Begell House Inc.
International Journal for Multiscale Computational Engineering
JMC
1543-1649
4
5-6
2006
Asymptotic Approximation of the Solution to the Robin Problem in a Thick Multistructure
545-558
C.
D'Apice
Department of Information Engineering and Applied Mathematics University of Salerno, via Ponte Don Melillo, 84084 Fisciano (SA), Italia
Taras A.
Mel'nik
Faculty of Mathematics & Mechanics, Taras Shevchenko University of Kyiv, Volodymyrska str. 64, 01033 Kyiv, Ukraine
U.
De Maio
Department of Applied Mathematics "R. Caccioppoli", Federico II University of Naples, Complesso Monte S. Angelo - Edificio "T", Via Cintia, 80126 Naples, Italy
We consider a mixed boundary value problem for the Poisson equation in a thick multistructure Ωε, which is the union of a domain Ω0 and a large number N of ε-periodically situated thin rings with variable thickness of order ε = O(N−1). The Robin conditions are given on the lateral boundaries of the thin rings. The leading terms of the asymptotic expansion for the solution are constructed and the corresponding estimates in the Sobolev space H1 (Ωε) are proved (as ε → 0) with minimal conditions for the right-hand side.
Consistent Loading in Structural Reduction Procedures for Beam Models
559-584
Slava
Krylov
School of Mechanical Engineering, Tel Aviv University, Israel
Isaac
Harari
Tel-Aviv University
D.
Gadasi
Department of Solid Mechanics Materials and Systems, Tel Aviv University, Ramat Aviv 69978, Tel Aviv, Israel
In multiphysics problems, a solid body is in interaction with various three-dimensional fields that generate complex patterns of rapidly varying distributed loading on the solid. Since three-dimensional computation requires excessive resources, methods of reduction to structural models are traditionally exploited in mechanics for the analysis of slender bodies. Although such procedures are well established, the reduction of loads is often performed in an ad hoc manner, which is not sufficient for many coupled problems. In the present work, we develop rigorous structural reduction (SR) procedures by using a variational framework to consistently convert three-dimensional data to the form required by structural representations. The approach is illustrated using the Euler-Bernoulli and Timoshenko beam theories. Some of the loading terms and boundary conditions of the four resulting structural problems (namely, tension, torsion, and two bending problems), which are formulated in terms of the original three-dimensional problem, could not be derived by ad hoc considerations. Numerical results show that the use of the SR procedures greatly economizes computation and provides insight into the mechanical behavior while preserving a level of accuracy comparable with the fully three-dimensional solution.
A Prototype Homogenization Model for Acoustics of Granular Materials
585-600
Xuming
Xie
Department of Mathematics, Morgan State University, Baltimore, MD, USA
Robert P
Gilbert
Alexander
Panchenko
Department of Mathematics, Washington State University, Pullman, WA 99164, USA
This paper introduces a homogenization approach to modeling acoustic vibrations of composite materials with internal friction. The model medium studied in the paper consists of a consolidated viscoelastic solid matrix with a large number of periodically arranged pores containing rigid solid particles. The particles are in frictional contact with the matrix. At the length scale of particles, the frictional forces are modeled initially by the Coulomb's law with normal compliance. These inequality-type conditions are approximated by nonlinear equations. The resulting microscale problem is averaged using formal two-scale homogenization. The effective acoustic equations are, in general, nonlinear and history dependent, and contain both effective stress and the effective drag force. The constitutive equations for the effective quantities are obtained explicitly for three different approximate models of contact conditions.
Discrete Bubble Modeling of Unsteady Cavitating Flow
601-616
Zhiliang
Xu
Computational Science Center, Brookhaven National Laboratory, Upton, NY 11973-5000
Myoungnyoun
Kim
Computational Science Center, Brookhaven National Laboratory, Upton, NY 11973-5000
Tianshi
Lu
Computational Science Center, Brookhaven National Laboratory, Upton, NY 11973-5000, USA
Wonho
Oh
Department of Applied Mathematics and Statistics, University at Stony Brook, Stony Brook, NY 11794-3600
James
Glimm
Computational Science Center, Brookhaven National Laboratory, Upton, NY 11973; and Department of Applied Mathematics and Statistics, SUNY at Stony Brook, Stony Brook, NY 11794, USA
Roman
Samulyak
Computational Science Center, Brookhaven National Laboratory, Upton, NY 11973, USA
Xiaolin
Li
Department of Applied Mathematics and Statistics, University at Stony Brook, Stony Brook, NY 11794-3600
Constantine
Tzanos
Department of Nuclear Engineering, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL 60439, USA
A discrete vapor bubble model is developed to simulate unsteady cavitating flows. In this model, the mixed vapor-liquid mixture is modeled as a system of pure phase domains (vapor and liquid) separated by free interfaces. On the phase boundary, a numerical solution for the phase transition is developed for compressible flows. This model is used to study the effect of cavitation bubbles on atomization, i.e., the breakup of a high-speed jet and spray formation. The major conclusion is that a multiscale (three-scale) model is sufficient to achieve agreement with quantitative macroscale flow parameters, such as spray opening angle and spray volume fraction or density, or as a qualitative measure, the occurrence of spray formation. The authors believe this to be the first numerical study of the atomization process at such a level of detail in modeling of the related physics.
The Global-Regional Model Interaction Problem: Analysis of Carpenter's Scheme and Related Issues
617-646
Assaf
Mar-Or
Inter-Departmental Program for Applied Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel
Dan
Givoli
Department of Aerospace Engineering, Technion−Israel Institute of Technology, Haifa 32000, Israel; Faculty of Civil Engineering & Geosciences, Technical University of Delft, 2600 GA Delft, The Netherlands
The multiscale global-regional model interaction problem for linear time-dependent waves is considered. The setup, which is sometimes called "nesting," arises in numerical weather prediction as well as in other fields concerning waves in very large domains. It involves the interaction of a crude global model and a fine limited-area (regional) model through an "open boundary." The multiscale nature of this general problem is described. A fundamental difficulty related to spurious modes, which prevents a trivial treatment of the problem, is discussed. The Carpenter scheme, originally proposed in a Note by K. M. Carpenter (Q. J. R. Met. Soc. 108:717−719,1982) for this type of problem, is then revisited in the context of the linear scalar wave equation. This scheme is analyzed here in the one-dimensional case. It is shown that the accuracy of the scheme hinges mainly on the numerical dispersion generated by the global model. Extension of the analysis to two dimensions is also discussed. Numerical experiments are presented for the Carpenter scheme in one dimension via some example problems, and conclusions are drawn about its performance. Ways of improving the scheme are indicated.
Error Control for Molecular Statics Problems
647-662
Serge
Prudhomme
The University of Texas at Austin
Paul T.
Bauman
Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712
J. Tinsley
Oden
Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712, USA
In this paper, we present an extension of goal-oriented error estimation and adaptation to the simulation of multiscale problems of molecular statics. Computable error estimates for the quasicontinuum method are developed with respect to specific quantities of interest, and an adaptive strategy based on these estimates is proposed for error control. The theoretical results are illustrated on a nanoindentation problem in which the quantity of interest is the force acting on the indenter. The promising capability of such error estimates and adaptive procedure for the solution of multiscale problems is demonstrated on numerical examples.
Microstructure-Based Multiscale Constitutive Modeling of γ — γ′ Nickel-Base Superalloys
663-692
A.-J.
Wang
Onity, Inc., Norcross, GA
R. S.
Kumar
ABAQUS, Inc., Providence, RI
M. M.
Shenoy
G.W.W. School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405
David L.
McDowell
School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA; GWW School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
A hierarchical system of microstructure-based constitutive models for cyclic deformation and low cycle fatigue (LCF) behavior of two-phase nickel-base superalloys is developed and implemented. Both precipitate (γ′) and matrix (γ) phases with or without smaller dispersed precipitates are explicitly modeled using crystal viscoplasticity theory with dislocation density as an internal state variable. The constitutive models are capable of capturing most of the important features of the deformation behavior of Ni-base superalloys, namely, (i) anomalous yield stress behaviors with respect to temperature and tension-compression asymmetry of flow stress due to non-Schmid effects; (ii) crystallographic orientation dependence represented by the crystal plasticity model; (iii) effects of γ′ precipitate size and spacing on initial yield strength and work hardening; and (iv) effects of precipitate distribution and morphology on localized cyclic plastic shear strain. The physically based hardening laws are employed to evolve dislocation densities for both phases in each slip system with consideration of dislocation interaction mechanisms. This type of' microstructure-sensitive constitutive model is applicable to the study of the effects of variability of microstructure on variability of LCF and creep behaviors or properties. A combined bottom-up and top-down strategy is used to determine model parameters to support simulations at length scales ranging from approximately 100 nm up to hundreds of microns. Models for γ′ precipitates are calibrated using bulk Ni3Al single-crystal data. The γ — γ′ two-phase models are calibrated for several superalloys and strain histories, providing good agreement with experiments. The utility of such models in modeling behavior at several characteristic length scales is briefly discussed.
Computational Homogenization of Nonlinear Hydromechanical Coupling in Poroplasticity
693-732
Fernando A.
Rochinha
Mechanical Engineering Department, Federal University of Rio de Janeiro, Brazil
Marcio A.
Murad
Laboratório Nacional de Computação Cientifica LNCC/MCT, Av. Getúlio Vargas 333,25651-070, Petrópolis, RJ, Brazil
Jesus A.
Luizar-Obregon
Universidade Federal Fluminense, Rua Tiradentes 17 Ingá, 24.210-510, Niterói, RJ, Brazil
In this paper, we propose a new two-scale model of fluid-saturated elastoplastic porous media based on micromechanical considerations. A formal nonlinear homogenization procedure using asymptotic expansion techniques is adopted to up-scale the microscopic constitutive behavior of an elastoplastic solid coupled with the movement of a Stokesian fluid. Considering the yield criterion at the microscale governed by the Mohr-Coulomb function and that the plastic deformation obeys the principle of maximum dissipation, we build up, computationally, a sharper macroscopic yield criterion and provide precise two-scale computations for the effective parameters of the homogenized medium. Within this context, we show that the homogenized results incorporate additional features inherent to the nonlinear hydromechanical coupling that have been overlooked by the purely macroscopic approaches. Variational principles along with the corresponding Galerkin approximations are proposed to discretize the local nonlinear closure problems leading to numerical effective constitutive laws. The influence of the new constitutive features obtained at the Darcy-scale effective model is propagated to the field-scale and illustrated numerically in a example of land subsidence caused by oil extraction of a weak heterogeneous reservoir with hydraulic conductivity characterized by long-range correlations displaying fractal character.
Statistical Properties of Local Residual Microstresses in Elastically Homogeneous Composite Half-Space
733-754
Valeriy A.
Buryachenko
Civil Engineering Department, University of Akron, Akron, Ohio 44325-3901, USA and Micromechanics and Composites LLC, 2520 Hingham Lane, Dayton, Ohio 45459, USA
V. I.
Kushch
Institute for Superhard Materials of the National Academy of Sciences, 04074 Kiev, Ukraine
We consider a linear elastic homogeneous composite half-space, which consists of a homogeneous matrix containing a random array of inclusions. The elastic properties of the matrix and the inclusions are the same, but the stress-free strains are different. A method of integral equations is proposed for the estimation of the first and second moments of residual microstresses in the constituents of elastically homogeneous composites in a half-space with a free edge. Explicit relations for these statistical moments are obtained using a modified superposition technique and taking the binary interactions of the inclusions into account, which is expressed through the numerical solution for one inclusion in the half-space. The statistical averages of stress fluctuations varying along the inclusion cross sections are completely defined by the random locations of surrounding inclusions. The numerical results are presented for a half-plane containing random distribution of circular identical inclusions. The solution for one inclusion in the half-plane is obtained by the method of complex potential.
Wavelet-based Spatiotemporal Multiscaling in Diffusion Problems with Chemically Reactive Boundary
755-770
George
Frantziskonis
Department of Civil Engineering and Engineering Mechanics, and Department of Material Science and Engineering, University of Arizona, USA
Sudib Kumar
Mishra
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
C. Stuart
Daw
Computer Science & Mathematics Division, Oak Ridge National Laboratory
Phani
Nukala
Computer Science & Mathematics Division, Oak Ridge National Laboratory
Rodney O.
Fox
Department of Chemical and Biological Engineering
Iowa State University Ames, IA USA
Pierre A.
Deymier
Department of Material Science and Engineering, University of Arizona, Tucson, Arizona 85721, USA
Chemically reacting flows over catalytic and noncatalytic surfaces are one of the elementary operations in chemical processing plants. The underlying physical phenomena span time and length scales over several orders of magnitude, which a robust and flexible modeling framework must efficiently account for. With this purpose as the eventual goal, we propose a wavelet-based multiscale numerical framework and demonstrate it on the coupling of two prototype methods for the problem of species generated on a chemically reactive boundary and diffusing through the bulk. The two methods consider different time and length scales. The first method in this coupling, termed "fine," models the chemical reactions on the reactive boundary stochastically by the kinetic Monte Carlo method and the diffusion in the medium deterministically using relatively small time increments and small spatial discretization mesh size. The second method, termed "coarse," models both the reaction and the diffusion deterministically and uses drastically larger time increments and spatial discretization size than the fine model. The two methods are coupled by forming a spatiotemporal compound wavelet matrix that combines information about the time and spatial scales contained in them.
On the Implementation of Plane Stress in Computational Multiscale Modeling
771-790
Robert
Lillbacka
FS Dynamics, Molndalsvagen 24, SE-412 63 Goteborg; Chalmers University of Technology, Department of Applied Mechanics, SE-412 96 Göteborg; and Swedish National Testing and Research Institute (SP), Brinellgatan 4, Box 857, SE-50115 Borås, Sweden
Fredrik
Larsson
Department of Applied Mechanics, Chalmers University of Technology, S-412 96 Gothenburg
Kenneth
Runesson
Chalmers University of Technology
Different aspects of the plane stress condition in concurrent two-scale computational (first-order) homogenization are discussed. The basic ingredient in computational homogenization is the calculation of the macroscale stress, for given macroscale deformation, via computations on a representative volume element (RVE). Two modeling assumptions are compared: The subscale (Hill-type) and macroscale-type (Taylor-type) plane stress conditions. The corresponding iterative strategies and the macroscale algorithmic tangent operators are derived using the primal (conventional) approach. The performance of the various iterative strategies are compared for a single RVE problem as well as in a fully concurrent analysis of a complex substructure (duplex stainless steel) under realistic subscale modeling based on crystal plasticity with hardening.
A Space-Time Multiscale Method for Molecular Dynamics Simulations of Biomolecules
791-802
Aiqin
Li
Scientific Computational Research Center, Rensselaer Polytechnic Institute, Troy, NY 12180, USA
Haim
Waisman
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, New York 10027, USA
Jacob
Fish
Civil Engineering and Engineering Mechanics, Columbia University, New York, New York
10027, USA
A novel multiscale approach for molecular-dynamics simulations is developed. The goal of this method is to reduce the time cost of molecular-dynamics simulations without loss of accuracy in the quantities of interest. The proposed approach consists of the waveform relaxation scheme aimed at capturing the high-frequency motions and a coarse-scale solution in space and time aimed at resolving smooth features (in both space and time domains) of the system. The use of proper orthogonal decomposition (POD) modes at the coarse-grained level has been found to accelerate convergence of the waveform relaxation scheme. The accuracy and efficiency of this method are reported by applying it to a model problem of chain of α-D-glucopyranose monomers.
INDEX to Volume 4
803-809