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A quadrature rule for the singular kernels of Laplace and Stokes equations over aclass of parameterizable surfaces
KTH, School of Engineering Sciences (SCI). (Numerical analysis)
##### Abstract [en]

A third order accurate corrected trapezoidal is developed to integrate numerically the singular kernels of Laplace and Stokes equations over a class of parameterizable surfaces with special focus on cylindrical surfaces. The corrected trapezoidal rule has so far been applied to flat surfaces on equidistant grids for the kernels of Laplace and Stokes. Corrected trapezoidal rules are based on the standard trapezoidal rule where the singular point, which is assumed to be a discretization point, is omitted. To account for the omitted point corrected weights are computed which are applied locally in a vicinity of, and at, the singular point. For general surfaces the weights depend on the position of the singular point on the surface leading to specific weights for each grid point. However we identify a special class of manifolds for which universal weights, independent of the position relative to the surface, can be computed. We select from this class of surfaces the model problem of a cylinder for which we explicitly develop and validate quadrature rules for both the kernel of Stokes and Laplace equations. This quadrature rule can be applied to simulations of pipe flows in conjunction with e.g. particle suspensions, fiber suspensions, swimming micro-organisms. Here we validate the obtained quadrature rule by computing the drag on a spheroidal particle positioned on the inner axis of a cylinder.

##### National Category
Computational Mathematics
##### Identifiers
OAI: oai:DiVA.org:kth-105562DiVA: diva2:571455
##### Note

QS 2012

Available from: 2012-11-22 Created: 2012-11-22 Last updated: 2012-11-22Bibliographically approved
##### In thesis
1. 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 $1/|\mathbf{x}|$. 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 $1/|\mathbf{x}|$. 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 $\mathcal{O}(N^{2})$. 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)
##### 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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