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2006

Volume 5, Issue 2, pp. 337-694


A Hierarchical Multiscale Method for Two-Phase Flow Based upon Mixed Finite Elements and Nonuniform Coarse Grids

Jorg E. Aarnes, Stein Krogstad, and Knut-Andreas Lie

Multiscale Model. Simul. 5, pp. 337-363 (27 pages) | Cited 23 times

Online Publication Date: August 03, 2006

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We analyze and further develop a hierarchical multiscale method for the numerical simulation of two-phase flow in highly heterogeneous porous media. The method is based upon a mixed finite-element formulation, where fine-scale features are incorporated into a set of coarse-grid basis functions for the flow velocities. By using the multiscale basis functions, we can retain the efficiency of an upscaling method by solving the pressure equation on a (moderate-sized) coarse grid, while at the same time produce a detailed and conservative velocity field on the underlying fine grid.
Earlier work has shown that the multiscale method performs excellently on highly heterogeneous cases using uniform coarse grids. In this paper, we extend the methodology to nonuniform and unstructured coarse grids and discuss various formulations for generating the coarse-grid basis functions. Moreover, we focus on the impact of large-scale features such as barriers or high-permeable channels and discuss potentially problematic flow cases. To improve the accuracy of the multiscale solution, we introduce adaptive strategies for the coarse grids, based on either local hierarchical refinement or adaptation of the coarse grid more directly to large-scale permeability structures of arbitrary shape. The resulting method is very flexible with respect to the size and the geometry of coarse-grid cells, meaning that grid refinement/adaptation can be performed in a straightforward manner. The suggested strategies are illustrated in several numerical experiments.

A Boltzmann Model for Trapped Particles in a Surface Potential

Pierre Degond, Céline Parzani, and Marie-Hélène Vignal

Multiscale Model. Simul. 5, pp. 364-392 (29 pages) | Cited 2 times

Online Publication Date: August 03, 2006

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In this article, we propose a model describing the transport of trapped particles in a surface potential. The potential confines particles close to the surface, increasing the number of surface collisions. First, we consider the case of noncharged particles. From a kinetic description, we rigorously derive a two dimensional Boltzmann equation. In the case of charged particles we introduce the coupling with the Poisson equation. We perform a formal asymptotic analysis which leads to a two dimensional Boltzmann equation coupled with a three dimensional Poisson equation. We illustrate the charged particle model with some numerical simulations of a gas discharge on a satellite solar array. We use a particle in cell (P.I.C.) scheme that is a particle discretization for the Boltzmann equation and a Fourier approximation for the Poisson equation.

Computing Best Transition Pathways in High-Dimensional Dynamical Systems: Application to the AlphaL \leftrightharpoons Beta \leftrightharpoons AlphaR Transitions in Octaalanine

Frank Noé, Marcus Oswald, Gerhard Reinelt, Stefan Fischer, and Jeremy C. Smith

Multiscale Model. Simul. 5, pp. 393-419 (27 pages) | Cited 7 times

Online Publication Date: August 03, 2006

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The direct computation of rare transitions in high-dimensional dynamical systems such as biomolecules via numerical integration or Monte Carlo is limited by the sampling problem. Alternatively, the dynamics of these systems can be modeled by transition networks (TNs) which are weighted graphs whose edges represent transitions between stable states of the system. The computation of the globally best transition paths connecting two selected stable states is straightforward with available graph-theoretical methods. However, these methods require that the energy barriers of all TN edges be determined, which is often computationally infeasible for large systems. Here, we introduce energy-bounded TNs, in which the transition barriers are specified in terms of lower and upper bounds. We present algorithms permitting the determination of the globally best paths on these TNs while requiring the computation of only a small subset of the true transition barriers. Several variants of the algorithm are given which achieve improved performance, including a parallel version. The effectiveness of the approach is demonstrated by various benchmarks on random TNs and by computing the refolding pathways of a polypeptide: the best transition pathways between the alphaL helix, alphaR helix, and beta-hairpin conformations of the octaalanine (Ala8) molecule in aqueous solution.

Towards Wall-Normal Filtering for Large-Eddy Simulation

Jeremy A. Templeton and Mohammad Shoeybi

Multiscale Model. Simul. 5, pp. 420-444 (25 pages) | Cited 1 time

Online Publication Date: August 03, 2006

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Large-eddy simulation (LES) is currently limited in applicability due to the presence of commutation errors and indeterminate boundary conditions when using the large filter widths necessary to efficiently compute high Reynolds number flows. A solution is proposed which involves defining the flow on an infinite domain such that homogeneous filters can be used. This formulation allows the LES equations, including boundary conditions, to be precisely defined. Simulations of both the forced and the unforced Burgers equations demonstrate that this method can accurately calculate the filtered velocity field.

Efficient Incorporation of Global Effects in Upscaled Models of Two-Phase Flow and Transport in Heterogeneous Formations

Yuguang Chen and Louis J. Durlofsky

Multiscale Model. Simul. 5, pp. 445-475 (31 pages) | Cited 4 times

Online Publication Date: August 03, 2006

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Geological variability occurring over multiple length scales can significantly affect fluid flow in subsurface formations. In this paper we explore the impact of large scale (global) information on the accuracy of coarse scale models for two-phase flow and transport. Based on these findings, a new methodology for generating coarse models is introduced. This approach, which avoids any global fine scale calculations, entails adaptive local-global single-phase parameter upscaling coupled with subgrid models for two-phase parameters. Two related subgrid treatments for two-phase flow effects---a pseudorelative permeability model and a generalized convection-diffusion model---are investigated and applied. The upscaled single-phase parameters (transmissibilities), computed using the adaptive local-global procedure, account explicitly for global boundary conditions as they are adapted for a specific flow scenario. The two-phase coarse scale functions are computed from local simulations using effective flux boundary conditions (EFBCs), which account approximately for global effects in the local computations. The advantages of the adaptive local-global upscaled single-phase parameters, as well as the superior accuracy of EFBCs relative to standard treatments for two-phase parameters, are demonstrated for several challenging two-dimensional example cases. The overall method is also applied to highly heterogeneous systems with global boundary conditions that vary in time. Using standard upscaling methods, errors for some of these cases are very large, though the new methodology is shown to consistently provide reasonably accurate coarse models.

Semigeostrophic Particle Motion and Exponentially Accurate Normal forms

Colin J. Cotter and Sebastian Reich

Multiscale Model. Simul. 5, pp. 476-496 (21 pages) | Cited 5 times

Online Publication Date: August 03, 2006

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We give an exponentially accurate normal form for a Lagrangian particle moving in a rotating shallow-water system in the semigeostrophic limit, which describes the motion in the region of an exponentially accurate slow manifold (a region of phase space for which dynamics on the fast scale are exponentially small in the Rossby number). We show how this result is related to the variational asymptotics approach of [M. Oliver, J. Fluid Mech., 551 (2006), pp. 197-234]; the difference is that on the Hamiltonian side it is possible to obtain strong bounds on the growth of fast motion away from (but close to) the slow manifold. Our normal form approach extends to numerical approximations via backward error analysis and extends to particle methods for the shallow-water equations, where the result shows that particles stay close to balance over long times in the semigeostrophic limit.

Averaging Methods for Stochastic Dynamics of Complex Reaction Networks: Description of Multiscale Couplings

Sergey Plyasunov and Adam P. Arkin

Multiscale Model. Simul. 5, pp. 497-513 (17 pages)

Online Publication Date: August 03, 2006

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This paper is concerned with classes of models of stochastic reaction dynamics with time-scale separation. We demonstrate that the existence of the time-scale separation naturally leads to the application of the averaging principles and elimination of degrees of freedom via renormalization of transition rates of slow reactions. The method suggested in this work is more general than other approaches presented previously in the literature and closely follows ideas of dynamical disorder in relaxation processes. The method is not limited to a particular type of stochastic process and can be applied to different types of processes describing fast dynamics; it also provides crossover to the case when separation of time scales is not well pronounced. We derive a family of exact fluctuation-dissipation relations which establish the connection between effective rates and the statistics of the reaction events in fast reaction channels. An illustration of the technique is provided. Examples show that renormalized transition rates exhibit, in general, nonexponential relaxation behavior with a broad range of possible scenarios.

Long Term Object Drift Forecast in the Ocean with Tide and Wind

P. Ailliot, E. Frénod, and V. Monbet

Multiscale Model. Simul. 5, pp. 514-531 (18 pages) | Cited 1 time

Online Publication Date: August 03, 2006

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In this paper, we propose a new method to forecast the drift of objects in the near coastal ocean over a period of several weeks. The proposed approach consists in estimating the probability of events linked to the drift using Monte Carlo simulations. It couples an averaging method which permits us to decrease the computational cost with a statistical method in order to take into account the variability of meteorological loading factors.

Heterogeneous Multiscale Methods for Interface Tracking of Combustion Fronts

Yi Sun and Bjorn Engquist

Multiscale Model. Simul. 5, pp. 532-563 (32 pages) | Cited 3 times

Online Publication Date: August 03, 2006

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In this paper we investigate the heterogeneous multiscale methods (HMM) for interface tracking and apply the technique to the simulation of combustion fronts. Our goal is to overcome the numerical difficulties, which are caused by different time scales between the transport part and the reactive part in the model equations of some interface tracking problems, such as combustion processes. HMM relies on an efficient coupling between the macroscale and microscale models. When the macroscale model is not fully known explicitly or not valid in localized regions, HMM provides a procedure for supplementing the missing data from a microscale model. Here we design and analyze a multiscale scheme in which a localized microscale model resolves the details in the model and a phase field or a front tracking method defines the interface on the macroscale. This multiscale technique overcomes the difficulty of stiffness of common problems in combustion processes. Numerical results for Majda's model and reactive Euler equations in one and two dimensions show substantially improved efficiency over traditional methods.

Mesoscale Averaging of Nucleation and Growth Models

Martin Burger, Vincenzo Capasso, and Livio Pizzocchero

Multiscale Model. Simul. 5, pp. 564-592 (29 pages) | Cited 6 times

Online Publication Date: August 03, 2006

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The aim of this paper is to derive a general theory for the averaging of heterogeneous processes with stochastic nucleation and deterministic growth. We start by generalizing the classical Johnson--Mehl--Avrami--Kolmogorov theory based on the causal cone to heterogeneous growth situations. Moreover, we relate the computation of the causal cone to a Hopf--Lax formula for Hamilton--Jacobi equations describing the growth of grains. As an outcome of the approach we obtain formulae for the expected values of geometric densities describing the growth processes; in particular we generalize the standard Avrami--Kolmogorov relations for the degree of crystallinity. By relating the computation of expected values to mesoscale averaging, we obtain a suitable description of the process at the mesoscale. We show how the variance of these mesoscale averages can be estimated in terms of quotients of the typical length on the microscale and on the mesoscale. Moreover, we discuss the efficient computation of the mesoscale averages in the typical case when the nucleation and growth rates are obtained from mesoscopic fields (such as, e.g., temperature). Finally, we give a brief outlook to possible extensions such as polycrystalline growth, which turns out to be rather straightforward when starting from our general framework.

Dimensional Reduction of the Fokker–Planck Equation for Stochastic Chemical Reactions

Per Lötstedt and Lars Ferm

Multiscale Model. Simul. 5, pp. 593-614 (22 pages) | Cited 6 times

Online Publication Date: July 31, 2006

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The Fokker–Planck equation models chemical reactions on a mesoscale. The solution is a probability density function for the copy number of the different molecules. The number of dimensions of the problem can be large, making numerical simulation of the reactions computationally intractable. The number of dimensions is reduced here by deriving partial differential equations for the first moments of some of the species and coupling them with a Fokker–Planck equation for the remaining species. With more simplifying assumptions, another system of equations is derived consisting of integrodifferential equations and a Fokker–Planck equation. In this way, the simulation of the chemical networks is possible without the exponential growth in computational work and memory of the original equation and with better modeling accuracy than the macroscopic reaction rate equations. Some terms in the equations are small and are ignored. Conditions are given for the influence of these terms to be small on the equations and the solutions. The difference between different models is illustrated in a numerical example.

An Optimization‐Based Multilevel Algorithm for Total Variation Image Denoising

Tony F. Chan and Ke Chen

Multiscale Model. Simul. 5, pp. 615-645 (31 pages) | Cited 20 times

Online Publication Date: July 31, 2006

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This paper proposes a fast multilevel method using primal relaxations for the total variation image denoising and analyzes its convergence. The basic primal relaxation is known to get stuck at a nonstationary point (nearly a local minimum) of the minimization, whose solution is known to be “nonsmooth” in the space of functions with bounded variation. Our idea is to use coarse level corrections, overcoming the deadlock in a basic primal relaxation scheme and achieving much improvement over relaxation. Moreover, to reach a global minimizer, further refinement of the multilevel method is needed, and we propose a nonregular coarse level based on a patch‐detection idea (relating to hemivariateness) to correct and improve the standard multilevel method. Both algorithmic and analytical results together with numerical experiments on both one‐ and two‐dimensional images are presented.

Layer Potential Techniques in Spectral Analysis. Part II: Sensitivity Analysis of Spectral Properties of High Contrast Band‐Gap Materials

Habib Ammari, Hyeoenbae Kang, Sofiane Soussi, and Habib Zribi

Multiscale Model. Simul. 5, pp. 646-663 (18 pages)

Online Publication Date: July 31, 2006

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We investigate the band‐gap structure of the frequency spectrum for waves in a high‐contrast, two‐component periodic medium. We consider two‐dimensional photonic crystals consisting of a background medium which is perforated by an array of holes periodic along each of the two orthogonal coordinate axes. We perform a high‐order sensitivity analysis with respect to the index ratio and small perturbations in the geometry of the holes. Our method, which is based on a boundary integral perturbation theory, gives a new tool for the optimal design problem in photonic crystals.

A Derivation of Continuum Nonlinear Plate Theory from Atomistic Models

Bernd Schmidt

Multiscale Model. Simul. 5, pp. 664-694 (31 pages) | Cited 7 times

Online Publication Date: September 05, 2006

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We derive plate theory from atomistic models in the spirit of [G. Friesecke, R. D. James, and S. Müller, Comm. Pure Appl. Math., 55 (2002), pp. 1461–1506] as a Γ‐limit as the number of atoms tends to infinity. While in the “thick film regime,” i.e., when the film consists of many layers of atoms, we recover the well‐known plate theory derived from three‐dimensional elasticity in [G. Friesecke, R. D. James, and S. Müller, Comm. Pure Appl. Math., 55 (2002), pp. 1461–1506]; for “thin films” new terms in the limit functional are obtained. These terms are due to the discrete nature of atomic models and surface effects and cannot be detected from continuum elasticity.
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