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Fast and accurate integral equation methods with applications in microfluidics
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.ORCID iD: 0000-0001-7425-8029
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is concerned with computational methods for fluid flows on the microscale, also known as microfluidics. This is motivated by current research in biological physics and miniaturization technology, where there is a need to understand complex flows involving microscale structures. Numerical simulations are an important tool for doing this.

The first, and smaller, part of the thesis presents a numerical method for simulating multiphase flows involving insoluble surfactants and moving contact lines. The method is based on an interface decomposition resulting in local, Eulerian grid representations. This provides a natural setting for solving the PDE governing the surfactant concentration on the interface.

The second, and larger, part of the thesis is concerned with a framework for simulating large systems of rigid particles in three-dimensional, periodic viscous flow using a boundary integral formulation. This framework can solve the underlying flow equations to high accuracy, due to the accurate nature of surface quadrature. It is also fast, due to the natural coupling between boundary integral methods and fast summation methods.

The development of the boundary integral framework spans several different fields of numerical analysis. For fast computations of large systems, a fast Ewald summation method known as Spectral Ewald is adapted to work with the Stokes double layer potential. For accurate numerical integration, a method known as Quadrature by Expansion is developed for this same potential, and also accelerated through a scheme based on geometrical symmetries. To better understand the errors accompanying this quadrature method, an error analysis based on contour integration and calculus of residues is carried out, resulting in highly accurate error estimates.

Abstract [sv]

Denna avhandling behandlar beräkningsmetoder för strömning på mikroskalan, även känt som mikrofluidik. Detta val av ämne motiveras av aktuell forskning inom biologisk fysik och miniatyrisering, där det ofta finns ett behov av att förstå komplexa flöden med strukturer på mikroskalan. Datorsimuleringar är ett viktigt verktyg för att öka den förståelsen.

Avhandlingens första, och mindre, del beskriver en numerisk metod för att simulera flerfasflöden med olösliga surfaktanter och rörliga kontaktlinjer. Metoden är baserad på en uppdelning av gränsskiktet, som tillåter det att representeras med lokala, Euleriska nät. Detta skapar naturliga förutsättningar för lösning av den PDE som styr surfaktantkoncentrationen på gränsskiktets yta.

Avhandlingens andra, och större, del beskriver ett ramverk för att med hjälp av en randintegralformulering simulera stora system av styva partiklar i tredimensionellt, periodiskt Stokesflöde. Detta ramverk kan lösa flödesekvationerna mycket noggrant, tack vare den inneboende höga noggrannheten hos metoder för numerisk integration på släta ytor. Metoden är också snabb, tack vare den naturliga kopplingen mellan randintegralmetoder och snabba summeringsmetoder.

Utvecklingen av ramverket för partikelsimuleringar täcker ett brett spektrum av ämnet numerisk analys. För snabba beräkningar på stora system används en snabb Ewaldsummeringsmetod vid namn spektral Ewald. Denna metod har anpassats för att fungera med den randintegralformulering för Stokesflöde som används. För noggrann numerisk integration används en metod kallad expansionskvadratur (eng. Quadrature by Expansion), som också har utvecklats för att passa samma Stokesformulering. Denna metod har även gjorts snabbare genom en nyutvecklad metod baserad på geometriska symmetrier. För att bättre förstå kvadraturmetodens inneboende fel har en analys baserad på konturintegraler och residykalkyl utförts, vilket har resulterat i väldigt noggranna felestimat.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2016. , 51 p.
Series
TRITA-MAT-A, 2016:03
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-185758ISBN: 978-91-7595-962-7 (print)OAI: oai:DiVA.org:kth-185758DiVA: diva2:923458
Public defence
2016-06-02, F3, Lindstedtsvägen 26, Stockholm, 10:00 (English)
Opponent
Supervisors
Funder
Swedish Research Council, 2011-3178Swedish Research Council, 2007-6375
Note

QC 20160427

Available from: 2016-04-27 Created: 2016-04-26 Last updated: 2016-04-27Bibliographically approved
List of papers
1. An explicit Eulerian method for multiphase flow with contact line dynamics and insoluble surfactant
Open this publication in new window or tab >>An explicit Eulerian method for multiphase flow with contact line dynamics and insoluble surfactant
2014 (English)In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 101, 50-63 p.Article in journal (Refereed) Published
Abstract [en]

The flow behavior of many multiphase flow applications is greatly influenced by wetting properties and the presence of surfactants. We present a numerical method for two-phase flow with insoluble surfactants and contact line dynamics in two dimensions. The method is based on decomposing the interface between two fluids into segments, which are explicitly represented on a local Eulerian grid. It provides a natural framework for treating the surfactant concentration equation, which is solved locally on each segment. An accurate numerical method for the coupled interface/surfactant system is given. The system is coupled to the Navier-Stokes equations through the immersed boundary method, and we discuss the issue of force regularization in wetting problems, when the interface touches the boundary of the domain. We use the method to illustrate how the presence of surfactants influences the behavior of free and wetting drops.

Keyword
Multiphase flow, Insoluble surfactant, Marangoni force, Moving contact line, Immersed boundary method
National Category
Fluid Mechanics and Acoustics
Identifiers
urn:nbn:se:kth:diva-48763 (URN)10.1016/j.compfluid.2014.05.029 (DOI)000340851500005 ()2-s2.0-84903152815 (Scopus ID)
Funder
Swedish Research Council, 621-2007-6375
Note

QC 20140919. Updated from accepted to published.

Available from: 2011-11-23 Created: 2011-11-23 Last updated: 2017-12-08Bibliographically approved
2. Fast Ewald summation for Stokesian particle suspensions
Open this publication in new window or tab >>Fast Ewald summation for Stokesian particle suspensions
2014 (English)In: International Journal for Numerical Methods in Fluids, ISSN 0271-2091, E-ISSN 1097-0363, Vol. 76, no 10, 669-698 p.Article in journal (Refereed) Published
Abstract [en]

We present a numerical method for suspensions of spheroids of arbitrary aspect ratio, which sediment under gravity. The method is based on a periodized boundary integral formulation using the Stokes double layer potential. The resulting discrete system is solved iteratively using generalized minimal residual accelerated by the spectral Ewald method, which reduces the computational complexity to O(N log N), where N is the number of points used to discretize the particle surfaces. We develop predictive error estimates, which can be used to optimize the choice of parameters in the Ewald summation. Numerical tests show that the method is well conditioned and provides good accuracy when validated against reference solutions. 

Place, publisher, year, edition, pages
John Wiley & Sons, 2014
Keyword
viscous flows, integral equations, error estimation, microfluidics, multibody dynamics, spectral, double layer, boundary integral, ewald summation
National Category
Fluid Mechanics and Acoustics Computational Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-116383 (URN)10.1002/fld.3953 (DOI)000344349000004 ()
Funder
Swedish Research Council, 2011-3178
Note

QC 20141119

Available from: 2013-01-17 Created: 2013-01-17 Last updated: 2017-12-06Bibliographically approved
3. Estimation of quadrature errors in layer potential evaluation using quadrature by expansion
Open this publication in new window or tab >>Estimation of quadrature errors in layer potential evaluation using quadrature by expansion
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In boundary integral methods it is often necessary to evaluate layer potentials on or close to the boundary, where the underlying integral is difficult to evaluate numerically. Quadrature by expansion (QBX) is a new method for dealing with such integrals, and it is based on forming a local expansion of the layer potential close to the boundary. In doing so, one introduces a new quadrature error due to nearly singular integration in the evaluation of expansion coefficients. Using a method based on contour integration and calculus of residues, the quadrature error of nearly singular integrals can be accurately estimated. This makes it possible to derive accurate estimates for the quadrature errors related to QBX, when applied to layer potentials in two and three dimensions. As examples we derive estimates for the Laplace and Helmholtz single layer potentials. These results can be used for parameter selection in practical applications.

National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-185511 (URN)
Funder
Swedish Research Council, 2011-3178
Note

QC 20160426

Available from: 2016-04-21 Created: 2016-04-21 Last updated: 2016-04-27Bibliographically approved
4. A fast integral equation method for solid particles in viscous flow using quadrature by expansion
Open this publication in new window or tab >>A fast integral equation method for solid particles in viscous flow using quadrature by expansion
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Boundary integral methods are advantageous when simulating viscous flow around rigid particles, due to the reduction in number of unknowns and straightforward handling of the geometry. In this work we present a fast and accurate framework for simulating spheroids in periodic Stokes flow, which is based on the completed double layer boundary integral formulation. The framework implements a new method known as quadrature by expansion (QBX), which uses surrogate local expansions of the layer potential to evaluate it to very high accuracy both on and off the particle surfaces. This quadrature method is accelerated through a newly developed precomputation scheme. The long range interactions are computed using the spectral Ewald (SE) fast summation method, which after integration with QBX allows the resulting system to be solved in M log M time, where M is the number of particles. This framework is suitable for simulations of large particle systems, and can be used for studying e.g. porous media models.

National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-185753 (URN)
Funder
Swedish Research Council, 2011-3178
Note

QC 20160426

Available from: 2016-04-26 Created: 2016-04-26 Last updated: 2016-04-27Bibliographically approved
5. Ewald summation for the rotlet singularity of Stokes flow
Open this publication in new window or tab >>Ewald summation for the rotlet singularity of Stokes flow
2016 (English)Report (Other academic)
Abstract [en]

Ewald summation is an efficient method for computing the periodic sums that appear when considering the Green's functions of Stokes flow together with periodic boundary conditions. We show how Ewald summation, and accompanying truncation error estimates, can be easily derived for the rotlet, by considering it as a superposition of electrostatic force calculations.

Publisher
9 p.
National Category
Fluid Mechanics and Acoustics Computational Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-184125 (URN)
Note

QC20160407

Available from: 2016-03-28 Created: 2016-03-28 Last updated: 2016-04-27Bibliographically approved

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