Award Abstract # 0927587
DynSyst_Special_Topics: Collaborative Research: Reduced Dynamical Descriptions of Infinite-Dimensional Nonlinear systems via a-Priori Basis Functions from Upper Bound Theories

NSF Org: DMS
Division Of Mathematical Sciences
Recipient: REGENTS OF THE UNIVERSITY OF MICHIGAN
Initial Amendment Date: September 11, 2009
Latest Amendment Date: February 14, 2011
Award Number: 0927587
Award Instrument: Standard Grant
Program Manager: Victor Roytburd
DMS
 Division Of Mathematical Sciences
MPS
 Directorate for Mathematical and Physical Sciences
Start Date: September 1, 2009
End Date: August 31, 2013 (Estimated)
Total Intended Award Amount: $240,000.00
Total Awarded Amount to Date: $264,000.00
Funds Obligated to Date: FY 2009 = $240,000.00
FY 2011 = $24,000.00
History of Investigator:
  • Charles Doering (Principal Investigator)
    doering@umich.edu
Recipient Sponsored Research Office: Regents of the University of Michigan - Ann Arbor
1109 GEDDES AVE STE 3300
ANN ARBOR
MI  US  48109-1015
(734)763-6438
Sponsor Congressional District: 06
Primary Place of Performance: Regents of the University of Michigan - Ann Arbor
1109 GEDDES AVE STE 3300
ANN ARBOR
MI  US  48109-1015
Primary Place of Performance
Congressional District:
06
Unique Entity Identifier (UEI): GNJ7BBP73WE9
Parent UEI:
NSF Program(s): APPLIED MATHEMATICS,
DYNAMICAL SYSTEMS
Primary Program Source: 01000910DB NSF RESEARCH & RELATED ACTIVIT
01001112DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s): 0000, 9251, OTHR
Program Element Code(s): 126600, 747800
Award Agency Code: 4900
Fund Agency Code: 4900
Assistance Listing Number(s): 47.049

ABSTRACT

The aim of this interdisciplinary collaborative research project is to develop a novel model reduction technique for forced dissipative infinite-dimensional dynamical systems by employing basis functions computed using upper bound theories. Like popular Proper Orthogonal Decomposition (POD) based methods, this approach associates the condensed variables needed for model reduction with coherent structures and captures nonlinear interactions between these linear modes via Galerkin projection and finite-dimensional truncation. Unlike empirical POD methods, however, this new method does not require extensive data sets from experiments or direct numerical simulations of the governing partial differential equations (PDEs) and thus yields truly predictive reduced models. The theoretical and computational methodology will be developed in the context of a particular physical system, thermal convection in fluid saturated porous media, that is of considerable environmental and technological importance and an ideal testbed for new ideas.

This research will contribute to the development of a general methodology for deriving simplified mathematical models of highly complex dynamical systems arising in diverse areas of science and engineering. In many applications of interest (e.g., control of various fluid flows to achieve drag reduction for oil pumped in pipelines or for air flowing past commercial jets, or for estimation of carbon dioxide sequestration by porous rock material for reducing global warming), direct numerical simulations based on the complete governing mathematical equations are infeasible using even the world's fastest high-performance supercomputers. This project will address these challenges using novel mathematical techniques to derive simplified equations directly from the governing physical laws that are amenable to practical computation and analysis.

PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH

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Baole Wen, Navid Dianati, Evelyn Lunasin, Gregory P. Chini, Charles R. Doering "New upper bounds and reduced dynamical modeling for Rayleigh-Benard convection in a fluid saturated porous layer" Communications in Nonlinear Science and Numerical Simulation , v.17 , 2012 , p.2191 10.1016/j.cnsns.2011.06.039
B. Wen, G.P. Chini, N. Dianti, and C.R. Doering "Computational approaches to aspect-ratio-dependent upper bounds and heat flux in porousmedium convection" Physics Letters A , v.377 , 2013 , p.2931
Chini, GP; Dianati, N; Zhang, ZX; Doering, CR "Low-dimensional models from upper bound theory" PHYSICA D-NONLINEAR PHENOMENA , v.240 , 2011 , p.241 View record at Web of Science 10.1016/j.physd.2010.06.01

PROJECT OUTCOMES REPORT

Disclaimer

This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.

This project focused on the development and application of mathematical model reduction (simplification) methods for scientific problems in select areas of theoretical fluid dynamics, ecology, and evolutionary biology.  The goal was to both investigate the accuracy and efficacy of a novel approach based on previous analysis in mathematical fluid mechanics, and to study some particular problems of current scientific interest.  The fluid mechanical problem we focused on was that of buoyancy-driven currents in a fluid saturated prorous medium, a classical problem that has seen renewed interest in connection with CO2 sequestration (climate science and technology).  New results concerning the maximal rate of sequestration were obtained.  Problems addressed in ecology and evolutionary biology involved reduced modeling studies of age-structured trophic dynamics and the effect of demographic stochasticity on evolutionary selection mechanisms.  In all cases significant scientific insights were accompanied by the development of new mathematical methods for the analysis.  Research results supported by this funding have been written up and published and/or submitted for publication in refereed scientific journals.  Moreover, the project supported the doctoral dissertation research for two PhD students, the masters thesis research for another graduate student, and provided original research experiences for one postdoc and five undergraduate students.


Last Modified: 11/04/2013
Modified by: Charles R Doering

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