Begell House Inc.
International Journal for Multiscale Computational Engineering
JMC
1543-1649
6
4
2008
Wavelet-Based Spatial Scaling of Coupled Reaction-Diffusion Fields
281-297
Sudib Kumar
Mishra
Krishna
Muralidharan
Department of Material Science and Engineering, University of Arizona, Tucson, Arizona 85721, USA
Pierre A.
Deymier
Department of Material Science and Engineering, University of Arizona, Tucson, Arizona 85721, USA
George
Frantziskonis
Department of Civil Engineering and Engineering Mechanics, and Department of Material Science and Engineering, University of Arizona, USA
Sreekanth
Pannala
Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA
Srdjan
Simunovic
Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA
Multiscale schemes for transferring information from fine to coarse scales are typically based on homogenization techniques. Such schemes smooth the fine scale features of the underlying fields, often resulting in the inability to accurately retain the fine scale correlations. In addition, higher-order statistical moments (beyond mean) of the relevant field variables are not necessarily preserved. As a superior alternative to averaging homogenization methods, a wavelet-based scheme for the exchange of information between a reactive and diffusive field in the context of multiscale reaction-diffusion problems is proposed and analyzed. The scheme is shown to be efficient in passing information along scales, from fine to coarse, i.e., upscaling as well as from coarse to fine, i.e., downscaling. It incorporates fine scale statistics (higher-order moments beyond mean), mainly due to the capability of wavelets to represent fields hierarchically. Critical to the success of the scheme is the identification of dominant scales containing the majority of the useful information. The dominant scales in effect specify the coarsest resolution possible. The scheme is applied in detail to the analysis of a diffusive system with a chemically reacting boundary. Reactions are simulated using kinetic Monte Carlo (kMC) and diffusion is solved by finite differences (FDs). Spatial scale differences are present at the interface of the kMC sites and the diffusion grid. The computational efficiency of the scheme is compared to results obtained by averaging homogenization, and to results from a benchmark scheme that ensures spatial scale parity between kMC and FD.
Analytical Estimates of the Subgrid Model for Burgers Equation: Ramifications for Spectral Methods for Conservation Laws
299-307
Assad A.
Oberai
Department of Mechanical Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, USA
Carlos E.
Colosqui
Department of Aerospace and Mechanical Engineering, Boston University, Boston, MA 02215, USA
John
Wanderer
Department of Aerospace and Mechanical Engineering, Boston University, Boston, MA 02215, USA
In several numerical methods used for simulating an inviscid, or nearly inviscid, nonlinear conservation law, a wavenumber-dependent viscosity is often employed as a subgrid model. In particular, in the spectral vanishing viscosity and hyperviscosity methods, the viscosity at low wavenumbers is set to zero. In this note, we verify whether this choice is consistent with the wavenumber dependence of the energy transfer to the subgrid scales. We evaluate this transfer for different choices of a desired numerical solution that are made precise by the choice of a restriction operator. We discover that, for the simple model system of Burgers equation, the exact subgrid viscosity is nonzero at low wavenumbers and, hence, the spectral vanishing viscosity and hyperviscosity methods are at odds with the exact subgrid model. We also observe that the exact subgrid viscosity is well described by a nonzero plateau at low wavenumbers, a cusp at the high wavenumbers, and is remarkably similar to the wavenumber-dependent viscosity observed in three-dimensional turbulence. We attribute this similarity to the locality of energy transfer in wavenumber space in both of these systems.
Scaling Up of an Underground Waste Disposal Model with Random Source Terms
309-325
Farid
Smai
Université de Lyon, CNRS, Institut Camille Jordan, Batiment ISTIL, France
Olivier
Gipouloux
Université de Lyon, CNRS, Institut Camille Jordan, Batiment ISTIL; and Département de Mathématiques, Faculté des Sciences et Techniques, Université de Saint Etienne, France
In this paper, we study the global behavior of an underground waste disposal in order to obtain an accurate upscaled model suitable for the computations involved in safety assessment processes. The disposal is made of a high number of modules containing the waste material. One supposes that the modules or disposal units are periodicaly distributed and that each disposal unit leaks randomly. Recent theoretical works, by use of a combination of classical homogenization and probability theories, allow to derive a global model where only one deterministic leaking source is considered. We propose, using these theoretical results, to give a numerical approach for a such multiscale problem to obtain solutions of the deterministic problem and the first-order statistical moments.
Multiple Time Scale Modeling of Stick-Slip Dynamics of Atomistic Systems
327-338
Erman
Guleryuz
School of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99164-2920, USA
Sergey N.
Medyanik
School of Mechanical and Materials Engineering, Washington State University, USA
Temporal evolution of an atomistic system may often be classified as a so-called stick-slip process. Such a process is characterized by the presence of two distinct phases—a slow phase when the system's configuration is evolving at a relatively slow pace and a fast phase, when some sudden dramatic changes occur. In this case, there are two disparate time scales involved: the fine scale associated with the fast phase and the coarse scale associated with the slow phase. When the system's evolution happens in a stick-slip manner, atomistic modeling techniques may be developed to take advantage of this multiscale nature of the system's dynamics. Thus, the slow phase may be effectively modeled using a quasi-static energy-minimization procedure, while the fast phase can be modeled dynamically. Recently, one such method was proposed [Medyanik, S. N., and Liu, W. K., Multiple time-scale method for atomistic simulations. Comp. Mech. 2008.] that explores the idea of a sequential coupling between static and dynamic formulations for an idealized one-dimensional model. In the current work, the idea is further developed and validated by applying the method to an actual atomistic system. This has allowed for estimation of the potential CPU time savings due to the method. In addition, the influence of the loading rate on the qualitative behavior of an atomistic system has been explored and the importance of modeling realistic loading rates has been identified. This further justifies the practical importance of the new method that may help to model more realistic loading velocities and strain rates and thus capture the correct physics. Computational savings for a range of loading velocities are reported, and future prospects of the method's development and applications are outlined.
Analytical and Numerical Study of the Size Effect on the Failure Response of Hierarchical Structures
339-348
J. F.
Labuz
Department of Civil Engineering, University of Minnesota, Minneapolis, MN 55455-0220, USA
Yong
Gan
Department of Civil and Environmental Engineering, University of Missouri-Columbia, USA
Zhen
Chen
Department of Civil & Environmental Engineering, University of Missouri, USA; Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, P. R. China
Recent interests in developing bio-inspired structures necessitate a better understanding of the onset and evolution of failure in hierarchical structures. With the use of different constitutive models, analytical and numerical studies are performed to explore the size effect on the structural failure response of bar members in a parallel arrangement. It is found from the analytical and numerical results that the shorter the member, the more likely for stable softening to occur if there is only one member in the system, and that the larger the member number in the system, the more energy dissipation is associated with the postlimit regime.
Role of Dispersion and Optical Phonons in a Lattice-Boltzmann Finite-Difference Model for Nanoscale Thermal Conduction
349-360
Pekka
Heino
Tampere University of Technology, Department of Electronics
A recently presented multiscale model for nanoscale thermal conduction is further developed to include dispersion and optical phonons in the nanoscale. Optical phonons are included as nonmoving heat reservoirs, and dispersion of the acoustic phonons is included by dividing the acoustic phonons into four categories with different group velocities. In the model, only the optical phonons are heated and the energy transfer rate from optical to acoustic phonons is described with a relaxation time. As a test case, a nanoscale hot spot is introduced into a silicon system and thermal conduction to ambient is calculated. The results show a temperature step at the spot boundary, while elsewhere the results agree with thermal diffusion. Optical phonons and dispersion are seen to increase the spot boundary thermal resistance. Moreover, heat transfer near hot spots is seen to be dominated by high heat capacity phonons, while further away from the spot heat conduction is dominated by fast phonons.
Homogenization of Fiber-Reinforced Composites under the Stochastic Aging Process
361-370
Marcin
Kaminski
Faculty of Civil Engineering, Architecture and Environmental Engineering, Technical University of Lodz, Poland
The main aim of this work is to develop a homogenization method for the periodic fiber-reinforced composite materials whose components are subjected to the stochastic aging processes. This homogenization method is entirely based on the effective modules method and the stochastic aging processes are modeled as the linear decay of the Young's moduli for the fiber and the matrix from their initial expectations and standard deviations using the aging velocity defined as the random variable; no stochastic structural or interface defects are considered. Both initial values of those parameters as well as their aging velocities are assumed to be the Gaussian random variables with the specified first two probabilistic moments. Computational implementation of the stochastic composite model is provided using the Monte Carlo simulation method implemented on the homogenization-oriented finite element method code MCCEFF, where the stochastic behavior of the homogenized tensor probabilistic moments are modeled in certain time moments through the aging history. The specific case of glass fibers embedded into epoxy resin shows that the governing influence on this composite effective characteristics has the aging process in the matrix; the fiber aging can be of secondary importance depending only on the particular data.
Adaptive Bridging of Scales in Continuum Modeling Based on Error Control
371-392
Kenneth
Runesson
Chalmers University of Technology
Fredrik
Larsson
Department of Applied Mechanics, Chalmers University of Technology, S-412 96 Gothenburg
The common approach of spatial homogenization for resolving strong material heterogeneity is based on complete scale separation. The other extreme approach is to completely resolve the fine scale(s) in the macroscale computation. In this paper, we propose a novel algorithm for scaletransition such that the two extremes presented above are bridged in a "seamless" fashion. An important ingredient is a generalized macrohomogeneity condition. As part of the algorithm, the approach to the subscale modeling is chosen adaptively based on the relation of the macroscale mesh diameter to the typical length scale of the subscale structure. Moreover, the macroscale mesh adaptivity is driven by an estimation of discretization errors, which is an absolutely essential feature. Numerical examples, although quite simple, illustrate the principle and the effectivity of the adaptive procedure.