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Quadrature rules for boundary integral methods applied to Stokes flow
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
2011 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

 

Fluid phenomena dominated by viscous effects can, in many cases, be modeled by the Stokes equations. The boundary integral form of the Stokes equations reduces the number of degrees of freedom in a numerical discretization by reformulating the threedimensional problem to two-dimensional integral equations to be discretized over the boundaries of the domain. Hence for the study of objects immersed in a fluid, such as drops or elastic or solid particles, integral equations are to be discretized over the surfaces of these objects only. As outer boundaries or confinements are added these must also be included in the formulation. This work is focused on the development and validation of such a wall treatment. An inherent difficulty in the numerical treatment of boundary integrals for Stokes flow is the integration of the singular fundamental solution of the Stokes equations – the so called Stokeslet. To alleviate this problem we developed a set of high-order quadrature rules for the numerical integration of the Stokeslet over a flat surface. Such a quadrature rule was first designed for singularities of the type 1/|x|. To assess the convergence properties of this quadrature rule a theoretical analysis has been performed. The slightly more complicated singularity of the Stokeslet required certain modifications of the integration rule developed for 1/|x|. To validate the quadrature rule developed for the Stokeslet against a physical model we use it in a classical problem in fluid dynamics, the sedimentation of a sphere onto a flat plate. This involves a direct discretization of the plane wall and at the same time of the immersed sphere. Without any special treatment the algebraic system given by the discrete problem is quite memory consuming since matrix blocks are full. By exploring the structure of the block matrices that build up the system we have found that the wall discretization leads to a matrix which is generated by only three of its columns. This information together with certain preconditioning considerations allowed us to use the Schur complement method thus leading to a less memory expensive solution to the algebraic system. As a final step it is shown that the numerical simulations match the analytical solution, within the limitations of the model. This wall treatment can be easily extended to the problem of two parallel walls, and it is also shown that the simulation is in good agreement with some known results for the two parallel walls problem.

 

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology , 2011. , vii, 30 p.
Series
Trita-CSC-A, ISSN 1653-5723 ; 2011:01
National Category
Computer Science
Identifiers
URN: urn:nbn:se:kth:diva-29763ISBN: 978-91-7415-862-5 (print)OAI: oai:DiVA.org:kth-29763DiVA: diva2:397518
Presentation
2011-02-15, Sal E32, KTH, Lindstedtsvägen 3, Stockholm, 13:00 (English)
Opponent
Supervisors
Funder
Swedish e‐Science Research Center
Note
QC 20110215Available from: 2011-02-15 Created: 2011-02-15 Last updated: 2012-05-24Bibliographically approved
List of papers
1. Corrected trapezoidal rules for a class of singular functions
Open this publication in new window or tab >>Corrected trapezoidal rules for a class of singular functions
2014 (English)In: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 34, no 4, 1509-1540 p.Article in journal (Refereed) Published
Abstract [en]

A set of accurate quadrature rules applicable to a class of integrable functions with isolated singularities is designed and analysed theoretically in one and two dimensions. These quadrature rules are based on the trapezoidal rule with corrected quadrature weights for points in the vicinity of the singularity. To compute the correction weights, small-size ill-conditioned systems have to be solved. The convergence of the correction weights is accelerated by the use of compactly supported functions that annihilate boundary errors. Convergence proofs with error estimates for the resulting quadrature rules are given in both one and two dimensions. The tabulated weights are specific for the singularities under consideration, but the methodology extends to a large class of functions with integrable isolated singularities. Furthermore, in one dimension we have obtained a closed form expression based on which the modified weights can be computed directly.

Keyword
singular functions, quadrature methods, high order
National Category
Computer Science
Identifiers
urn:nbn:se:kth:diva-29760 (URN)10.1093/imanum/drt046 (DOI)000343320900008 ()2-s2.0-84904197227 (Scopus ID)
Note

QC 20141121. Updated from submitted to published.

Available from: 2011-02-15 Created: 2011-02-15 Last updated: 2017-12-11Bibliographically approved
2. A wall treatment for confined Stokes flow
Open this publication in new window or tab >>A wall treatment for confined Stokes flow
(English)Article in journal (Other academic) Submitted
Abstract [en]

 

The study of bodies immersed in Stokes flow is crucial in various microfluidic applications. Recasting the governing equations in a boundary integral formulation reduces the three-dimensional problem to two-dimensional integral equations to be discretized over the surface of the submerged objects. The present work focuses on the development and validation of a wall treatment where the wall is discretized in the same fashion as the immersed bodies. For this purpose, a set of high-order quadrature rules for the numerical integration of integrals containing the singular Green’s function-the so-called Stokeslet - has been developed. By coupling the wall discretization to the discretization of immersed objects, we exploit the structure of the block matrix corresponding to the wall discretization in order to substantially reduce the memory usage. For validation, the classical problem of a sedimenting sphere in the vicinity of solid walls is studied.

National Category
Computer Science
Identifiers
urn:nbn:se:kth:diva-29762 (URN)
Note
QS 20120328Available from: 2011-02-15 Created: 2011-02-15 Last updated: 2012-11-22Bibliographically approved

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