Award Abstract # 2012371
RUI: Computational Models for Coupled Free/Porous Media Flow

NSF Org: DMS
Division Of Mathematical Sciences
Recipient: THE RESEARCH FOUNDATION FOR THE STATE UNIVERSITY OF NEW YORK
Initial Amendment Date: June 1, 2020
Latest Amendment Date: July 22, 2022
Award Number: 2012371
Award Instrument: Continuing Grant
Program Manager: Yuliya Gorb
ygorb@nsf.gov
 (703)292-2113
DMS
 Division Of Mathematical Sciences
MPS
 Directorate for Mathematical and Physical Sciences
Start Date: September 1, 2020
End Date: August 31, 2024 (Estimated)
Total Intended Award Amount: $160,114.00
Total Awarded Amount to Date: $160,114.00
Funds Obligated to Date: FY 2020 = $54,479.00
FY 2021 = $52,375.00

FY 2022 = $53,260.00
History of Investigator:
  • Svetlana Tlupova (Principal Investigator)
    tlupovs@farmingdale.edu
Recipient Sponsored Research Office: SUNY College of Technology Farmingdale
2350 BROADHOLLOW RD
FARMINGDALE
NY  US  11735-1006
(631)420-2687
Sponsor Congressional District: 02
Primary Place of Performance: FARMINGDALE STATE COLLEGE
2350 Broadhollow Road
Farmingdale
NY  US  11735-1021
Primary Place of Performance
Congressional District:
02
Unique Entity Identifier (UEI): NK5CPKU3K7T8
Parent UEI:
NSF Program(s): COMPUTATIONAL MATHEMATICS
Primary Program Source: 01002021DB NSF RESEARCH & RELATED ACTIVIT
01002122DB NSF RESEARCH & RELATED ACTIVIT

01002223DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s): 9263
Program Element Code(s): 127100
Award Agency Code: 4900
Fund Agency Code: 4900
Assistance Listing Number(s): 47.049

ABSTRACT

This project will study fluid flow models with applications in human health and the environment. The focus is on models that combine a free fluid part and a porous media part. For example, in physiology, such fluid flow models help to understand transport of oxygen and nutrients between blood vessels and tissue. In hydrology, such models can be used to study and predict contamination by toxic chemicals from leaky underground storage tanks or landfills mixing with groundwater. In industrial filtration, the flow models studied in this project can be incorporated into more complex models to optimize the design of the three-way catalytic converter used to reduce vehicle emission levels. Efficient computer simulation of these coupled free/porous flows remains challenging due to the multi-physics nature of the systems. This research aims to develop accurate and efficient numerical simulation techniques for such models. The project provides training through undergraduate research experiences.

This project studies models of coupled flow systems using the incompressible Stokes equations in the free domain and the Darcy equations in the porous domain. An advection-diffusion equation is added to the system to model the transport phenomena at the free/porous interface. The research aims to develop, analyze, and implement an integrated numerical model of this system of equations in three dimensions. The main goals are to (i) address severe limitations of the direct solution, due to incompatibilities in the differential operators in the free and porous subdomains, (ii) develop robust, high accuracy, and well-conditioned integral equation formulations, (iii) apply rigorous analysis on the iterative methods to deduce optimal convergence, and (iv) add the ability to combine different methods of discretization to effectively model heterogeneous porous media. The project will develop a new boundary integral formulation, with regularization and correction for high accuracy, and a simple quadrature based on implicit representation of the surface. Efficient domain decomposition methods with new transmission conditions based on non-local operators will be used to speed up the iteration process. High-performance computing techniques and a kernel-independent treecode algorithm will be implemented to increase the speed of computations.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH

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Beale, J. Thomas and Jones, Christina and Reale, Jillian and Tlupova, Svetlana "A novel regularization for higher accuracy in the solution of the 3-dimensional Stokes flow" Involve, a Journal of Mathematics , v.15 , 2022 https://doi.org/10.2140/involve.2022.15.515 Citation Details
Beale, J Thomas and Storm, Michael and Tlupova, Svetlana "The adjoint double layer potential on smooth surfaces in $\mathbb {R}^3$ and the Neumann problem" Advances in Computational Mathematics , v.50 , 2024 https://doi.org/10.1007/s10444-024-10111-0 Citation Details
Beale, J Thomas and Tlupova, Svetlana "Extrapolated regularization of nearly singular integrals on surfaces" Advances in Computational Mathematics , v.50 , 2024 https://doi.org/10.1007/s10444-024-10161-4 Citation Details
Boateng, Henry A and Tlupova, Svetlana "The Effect of Global Smoothness on the Accuracy of Treecodes" Communications in Computational Physics , v.32 , 2022 https://doi.org/10.4208/cicp.OA-2022-0153 Citation Details
Boateng, Henry A. and Tlupova, Svetlana "A treecode algorithm based on tricubic interpolation" Journal of Computational Mathematics and Data Science , v.5 , 2022 https://doi.org/10.1016/j.jcmds.2022.100068 Citation Details
Tlupova, Svetlana "A domain decomposition solution of the Stokes-Darcy system in 3D based on boundary integrals" Journal of Computational Physics , v.450 , 2022 https://doi.org/10.1016/j.jcp.2021.110824 Citation Details

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.

The aim of this project was the development, analysis, and implementation of accurate and efficient numerical methods to solve multiphysics problems where a fluid flowing in a free domain (modeled by the Stokes equations) is coupled with flow in a porous medium (modeled by Darcy's law). The PI developed the first solution for this system based on the boundary integral equation method in three dimensions. A natural approach for such multiphysics problems is to use a domain decomposition algorithm where the Darcy and Stokes problems are solved independently through an iterative process, while exchanging information through the interface. Efficient and highly accurate boundary integral equations were formulated in both subdomains. We also conducted a convergence analysis using spherical harmonics representations of solutions. Providing such an analysis ensures that the algorithm has guaranteed convergence and provides guidance on the design of alternative transmission conditions with optimal convergence properties.

Boundary integral equation methods are extremely powerful in solving many problems, with the main challenges being the accurate evaluation on and off the boundary, and the computational cost of solving dense matrix systems that result from discretizing the integrals. In addressing the issue of accuracy, J. Thomas Beale and the PI developed a remarkably straightforward method based on extrapolated regularization. The kernels in the integrals are smoothed out using a length parameter, then evaluated for several values of this parameter to obtain an extrapolated value that achieves high accuracy uniformly for targets on and off the boundary. This method is expected to be useful in a very wide range of applications in fluid dynamics and electromagnetics where high accuracy is required particularly when several boundaries are near touching, or quantities are needed on a background grid.

When the boundary integrals are discretized, the resulting large dense matrix systems are typically solved using an iterative method, and this can be prohibitively slow for large systems. Together with Henry A. Boateng, the PI developed a treecode method based on local tricubic interpolation. Such an algorithm can be applied to evaluate the discretized integral equations in nearly linear time. We also showed that the global smoothness properties of the kernel approximations have an effect on the overall accuracy of the treecode algorithm. Using a smoother approximation, higher accuracy can be obtained particularly when evaluating the derivatives of the discretized integral. This has the potential, for example, of better preserving the divergence-free properties, or allowing larger time steps when evolving a surface.

Six undergraduate students from SUNY Farmingdale were involved in various research projects related to this project; 2 out of the total 6 published papers were done in collaboration with undergraduates, with another paper in preparation. One of the students has gone on to a Ph.D. program at the New Jersey Institute of Technology. The codes created as part of this work have been made available on GitHub.

 


Last Modified: 12/04/2024
Modified by: Svetlana Tlupova

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