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An accurate integral equation method for Stokes flow with piecewise smooth boundaries
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Two-dimensional Stokes flow through a periodic channel is considered. The channel walls need only be Lipschitz continuous, in other words they are allowed to have corners. Boundary integral methods are an attractive numerical method to solve the Stokes equations, as the problem can be reformulated into a problem that must be solved only over the boundary of the domain. When the boundary is at least C1 smooth, the boundary integral kernel is a compact operator, and traditional Nyström methods can be used to obtain highly accurate solutions. In the case of Lipschitz continuous boundaries however, obtaining accurate solutions using the standard Nyström method can require high resolution. We adapt a technique known as recursively compressed inverse preconditioning to accurately solve the Stokes equations without requiring any more resolution than is needed to resolve the boundary. Combined with a periodic fast summation method we construct a method that is O(N log N ) where N is the number of quadrature points on the boundary. We demonstrate the robustness of this method by extending an existing boundary integral method for viscous drops to handle the movement of drops near corners.

National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-264366OAI: oai:DiVA.org:kth-264366DiVA, id: diva2:1373100
Note

QC 20191126

Available from: 2019-11-26 Created: 2019-11-26 Last updated: 2019-11-26Bibliographically approved
In thesis
1. Boundary integral methods for fast and accurate simulation of droplets in two-dimensional Stokes flow
Open this publication in new window or tab >>Boundary integral methods for fast and accurate simulation of droplets in two-dimensional Stokes flow
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Accurate simulation of viscous fluid flows with deforming droplets creates a number of challenges. This thesis identifies these principal challenges and develops a numerical methodology to overcome them. Two-dimensional viscosity-dominated fluid flows are exclusively considered in this work. Such flows find many applications, for example, within the large and growing field of microfluidics; accurate numerical simulation is of paramount importance for understanding and exploiting them.

A boundary integral method is presented which enables the simulation of droplets and solids with a very high fidelity. The novelty of this method is in its ability to accurately handle close interactions of drops, and of drops and solid boundaries, including boundaries with sharp corners. The boundary integral method is coupled with a spectral method to solve a PDE for the time-dependent concentration of surfactants on each of the droplet interfaces. Surfactants are molecules that change the surface tension and are therefore highly influential in the types of flow problems which are considered herein.

A method’s usefulness is not dictated by accuracy alone. It is also necessary that the proposed method is computationally efficient. To this end, the spectral Ewald method has been adapted and applied. This yields solutions with computational cost O(N log N ), instead of O(N^2), for N source and target points.

Together, these innovations form a highly accurate, computationally efficient means of dealing with complex flow problems. A theoretical validation procedure has been developed which confirms the accuracy of the method.

Abstract [sv]

Att noggrant simulera viskösa flöden med deformerande droppar medför flera utmaningar. Denna avhandling identifierar de viktigaste utmaningarna och utvecklar numeriska metoder för att övervinna dem. Visköst dominerade tvådimensionella flöden studeras. Sådana flöden har många tillämpningar till exempel inom mikrofluidik och noggrann beräkning är av största vikt för att förstå och utnyttja dem.

En randintegralsmetod som möjliggör simulering av droppar och fasta ränder med en mycket hög noggrannhet presenteras. Metoden särskiljer sig från andra genom dess förmåga att hantera nära samspel mellan droppar och förekomst av hörn på de fasta ränderna. Randin- tegralsmetoden är kopplad till en spektral metod som möjliggör inkluderandet av surfaktanter i flödesproblemet. Surfaktanter är molekyler som förändrar ytspänningen och de är därför betydelsefulla för de typer av flödesproblem som beaktas här.

En metods användbarhet bestäms inte endast av dess noggrannhet. Det är också nödvändigt att den föreslagna metoden är effektiv. För detta ändamål har metoden spektral Ewald anpassats och tillämpats. Detta ger lösningar med beräkningskostnaden O(N log N ) istället för O(N^2), där N är antalet diskreta punkter i systemet.

Tillsammans utgör dessa innovationer ett mycket noggrant, be- räkningseffektivt sätt att hantera komplexa flödesproblem. Ett teoretiskt valideringsförfarande har utvecklats som bekräftar metodens noggrannhet.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2019. p. 49
Series
TRITA-SCI-FOU ; 2019;50
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics, Numerical Analysis
Identifiers
urn:nbn:se:kth:diva-264369 (URN)978-91-7873-355-2 (ISBN)
Public defence
2019-12-18, F3, Lindstedtsvägen 26, Stockholm, 10:00 (English)
Opponent
Supervisors
Funder
Göran Gustafsson Foundation for Research in Natural Sciences and Medicine
Available from: 2019-11-27 Created: 2019-11-26 Last updated: 2019-11-27Bibliographically approved

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Bystricky, LukasTornberg, Anna-Karin

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