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Fast Ewald summation for Stokesian particle suspensions
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.ORCID iD: 0000-0001-7425-8029
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
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. Vol. 76, no 10, 669-698 p.
Keyword [en]
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: urn:nbn:se:kth:diva-116383DOI: 10.1002/fld.3953ISI: 000344349000004OAI: oai:DiVA.org:kth-116383DiVA: diva2:589279
Funder
Swedish Research Council, 2011-3178
Note

QC 20141119

Available from: 2013-01-17 Created: 2013-01-17 Last updated: 2017-12-06Bibliographically approved
In thesis
1. Computational methods for microfluidics
Open this publication in new window or tab >>Computational methods for microfluidics
2013 (English)Licentiate 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 paper of the thesis presents a numerical method for simulating multiphase flows involving insoluble surfactants and moving contact lines. The method is based on an explicit interface tracking method, wherein the interface between two fluids is decomposed into segments, which are represented locally on an Eulerian grid. The framework of this method provides a natural setting for solving the advection-diffusion equation governing the surfactant concentration on the interface. Open interfaces and moving contact lines are also incorporated into the method in a natural way, though we show that care must be taken when regularizing interface forces to the grid near the boundary of the computational domain.

In the second paper we present a boundary integral formulation for sedimenting particles in periodic Stokes flow, using the completed double layer boundary integral formulation. The long-range nature of the particle-particle interactions lead to the formulation containing sums which are not absolutely convergent if computed directly. This is solved by applying the method of Ewald summation, which in turn is computed in a fast manner by using the FFT-based spectral Ewald method. The complexity of the resulting method is O(N log N), as the system size is scaled up with the number of discretization points N. We apply the method to systems of sedimenting spheroids, which are discretized using the Nyström method and a basic quadrature rule.

The Ewald summation method used in the boundary integral method of the second paper requires a decomposition of the potential being summed. In the introductory chapters of the thesis we present an overview of the available methods for creating Ewald decompositions, and show how the methods and decompositions can be related to each other.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2013. viii, 39 p.
Series
Trita-NA, ISSN 0348-2952 ; 2013:01
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-116384 (URN)978-91-7501-625-2 (ISBN)
Presentation
2013-02-19, F3, Lindstedtsvägen 26, Kungliga Tekniska Högskolan, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

QC 20130124

Available from: 2013-01-24 Created: 2013-01-17 Last updated: 2013-01-24Bibliographically approved
2. Fast and accurate integral equation methods with applications in microfluidics
Open this publication in new window or tab >>Fast and accurate integral equation methods with applications in microfluidics
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:nbn:se:kth:diva-185758 (URN)978-91-7595-962-7 (ISBN)
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

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