Wavelet-based spatial and temporal multiscaling: Bridging the atomistic and continuum space and time scales G. Frantziskonis 1, * and P. Deymier 2 1 Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, Arizona 85721, USA 2 Department of Materials Science and Engineering, University of Arizona, Tucson, Arizona 85721, USA ~Received 21 January 2003; revised manuscript received 4 April 2003; published 25 July 2003! A wavelet-based multiscale methodology is presented that naturally addresses time scaling in addition to spatial scaling. The method combines recently developed atomistic-continuum models and wavelet analysis. An atomistic one-dimensional harmonic crystal is coupled to a one-dimensional continuum. The methodology is illustrated through analysis of the dispersion relation, which is highly dispersive at small spatial scales and, as usual, nondispersive at large ~continuum! scales. It is feasible to obtain the complete dispersion relation through the combination of the atomistic and the continuum analyses. Wavelet analysis in this work is not only used for bridging the atomistic and continuum scales but also for efficiently extracting the dispersion relation from the solution of wave propagation problems. DOI: 10.1103/PhysRevB.68.024105 PACS number~s!: 71.15.Pd, 31.15.Qg, 46.40.Cd I. INTRODUCTION Multiscaling has recently received increasing attention in several branches of physical science. In materials, a large part of the work is devoted to modern simulation methods involving coupling of length scales and sometimes time scales. Simulation methods for coupling length scales can be characterized as either serial or concurrent. In serial methods a set of calculations at a fundamental level ~small length scale! is used to evaluate parameters for use in a phenom- enological model at a longer length scale. For example, ato- mistic simulations can be used to deduce constitutive behav- ior of finite elements, which are then used to simulate larger- scale problems. 1 Several research groups are presently working productively on such methods, and several applica- tions can be found. 2–5 Concurrent methods rely on coupling seamlessly different computational methodologies applied to different regions of a material. For example, crack propagation is a problem that was tackled early on by multiscale methods. 1,6 Atomic simu- lation techniques ~molecular dynamics! were used to model the crack tip where large deformations ~even bond breakage! occur and continuum approaches @finite-element ~FE! meth- ods# were used to model the region far away from the crack tip. Time scaling is of fundamental importance for many physical processes including diffusion or dynamics of mac- romolecular systems. These are systems with relaxation pro- cesses with vastly different scales ~e.g., bond vibration ver- sus macromolecule conformational change!. Standard atomistic simulation methods are constrained by the shortest of these time scales. Surface diffusion has been addressed rigorously by Voter and co-workers ~review paper 7 !. 8,9 Here, based on transition-state theory, state-to-state transitions are obtained by several methodologies ~accelerated dynamics methods such as hyperdynamics, parallel replica dynamics, temperature-accelerated dynamics!, enabling the simulation of diffusion over extended time intervals. In Sec. V we briefly discuss possible connections of these works to the one reported herein. We also note attempts to use wavelet analy- sis for detecting transitions, metastable structures, and for compressing data in such time-scaling works 10 and for mo- lecular dynamics simulations of polymer chains. 11 A novel multiscale method based on wavelets has been examined, up to now addressing spatial scales. 12–15 Here, the inherent capabilities of wavelet analysis to represent objects in a multiscale fashion are fully taken advantage of. The wavelet-based approach establishes a bridge between phe- nomena at different scales. Let us explain the process with respect to a ‘‘simple’’ material-related problem, i.e., consider a material for which porosity is the ~only! source of hetero- geneity and it manifests itself differently at various scales. For illustration purposes, consider a two-scale heterogeneity; at a scale large enough only the spatial distribution of ‘‘large’’ pores is observable, while at small scales that of the ‘‘small’’ pores can be seen. Even if the structure of the pores at these two distinct cases is fully specified or observed, it is difficult to compare the role of the porosity at each of the two scales on overall material properties. In other words, for, say, mechanical or electrical breakdown ~failure!, based solely on the spatial distribution of pores, it is difficult to decide whether the large or the small pores are of most im- portance. Furthermore, when the spatial scales extend be- yond two orders of magnitude, numerical simulations be- come, in general, impossible ~with present computers!. In order to identify the role of microstructure ~two-scale porosity, extendable to more general cases! at each scale, consider the statistics of the wavelet transform of, say, the strain field for a deformation problem at one of the two scales. This information forms part of the wavelet transform of the entire medium, i.e., that consisting of pores at both distinct scales. By obtaining the wavelet transform at both scales and compounding them together, we have the statistics of the wavelet transform of the medium at all scales. This ‘‘compound’’ wavelet transform contains information on the entire medium, which can be used, for example, for crack initiation studies; it accounts for full interaction between scales. This process is used for identification of dominant scales 13 and is extended to a medium with pores and inclu- PHYSICAL REVIEW B 68, 024105 ~2003! 0163-1829/2003/68~2!/024105~8!/$20.00 ©2003 The American Physical Society 68 024105-1