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High Performance Finite Element Methods with Application to Simulation of Vertical Axis Wind Turbines and Diffusion MRI
KTH, School of Electrical Engineering and Computer Science (EECS), Computer Science, Computational Science and Technology (CST).ORCID iD: 0000-0002-3213-0040
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Finite element methods have been developed over decades, and together with the growth of computer power, they become more and more important in dealing with large-scale simulations in science and industry.The objective of this thesis is to develop high-performance finite element methods, with two concrete applications: computational fluid dynamics (CFD) with simulation of turbulent flow past a vertical axis wind turbine (VAWT), and computational diffusion magnetic resonance imaging (CDMRI). The thesis presents contributions in the form of both new numerical methods for high-performance computing frameworks and efficient, tested software, published open source as part of the FEniCS/FEniCS-HPC platform. More specifically, we have four main contributions through the thesis work.

First, we develop a DFS-ALE method which combines the Direct finite element simulation method (DFS) with the Arbitrary Lagrangian-Eulerian method (ALE) to solve the Navier-Stokes equations for a rotating turbine. This method is enhanced with dual-based a posteriori error control and automated mesh adaptation. Turbulent boundary layers are modeled by a slip boundary condition to avoid a full resolution which is impossible even with the most powerful computers available today. The method is validated against experimental data with a good agreement.

Second, we propose a partition of unity finite element method to tackle interface problems. In CFD, it allows for imposing slip velocity boundary conditions on conforming internal interfaces for a fluid-structure interaction model. In CDMRI, it helps to overcome the difficulties that the standard approaches have when imposing the microscopic heterogeneity of the biological tissues and allows for efficient solutions of the Bloch-Torrey equation in heterogeneous domains. The method facilitates a straightforward implementation on the FEniCS/ FEniCS-HPC platform. The method is validated against reference solutions, and the implementation shows a strong parallel scalability.

Third, we propose a finite element discretization on manifolds in order to efficiently simulate the diffusion MRI signal in domains that have a thin layer or a thin tube geometrical structure. The method helps to significantly reduce the required simulation time, computer memory, and difficulties associated with mesh generation, while maintaining the accuracy. Thus, it opens the possibility to simulate complicated structures at a low cost, for a better understanding of diffusion MRI in the brain.

Finally, we propose an efficient portable simulation framework that integrates recent advanced techniques in both mathematics and computer science to enable the users to perform simulations with the Cloud computing technology. The simulation framework consists of Python, IPython and C++ solvers working either on a web browser with Google Colaboratory notebooks or on the Google Cloud Platform with MPI parallelization.

Abstract [sv]

Finita elementmetoder har utvecklats under årtionden, och har, till- sammans med tillväxten i datorkraft, blivit allt viktigare för att utföra storskaliga simuleringar inom både akademin och industrin. Målet med denna avhandling är att utveckla finita elementmetoder med högprestanda, med särskilt fokus på två konkreta applikationer; beräknings- strömningsdynamik (eng. Computational Fluid Dynamics (CFD)) för simulering av turbulent flöde runt en vindturbin, och beräkningar inom diffusionsmagnetresonanstomografi (eng. Computational diffusion magnetic resonance imaging (CDMRI)). Denna avhandling innehåller bidrag till ovanstående områden i form av såväl nya numeriska metoder för högprestandaberäkningsramverk och testad effektiv programvara vilken publicerats som öppen källkod som del av plattformen FEniCS/FEniCS-HPC. Mer specifikt presenterar vi fyra huvudbidrag i detta avhandlingsarbete.

Först utvecklar vi en DFS-ALE-metod som kombinerar Direkt Fini- ta Elementsimulering (DFS) med den Arbiträra Lagrange-Eulermetoden (ALE) för att lösa Navier-Stokes ekvationer för en roterande turbin. Vår metod är en förbättrad variant med dualbaserad a posteriori felkontroll och automatiserad adaptering av beräkningsnätet. Turbulenta gränsskikt modelleras med ett sliprandvillkor för att undvika full upplösning av problemet, vilket är omöjligt även med de mest kraftfulla datorer som finns att tillgå idag. Metoden valideras mot experimentell data, med god överensstämmelse.

Därnäst föreslår vi en enhetspartitions finita element metod för att tackla interfaceproblem. Inom CFD möjliggör detta att påtvinga ett sliprandvillkor på konforma inre interface för en fluidstrukturinter-kationsmodell. Inom CDMRI bidrar det med att överkomma svårigheterna med att påtvinga mikroskopisk heterogenitet av den biologiska vävnaden, och möjliggör effektiv lösning av Bloch-Torrey ekvationen i heterogena domäner. Metoden gör det enklare att göra en rättfram implementering i FEniCS/FEniCS-HPC. Metoden valideras mot referenslösnignar, och implementationen visar på stark parallel skalning (eng. strong parallel scaling).

Sedan föreslår vi en finita elementdisktretisering på mångfalder för att effektivt kunna simulera diffusions-MRI-signaler i områden med en tunn geometrisk struktur. Metoden bidrar med att signifikant reducera simuleringstiden, minnesåtgång och svårigheter associerade med genereringen av beräkningsnät, utan att påverka precisionen i beräkningarna. Detta öppnar för möjligheter att simulera komplicerade strukturer till låg kostnad, för att bättre förstå diffusionsmagnettomografi i hjärnan.

Tilll sist föreslår vi ett effektivt portabelt simuleringsramverk som integrear nya avancerade tekniker inom både matematik och datave- tenskap för att möjliggöra för användaren att utföra simuleringar med datormolnberäkningsteknologin. Simuleringsramverket består av Python, IPython och C++-lösare som används tillsammans antingen i en webbläsare med Google Colaboration notebooks eller på Google Cloud-plattformen med MPI-parallellisering.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2019. , p. 53
Series
TRITA-EECS-AVL ; 2019:76
Keywords [en]
High performance finite element method, computational diffusion MRI, turbulent flow, vertical axis wind turbine, Cloud computing.
Keywords [sv]
högprestanda finita elementmetod, beräkningsdiffusionsmagnetresonanstomografi, turbulent flöde, vertikalaxlad vindturbin, datormolnberäkning
National Category
Natural Sciences Medical and Health Sciences
Research subject
Applied and Computational Mathematics; Biological Physics; Computer Science; Engineering Mechanics; Mathematics; Physics, Biological and Biomedical Physics
Identifiers
URN: urn:nbn:se:kth:diva-263200ISBN: 978-91-7873-337-8 (electronic)OAI: oai:DiVA.org:kth-263200DiVA, id: diva2:1367279
Public defence
2019-12-04, F3, Lindstedtsvägen 26, Stockholm, 10:15 (English)
Opponent
Supervisors
Note

QC 20191105

Available from: 2019-11-05 Created: 2019-11-01 Last updated: 2019-11-11Bibliographically approved
List of papers
1. Direct Finite Element Simulation of the Turbulent Flow Past a Vertical Axis Wind Turbine
Open this publication in new window or tab >>Direct Finite Element Simulation of the Turbulent Flow Past a Vertical Axis Wind Turbine
2019 (English)In: Renewable energy, ISSN 0960-1481, E-ISSN 1879-0682, Vol. 135, p. 238-247Article in journal (Refereed) Published
Abstract [en]

There is today a significant interest in harvesting renewable energy, specifically wind energy, in offshore and urban environments. Vertical axis wind turbines get increasing attention since they are able to capture the wind from any direction. They are relatively easy to install and to transport, cheaper to build and maintain, and quite safe for humans and birds. Detailed computer simulations of the fluid dynamics of wind turbines provide an enhanced understanding of the technology and may guide design improvements. In this paper, we simulate the turbulent flow past a vertical axis wind turbine for a range of rotation angles in parked and rotating conditions. We propose the method of Direct Finite Element Simulation in a rotating ALE framework, abbreviated as DFS-ALE. The simulation results are validated against experimental data in the form of force measurements. We find that the simulation results are stable with respect to mesh refinement and that we capture well the general shape of the variation of force measurements over the rotation angles.

Place, publisher, year, edition, pages
Elsevier, 2019
Keywords
VAWT, Direct FEM simulation, ALE
National Category
Energy Systems
Research subject
Computer Science; Applied and Computational Mathematics; Vehicle and Maritime Engineering
Identifiers
urn:nbn:se:kth:diva-224801 (URN)10.1016/j.renene.2018.11.098 (DOI)000459365600021 ()2-s2.0-85058018814 (Scopus ID)
Note

QC 20180326

Available from: 2018-03-26 Created: 2018-03-26 Last updated: 2019-11-01Bibliographically approved
2. A partition of unity finite element method for computational diffusion MRI
Open this publication in new window or tab >>A partition of unity finite element method for computational diffusion MRI
2018 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 375, p. 271-290Article in journal (Refereed) Published
Abstract [en]

The Bloch–Torrey equation describes the evolution of the spin (usually water proton) magnetization under the influence of applied magnetic field gradients and is commonly used in numerical simulations for diffusion MRI and NMR. Microscopic heterogeneity inside the imaging voxel is modeled by interfaces inside the simulation domain, where a discontinuity in the magnetization across the interfaces is produced via a permeability coefficient on the interfaces. To avoid having to simulate on a computational domain that is the size of an entire imaging voxel, which is often much larger than the scale of the microscopic heterogeneity as well as the mean spin diffusion displacement, smaller representative volumes of the imaging medium can be used as the simulation domain. In this case, the exterior boundaries of a representative volume either must be far away from the initial positions of the spins or suitable boundary conditions must be found to allow the movement of spins across these exterior boundaries.

Many approaches have been taken to solve the Bloch–Torrey equation but an efficient high-performance computing framework is still missing. In this paper, we present formulations of the interface as well as the exterior boundary conditions that are computationally efficient and suitable for arbitrary order finite elements and parallelization. In particular, the formulations are based on the partition of unity concept which allows for a discontinuous solution across interfaces conforming with the mesh with weak enforcement of real (in the case of interior interfaces) and artificial (in the case of exterior boundaries) permeability conditions as well as an operator splitting for the exterior boundary conditions. The method is straightforward to implement and it is available in FEniCS for moderate-scale simulations and in FEniCS-HPC for large-scale simulations. The order of accuracy of the resulting method is validated in numerical tests and a good scalability is shown for the parallel implementation. We show that the simulated dMRI signals offer good approximations to reference signals in cases where the latter are available and we performed simulations for a realistic model of a neuron to show that the method can be used for complex geometries.

Place, publisher, year, edition, pages
Elsevier, 2018
Keywords
Computational diffusion MRI, Bloch–Torrey equation, Partition of unity finite element method, Interface conditions, Weak pseudo-periodic conditions, FEniCS/FEniCS-HPC
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics; Biological Physics; Computer Science
Identifiers
urn:nbn:se:kth:diva-234286 (URN)10.1016/j.jcp.2018.08.039 (DOI)000450907600014 ()2-s2.0-85054048672 (Scopus ID)
Funder
Swedish Energy Agency, P40435-1
Note

QC 20180906

Available from: 2018-09-06 Created: 2018-09-06 Last updated: 2019-11-01Bibliographically approved
3. A fluid-structure interaction model with weak slip velocity boundary conditions on conforming internal interfaces
Open this publication in new window or tab >>A fluid-structure interaction model with weak slip velocity boundary conditions on conforming internal interfaces
Show others...
2018 (English)Conference paper, Published paper (Other (popular science, discussion, etc.))
Abstract [en]

We develop a PUFEM–Partition of Unity Finite Element Method to impose slip velocity boundary conditions on conforming internal interfaces for a fluid-structure interaction model. The method facilitates a straightforward implementation on the FEniCS/FEniCS-HPC platform. We show two results for 2D model problems with the implementation on FEniCS: (1) optimal convergence rate is shown for a stationary Navier-Stokes flow problem, and (2) the slip velocity conditions give qualitatively the correct result for the Euler flow. 

Keywords
fluid-structure interaction, slip boundary conditions, conforming meshes, internal interfaces
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics; Computer Science
Identifiers
urn:nbn:se:kth:diva-225143 (URN)
Conference
6th European Conference on Computational Mechanics (ECCM), 7th European Conference on Computational Fluid Dynamics (ECFD 7), 1115 June 2018, Glasgow, UK
Note

QC 20190215

Available from: 2018-03-31 Created: 2018-03-31 Last updated: 2019-11-01Bibliographically approved
4. Diffusion MRI simulation in thin-layer and thin-tube media using a discretization on manifolds
Open this publication in new window or tab >>Diffusion MRI simulation in thin-layer and thin-tube media using a discretization on manifolds
Show others...
2019 (English)In: Journal of magnetic resonance, ISSN 1090-7807, E-ISSN 1096-0856, Vol. 299, p. 176-187Article in journal (Refereed) Published
Abstract [en]

The Bloch-Torrey partial differential equation can be used to describe the evolution of the transverse magnetization of the imaged sample under the influence of diffusion-encoding magnetic field gradients inside the MRI scanner. The integral of the magnetization inside a voxel gives the simulated diffusion MRI signal. This paper proposes a finite element discretization on manifolds in order to efficiently simulate the diffusion MRI signal in domains that have a thin layer or a thin tube geometrical structure. The variable thickness of the three-dimensional domains is included in the weak formulation established on the manifolds. We conducted a numerical study of the proposed approach by simulating the diffusion MRI signals from the extracellular space (a thin layer medium) and from neurons (a thin tube medium), comparing the results with the reference signals obtained using a standard three-dimensional finite element discretization. We show good agreements between the simulated signals using our proposed method and the reference signals for a wide range of diffusion MRI parameters. The approximation becomes better as the diffusion time increases. The method helps to significantly reduce the required simulation time, computational memory, and difficulties associated with mesh generation, thus opening the possibilities to simulating complicated structures at low cost for a better understanding of diffusion MRI in the brain.

Place, publisher, year, edition, pages
Academic Press, 2019
Keywords
Diffusion MRI; finite element method; Bloch-Torrey equation; FEniCS; thin layer; thin tube.
National Category
Medical and Health Sciences
Research subject
Applied and Computational Mathematics; Computer Science
Identifiers
urn:nbn:se:kth:diva-235070 (URN)10.1016/j.jmr.2019.01.002 (DOI)000460655200018 ()2-s2.0-85059768594 (Scopus ID)
Funder
Swedish Energy Agency, P40435-1
Note

QC 20180919

Available from: 2018-09-14 Created: 2018-09-14 Last updated: 2019-11-01Bibliographically approved
5. Portable simulation framework for diffusion MRI
Open this publication in new window or tab >>Portable simulation framework for diffusion MRI
Show others...
2019 (English)In: Journal of magnetic resonance, ISSN 1090-7807, E-ISSN 1096-0856, Vol. 309, article id 106611Article in journal (Refereed) Published
Abstract [en]

The numerical simulation of the diffusion MRI signal arising from complex tissue micro-structures is helpful for understanding and interpreting imaging data as well as for designing and optimizing MRI sequences. The discretization of the Bloch-Torrey equation by finite elements is a more recently developed approach for this purpose, in contrast to random walk simulations, which has a longer history. While finite elements discretization is more difficult to implement than random walk simulations, the approach benefits from a long history of theoretical and numerical developments by the mathematical and engineering communities. In particular, software packages for the automated solutions of partial differential equations using finite elements discretization, such as FEniCS, are undergoing active support and development. However, because diffusion MRI simulation is a relatively new application area, there is still a gap between the simulation needs of the MRI community and the available tools provided by finite elements software packages. In this paper, we address two potential difficulties in using FEniCS for diffusion MRI simulation. First, we simplified software installation by the use of FEniCS containers that are completely portable across multiple platforms. Second, we provide a portable simulation framework based on Python and whose code is open source. This simulation framework can be seamlessly integrated with cloud computing resources such as Google Colaboratory notebooks working on a web browser or with Google Cloud Platform with MPI parallelization. We show examples illustrating the accuracy, the computational times, and parallel computing capabilities. The framework contributes to reproducible science and open-source software in computational diffusion MRI with the hope that it will help to speed up method developments and stimulate research collaborations.

Place, publisher, year, edition, pages
Academic Press, 2019
Keywords
Cloud computing, diffusion MRI, Bloch-Torrey equation, interface conditions, pseudo-periodic conditions, FEniCS.
National Category
Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-256328 (URN)10.1016/j.jmr.2019.106611 (DOI)000497799500005 ()31574354 (PubMedID)2-s2.0-85072714990 (Scopus ID)
Note

QC 20190822

Available from: 2019-08-21 Created: 2019-08-21 Last updated: 2019-12-13Bibliographically approved
6. Technical Report -- Comparison of Direct Finite Element Simulation with Actuator Line Models and Vortex Models for Simulation of Turbulent Flow Past a Vertical Axis wind Turbine
Open this publication in new window or tab >>Technical Report -- Comparison of Direct Finite Element Simulation with Actuator Line Models and Vortex Models for Simulation of Turbulent Flow Past a Vertical Axis wind Turbine
2019 (English)Report (Other (popular science, discussion, etc.))
Abstract [en]

We compare three different methodologies for simulation of turbulent flow past a vertical axis wind turbine: (i) full resolution of the turbine blades in a Direct Finite Element Simulation (DFS), (ii) implicit representation of the turbine blades in a 3D Actuator Line Method (ALM), and (iii) implicit representation of the turbine blades as sources in a Vortex Model (VM). The integrated normal force on one blade is computed for a range of azimuthal angles, and is compared to experimental data for the different tip speed ratios, 2.55, 3.44 and 4.09.

Publisher
p. 11
Keywords
VAWT, DFS-ALE, Actuator Line Models, Vortex Models
National Category
Engineering and Technology Natural Sciences
Identifiers
urn:nbn:se:kth:diva-257890 (URN)
Note

QC 20190909

Available from: 2019-09-08 Created: 2019-09-08 Last updated: 2019-11-01Bibliographically approved

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Citation style
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  • harvard1
  • ieee
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  • vancouver
  • Other style
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  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
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