces acting on discrete systems notable examples are the configurational forces acting on material nodes in a finite element discretisation and the j integral at the tip of a crack in a finite element mesh

some remarks on brauer groups of k3 surfaces abstract we discuss the geometry of the genus one fibrations associated to an elliptic fibration on a k3 surface we show that the two torsion subgroup of the brauer group of a general elliptic fibration is naturally isomorphic to the two torsion of the jacobian of a curve associated to the fibration we remark that this is related to recillas trigonal construction finally we discuss the two torsion in the brauer group of a general k3 surface with a polarization of degree two

error analysis of coarse graining for stochastic lattice dynamics the coarse grained monte carlo cgmc algorithm was originally proposed in the series of works m a katsoulakis a j majda and d g vlachos j comput phys 186 2003 pp 250 278 m a katsoulakis a j majda and d g vlachos proc natl acad sci usa 100 2003 pp 782 787 m a katsoulakis and d g vlachos j chem phys 119 2003 pp 9412 9427 in this paper we further investigate the approximation properties of the coarse graining procedure and provide both analytical and numerical evidence that the hierarchy of the coarse models is built in a systematic way that allows for error control in both transient and long time simulations we demonstrate that the numerical accuracy of the cgmc algorithm as an approximation of stochastic lattice spin flip dynamics is of order two in terms of the coarse graining ratio and that the natural small parameter is the coarse graining ratio over the range of particle particle interactions the error estimate is shown to hold in the weak convergence sense we employ the derived analytical results to guide cgmc algorithms and demonstrate a cpu speed up in demanding computational regimes that involve nucleation phase transitions and metastability

probability that a band gap extremum is located on the irreducible brillouin zone contour for the 17 different plane crystallographic lattices abstract after recalling the first brillouin zone bz and the first irreducible brillouin zone ibz of a lattice in terms of its plane crystallographic group we investigate the danger of restricting a band gap detection to the contour of the ibz instead of its full ibz based on hundreds of porous phononic crystal simulations we provide for the 17 plane crystallographic groups i statistics of the band gap localizations ii probabilities to get non full band gaps and iii averages of the bandwidth error made when only the ibz contour is considered it is found that for phononic crystals the ibz contour provides accurate results only for highly symmetric lattices

coarse graining molecular dynamics models using an extended galerkin projection method summary we present a new framework for coarse graining molecular dynamics models for crystalline solids the reduction method is based on a galerkin projection to a subspace whose dimension is much smaller than that of the full atomistic model to effectively reduce artificial reflections of phonons at the interface we construct extended subspaces with increasing accuracy by adding more coarse grained variables near the interface between lattice defects and the surrounding region this approach is equivalent to the generalized langevin model but it eliminates the need to precompute the memory function a well known practical difficulty further the variational formulation preserves the stability of the molecular models copyright 2014 john wiley sons ltd