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  • 1.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Lindbo, Dag
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    An explicit Eulerian method for multiphase flow with contact line dynamics and insoluble surfactant2014In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 101, p. 50-63Article in journal (Refereed)
    Abstract [en]

    The flow behavior of many multiphase flow applications is greatly influenced by wetting properties and the presence of surfactants. We present a numerical method for two-phase flow with insoluble surfactants and contact line dynamics in two dimensions. The method is based on decomposing the interface between two fluids into segments, which are explicitly represented on a local Eulerian grid. It provides a natural framework for treating the surfactant concentration equation, which is solved locally on each segment. An accurate numerical method for the coupled interface/surfactant system is given. The system is coupled to the Navier-Stokes equations through the immersed boundary method, and we discuss the issue of force regularization in wetting problems, when the interface touches the boundary of the domain. We use the method to illustrate how the presence of surfactants influences the behavior of free and wetting drops.

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  • 2.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Saffar Shamshirgar, Davoud
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Fast Ewald summation for free-space Stokes potentials2017In: Research in the Mathematical Sciences, ISSN 2197-9847, Vol. 4, no 1Article in journal (Refereed)
    Abstract [en]

    We present a spectrally accurate method for the rapid evaluation of free-space Stokes potentials, i.e., sums involving a large number of free space Green’s functions. We consider sums involving stokeslets, stresslets and rotlets that appear in boundary integral methods and potential methods for solving Stokes equations. The method combines the framework of the Spectral Ewald method for periodic problems (Lindbo and Tornberg in J Comput Phys 229(23):8994–9010, 2010. doi: 10.1016/j.jcp.2010.08.026 ), with a very recent approach to solving the free-space harmonic and biharmonic equations using fast Fourier transforms (FFTs) on a uniform grid (Vico et al. in J Comput Phys 323:191–203, 2016. doi: 10.1016/j.jcp.2016.07.028 ). Convolution with a truncated Gaussian function is used to place point sources on a grid. With precomputation of a scalar grid quantity that does not depend on these sources, the amount of oversampling of the grids with Gaussians can be kept at a factor of two, the minimum for aperiodic convolutions by FFTs. The resulting algorithm has a computational complexity of $$O(N \log N)$$ O ( N log N ) for problems with N sources and targets. Comparison is made with a fast multipole method to show that the performance of the new method is competitive.

  • 3.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Sorgentone, Chiara
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Quadrature error estimates for layer potentials evaluated near curved surfaces in three dimensions2022In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 111, p. 1-19Article in journal (Refereed)
    Abstract [en]

    The quadrature error associated with a regular quadrature rule for evaluation of a layer potential increases rapidly when the evaluation point approaches the surface and the integral becomes nearly singular. Error estimates are needed to determine when the accuracy is insufficient and a more costly special quadrature method should be utilized.& nbsp;The final result of this paper are such quadrature error estimates for the composite Gauss-Legendre rule and the global trapezoidal rule, when applied to evaluate layer potentials defined over smooth curved surfaces in R-3. The estimates have no unknown coefficients and can be efficiently evaluated given the discretization of the surface, invoking a local one-dimensional root-finding procedure. They are derived starting with integrals over curves, using complex analysis involving contour integrals, residue calculus and branch cuts. By complexifying the parameter plane, the theory can be used to derive estimates also for curves in R3. These results are then used in the derivation of the estimates for integrals over surfaces. In this procedure, we also obtain error estimates for layer potentials evaluated over curves in R2. Such estimates combined with a local root-finding procedure for their evaluation were earlier derived for the composite Gauss-Legendre rule for layer potentials written in complex form [4]. This is here extended to provide quadrature error estimates for both complex and real formulations of layer potentials, both for the Gauss-Legendre and the trapezoidal rule.& nbsp;Numerical examples are given to illustrate the performance of the quadrature error estimates. The estimates for integration over curves are in many cases remarkably precise, and the estimates for curved surfaces in R-3 are also sufficiently precise, with sufficiently low computational cost, to be practically useful.

  • 4.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    A fast integral equation method for solid particles in viscous flow using quadrature by expansionManuscript (preprint) (Other academic)
    Abstract [en]

    Boundary integral methods are advantageous when simulating viscous flow around rigid particles, due to the reduction in number of unknowns and straightforward handling of the geometry. In this work we present a fast and accurate framework for simulating spheroids in periodic Stokes flow, which is based on the completed double layer boundary integral formulation. The framework implements a new method known as quadrature by expansion (QBX), which uses surrogate local expansions of the layer potential to evaluate it to very high accuracy both on and off the particle surfaces. This quadrature method is accelerated through a newly developed precomputation scheme. The long range interactions are computed using the spectral Ewald (SE) fast summation method, which after integration with QBX allows the resulting system to be solved in M log M time, where M is the number of particles. This framework is suitable for simulations of large particle systems, and can be used for studying e.g. porous media models.

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  • 5.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    A fast integral equation method for solid particles in viscous flow using quadrature by expansion2016In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 326, p. 420-445Article in journal (Refereed)
    Abstract [en]

    Boundary integral methods are advantageous when simulating viscous flow around rigid particles, due to the reduction in number of unknowns and straightforward handling of the geometry. In this work we present a fast and accurate framework for simulating spheroids in periodic Stokes flow, which is based on the completed double layer boundary integral formulation. The framework implements a new method known as quadrature by expansion (QBX), which uses surrogate local expansions of the layer potential to evaluate it to very high accuracy both on and off the particle surfaces. This quadrature method is accelerated through a newly developed precomputation scheme. The long range interactions are computed using the spectral Ewald (SE) fast summation method, which after integration with QBX allows the resulting system to be solved in M log M time, where M is the number of particles. This framework is suitable for simulations of large particle systems, and can be used for studying e.g. porous media models.

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  • 6.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Adaptive Quadrature by Expansion for Layer Potential Evaluation in Two Dimensions2018In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 40, no 3, p. A1225-A1249Article in journal (Refereed)
    Abstract [en]

    When solving partial differential equations using boundary integral equation methods, accurate evaluation of singular and nearly singular integrals in layer potentials is crucial. A recent scheme for this is quadrature by expansion (QBX), which solves the problem by locally approximating the potential using a local expansion centered at some distance from the source boundary. In this paper we introduce an extension of the QBX scheme in two dimensions (2D) denoted AQBX—adaptive quadrature by expansion—which combines QBX with an algorithm for automated selection of parameters, based on a target error tolerance. A key component in this algorithm is the ability to accurately estimate the numerical errors in the coefficients of the expansion. Combining previous results for flat panels with a procedure for taking the panel shape into account, we derive such error estimates for arbitrarily shaped boundaries in 2D that are discretized using panel-based Gauss–Legendre quadrature. Applying our scheme to numerical solutions of Dirichlet problems for the Laplace and Helmholtz equations, and also for solving these equations, we find that the scheme is able to satisfy a given target tolerance to within an order of magnitude, making it useful for practical applications. This represents a significant simplification over the original QBX algorithm, in which choosing a good set of parameters can be hard.

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  • 7.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Error estimation for quadrature by expansion in layer potential evaluation2017In: Advances in Computational Mathematics, ISSN 1019-7168, E-ISSN 1572-9044, Vol. 43, no 1, p. 195-234Article in journal (Refereed)
    Abstract [en]

    In boundary integral methods it is often necessary to evaluate layer potentials on or close to the boundary, where the underlying integral is difficult to evaluate numerically. Quadrature by expansion (QBX) is a new method for dealing with such integrals, and it is based on forming a local expansion of the layer potential close to the boundary. In doing so, one introduces a new quadrature error due to nearly singular integration in the evaluation of expansion coefficients. Using a method based on contour integration and calculus of residues, the quadrature error of nearly singular integrals can be accurately estimated. This makes it possible to derive accurate estimates for the quadrature errors related to QBX, when applied to layer potentials in two and three dimensions. As examples we derive estimates for the Laplace and Helmholtz single layer potentials. These results can be used for parameter selection in practical applications.

  • 8.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Estimation of quadrature errors in layer potential evaluation using quadrature by expansionManuscript (preprint) (Other academic)
    Abstract [en]

    In boundary integral methods it is often necessary to evaluate layer potentials on or close to the boundary, where the underlying integral is difficult to evaluate numerically. Quadrature by expansion (QBX) is a new method for dealing with such integrals, and it is based on forming a local expansion of the layer potential close to the boundary. In doing so, one introduces a new quadrature error due to nearly singular integration in the evaluation of expansion coefficients. Using a method based on contour integration and calculus of residues, the quadrature error of nearly singular integrals can be accurately estimated. This makes it possible to derive accurate estimates for the quadrature errors related to QBX, when applied to layer potentials in two and three dimensions. As examples we derive estimates for the Laplace and Helmholtz single layer potentials. These results can be used for parameter selection in practical applications.

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  • 9.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Fast Ewald summation for Stokesian particle suspensions2014In: International Journal for Numerical Methods in Fluids, ISSN 0271-2091, E-ISSN 1097-0363, Vol. 76, no 10, p. 669-698Article in journal (Refereed)
    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. 

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  • 10.
    Bagge, Joar
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Rosén, Tomas
    KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, School of Engineering Sciences in Chemistry, Biotechnology and Health (CBH), Centres, Wallenberg Wood Science Center. KTH, School of Engineering Sciences (SCI), Engineering Mechanics.
    Lundell, Fredrik
    KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, School of Engineering Sciences in Chemistry, Biotechnology and Health (CBH), Centres, Wallenberg Wood Science Center. KTH, School of Engineering Sciences (SCI), Engineering Mechanics.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Parabolic velocity profile causes shape-selective drift of inertial ellipsoids2021In: Journal of Fluid Mechanics, ISSN 0022-1120, E-ISSN 1469-7645, Vol. 926, article id A24Article in journal (Refereed)
    Abstract [en]

    Understanding particle drift in suspension flows is of the highest importance in numerous engineering applications where particles need to be separated and filtered out from the suspending fluid. Commonly known drift mechanisms such as the Magnus force, Saffman force and Segre-Silberberg effect all arise only due to inertia of the fluid, with similar effects on all non-spherical particle shapes. In this work, we present a new shape-selective lateral drift mechanism, arising from particle inertia rather than fluid inertia, for ellipsoidal particles in a parabolic velocity profile. We show that the new drift is caused by an intermittent tumbling rotational motion in the local shear flow together with translational inertia of the particle, while rotational inertia is negligible. We find that the drift is maximal when particle inertial forces are of approximately the same order of magnitude as viscous forces, and that both extremely light and extremely heavy particles have negligible drift. Furthermore, since tumbling motion is not a stable rotational state for inertial oblate spheroids (nor for spheres), this new drift only applies to prolate spheroids or tri-axial ellipsoids. Finally, the drift is compared with the effect of gravity acting in the directions parallel and normal to the flow. The new drift mechanism is stronger than gravitational effects as long as gravity is less than a critical value. The critical gravity is highest (i.e. the new drift mechanism dominates over gravitationally induced drift mechanisms) when gravity acts parallel to the flow and the particles are small.

  • 11.
    Bagge, Joar
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Accurate quadrature methods with application to Stokes flow with particles in confined geometries2017In: Proceedings of the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11) / [ed] David J. Chappell, Nottingham: Nottingham Trent University, 2017, p. 15-24Conference paper (Refereed)
    Abstract [en]

    Boundary integral methods are attractive for simulating Stokes flow with particles or droplets due to the reduction in dimensionality and natural handling of the geometry. In many problems walls are present, and it becomes necessary to evaluate singular or nearly singular layer potentials over the wall. In this paper we show how this can be done using quadrature by expansion (QBX), a relatively new method based on local expansions of the layer potential. We present results for the Laplace single layer potential and the Stokes double layer potential. QBX can be used to evaluate the potentials to high accuracy arbitrarily close to the wall and on the wall. We also discuss how some quantities can be precomputed and how geometric symmetries can be used to reduce precomputation and storage.

  • 12.
    Bagge, Joar
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Accurate quadrature via line extrapolation and rational approximation with application to boundary integral methods for Stokes flow2023Report (Other academic)
    Abstract [en]

    In boundary integral methods, special quadrature methods are needed to approximate layer potentials, integrals where the integrand is singular or sharply peaked for evaluation points on or close to the boundaries. In this paper, we study a method based on extrapolation or interpolation along a line, sometimes called the Hedgehog method. In this method, the layer potential is evaluated with a regular quadrature method for evaluation points along a line, and an approximant is constructed and evaluated in an area of interest where the original layer potential is difficult to evaluate due to it being singular or sharply peaked.

    We analyze the errors in the Hedgehog method with polynomial approximation, and use this to construct optimal distributions of sample points. Furthermore, rational approximation is introduced in the Hedgehog method, and compared with polynomial approximation. It is found that rational approximation can typically achieve a lower error than polynomial approximation, and does not increase the computational cost of the method significantly. Strategies for avoiding and dealing with spurious poles in rational approximation are discussed.

    We compare extrapolation (no sample point on the boundary) with interpolation (sample point present) in the Hedgehog method, and find that the error in our example is lower in the interpolation case by around one order of magnitude, compared to the extrapolation case.

    We consider a specific test case, consisting of two rigid rodlike particles in Stokes flow. Parameter selection and error estimation for the Hedgehog method is discussed for this test case. The accuracy and computational cost of the Hedgehog method is examined, and compared with another special quadrature method, namely quadrature by expansion (QBX). We find that the Hedgehog method should be able to compete with QBX in this context, but further investigation is needed for strict tolerances.

  • 13.
    Bagge, Joar
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Fast Ewald summation for Stokes flow with arbitrary periodicityManuscript (preprint) (Other academic)
    Abstract [en]

    A fast and spectrally accurate Ewald summation method for the evaluation of stokeslet, stresslet and rotlet potentials of three-dimensional Stokes flow is presented. This work extends the previously developed Spectral Ewald method for Stokes flow to periodic boundary conditions in any number (three, two, one, or none) of the spatial directions, in a unified framework. The periodic potential is split into a short-range and a long-range part, where the latter is treated in Fourier space using the fast Fourier transform. A crucial component of the method is the modified kernels used to treat singular integration. We derive new modified kernels, and new improved truncation error estimates for the stokeslet and stresslet. An automated procedure for selecting parameters based on a given error tolerance is designed and tested. Analytical formulas for validation in the doubly and singly periodic cases are presented. We show that the computational time of the method scales like O(N log N) for N sources and targets, and investigate how the time depends on the error tolerance and window function, i.e. the function used to smoothly spread irregular point data to a uniform grid. The method is fastest in the fully periodic case, while the run time in the free-space case is around three times as large. Furthermore, the highest efficiency is reached when applying the method to a uniform source distribution in a primary cell with low aspect ratio. The work presented in this paper enables efficient and accurate simulations of three-dimensional Stokes flow with arbitrary periodicity using e.g. boundary integral and potential methods.

  • 14.
    Bagge, Joar
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Fast Ewald summation for Stokes flow with arbitrary periodicity2023In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 493, p. 112473-, article id 112473Article in journal (Refereed)
    Abstract [en]

    A fast and spectrally accurate Ewald summation method for the evaluation of stokeslet, stresslet and rotlet potentials of three-dimensional Stokes flow is presented. This work extends the previously developed Spectral Ewald method for Stokes flow to periodic boundary conditions in any number (three, two, one, or none) of the spatial directions, in a unified framework. The periodic potential is split into a short-range and a long-range part, where the latter is treated in Fourier space using the fast Fourier transform. A crucial component of the method is the modified kernels used to treat singular integration. We derive new modified kernels, and new improved truncation error estimates for the stokeslet and stresslet. An automated procedure for selecting parameters based on a given error tolerance is designed and tested. Analytical formulas for validation in the doubly and singly periodic cases are presented. We show that the computational time of the method scales like O(Nlog⁡N) for N sources and targets, and investigate how the time depends on the error tolerance and window function, i.e. the function used to smoothly spread irregular point data to a uniform grid. The method is fastest in the fully periodic case, while the run time in the free-space case is around three times as large. Furthermore, the highest efficiency is reached when applying the method to a uniform source distribution in a primary cell with low aspect ratio. The work presented in this paper enables efficient and accurate simulations of three-dimensional Stokes flow with arbitrary periodicity using e.g. boundary integral and potential methods.

  • 15.
    Bagge, Joar
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Highly accurate special quadrature methods for Stokesian particle suspensions in confined geometries2021In: International Journal for Numerical Methods in Fluids, ISSN 0271-2091, E-ISSN 1097-0363, Vol. 93, no 7, p. 2175-2224Article in journal (Refereed)
    Abstract [en]

    Boundary integral methods are highly suited for problems with complicated geometries, but require special quadrature methods to accurately compute the singular and nearly singular layer potentials that appear in them. This article presents a boundary integral method that can be used to study the motion of rigid particles in three-dimensional periodic Stokes flow with confining walls. A centerpiece of our method is the highly accurate special quadrature method, which is based on a combination of upsampled quadrature and quadrature by expansion, accelerated using a precomputation scheme. The method is demonstrated for rodlike and spheroidal particles, with the confining geometry given by a pipe or a pair of flat walls. A parameter selection strategy for the special quadrature method is presented and tested. Periodic interactions are computed using the spectral Ewald fast summation method, which allows our method to run in O(n log n) time for n grid points in the primary cell, assuming the number of geometrical objects grows while the grid point concentration is kept fixed.

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  • 16.
    Broms, Anna
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Sandberg, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    A locally corrected multiblob method with hydrodynamically matched grids for the Stokes mobility problem2023In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 487, p. 112172-112172, article id 112172Article in journal (Refereed)
    Abstract [en]

    Inexpensive numerical methods are key to enabling simulations of systems of a large number of particles of different shapes in Stokes flow and several approximate methods have been introduced for this purpose. We study the accuracy of the multiblob method for solving the Stokes mobility problem in free space, where the 3D geometry of a particle surface is discretised with spherical blobs and the pair-wise interaction between blobs is described by the RPY-tensor. The paper aims to investigate and improve on the magnitude of the error in the solution velocities of the Stokes mobility problem using a combination of two different techniques: an optimally chosen grid of blobs and a pair-correction inspired by Stokesian dynamics. Different optimisation strategies to determine a grid with a given number of blobs are presented with the aim of matching the hydrodynamic response of a single accurately described ideal particle, alone in the fluid. It is essential to obtain small errors in this self-interaction, as they determine the basic error level in a system of well-separated particles. With an optimised grid, reasonable accuracy can be obtained even with coarse blob-resolutions of the particle surfaces. The error in the self-interaction is however sensitive to the exact choice of grid parameters and simply hand-picking a suitable geometry of blobs can lead to errors several orders of magnitude larger in size. The pair-correction is local and cheap to apply, and reduces the error for moderately separated particles and particles in close proximity. Two different types of geometries are considered: spheres and axisymmetric rods with smooth caps. The error in solutions to mobility problems is quantified for particles of varying inter-particle distances for systems containing a few particles, comparing to an accurate solution based on a second kind BIE-formulation where the quadrature error is controlled by employing quadrature by expansion (QBX).

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  • 17.
    Broms, Anna
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    A barrier method for contact avoiding particles in Stokes flow2024In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 497, p. 112648-112648, article id 112648Article in journal (Refereed)
    Abstract [en]

    Rigid particles in a Stokesian fluid experience an increasingly strong lubrication resistance as particle gaps narrow. Numerically, resolving these lubrication forces comes at an intractably large cost, even for moderate system sizes. Hence, it can typically not be guaranteed that artificial particle collisions and overlaps do not occur in a dynamic simulation, independently of the choice of method to solve the Stokes equations. In this work, the potentially large set of non-overlap constraints, in terms of the Euclidean distance between boundary points on disjoint particles, are efficiently represented via a barrier energy. We solve for the minimum magnitudes of repelling contact forces and torques between any particle pair in contact to correct for overlaps by enforcing a zero barrier energy at the next time level, given a contact-free configuration at a previous instance in time. Robustness for the method is illustrated using a multiblob method to solve the mobility problem in Stokes flow, applied to suspensions of spheres, rods and boomerang shaped particles. Collision free configurations are obtained at all instances in time, and considerably larger time-steps can be taken than without the technique. The effect of the contact forces on the collective order of a set of rods in a background flow that naturally promote particle interactions is also illustrated.

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    fulltext
  • 18.
    Bystricky, Lukas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Pålsson, Sara
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    An accurate integral equation method for Stokes flow with piecewise smooth boundariesManuscript (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.

  • 19.
    Bystricky, Lukas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Pålsson, Sara
    KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    An accurate integral equation method for Stokes flow with piecewise smooth boundaries2020In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125Article in journal (Refereed)
    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 tool for numerically solving the Stokes equations, as the partial differential equation can be reformulated into an integral equation 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(Nlog 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. 

  • 20. Engblom, S.
    et al.
    Do-Quang, Minh
    KTH, School of Engineering Sciences (SCI), Mechanics, Physicochemical Fluid Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Amberg, Gustav
    KTH, School of Engineering Sciences (SCI), Mechanics, Physicochemical Fluid Mechanics. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    On diffuse interface modeling and simulation of surfactants in two-phase fluid flow2013In: Communications in Computational Physics, ISSN 1815-2406, E-ISSN 1991-7120, Vol. 14, no 4, p. 879-915Article in journal (Refereed)
    Abstract [en]

    An existing phase-fieldmodel of two immiscible fluids with a single soluble surfactant present is discussed in detail. We analyze the well-posedness of the model and provide strong evidence that it is mathematically ill-posed for a large set of physically relevant parameters. As a consequence, critical modifications to the model are suggested that substantially increase the domain of validity. Carefully designed numerical simulations offer informative demonstrations as to the sharpness of our theoretical results and the qualities of the physical model. A fully coupled hydrodynamic test-case demonstrates the potential to capture also non-trivial effects on the overall flow.

  • 21.
    Engquist, Björn
    et al.
    Department of Mathematics and Institute for Computational Engineering and Sciences, The University of Texas at Austin, USA.
    Häggblad, Jon
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Runborg, Olof
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    On Consistent Boundary Conditions for the Yee Scheme in 3DManuscript (preprint) (Other academic)
    Abstract [en]

    The standard staircase approximation of curved boundaries in the Yee scheme is inconsistent. Consistency can however be achieved by modifying the algorithm close to the boundary.  We consider a technique to consistently model curved boundaries where the coefficients of the update stencil is modified, thus preserving the Yee structure.  The method has previously been successfully applied to acoustics in two and three dimension, as well as electromagnetics in two dimensions.  In this paper we generalize to electromagnetics in three dimensions.  Unlike in previous cases there is a non-zero divergence growth along the boundary that needs to be projected away.  We study the convergence and provide numerical examples that demonstrates the improved accuracy.

    Download full text (pdf)
    consistentyee3d
  • 22.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Runborg, Olof
    KTH, Superseded Departments (pre-2005), Numerical Analysis and Computer Science, NADA.
    Tornberg, Anna Karin
    High-frequency wave propagation by the segment projection method2002In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 178, no 2, p. 373-390Article in journal (Refereed)
    Abstract [en]

    Geometrical optics is a standard technique used for the approximation of high-frequency wave propagation. Computational methods based on partial differential equations instead of the traditional ray tracing have recently been applied to geometrical optics. These new methods have a number of advantages but typically exhibit difficulties with linear superposition of waves. In this paper we introduce a new partial differential technique based on the segment projection method in phase space. The superposition problem is perfectly resolved and so is the problem of computing amplitudes in the neighborhood of caustics. The computational complexity is of the same order as that of ray tracing. The new algorithm is described and a number of computational examples are given. including a simulation of waveguides.

  • 23.
    Engquist, Björn
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Tornberg, Anna Karin
    A finite element based level-set method for multi-phase flow, Proceedings of Conference on Progress in Numerical Solutions of Partial Differential Equations2002In: Scientific World Journal, E-ISSN 1537-744X, p. 86-110Article in journal (Refereed)
  • 24. Engquist, Björn
    et al.
    Tornberg, Anna-Karin
    Tsai, R.
    Discretization of Dirac delta functions in level set methods2005In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 207, no 1, p. 28-51Article in journal (Refereed)
    Abstract [en]

    Discretization of singular functions is an important component in many problems to which level set methods have been applied. We present two methods for constructing consistent approximations to Dirac delta measures concentrated on piecewise smooth curves or surfaces. Both methods are designed to be convenient for level set simulations and are introduced to replace the commonly used but inconsistent regularization technique that is solely based on a regularization parameter proportional to the mesh size. The first algorithm is based on a tensor product of regularized one-dimensional delta functions. It is independent of the irregularity relative to the grid. In the second method, the regularization is constructed from a one-dimensional regularization that is extended to multi-dimensions with a variable support depending on the orientation of the singularity relative to the computational grid. Convergence analysis and numerical results are given. © 2005 Published by Elsevier Inc.

  • 25.
    Fryklund, Fredrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    af Klinteberg, Ludvig
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    An adaptive kernel-split quadrature method for parameter-dependent layer potentials2022In: Advances in Computational Mathematics, ISSN 1019-7168, E-ISSN 1572-9044, Vol. 48, no 2, article id 12Article in journal (Refereed)
    Abstract [en]

    Panel-based, kernel-split quadrature is currently one of the most efficient methods available for accurate evaluation of singular and nearly singular layer potentials in two dimensions. However, it can fail completely for the layer potentials belonging to the modified Helmholtz, modified biharmonic, and modified Stokes equations. These equations depend on a parameter, denoted alpha, and kernel-split quadrature loses its accuracy rapidly when this parameter grows beyond a certain threshold. This paper describes an algorithm that remedies this problem, using per-target adaptive sampling of the source geometry. The refinement is carried out through recursive bisection, with a carefully selected rule set. This maintains accuracy for a wide range of the parameter alpha, at an increased cost that scales as log alpha. Using this algorithm allows kernel-split quadrature to be both accurate and efficient for a much wider range of problems than previously possible.

  • 26.
    Fryklund, Fredrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    af Klinteberg, Ludvig
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    An adaptive kernel-split quadrature method for parameter-dependent layer potentialsManuscript (preprint) (Other academic)
  • 27.
    Fryklund, Fredrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Kropinski, Mary Catherine A.
    Simon Fraser Univ, Dept Math, Burnaby, BC, Canada..
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    An integral equation-based numerical method for the forced heat equation on complex domains2020In: Advances in Computational Mathematics, ISSN 1019-7168, E-ISSN 1572-9044, Vol. 46, no 5, article id 69Article in journal (Refereed)
    Abstract [en]

    Integral equation-based numerical methods are directly applicable to homogeneous elliptic PDEs and offer the ability to solve these with high accuracy and speed on complex domains. In this paper, such a method is extended to the heat equation with inhomogeneous source terms. First, the heat equation is discretised in time, then in each time step we solve a sequence of so-called modified Helmholtz equations with a parameter depending on the time step size. The modified Helmholtz equation is then split into two: a homogeneous part solved with a boundary integral method and a particular part, where the solution is obtained by evaluating a volume potential over the inhomogeneous source term over a simple domain. In this work, we introduce two components which are critical for the success of this approach: a method to efficiently compute a high-regularity extension of a function outside the domain where it is defined, and a special quadrature method to accurately evaluate singular and nearly singular integrals in the integral formulation of the modified Helmholtz equation for all time step sizes.

  • 28.
    Fryklund, Fredrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Lehto, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Partition of unity extension of functions on complex domains2018In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 375, p. 57-79Article in journal (Refereed)
    Abstract [en]

    We introduce an efficient algorithm, called partition of unity extension or PUX, to construct an extension of desired regularity of a function given on a complex multiply connected domain in 2D. Function extension plays a fundamental role in extending the applicability of boundary integral methods to inhomogeneous partial differential equations with embedded domain techniques. Overlapping partitions are placed along the boundaries, and a local extension of the function is computed on each patch using smooth radial basis functions; a trivially parallel process. A partition of unity method blends the local extrapolations into a global one, where weight functions impose compact support. The regularity of the extended function can be controlled by the construction of the partition of unity function. We evaluate the performance of the PUX method in the context of solving the Poisson equation on multiply connected domains using a boundary integral method and a spectral solver. With a suitable choice of parameters the error converges as a tenth order method down to 10−14.

  • 29.
    Fryklund, Fredrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Pålsson, Sara
    KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    An adaptive integral equation method for the two-dimensional advection-diffusion equation on time-dependent domainsManuscript (preprint) (Other academic)
  • 30.
    Fryklund, Fredrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Pålsson, Sara
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    An integral equation method for the advection-diffusion equation on time-dependent domains in the plane2023In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 475, article id 111856Article in journal (Refereed)
    Abstract [en]

    Boundary integral methods are attractive for solving homogeneous elliptic partial differen-tial equations on complex geometries, since they can offer accurate solutions with a com-putational cost that is linear or close to linear in the number of discretization points on the boundary of the domain. However, these numerical methods are not straightforward to ap-ply to time-dependent equations, which often arise in science and engineering. We address this problem with an integral equation-based solver for the advection-diffusion equation on moving and deforming geometries in two space dimensions. In this method, an adap-tive high-order accurate time-stepping scheme based on semi-implicit spectral deferred correction is applied. One time-step then involves solving a sequence of non-homogeneous modified Helmholtz equations, a method known as elliptic marching. Our solution method-ology utilizes several recently developed methods, including special purpose quadrature, a function extension technique and a spectral Ewald method for the modified Helmholtz kernel. Special care is also taken to handle the time-dependent geometries. The numerical method is tested through several numerical examples to demonstrate robustness, flexibility and accuracy.

  • 31.
    Fryklund, Fredrik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Partition of unity extension for high-order approximationsManuscript (preprint) (Other academic)
  • 32.
    Gustavsson, Katarina
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Tornberg, Anna Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Numerical Simulations of Rigid Fiber Suspensions2008Conference paper (Refereed)
    Abstract [en]

    In this paper, we present a numerical method designed to simulate the

    challenging problem of the dynamics of slender fibers immersed in an incompressible

    fluid. Specifically, we consider microscopic, rigid fibers, that

    sediment due to gravity. Such fibers make up the micro-structure of many

    suspensions for which the macroscopic dynamics are not well understood.

    Our numerical algorithm is based on a non-local slender body approximation

    that yields a system of coupled integral equations, relating the forces

    exerted on the fibers to their velocities, which takes into account the hydrodynamic

    interactions of the fluid and the fibers. The system is closed by

    imposing the constraints of rigid body motions.

    The fact that the fibers are straight have been further exploited in the

    design of the numerical method, expanding the force on Legendre polynomials

    to take advantage of the specific mathematical structure of a finite-part

    integral operator, as well as introducing analytical quadrature in a manner

    possible only for straight fibers.

    We have carefully treated issues of accuracy, and present convergence

    results for all numerical parameters before we finally discuss the results from

    simulations including a larger number of fibers.

  • 33.
    Gustavsson, Katarina
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Gravity induced sedimentation of slender fibers2009In: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 21, no 12Article in journal (Refereed)
    Abstract [en]

    Gravity induced sedimentation of slender, rigid fibers in a highly viscous fluid is investigated by large scale numerical simulations. The fiber suspension is considered on a microscopic level and the flow is described by the Stokes equations in a three dimensional periodic domain. Numerical simulations are performed to study in great detail the complex dynamics of a cluster of fibers. A repeating cycle is identified. It consists of two main phases: a densification phase, where the cluster densifies and grows, and a coarsening phase, during which the cluster becomes smaller and less dense. The dynamics of these phases and their relation to fluctuations in the sedimentation velocity are analyzed. Data from the simulations are also used to investigate how average fiber orientations and sedimentation velocities are influenced by the microstructure in the suspension. The dynamical behavior of the fiber suspension is very sensitive to small random differences in the initial configuration and a number of realizations of each numerical experiment are performed. Ensemble averages of the sedimentation velocity and fiber orientation are presented for different values of the effective concentration of fibers and the results are compared to experimental data. The numerical code is parallelized using the Message Passing Instructions (MPI) library and numerical simulations with up 800 fibers can be run for very long times which is crucial to reach steady levels of the averaged quantities. The influence of the periodic boundary conditions on the process is also carefully investigated.

  • 34. Jung, Sunghwan
    et al.
    Spagnolie, S. E.
    Parikh, K.
    Shelley, M.
    Tornberg, Anna-Karin
    Periodic sedimentation in a Stokesian fluid2006In: Physical Review E, ISSN 1539-3755, Vol. 74, no 3Article in journal (Refereed)
    Abstract [en]

    We study the sedimentation of two identical but nonspherical particles sedimenting in a Stokesian fluid. Experiments and numerical simulations reveal periodic orbits wherein the bodies mutually induce an in-phase rotational motion accompanied by periodic modulations of sedimentation speed and separation distance. We term these tumbling orbits and find that they appear over a broad range of body shapes.

  • 35. Kanevsky, Alex
    et al.
    Shelley, Michael J.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Modeling simple locomotors in Stokes flow2010In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 229, no 4, p. 958-977Article in journal (Refereed)
    Abstract [en]

    Motivated by the locomotion of flagellated micro-organisms and by recent experiments of chemically driven nanomachines, we study the dynamics of bodies of simple geometric shape that are propelled by specified tangential surface stresses. We develop a mathematical description of the body dynamics based on a mixed-type boundary integral formulation. We also derive analytic axisymmetric solutions for the case of a single locomoting sphere and ellipsoid based on spherical and ellipsoidal harmonics, and compare our numerical results to these. The hydrodynamic interactions between two spherical and ellipsoidal swimmers in an infinite fluid are then simulated using second-order accurate spatial and temporal discretizations. We find that the near-field interactions result in complex and interesting changes in the locomotors' orientations and trajectories. Stable as well as unstable pairwise swimming motions are observed, similar to the recent findings of Pooley et al. [C.M. Pooley, G.P. Alexander, J.M. Yeomans, Hydrodynamic interaction between two swimmers at low Reynolds number, Phys. Rev. Lett. 99 (2007) 228103].

  • 36. Khatri, Shilpa
    et al.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    A numerical method for two phase flows with insoluble surfactants2011In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 49, no 1, p. 150-165Article in journal (Refereed)
    Abstract [en]

    In many practical multiphase flow problems, i.e. treatment of gas emboli and various microfluidic applications, the effect of interfacial surfactants, or surface reacting agents, on the surface tension between the fluids is important. The surfactant concentration on an interface separating the fluids can be modeled with a time dependent differential equation defined on the moving and deforming interface. The equations for the location of the interface and the surfactant concentration on the interface are coupled with the Navier-Stokes equations. These equations include the singular surface tension forces from the interface on the fluid, which depend on the interfacial surfactant concentration. A new accurate and inexpensive numerical method for simulating the evolution of insoluble surfactants is presented in this paper. It is based on an explicit yet Eulerian discretization of the interface, which for two dimensional flows allows for the use of uniform one dimensional grids to discretize the equation for the interfacial surfactant concentration. A finite difference method is used to solve the Navier-Stokes equations on a regular grid with the forces from the interface spread to this grid using a regularized delta function. The timestepping is based on a Strang splitting approach. Drop deformation in shear flows in two dimensions is considered. Specifically, the effect of surfactant concentration on the deformation of the drops is studied for different sets of flow parameters.

  • 37. Khatri, Shilpa
    et al.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    An embedded boundary method for soluble surfactants with interface tracking for two-phase flows2014In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 256, p. 768-790Article in journal (Refereed)
    Abstract [en]

    Surfactants, surface reacting agents, lower the surface tension of the interface between fluids in multiphase flow. This capability of surfactants makes them ideal for many applications, including wetting, foaming, and dispersing. Due to their molecular composition, surfactants are adsorbed from the bulk fluid to the interface between the fluids, leading to different concentrations on the interface and in the fluid. In a previous paper [21], we introduced a new second order method using uniform grids to simulate insoluble surfactants in multiphase flow. This method used Strang splitting allowing for a fully second order treatment in time. Here, we use the same numerical methods to explicitly represent the singular interface, treat the interfacial surfactant concentration, and couple with the Navier-Stokes equations. Now, we introduce a second order method for the surfactants in the bulk that continues to allow the use of regular grids for the full problem. Difficulties arise since the boundary condition for the bulk concentration, which handles the flux of surfactant between the interface and bulk fluid, is applied at the interface which cuts arbitrarily through the regular grid. We extend the embedded boundary method, introduced in [22], to handle this challenge. Through our results, we present the effect of the solubility of the surfactants. We show results of drop dynamics due to resulting Marangoni stresses and of drop deformations in shear flow in the presence of soluble surfactants. There is a large nondimensional parameter space over which we try to understand the drop dynamics.

  • 38.
    Lindbo, Dag
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Fast and spectrally accurate Ewald summation for 2-periodic electrostatic systems2012In: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 136, no 16, p. 164111-1-164111-16Article in journal (Refereed)
    Abstract [en]

    A new method for Ewald summation in planar/slablike geometry, i.e., systems where periodicity applies in two dimensions and the last dimension is "free" (2P), is presented. We employ a spectral representation in terms of both Fourier series and integrals. This allows us to concisely derive both the 2P Ewald sum and a fast particle mesh Ewald (PME)-type method suitable for large-scale computations. The primary results are: (i) close and illuminating connections between the 2P problem and the standard Ewald sum and associated fast methods for full periodicity; (ii) a fast, O(N log N), and spectrally accurate PME-type method for the 2P k-space Ewald sum that uses vastly less memory than traditional PME methods; (iii) errors that decouple, such that parameter selection is simplified. We give analytical and numerical results to support this.

  • 39.
    Lindbo, Dag
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Fast and spectrally accurate summation of 2-periodic Stokes potentialsManuscript (preprint) (Other academic)
    Abstract [en]

    We derive a Ewald decomposition for the Stokeslet in planar periodicity and a novel PME-type O(N log N) method for the fast evaluation of the resulting sums. The decomposition is the natural 2P counterpart to the classical 3P decomposition by Hasimoto, and is given in an explicit form not found in the literature. Truncation error estimates are provided to aid in selecting parameters. The fast, PME-type, method appears to be the first fast method for computing Stokeslet Ewald sums in planar periodicity, and has three attractive properties: it is spectrally accurate; it uses the minimal amount of memory that a gridded Ewald method can use; and provides clarity regarding numerical errors and how to choose parameters. Analytical and numerical results are give to support this. We explore the practicalities of the proposed method, and survey the computational issues involved in applying it to 2-periodic boundary integral Stokes problems.

  • 40.
    Lindbo, Dag
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Interface tracking using patches2011Manuscript (preprint) (Other academic)
  • 41.
    Lindbo, Dag
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Spectral accuracy in fast Ewald-based methods for particle simulations2011In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 230, no 24, p. 8744-8761Article in journal (Refereed)
    Abstract [en]

    A spectrally accurate fast method for electrostatic calculations under periodic boundary conditions is presented. We follow the established framework of FFT-based Ewald summation, but obtain a method with an important decoupling of errors: it is shown, for the proposed method, that the error due to frequency domain truncation can be separated from the approximation error added by the fast method. This has the significance that the truncation of the underlying Ewald sum prescribes the size of the grid used in the FFT-based fast method, which clearly is the minimal grid. Both errors are of exponential-squared order, and the latter can be controlled independently of the grid size. We compare numerically to the established SPME method by Essmann et al. and see that the memory required can be reduced by orders of magnitude. We also benchmark efficiency (i.e. error as a function of computing time) against the SPME method, which indicates that our method is competitive. Analytical error estimates are proven and used to select parameters with a great degree of reliability and ease.

  • 42.
    Lindbo, Dag
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Spectrally accurate fast summation for periodic Stokes potentials2010In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 229, no 23, p. 8994-9010Article in journal (Refereed)
    Abstract [en]

    A spectrally accurate method for the fast evaluation of N-particle sums of the periodic Stokeslet is presented. Two different decomposition methods, leading to one sum in real space and one in reciprocal space, are considered. An FFT based method is applied to the reciprocal part of the sum, invoking the equivalence of multiplications in reciprocal space to convolutions in real space, thus using convolutions with a Gaussian function to place the point sources on a grid. Due to the spectral accuracy of the method, the grid size needed is low and also in practice, for a fixed domain size, independent of N. The leading cost, which is linear in N, arises from the to-grid and from-grid operations. Combining this FFT based method for the reciprocal sum with the direct evaluation of the real space sum, a spectrally accurate algorithm with a total complexity of 0(N log N) is obtained. This has been shown numerically as the system is scaled up at constant density. (C) 2010 Elsevier Inc. All rights reserved.

  • 43.
    Marin, Oana
    et al.
    KTH, School of Engineering Sciences (SCI).
    Gustavsson, Katarina
    KTH, School of Engineering Sciences (SCI).
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI).
    A fast summation method for fiber simulationsManuscript (preprint) (Other academic)
  • 44.
    Marin, Oana
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Gustavsson, Katarina
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    A highly accurate boundary treatment for confined Stokes flow2012In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 66, p. 215-230Article in journal (Refereed)
    Abstract [en]

    Fluid flow phenomena in the Stokesian regime abounds in nature as well as in microfluidic applications. Discretizations based on boundary integral formulations for such flow problems allow for a reduction in dimensionality but have to deal with dense matrices and the numerical evaluation of integrals with singular kernels. The focus of this paper is the discretization of wall confinements, and specifically the numerical treatment of flat solid boundaries (walls), for which a set of high-order quadrature rules that accurately integrate the singular kernel of the Stokes equations are developed. Discretizing by Nystrom's method, the accuracy of the numerical integration determines the accuracy of the solution of the boundary integral equations, and a higher order quadrature method yields a large gain in accuracy at negligible cost. The structure of the resulting submatrix associated with each wall is exploited in order to substantially reduce the memory usage. The expected convergence of the quadrature rules is validated through numerical tests, and this boundary treatment is further applied to the classical problem of a sedimenting sphere in the vicinity of solid walls.

  • 45.
    Marin, Oana
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
    Gustavsson, Katarina
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Tornberg, Anna-Karin
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    A wall treatment for confined Stokes flowArticle in journal (Other academic)
    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.

  • 46.
    Marin, Oana
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Runborg, Olof
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Corrected trapezoidal rules for a class of singular functions2014In: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 34, no 4, p. 1509-1540Article in journal (Refereed)
    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.

  • 47.
    Ojala, Rikard
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    An accurate integral equation method for simulating multi-phase Stokes flow2015In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 298, p. 145-160Article in journal (Refereed)
    Abstract [en]

    We introduce a numerical method based on an integral equation formulation for simulating drops in viscous fluids in the plane. It builds upon the method introduced by Kropinski in 2001 [17], but improves on it by adding an interpolatory quadrature approach for handling near-singular integrals. Such integrals typically arise when drop boundaries come close to one another, and are difficult to compute accurately using standard quadrature rules. Adapting the interpolatory quadrature method introduced by Helsing and Ojala in 2008 [11] to the current application, very general drop configurations can be handled while still maintaining stability and high accuracy. The performance of the new method is demonstrated by some challenging numerical examples.

  • 48.
    Pålsson, Sara
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Siegel, Michael
    New Jersey Inst Technol, Dept Math Sci, Newark, NJ 07102 USA..
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Simulation and validation of surfactant-laden drops in two-dimensional Stokes flow2019In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 386, p. 218-247Article in journal (Refereed)
    Abstract [en]

    Performing highly accurate simulations of droplet systems is a challenging problem. This is primarily due to the interface dynamics which is complicated further by the addition of surfactants. This paper presents a boundary integral method for computing the evolution of surfactant-covered droplets in 2D Stokes flow. The method has spectral accuracy in space and the adaptive time-stepping scheme allows for control of the temporal errors. Previously available semi-analytical solutions (based on conformal-mapping techniques) are extended to include surfactants, and a set of algorithms is introduced to detail their evaluation. These semi-analytical solutions are used to validate and assess the accuracy of the boundary integral method, and it is demonstrated that the presented method maintains its high accuracy even when droplets are in close proximity. 

  • 49.
    Pålsson, Sara
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Sorgentone, Chiara
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW. KTH, Centres, SeRC - Swedish e-Science Research Centre.
    Adaptive time-stepping for surfactant-laden drops2017In: Proceedings of the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11) / [ed] Chappell, D.J., 2017Conference paper (Refereed)
    Abstract [en]

    An adaptive time-stepping scheme is presented aimed at computing the dynamics of surfactant-covered deforming droplets. This involves solving a coupled system, where one equation corresponds to the evolution of the drop interfaces and one to the surfactant concentration. The first is discretised in space using a boundary integral formulation which can be treated explicitly in time. The latter is a convection-diffusion equation solved with a spectral method and is advantageously solved with a semi-implicit method in time. The scheme is adaptive with respect to drop deformation as well as surfactant concentration and the adjustment of time-steps takes both errors into account. It is applied and demonstrated for simulation of the deformation of surfactant-covered droplets, but can easily be applied to any system of equations with similar structure. Tests are performed for both 2D and 3D formulations and the scheme is shown to meet set error tolerances in an efficient way.

  • 50.
    Pålsson, Sara
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    An integral equation method for closely interacting surfactant-covered droplets in wall-confined Stokes flowManuscript (preprint) (Other academic)
    Abstract [en]

    A highly accurate method for simulating surfactant-covered droplets in two-dimensional Stokes flow with solid boundaries is presented. The method handles both periodic channel flows of arbitrary shape and stationary solid constrictions. A boundary integral method together with a special quadrature scheme is applied to solve the Stokes equations to high accuracy, also for droplets in close interaction. The problem is considered in a periodic setting and Ewald decompositions for the Stokeslet and stresslet are derived to make the periodic sums convergent. Computations are sped up using the spectral Ewald method. The time evolution is handled with a fourth order, adaptive, implicit-explicit time-stepping scheme. The numerical method is tested through several convergence studies and other challenging examples and is shown to handle drops in close proximity both to other drops and solid objects to a high accuracy.

12 1 - 50 of 89
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