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Corrected trapezoidal rules for a class of singular functions
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.ORCID iD: 0000-0002-6321-8619
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
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.

Place, publisher, year, edition, pages
2014. Vol. 34, no 4, 1509-1540 p.
Keyword [en]
singular functions, quadrature methods, high order
National Category
Computer Science
Identifiers
URN: urn:nbn:se:kth:diva-29760DOI: 10.1093/imanum/drt046ISI: 000343320900008Scopus ID: 2-s2.0-84904197227OAI: oai:DiVA.org:kth-29760DiVA: diva2:397500
Note

QC 20141121. Updated from submitted to published.

Available from: 2011-02-15 Created: 2011-02-15 Last updated: 2017-12-11Bibliographically approved
In thesis
1. Quadrature rules for boundary integral methods applied to Stokes flow
Open this publication in new window or tab >>Quadrature rules for boundary integral methods applied to Stokes flow
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:nbn:se:kth:diva-29763 (URN)978-91-7415-862-5 (ISBN)
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
2. Boundary integral methods for Stokes flow: Quadrature techniques and fast Ewald methods
Open this publication in new window or tab >>Boundary integral methods for Stokes flow: Quadrature techniques and fast Ewald methods
2012 (English)Doctoral 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 three-dimensional 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/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.

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, e.g. 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 . 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 . An extension of this type of quadrature rule to a cylindrical surface is also developed. These quadrature rules are tested also on physical problems that have an analytic solution in the literature.

Another difficulty associated with boundary integral problems is introduced by periodic boundary conditions. For a set of particles in a periodic domain periodicity is imposed by requiring that the motion of each particle has an added contribution from all periodic images of all particles all the way up to infinity. This leads to an infinite sum which is not absolutely convergent, and an additional physical constraint which removes the divergence needs to be imposed. The sum is decomposed into two fast converging sums, one that handles the short range interactions in real space and the other that sums up the long range interactions in Fourier space. Such decompositions are already available in the literature for kernels that are commonly used in boundary integral formulations. Here a decomposition in faster decaying sums than the ones present in the literature is derived for the periodic kernel of the stress tensor.

However the computational complexity of the sums, regardless of the decomposition they stem from, is . This complexity can be lowered using a fast summation method as we introduced here for simulating a sedimenting fiber suspension. The fast summation method was initially designed for point particles, which could be used for fibers discretized numerically almost without any changes. However, when two fibers are very close to each other, analytical integration is used to eliminate numerical inaccuracies due to the nearly singular behavior of the kernel and the real space part in the fast summation method was modified to allow for this analytical treatment. The method we have developed for sedimenting fiber suspensions allows for simulations in large periodic domains and we have performed a set of such simulations at a larger scale (larger domain/more fibers) than previously feasible.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2012. ix, 59 p.
Series
Trita-NA, ISSN 0348-2952 ; 2012:14
Keyword
boundary integral, Stokes flow, quadrature rule, Ewald decomposition
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-105540 (URN)978-91-7501-578-1 (ISBN)
Public defence
2012-12-14, D2, Lindstedtsvägen, 5, Stockholm, 10:15 (English)
Opponent
Supervisors
Funder
Swedish e‐Science Research Center
Note

QC 20121122

Available from: 2012-11-22 Created: 2012-11-22 Last updated: 2013-04-09Bibliographically approved

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