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Multiscale Modeling & Simulation

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2007

Volume 6, Issue 4, pp. 1059-1381


Finite Element Heterogeneous Multiscale Methods with Near Optimal Computational Complexity

Assyr Abdulle and Bjorn Engquist

Multiscale Model. Simul. 6, pp. 1059-1084 (26 pages) | Cited 3 times

Online Publication Date: December 21, 2007

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This paper is concerned with a numerical method for multiscale elliptic problems. Using the framework of the heterogeneous multiscale methods (HMM), we propose a micro-macro approach which combines the finite element method (FEM) for the macroscopic solver and the pseudospectral method for the microsolver. Unlike the micro-macro methods based on the standard FEM proposed so far, in the HMM we obtain, for periodic homogenization problems, a method that has almost-linear complexity in the number of degrees of freedom of the discretization of the macro- (slow) variable.

Young Measure Approach to Computing Slowly Advancing Fast Oscillations

Zvi Artstein, Jasmine Linshiz, and Edriss S. Titi

Multiscale Model. Simul. 6, pp. 1085-1097 (13 pages) | Cited 2 times

Online Publication Date: December 21, 2007

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We offer a multiscale and averaging strategy to compute the solution of a singularly perturbed system when the fast dynamics oscillates rapidly; namely, the fast dynamics, rather than settling on a manifold of smaller order, forms cycle-like limits which advance along with the slow dynamics. We describe the limit as a Young measure with values being supported on the limit cycles, averaging with respect to which induces the equation for the slow dynamics. In particular, computing the tube of limit cycles establishes a good approximation for arbitrarily small singular parameters. Possible algorithms are displayed and concrete numerical examples are exhibited.

A 1D Macroscopic Phase Field Model for Dislocations and a Second Order $\Gamma$-Limit

Matteo Focardi and Adriana Garroni

Multiscale Model. Simul. 6, pp. 1098-1124 (27 pages)

Online Publication Date: December 21, 2007

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We study the asymptotic behavior in terms of $\Gamma$-convergence of the one dimensional energy $F_{\varepsilon}(u) = \mu_{\varepsilon} \int_I \int_I \frac{|u(x)-u(y)|^2}{|x-y|^2}\,dx\,dy + \eta_{\varepsilon} \int_I W(\frac{u(x)}{\varepsilon})dx,$ where $I$ is a given interval, and $W$ is a one-periodic potential that vanishes exactly on ${\bf Z}$. Different regimes for the asymptotic behavior of the parameters $\mu_{\varepsilon}$ and $\eta_{\varepsilon}$ are considered. In a very diluted regime we get a limit defined on $BV(I)$ and proportional to the total variation of $u$. In this particular case we also consider the limit of a suitable boundary value problem for which we characterize the second order $\Gamma$-limit. The study under consideration is motivated by the analysis of a variational model for a very important class of defects in crystals, the dislocations, and the derivation of macroscopic models for plasticity.

Automated Generation of Reduced Stochastic Weather Models I: Simultaneous Dimension and Model Reduction for Time Series Analysis

Illia Horenko, Rupert Klein, Stamen Dolaptchiev, and Christof Schütte

Multiscale Model. Simul. 6, pp. 1125-1145 (21 pages) | Cited 6 times

Online Publication Date: January 09, 2008

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We present a method for simultaneous dimension reduction, model fitting, and metastability analysis of high-dimensional time series. The approach is based on the combination of hidden Markov models (HMMs) with localized principal component analysis (PCA) (which is used to identify the essential dimensions in the form of empirical orthogonal functions (EOFs) for each of the hidden states) and fitting of multidimensional stochastic differential equations (SDEs). This means that the analyzed data is clustered according to differences in essential dimensions and SDE models specific to each of the hidden states. We derive explicit estimators for PCA-SDE model parameters in the case of fixed sequences of HMM states and employ the expectation-maximization algorithm for numerical optimization of HMM-PCA-SDE parameters. We demonstrate the performance of the method by application to historical temperature data in Europe during 1976–2002. In a comparison with the standard SARMA (seasonal autoregressive moving average model) technique for time series analysis, the HMM-PCA-SDE method exhibits better numerical performance and efficiency, especially on high-dimensional data sets and for a 20-dimensional reduced state space. We also compare the results of both models w.r.t. errors of one-day temperature predictions.

Multiscale Modeling of Chemical Kinetics via the Master Equation

Shev MacNamara, Kevin Burrage, and Roger B. Sidje

Multiscale Model. Simul. 6, pp. 1146-1168 (23 pages) | Cited 7 times

Online Publication Date: January 11, 2008

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We present numerical methods for both the direct solution and simulation of the chemical master equation (CME), and, compared to popular methods in current use, such as the Gillespie stochastic simulation algorithm (SSA) and $\tau$-Leap approximations, this new approach has the advantage of being able to detect when the system has settled down to equilibrium. This improved performance is due to the incorporation of information from the associated CME, a valuable complementary approach to the SSA that has often been felt to be too computationally inefficient. Hybrid methods, that combine these complementary approaches and so are able to detect equilibrium while maintaining the efficiency of the leap methods, are also presented. Amongst CME-solvers the recently suggested finite state projection algorithm is especially well suited to this purpose and has been adapted here for the task, leading to a type of “exact $\tau$-Leap.” It is also observed that a CME-solver is often more efficient than an SSA or even a $\tau$-Leap approach for computing moments of the solution such as the mean and variance. These techniques are demonstrated on a test suite of five biologically inspired models, namely, stochastic models of the genetic toggle, receptor oligomerization, the Schlögl reactions, Goutsias' model of regulated gene transcription, and a decaying-dimerizing reaction set. For the gene toggle it is observed that important experimentally measurable traits such as the percentage of cells that undergo so-called switching may also be more efficiently approximated via a CME-based approach.

Hybrid Multiscale Methods II. Kinetic Equations

Giacomo Dimarco and Lorenzo Pareschi

Multiscale Model. Simul. 6, pp. 1169-1197 (29 pages) | Cited 3 times

Online Publication Date: January 11, 2008

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In this work we consider the development of a new family of hybrid numerical methods for the solution of kinetic equations which involves different scales. The basic idea is to couple macroscopic and microscopic models in all cases in which the macroscopic model does not provide correct results. The key aspect in the development of the algorithms is the choice of a suitable hybrid representation of the solution and a merging of Monte Carlo methods in nonequilibrium regimes with deterministic methods in equilibrium ones. This approach permits us to treat efficiently both the microscopic and the macroscopic scales. Applications to the Boltzmann–BGK equation are presented to show the performance of the new methods.

Exact Bounds on the Effective Behavior of a Conducting Discrete Polycrystal

Andrea Braides and Antoine Gloria

Multiscale Model. Simul. 6, pp. 1198-1216 (19 pages)

Online Publication Date: January 11, 2008

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In a recent paper by Braides and Francfort, the problem of the characterization of the overall properties of lattice energies describing networks with arbitrary mixtures of two types of linear conductors has been addressed in a two-dimensional setting. In this paper we investigate the connection between that discrete optimization process and the theory of bounds for mixtures of continuum energies, for which the choice of the relationships between the different phases of the mixture is unusual and leads to remarkably simple results in terms of $G$-closure.

Uncertainty Quantification for Atomistic Reaction Models: An Equation-Free Stochastic Simulation Algorithm Example

Yu Zou and Ioannis G. Kevrekidis

Multiscale Model. Simul. 6, pp. 1217-1233 (17 pages)

Online Publication Date: January 11, 2008

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We describe a computational framework linking uncertainty quantification (UQ) methods for continuum problems depending on random parameters with equation-free (EF) methods for performing continuum deterministic numerics by acting directly on atomistic/stochastic simulators. Our illustrative example is a heterogeneous catalytic reaction mechanism with an uncertain atomistic kinetic parameter; the “inner” dynamic simulator of choice is a Gillespie stochastic simulation algorithm (SSA). We demonstrate UQ computations at the coarse-grained level, in a nonintrusive way, through the design of brief, appropriately initialized computational experiments with the SSA code. The system is thus observed at three levels: (a) a fine scale for each stochastic simulation at each value of the uncertain parameter; (b) an intermediate coarse-grained state for the expected behavior of the SSA at each value of the uncertain parameter; and (c) the desired fully coarse-grained level: distributions of the coarse-grained behavior over the range of uncertain parameter values. The latter are computed in the form of generalized polynomial chaos (gPC) coefficients in terms of the random parameter. Coarse projective integration and coarse fixed point computation are employed to accelerate the computational evolution of these desired observables, to converge on random stable/unstable steady states, and to perform parametric studies with respect to other (nonrandom) system parameters.

Mesoscale Analysis of the Equation-Free Constrained Runs Initialization Scheme

Pieter Van Leemput, Wim Vanroose, and Dirk Roose

Multiscale Model. Simul. 6, pp. 1234-1255 (22 pages) | Cited 1 time

Online Publication Date: February 01, 2008

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In this article, we analyze the stability, convergence, and accuracy of the constrained runs initialization scheme for a mesoscale lattice Boltzmann model (LBM). This type of initialization scheme was proposed by Gear and Kevrekidis in [J. Sci. Comput., 25 (2005), pp. 17–28] in the context of both singularly perturbed ordinary differential equations and equation-free computing. It maps the given macroscopic initial variables to the higher-dimensional space of microscopic/mesoscopic variables. The scheme performs short runs with the microscopic/mesoscopic simulator and resets the macroscopic variables (typically the lower order moments of the microscopic/mesoscopic variables), while leaving the higher order moments unchanged. We use the LBM Bhatnagar–Gross–Krook (BGK) model for one-dimensional reaction-diffusion systems as the microscopic/mesoscopic model. For such systems, we prove that the constrained runs scheme is unconditionally stable and that it converges to an approximation of the slaved state, i.e., the mesoscopic state which is consistent with the macroscopic initial condition. This approximation is correct up to and including the first order terms in the Chapman–Enskog expansion of the LBM. The asymptotic convergence factor is $|1-\omega|$ with $\omega$ the BGK relaxation parameter. The results are illustrated numerically for the FitzHugh–Nagumo system. Furthermore, we use the constrained runs scheme to perform a coarse equation-free bifurcation analysis of this model. Finally, we show that the constrained runs scheme is very similar to the LBM initialization scheme proposed by Mei et al. in [Comput. Fluids, 35 (2006), pp. 855–862] when implemented for our model problem, and that our numerical analysis applies to the latter scheme also.

Algorithm Refinement for Fluctuating Hydrodynamics

Sarah A. Williams, John B. Bell, and Alejandro L. Garcia

Multiscale Model. Simul. 6, pp. 1256-1280 (25 pages) | Cited 8 times

Online Publication Date: February 01, 2008

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This paper introduces an adaptive mesh and algorithm refinement method for fluctuating hydrodynamics. This particle-continuum hybrid simulates the dynamics of a compressible fluid with thermal fluctuations. The particle algorithm is direct simulation Monte Carlo (DSMC), a molecular-level scheme based on the Boltzmann equation. The continuum algorithm is based on the Landau–Lifshitz Navier–Stokes (LLNS) equations, which incorporate thermal fluctuations into macroscopic hydrodynamics by using stochastic fluxes. It uses a recently developed solver for the LLNS equations based on third-order Runge–Kutta. We present numerical tests of systems in and out of equilibrium, including time-dependent systems, and demonstrate dynamic adaptive refinement by the computation of a moving shock wave. Mean system behavior and second moment statistics of our simulations match theoretical values and benchmarks well. We find that particular attention should be paid to the spectrum of the flux at the interface between the particle and continuum methods, specifically for the nonhydrodynamic (kinetic) time scales.

A Diffusion Model for Rarefied Flows in Curved Channels

K. Aoki, P. Degond, L. Mieussens, S. Takata, and H. Yoshida

Multiscale Model. Simul. 6, pp. 1281-1316 (36 pages) | Cited 2 times

Online Publication Date: February 22, 2008

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In this paper, we derive a one-dimensional convection-diffusion model for a rarefied gas flow in a two-dimensional curved channel on the basis of the Boltzmann (Bhatnagar–Gross–Krook) model. The flow is driven by the temperature gradient along the channel walls, which is known as the thermal creep phenomenon. This device can be used as a micropumping system without any moving part. Our derivation is based on the asymptotic technique of the diffusion approximation. It gives a macroscopic (fluid) approximation of the microscopic (kinetic) equation. We also derive the connection conditions at the junction where the curvature is not continuous. The pumping device is simulated by using a numerical approximation of our convection-diffusion model which turns out to agree very well with full two-dimensional kinetic simulations. It is then used to obtain very fast computations on long pumping devices, while the computational cost of full kinetic computations nowadays is still prohibitive for such cases.

Multiscale Analysis and Computation for the Three-Dimensional Incompressible Navier–Stokes Equations

Thomas Y. Hou, Danping Yang, and Hongyu Ran

Multiscale Model. Simul. 6, pp. 1317-1346 (30 pages)

Online Publication Date: March 26, 2008

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In this paper, we perform a systematic multiscale analysis for the three-dimensional incompressible Navier–Stokes equations with multiscale initial data. There are two main ingredients in our multiscale method. The first one is that we reparameterize the initial data in the Fourier space into a formal two-scale structure. The second one is the use of a nested multiscale expansion together with a multiscale phase function to characterize the propagation of the small-scale solution dynamically. By using these two techniques and performing a systematic multiscale analysis, we derive a multiscale model which couples the dynamics of the small-scale subgrid problem to the large-scale solution without a closure assumption or unknown parameters. Furthermore, we propose an adaptive multiscale computational method which has a complexity comparable to a dynamic Smagorinsky model. We demonstrate the accuracy of the multiscale model by comparing with direct numerical simulations for both two- and three-dimensional problems. In the two-dimensional case we consider decaying turbulence, while in the three-dimensional case we consider forced turbulence. Our numerical results show that our multiscale model not only captures the energy spectrum very accurately, it can also reproduce some of the important statistical properties that have been observed in experimental studies for fully developed turbulent flows.

Numerical Solutions of the Complex Langevin Equations in Polymer Field Theory

Erin M. Lennon, George O. Mohler, Hector D. Ceniceros, Carlos J. García-Cervera, and Glenn H. Fredrickson

Multiscale Model. Simul. 6, pp. 1347-1370 (24 pages) | Cited 5 times

Online Publication Date: March 26, 2008

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Using a diblock copolymer melt as a model system, we show that complex Langevin (CL) simulations constitute a practical method for sampling the complex weights in field theory models of polymeric fluids. Prior work has primarily focused on numerical methods for obtaining mean-field solutions—the deterministic limit of the theory. This study is the first to go beyond Euler–Maruyama integration of the full stochastic CL equations. Specifically, we use analytic expressions for the linearized forces to develop improved time integration schemes for solving the nonlinear, nonlocal stochastic CL equations. These methods can decrease the computation time required by orders of magnitude. Further, we show that the spatial and temporal multiscale nature of the system can be addressed by the use of Fourier acceleration.

Identifying Strain Localization in Heterogeneous Microstructures via an Information Passing Modeling Approach

Ke-Shen Cheong

Multiscale Model. Simul. 6, pp. 1371-1381 (11 pages)

Online Publication Date: March 26, 2008

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A multiscale modeling approach employing crystal plasticity theory to identify microstructure-related strain localization is demonstrated in this work. The proposed methodology applies an information passing procedure where orientation-dependent single crystal stress-strain data, generated from a tested and validated dislocation mechanics-based constitutive model, is assigned to individual grains in a sample microstructure. Within a finite element framework, this approach was tested and compared with an actual deformed tensile test specimen with a quasi-two-dimensional microstructure. Qualitatively, the numerical results from the simulation demonstrate that the proposed approach is capable of reproducing the macroscopic shear band physically observed in the actual experiment without recourse to running full crystal plasticity simulations and at reduced computational time. The study also shows that elastic anisotropy due to the different grain orientations has a negligible effect on the initiation and development of strain localization. From a practical standpoint, such an approach will be useful in serving as an indicator for microstructure-related damage, especially in material systems where the component and microstructure dimensions are comparable and plasticity is intrinsically localized.
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