Stochastic mixed multiscale finite element methods and their applications in random porous media

We present a framework for stochastic mixed multiscale finite element methods (mixed MsFEMs) for elliptic equations with heterogeneous random coefficients. The use of some global information is necessary in multiscale simulations when there is no scale separation for the heterogeneity. The methods in the proposed framework for the stochastic mixed MsFEMs use some global information. The media properties in a stochastic environment drastically vary among realizations and, thus, many global fields are needed for multiscale simulation. The computations of these global fields on a fine grid can be very expensive. One can utilize upscaling methods to compute the global information on an intermediate coarse grid that reduces the computational cost. We investigate two approaches of stochastic mixed MsFEMs in the framework. First approach entails no stochastic interpolation and the second approach uses stochastic interpolation. If the random media have deterministic features that play significant roles in the flow, we can use the deterministic features of the random media as the global information. This reduces the computational cost of the simulations. We make convergence analysis of the stochastic mixed MsFEMs and investigate their applications to incompressible two-phase flows in random porous media. The numerical results demonstrate the effectiveness of the proposed methods and confirm the convergence.     

AES for multiscale localization modeling in granular media

This work presents a multiscale strong discontinuity approach to tackle key challenges in modeling localization behavior in granular media: accommodation of discontinuities in the kinematic fields, and direct linkage to the underlying grain-scale information. Assumed enhanced strain (AES) concepts are borrowed to enhance elements for post-localization analysis, but are reformulated within a recently-proposed hierarchical multiscale computational framework. Unlike classical AES methods, where material properties are usually constants or assumed to evolve with some arbitrary phenomenological laws, this framework provides a bridge to extract evolutions of key material parameters, such as friction and dilatancy, based on grain scale computational or experimental data. More importantly, the phenomenological softening modulus typically used in AES methods is no longer required. Numerical examples of plane strain compression tests are presented to illustrate the applicability of this method and to analyze its numerical performance.     

Parametrized invariant manifolds: a recipe for multiscale modeling?

Given a fine-scale physical theory characterized by an evolutionary system of equations and a set of quantities, defined from the variables of the fine theory, that serve as a coarse representation of the fine scale phenomena, a method for developing an evolutionary set of equations for the coarse quantities is presented. This set may be considered as governing equations for a coarse model corresponding to the fine one since its solutions can be formally shown to generate exact/approximate solutions to the fine theory in a well-defined sense. The method for generating the coarse representation is illustrated on model problems.     

Immersive structural biology: a new approach to hybrid modeling of macromolecular assemblies

Advanced biophysical imaging techniques, such as cryo-electron microscopy or tomography, enable 3D volumetric reconstructions of large macromolecular complexes in a near-native environment. However, pure volumetric data is insufficient for a detailed understanding of the underlying protein–protein interactions. This obstacle can be overcome by assembling an atomic model of the whole protein complex from known atomic structures, which are available from either X-ray crystallography or homology modeling. Due to many factors such as noise, conformational variability, experimental artifacts, and inexact model structures, existing automatic docking procedures are known to report false positives for a significant number of cases. The present paper focuses on a new technique to combine an offline exhaustive search algorithm with interactive visualization, collision detection, and haptic rendering. The resulting software system is highly immersive and allows the user to efficiently solve even difficult multi-resolution docking problems. Stereoscopic viewing, combined with head tracking and force feedback, generates an ideal virtual environment for true interaction with and solution of hybrid biomolecular modeling problems.     

Computer-aided multiscale modeling tools for composite materials and structures

This paper presents recent research efforts at Rensselaer Polytechnic Institute aimed at developing computer-aided multiscale modeling tools for composite materials and structures aimed at predicting the macromechanical (overall) structural response, such as critical deformation, vibration and buckling modes, as well as various failure modes on the mesomechanical (lamina) level, such as delamination and ply buckling, and on the micromechanical (the scale of microconstituents) level, such as debonding, microbuckling, etc.The building blocks of this technology are (i) idealization error estimators aimed at quantifying the quality of the numerical and mathematical models of composites, (ii) multigrid technology aimed at superconvergent solution of the multiscale computational models, (iii) mathematical homogenization theory aimed at constructing inter-scale transfer operators for rapid and reliable information flow between the scales, (iv) system identification for in situ characterization of the phases and their interface, and (v) multiscale model construction and visualization.     

Confidence modeling for handwriting recognition: algorithms and applications

Confidence scoring can assist in determining how to use imperfect handwriting-recognition output. We explore a confidence-scoring framework for post-processing recognition for two purposes: deciding when to reject the recognizer's output, and detecting when to change recognition parameters e.g., to relax a word-set constraint. Varied confidence scores, including likelihood ratios and posterior probabilities, are applied to an Hidden-Markov-Model (HMM) based on-line recognizer. Receiver-operating characteristic curves reveal that we successfully reject 90% of word recognition errors while rejecting only 33% of correctly-recognized words. For isolated digit recognition, we achieve 90% correct rejection while limiting false rejection to 13%.     

Predictive modeling for wastewater applications: Linear and nonlinear approaches

This study compares the multi-period predictive ability of linear ARIMA models to nonlinear time delay neural network models in water quality applications. Comparisons are made for a variety of artificially generated nonlinear ARIMA data sets that simulate the characteristics of wastewater process variables and watershed variables, as well as two real-world wastewater data sets. While the time delay neural network model was more accurate for the two real-world wastewater data sets, the neural networks were not always more accurate than linear ARIMA for the artificial nonlinear data sets. In some cases of the artificial nonlinear data, where multi-period predictions are made, the linear ARIMA model provides a more accurate result than the time delay neural network. This study suggests that researchers and practitioners should carefully consider the nature and intended use of water quality data if choosing between neural networks and other statistical methods for wastewater process control or watershed environmental quality management.     

Numerical modeling of multiphase flows in microfluidics and micro process engineering: a review of methods and applications

This article presents a comprehensive review of numerical methods and models for interface resolving simulations of multiphase flows in microfluidics and micro process engineering. The focus of the paper is on continuum methods where it covers the three common approaches in the sharp interface limit, namely the volume-of-fluid method with interface reconstruction, the level set method and the front tracking method, as well as methods with finite interface thickness such as color-function based methods and the phase-field method. Variants of the mesoscopic lattice Boltzmann method for two-fluid flows are also discussed, as well as various hybrid approaches. The mathematical foundation of each method is given and its specific advantages and limitations are highlighted. For continuum methods, the coupling of the interface evolution equation with the single-field Navier–Stokes equations and related issues are discussed. Methods and models for surface tension forces, contact lines, heat and mass transfer and phase change are presented. In the second part of this article applications of the methods in microfluidics and micro process engineering are reviewed, including flow hydrodynamics (separated and segmented flow, bubble and drop formation, breakup and coalescence), heat and mass transfer (with and without chemical reactions), mixing and dispersion, Marangoni flows and surfactants, and boiling.     

Generalized-ensemble algorithms: enhanced sampling techniques for Monte Carlo and molecular dynamics simulations

In complex systems with many degrees of freedom such as spin glass and biomolecular systems, conventional simulations in canonical ensemble suffer from the quasi-ergodicity problem. A simulation in generalized ensemble performs a random walk in potential energy space and overcomes this difficulty. From only one simulation run, one can obtain canonical ensemble averages of physical quantities as functions of temperature by the single-histogram and/or multiple-histogram reweighting techniques. In this article we review the generalized ensemble algorithms. Three well-known methods, namely, multicanonical algorithm (MUCA), simulated tempering (ST), and replica-exchange method (REM), are described first. Both Monte Carlo (MC) and molecular dynamics (MD) versions of the algorithms are given. We then present five new generalized-ensemble algorithms which are extensions of the above methods.     

On some multiscale algorithms for sector modeling in multiphase flow in porous media

A multiscale algorithm for a multiphase filtration problem is proposed. Filtration fluxes on a fine grid are determined from the solution of the pressure equation on a coarse grid. Further, the domain is decomposed into subdomains with an acceptable number of cells and the full second boundary condition filtration problem is solved using the fluxes. The support operator method has been improved for a complex structure cell for solution of the pressure equation on a coarse grid. This method is a high resolution one: a divergence operator has the approximation of the second order, while fluxes have the approximation of the first order. At the same time, the method allows revealing the solution’s properties related to a fine grid structure.     

Orthogonal parallel MCMC methods for sampling and optimization

Monte Carlo (MC) methods are widely used for Bayesian inference and optimization in statistics, signal processing and machine learning. A well-known class of MC methods are Markov Chain Monte Carlo (MCMC) algorithms. In order to foster better exploration of the state space, specially in high-dimensional applications, several schemes employing multiple parallel MCMC chains have been recently introduced. In this work, we describe a novel parallel interacting MCMC scheme, called orthogonal MCMC (O-MCMC), where a set of “vertical” parallel MCMC chains share information using some “horizontal” MCMC techniques working on the entire population of current states. More specifically, the vertical chains are led by random-walk proposals, whereas the horizontal MCMC techniques employ independent proposals, thus allowing an efficient combination of global exploration and local approximation. The interaction is contained in these horizontal iterations. Within the analysis of different implementations of O-MCMC, novel schemes in order to reduce the overall computational cost of parallel Multiple Try Metropolis (MTM) chains are also presented. Furthermore, a modified version of O-MCMC for optimization is provided by considering parallel Simulated Annealing (SA) algorithms. Numerical results show the advantages of the proposed sampling scheme in terms of efficiency in the estimation, as well as robustness in terms of independence with respect to initial values and the choice of the parameters.     

Continuation methods: Theory and applications

This paper surveys in a tutorial fashion theoretical and applications aspects of the confirmation method for the solution of large scale system engineering problems. The continuation method is motivated and defined. The existence of the method is formulated in terms of degree theory a la Algebraic Topology. Examples are given throughout the paper and applications to engineering and economic problems are cited and described.     

Liquid-infiltrated photonic crystals: enhanced light-matter interactions for lab-on-a-chip applications

Optical techniques are finding widespread use in analytical chemistry for chemical and bio-chemical analysis. During the past decade, there has been an increasing emphasis on miniaturization of chemical analysis systems and naturally this has stimulated a large effort in integrating microfluidics and optics in lab-on-a-chip microsystems. This development is partly defining the emerging field of optofluidics. Scaling analysis and experiments have demonstrated the advantage of micro-scale devices over their macroscopic counterparts for a number of chemical applications. However, from an optical point of view, miniaturized devices suffer dramatically from the reduced optical path compared to macroscale experiments, e.g. in a cuvette. Obviously, the reduced optical path complicates the application of optical techniques in lab-on-a-chip systems. In this paper we theoretically discuss how a strongly dispersive photonic crystal environment may be used to enhance the light-matter interactions, thus potentially compensating for the reduced optical path in lab-on-a-chip systems. Combining electromagnetic perturbation theory with full-wave electromagnetic simulations we address the prospects for achieving slow-light enhancement of Beer–Lambert–Bouguer absorption, photonic band-gap based refractometry, and high-Q cavity sensing. Invited paper for the “Optofluidics” special issue edited by Prof. David Erickson.     

On a multiscale computational strategy with time and space homogenization for structural mechanics

A new multiscale computational strategy was recently proposed for the analysis of structures described both on a fine space scale and a fine time scale. This strategy, which involves homogenization in space as well as in time, could replace in several domains of application the standard homogenization techniques, which are generally limited to the space domain. It is an iterative strategy which calls for the resolution of problems on both a micro (fine) scale and a macro (homogenized) scale. In this paper, we review the bases of this approach and present improved approximation techniques to solve the micro- and macro-problems.