| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | June 29, 2014 |
| Latest Amendment Date: | June 29, 2014 |
| Award Number: | 1419060 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Leland Jameson
DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | July 1, 2014 |
| End Date: | June 30, 2018 (Estimated) |
| Total Intended Award Amount: | $248,384.00 |
| Total Awarded Amount to Date: | $248,384.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
201 DOWMAN DR NE ATLANTA GA US 30322-1061 (404)727-2503 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
400 Dowman Dr Atlanta GA US 30322-1005 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | COMPUTATIONAL MATHEMATICS |
| Primary Program Source: |
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| Program Reference Code(s): |
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| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
Networks perfused by fluids are found in several engineering applications, ranging from hydrogeology, oil distribution, and internal combustion engines to hemodynamics. A quantitative analysis of these problems is of utmost interest for understanding fluid dynamics in the network, for predicting effects of local changes on the network (for instance, the effects of a surgical operation over the fluid dynamics in the arterial tree), and for optimizing flow distribution. Mathematical description and numerical approximation of these problems are challenging when coupling the accurate description of local dynamics with the large scale of the network. This proposal investigates a novel numerical method to undertake the quantitative analysis of fluid dynamics in complex networks called HiMod (Hierarchical Model Reduction). The primary (but not exclusive) application is the physiopathology of the arterial system, including in the mathematical model up to almost 2000 segments of the network. Several specific properties of this method need to be investigated for its development and engineering. The research provides a graduate student the opportunity of working on advanced mathematical and numerical techniques - including theoretical as well as practical aspects - in a truly interdisciplinary framework with frequent contacts with engineers and doctors expected to be the end users of these methodologies.
Network of pipes are often modeled by assembling simplified equations describing each segment, like the well known Euler equations. Originally proposed for blood flow (incompressible fluid in compliant pipes) they have been extensively used in gas dynamics - for instance - in internal combustion engines (compressible fluid in rigid pipes). These equations are the result of several approximations to reduce the fully 3D mathematical model to a 1D set of hyperbolic equations. Unfortunately, this model reduction prevents proper capture the local features of the network that affects the global dynamics. The HiMod approach moves from a different perspective. We couple different numerical approximation techniques along the mainstream and the transversal directions. We use a finite element approximation for the mainstream and a spectral or modal approximation transversally. The number of modes can be locally and adaptively tuned to get the best possible trade-off between accuracy and computational efficiency. The rationale is that a relatively small number of modes is enough to guarantee good accuracy for the transversal dynamics, leading to a system of 1D problems (called a "psychologically 1D" model). Moving from preliminary promising studies for advection-diffusion problems, in this proposal we aim at developing the method for the 3D incompressible Navier-Stokes equations and fluid-structure interaction problems. Inf-sup stability and accuracy of the HiMod discretization as well as its role as preconditioner of the full problem will be investigated, together with adaptive techniques for the appropriate (automatic) selection of the transversal modes.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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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.
Introduction
This project deals with the analysis and implementation of a novel technique for the solution of partial differential equations in domains with a prevalent direction. Specifically, the project was considering problems of incompressible fluid dynamics, with special emphasis on biomedical applications (blood flow problems). The method was new at the time of the proposal, with many conceptual aspects to understand, many theoretical features to investigate and rigorously analyze and a significant implementation and testing stage to complete. The inclusion of fluid-structure interaction problems was part of the tasks to achieve.
The method is now available as a conceptual framework to the scientific community as a customized, efficient alternative to purely one-dimensional approaches. The modularity, i.e., the relatively easy (and also, possibly, automated) selection of the modal discretization to calibrate the level of accuracy for the transverse dynamics makes this approach an exciting tool for a wide range of applications. In the experience of the PI, the analysis of networks of fluids has an interest in mechanical engineering (internal combustion engines), geophysical and environmental applications (rivers and pipes), biomedical applications (blood flow problems).
Most of the prospected tasks presented in the proposal were successfully accomplished.
Specific Goals
Specifically, we implemented our method in two research software libraries, including:
- rectangular/circular pipes
- polar/cartesian coordinates
- general boundary conditions
- Iso-Geometric Analysis discretization of the axial coordinate
- Direct and Inverse problems (parameter identification).
We performed the theoretical analysis of the method, obtaining several results.
- The convergence rate we proved in rectangular pipes shows that the accuracy scales with the expected dependence on the polynomial degree of the finite element component and the number of modes for the spectral component.
- The polar coordinate systems are generally more accurate than methods based on an adjustment of cartesian coordinates for cylindrical pipes; yet, they are more exposed to numerical errors and less versatile for real applications.
- While different pairs of finite dimensional pairs of HiMod spaces were tested to be Inf-Sup stable for the solution of the incompressible Navier-Stokes equations, we have been able of proving a rigorous proof for a special case (full Dirichlet problem in 2D with Legendre polynomials for the spectral basis and regular finite elements for the axial discretization).
On the side of numerical testing, we demonstrated numerically that coupling Iso-Geometric Analysis for the axial discretization with the spectral approximation for the transverse discretization yields excellent results, particularly for curved pipes.
We also implemented the method on real cases of patient-specific coronaries, demonstrating that HiMOD (and its Transversally Enriched Finite Element formulation) outperforms largely the numerical approximation based on Finite Elements, with computational costs more similar to one-dimensional solvers.
The method was also applied for the solution of inverse problems, in particular, parameter identification with the combination of Kalman filtering techniques.
The project achieved a substantial understanding of the potentiality of this technique and promoted its use to the scientific community.
Recently, we applied this methodology to a hot topic related to the uncertainty quantification in the simulation of blood flow problems. This application seems to be an extremely promising advancement not only for the mathematical aspects but also for the penetration of numerical tools in the clinical practice of cardiovascular diseases, which is one of the distinctive traits of the PI research. In fact, clinical practice is progressively relying on numerical tools (as the recent case of the company HeartFlow demonstrates), having effective and reliable solvers is crucial for the quality certification of numerical simulations.
Deliverables
The code implementing the method is made available to the community upon request.
The website https://sites.google.com/view/himod/home provides information and links.
The Emory branch of the library lifeV (www.lifev.org) incorporates the results of the project and will be linked to the web page after some final bug-fixing.
The research has produced several papers, peer-reviewed proceedings, and reports.
Three Ph.D. students have been involved in the project, one has already defended his dissertation. Sofia Guzzetti, the Ph.D. student supported entirely by the project, will defend her dissertation in Spring 2019.
Human resources and network
The project offered the opportunity for training young resources, for creating new collaborations and networks, for outreaching young (high school) students.
In addition to the collaborative framework already prospected in the project (P. Blanco and S. Perotto), new collaborations were originated by the project: - A. Reali with the University of Pavia, Italy, for the coupling of the IGA for the axial approximation with the spectral approximation for transverse dynamics; - K. Carlberg with Sandia at Livermore, for the DDUQ application of TEPEM in cardiovascular applications.
The dissemination of the activities led to the organization of several Minisymposia at national and international relevant conferences as listed in a subsequent section. In addition, with the support of the project, we organized two meetings at Emory.
Last Modified: 09/29/2018
Modified by: Alessandro Veneziani
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