Variational Multiscale Methods and Multiscale Finite
Elements for Heterogeneous Porous Media
Multiscale (i.e., "generalized")
finite elements were introduced by Babuska and Osborn
in the 80's, and they have seen a reemergence for handling problems with
natural heterogeneities. The method
involves solving the overall partial differential equation on a course grid by
incorporating system microstructure directly into the finite element basis
itself. This is accomplished by solving
local or subgrid problems that resolve the
microstructure. In this way, one can
improve the overall resolution of the finite element approximation. Multiscale and variational aspects of the
method have recently been put on a sound theoretical foundation. This session will explore various
developments that have tailored such methods to solving problems in simulating
flow and transport in porous media.
Pavel B. Bochev
Sandia National Laboratories
Variational Multiscale Analysis
Variational
multiscale analysis (VMA) is a systematic approach for modeling of multiscale
phenomena in computational sciences. VMA is based on an additive decomposition
of the solution spaces into resolved and
unresolved scales and a subsequent
derivation of exact governing
equations for each scale. In conjunction with appropriate modeling assumptions,
these equations serve to provide the basis for variational formulations capable
of representing multiscale phenomena. Most notably, the VMA framework allows
one to readily identify the appropriate scale-to-scale interactions and to
replace interactions that depend on unresolved physics by suitable model terms.
The VMA
formalism is applicable to a broad class of multiscale phenomena, ranging from
problems where scale separation is induced by choice of a discrete space (“resolved” scales), to problems where
resolved and unresolved scales may represent concurrent physical models
operating at different space and/or time scales.
The purpose of this session is to explore
further the potential of VMA as a unifying methodology for modeling and
analysis of multiscale phenomena. Speakers will focus on examples from
important application areas such as turbulence, multiscale wave and transport
problems where VMA has led to a variety of improved numerical methods.
Space-Time Adaptive Methods for Multiscale Problems
Contemporary multiscale problems arise in
several areas of modeling and simulation.
In this lecture we begin with some remarks on multiscale aspects of the
“end to end” design problem embracing adaptive modeling and meshing. We then recall classical multiscale ideas
from singular perturbation theory where different physics and mathematical
behavior dominates in different subregions and
discuss some very early adaptive meshing work on such problems. This leads
naturally to a discussion of imbedded subgrid and
adaptive mesh strategies to resolve multiscale behavior. Numerical studies of
fluid flow with transport and from biological transport will be shown to
illustrate the strategies. Some comments on adaptive space-time approaches
conclude the lecture.
Simulation and Analysis of Large Networks
In this talk, I will explore multiscale approaches in the
modeling, analysis, control, and simulation of large networks. In this setting,
multiscale approaches have been applied with respect to three different
dimensions: time, state, and space. In the temporal dimension, multiscale
methods have been of interest in the modeling of communication network traffic,
motivated by studies showing that network traffic exhibits similar behavior over
multiple time-scales (self-similarity). In the state domain, for some time there
has been an interest in using fluid state-approximations in modeling networks.
Multiscale approaches in the spatial domain have only recently been explored. An
example is the study of large wireless networks via scaling laws as the number
of nodes grows. Finally, we also explore the possibility of continuum models
(e.g., PDEs) for large networks.
Jacob Fish
Rensselaer Polytechnic Institute
Discrete-to-Continuum Multiscale Bridging Approaches
In this talk, I will present
information-passing and concurrent discrete-to-continuum multiscale bridging
approaches. In the concurrent approach both, the discrete and continuum scales
are simultaneously resolved, whereas in the information-passing schemes, the
discrete scale is modeled and its gross response is infused into the continuum
scale. Among the information-passing bridging techniques, I will present the
Generalized Mathematical Homogenization (GMH) theory and the Multiscale
Enrichment based on the Partition of Unity (MEPU) method. The MEPU approach
gives rise to the enriched quasi-continuum formulation, capable of dealing with
heterogeneous inter-atomic potentials as well as with high velocity impact
problems. Among the concurrent bridging techniques, attention is restricted to
multilevel-like methods.
M.
Complex Fluids and Soft Matter
Engineered and biological materials are rarely
described by the classical equations of mathematical physics, either during
production or in performance. Such materials are mixtures, where weak forces
& physical bonds conspire toward properties and behavior over wide ranges
of temporal & spatial scales. Transition phenomena become typical and
keys to devices & high-performance materials, as well as biological feedback
strategies. This session will focus on two model systems, macromolecular fluids
related to polymer nano-composites & biological liquids which perform
vital trapping and transport functions in mammals. Multiscale theory &
experimental phenomena are presented, with the challenge ahead to find synchrony
between them.
Multiscale Computational Challenges in Soft Materials
"Soft" materials have properties
that are determined largely by entropy and by weak effects such as van der Waals, electrostatic and
hydrodynamic forces -- they are often characterized by long-range interactions,
self-organization and slow dynamics.
These materials are important both scientifically and technologically;
examples include colloidal suspensions and aqueous solutions of genomic DNA.
Speakers in this session will address various issues that arise in the
computational study of of soft materials, including
computational fluid dynamics at both the macroscale
and mesoscale (e.g. as arises in microfluidic
devices) as well as efficient determination of equilibrium spatial structure in
inhomogeneous polymeric materials such as block copolymers, an important class
of self-assembling materials. Finally, an experimentally-oriented overview of
the dynamics of granular materials will be presented; these fascinating
materials embody many of the challenges facing modeling and simulation in this
area.
Reduced-order Modeling, Data Compression, and the Design of
Experiments
Reduced-order modeling for complex systems such as the Navier-Stokes equations can be viewed as simply being exercises in data compression and the design of experiments. One advantage of taking this viewpoint is that it naturally brings into play the vast statistical literature on these subjects. Although much of that literature is not relevant to the PDE setting, there are some aspects that bear examination. We review some techniques for reduced-order modeling, including reduced-basis methods, proper orthogonal decomposition, and centroidal Voronoi tessellations. We show the importance of intelligent snapshot generation and how the application of statistical techniques can improve the design of reduced-order models. We illustrate some reduced-order modeling techniques with some simple fluids calculations.
Andrew J. Majda
Courant Institute of Mathematical Sciences
Systematic Multi-Scale Stochastic Modeling and Quantifying
Uncertainty in Atmosphere/Ocean Science
Many of the central problems in improved seasonal and intraseasonal prediction involve the interaction of highly anisotropic physical processes across a wide range of spatio-temporal scales. The interaction and organization of clouds into cluster, supercluster and planetary scale dynamics is the most prominent example where contemporary GCM's are inadequate. The lead talk describes new systematic mathematical strategies for attacking these problems through multi-scale and stochastic techniques as well as new strategies for quantifying predictability through information theory. The supplementary talks will both expand the discussions and demonstrate how the methodologies developed for problems in the geosciences are also directly useful for diverse scientific applications ranging from systematic coarse-grained mesoscopic stochastic models in material science to systematic analysis of the effects of thermal noise for immersed structures in microfluids in cellular biology.
Mark S. Shepard
Rensselaer Polytechnic Institute
Adaptive Modeling and Simulation
The effective application of multiscale modeling technologies will require simulation systems capable of adaptively applying the appropriate models and discretizations over the space and time scales of the problem. This presentation will briefly review some of the methods under development for multiscale modeling discussing issues associated with their adaptive application. Attention will then turn to a discussion of the design and implementation of an interoperable component-based toolkit to support the implementation of adaptive multiscale simulations. This toolkit must support the (i) application of appropriate mathematical models for each relevant scale, (ii) coupling of information in the appropriate forms between scales, (iii) application of model error estimation and model up-grading, and (iv) application of adaptive discretization error control. Some examples of components tools that could be part of an adaptive multiscale simulation toolkit will be presented.
Edge Detections, Hierarchical Decompositions, and Multiscale Nonlinear Dynamics
We discuss three prototype problems which are dominated by the presence of different scales. We begin with data processing involving different regions of smoothness. The interfaces of such regions are detected from their noisy spectral data using separation of scales. We continue with different scales of resolution encountered in image processing. Here we present a novel hierarchical decomposition of texture into different scales of edges. We conclude with a discussion of nonlinear dynamics, involving the passage from kinetic to macroscopic scales. Regularizing effect quantified by the averaging lemma and magnetic reconnection governed by whistler waves demonstrate multiscale dispersive effects.
Joseph E. Flaherty
Rensselaer Polytechnic Institute
Adaptive Multiscale Modeling and Simulation with
Tissue Engineering Applications
We discuss the potential for an adaptive
multiscale modeling and simulation framework that would automate many of the
decisions associated with multiscale analyses while providing maximum flexibility
for experimentation and development. The
major components of such a system would be (i)
mathematical models and solution methodologies for each relevant scale, (ii) an ability to couple information between scales, (iii)
an ability to select appropriate models for space/time scales under
consideration, and (iv) an ability to appraise the accuracy of a model and its
computed solution. We envision a system
that would proceed by solving a system-wide model and invoke smaller scale
models when and where necessary to describe relevant information on smaller
scales. Model selection would be
indicated by estimates of the errors committed when neglecting certain
phenomena predicated by a finer-scale model.
The software and methodology should be applicable to problems in diverse
disciplines; however, we describe an application involving the fabrication of
bio-artificial tissue for use as replacement arteries for coronary disease.