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Tornberg, Anna-KarinORCID iD iconorcid.org/0000-0002-4290-1670
Publications (10 of 60) Show all publications
Broms, A., Barnett, A. H. & Tornberg, A.-K. (2025). Accurate close interactions of Stokes spheres using lubrication-adapted image systems. Journal of Computational Physics, 523, Article ID 113636.
Open this publication in new window or tab >>Accurate close interactions of Stokes spheres using lubrication-adapted image systems
2025 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 523, article id 113636Article in journal (Refereed) Published
Abstract [en]

Stokes flows with near-touching rigid particles induce near-singular lubrication forces under relative motion, making their accurate numerical treatment challenging. With the aim of controlling the accuracy with a computationally cheap method, we present a new technique that combines the method of fundamental solutions (MFS) with the method of images. For rigid spheres, we propose to represent the flow using Stokeslet proxy sources on interior spheres, augmented by lines of image sources adapted to each near-contact to resolve lubrication. Source strengths are found by a least-squares solve at contact-adapted boundary collocation nodes. We include extensive numerical tests, and validate against reference solutions from a well-resolved boundary integral formulation. With less than 60 additional image sources per particle per contact, we show controlled uniform accuracy to three relative digits in surface velocities, and up to five digits in particle forces and torques, for all separations down to a thousandth of the radius. In the special case of flows around fixed particles, the proxy sphere alone gives controlled accuracy. A one-body preconditioning strategy allows acceleration with the fast multipole method, hence close to linear scaling in the number of particles. This is demonstrated by solving problems of up to 2000 spheres on a workstation using only 700 proxy sources per particle.

Place, publisher, year, edition, pages
Elsevier BV, 2025
Keywords
Collocation, Elliptic PDE, Method of fundamental solutions, Method of images, Stokes flow
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-357915 (URN)10.1016/j.jcp.2024.113636 (DOI)001373648900001 ()2-s2.0-85211031187 (Scopus ID)
Note

Not duplicate with DiVA 1847430

QC 20241219

Available from: 2024-12-19 Created: 2024-12-19 Last updated: 2025-01-28Bibliographically approved
Broms, A. & Tornberg, A.-K. (2024). A barrier method for contact avoiding particles in Stokes flow. Journal of Computational Physics, 497, Article ID 112648.
Open this publication in new window or tab >>A barrier method for contact avoiding particles in Stokes flow
2024 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 497, article id 112648Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier, 2024
Keywords
Stokes flow, Contact problem, Rigid particles, Barrier method, Constrained minimisation
National Category
Computational Mathematics Fluid Mechanics
Research subject
Applied and Computational Mathematics, Numerical Analysis
Identifiers
urn:nbn:se:kth:diva-340423 (URN)10.1016/j.jcp.2023.112648 (DOI)001123740300001 ()2-s2.0-85177875562 (Scopus ID)
Funder
KTH Royal Institute of TechnologySwedish Research Council, 2016-06119Swedish Research Council, 2019-05206
Note

QC 20231205

Available from: 2023-12-05 Created: 2023-12-05 Last updated: 2025-02-20Bibliographically approved
Broms, A., Sandberg, M. & Tornberg, A.-K. (2023). A locally corrected multiblob method with hydrodynamically matched grids for the Stokes mobility problem. Journal of Computational Physics, 487, Article ID 112172.
Open this publication in new window or tab >>A locally corrected multiblob method with hydrodynamically matched grids for the Stokes mobility problem
2023 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 487, article id 112172Article in journal (Refereed) Published
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).

Place, publisher, year, edition, pages
Elsevier, 2023
Keywords
Accuracy, Axisymmetry, Grid optimisation, Pair-correction, Rigid multiblob, Stokes flow
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics, Numerical Analysis
Identifiers
urn:nbn:se:kth:diva-328324 (URN)10.1016/j.jcp.2023.112172 (DOI)001122361800001 ()2-s2.0-85156216042 (Scopus ID)
Funder
Swedish Research Council, 2019-05206Swedish Research Council, 2016-06119
Note

QC 20230619

Available from: 2023-06-07 Created: 2023-06-07 Last updated: 2025-02-20Bibliographically approved
Bagge, J. & Tornberg, A.-K. (2023). Accurate quadrature via line extrapolation and rational approximation with application to boundary integral methods for Stokes flow.
Open this publication in new window or tab >>Accurate quadrature via line extrapolation and rational approximation with application to boundary integral methods for Stokes flow
2023 (English)Report (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.

Publisher
p. 26
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics, Numerical Analysis
Identifiers
urn:nbn:se:kth:diva-326996 (URN)
Funder
Swedish Research Council, 2019-05206
Note

QC 20230522

Available from: 2023-05-16 Created: 2023-05-16 Last updated: 2023-05-22Bibliographically approved
Fryklund, F., Pålsson, S. & Tornberg, A.-K. (2023). An integral equation method for the advection-diffusion equation on time-dependent domains in the plane. Journal of Computational Physics, 475, Article ID 111856.
Open this publication in new window or tab >>An integral equation method for the advection-diffusion equation on time-dependent domains in the plane
2023 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 475, article id 111856Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier BV, 2023
Keywords
Modified Helmholtz equation, Advection-diffusion equation, Function extension, Integral equations, Time-dependent geometry
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-323750 (URN)10.1016/j.jcp.2022.111856 (DOI)000915977100001 ()2-s2.0-85145777278 (Scopus ID)
Note

QC 20230214

Available from: 2023-02-14 Created: 2023-02-14 Last updated: 2023-02-14Bibliographically approved
Sorgentone, C. & Tornberg, A.-K. (2023). Estimation of quadrature errors for layer potentials evaluated near surfaces with spherical topology. Advances in Computational Mathematics, 49(6), Article ID 87.
Open this publication in new window or tab >>Estimation of quadrature errors for layer potentials evaluated near surfaces with spherical topology
2023 (English)In: Advances in Computational Mathematics, ISSN 1019-7168, E-ISSN 1572-9044, Vol. 49, no 6, article id 87Article in journal (Refereed) Published
Abstract [en]

Numerical simulations with rigid particles, drops, or vesicles constitute some examples that involve 3D objects with spherical topology. When the numerical method is based on boundary integral equations, the error in using a regular quadrature rule to approximate the layer potentials that appear in the formulation will increase rapidly as the evaluation point approaches the surface and the integrand becomes sharply peaked. To determine when the accuracy becomes insufficient, and a more costly special quadrature method should be used, error estimates are needed. In this paper, we present quadrature error estimates for layer potentials evaluated near surfaces of genus 0, parametrized using a polar and an azimuthal angle, discretized by a combination of the Gauss-Legendre and the trapezoidal quadrature rules. The error estimates involve no unknown coefficients, but complex-valued roots of a specified distance function. The evaluation of the error estimates in general requires a one-dimensional local root-finding procedure, but for specific geometries, we obtain analytical results. Based on these explicit solutions, we derive simplified error estimates for layer potentials evaluated near spheres; these simple formulas depend only on the distance from the surface, the radius of the sphere, and the number of discretization points. The usefulness of these error estimates is illustrated with numerical examples.

Place, publisher, year, edition, pages
Springer Nature, 2023
Keywords
Close evaluation, Error estimate, Gaussian grid, Layer potentials, Nearly singular, Quadrature, Spherical topology
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-340840 (URN)10.1007/s10444-023-10083-7 (DOI)001121844500003 ()2-s2.0-85177680609 (Scopus ID)
Note

QC 20231218

Available from: 2023-12-18 Created: 2023-12-18 Last updated: 2024-01-03Bibliographically approved
Bagge, J. & Tornberg, A.-K. (2023). Fast Ewald summation for Stokes flow with arbitrary periodicity. Journal of Computational Physics, 493, 112473, Article ID 112473.
Open this publication in new window or tab >>Fast Ewald summation for Stokes flow with arbitrary periodicity
2023 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 493, p. 112473-, article id 112473Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier BV, 2023
Keywords
Boundary integral equations, Creeping flow, Fast summation, Fourier analysis, Reduced periodicity, Stokes potentials
National Category
Computational Mathematics Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-337457 (URN)10.1016/j.jcp.2023.112473 (DOI)001079170600001 ()2-s2.0-85170649224 (Scopus ID)
Note

Not duplicate with DiVA 1757235

QC 20231006

Available from: 2023-10-06 Created: 2023-10-06 Last updated: 2023-10-31Bibliographically approved
Fryklund, F., af Klinteberg, L. & Tornberg, A.-K. (2022). An adaptive kernel-split quadrature method for parameter-dependent layer potentials. Advances in Computational Mathematics, 48(2), Article ID 12.
Open this publication in new window or tab >>An adaptive kernel-split quadrature method for parameter-dependent layer potentials
2022 (English)In: Advances in Computational Mathematics, ISSN 1019-7168, E-ISSN 1572-9044, Vol. 48, no 2, article id 12Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Springer Nature, 2022
Keywords
Integral equations, Partial differential equations, Layer potentials, Modified Helmholtz equation, Modified Stokes equation
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-310254 (URN)10.1007/s10444-022-09927-5 (DOI)000766563600001 ()2-s2.0-85126240977 (Scopus ID)
Note

QC 20220325

Available from: 2022-03-25 Created: 2022-03-25 Last updated: 2022-06-25Bibliographically approved
af Klinteberg, L., Sorgentone, C. & Tornberg, A.-K. (2022). Quadrature error estimates for layer potentials evaluated near curved surfaces in three dimensions. Computers and Mathematics with Applications, 111, 1-19
Open this publication in new window or tab >>Quadrature error estimates for layer potentials evaluated near curved surfaces in three dimensions
2022 (English)In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 111, p. 1-19Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Elsevier BV, 2022
Keywords
Layer potential, Close evaluation, Quadrature, Nearly singular, Error estimate
National Category
Atom and Molecular Physics and Optics Applied Mechanics Fusion, Plasma and Space Physics
Identifiers
urn:nbn:se:kth:diva-312763 (URN)10.1016/j.camwa.2022.02.001 (DOI)000789919800001 ()2-s2.0-85124958447 (Scopus ID)
Note

QC 20220523

Available from: 2022-05-23 Created: 2022-05-23 Last updated: 2022-06-25Bibliographically approved
Bystricky, L., Pålsson, S. & Tornberg, A.-K. (2021). An accurate integral equation method for Stokes flow with piecewise smooth boundaries. BIT Numerical Mathematics, 61(1), 309-335
Open this publication in new window or tab >>An accurate integral equation method for Stokes flow with piecewise smooth boundaries
2021 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 61, no 1, p. 309-335Article in journal (Refereed) Published
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. 

Place, publisher, year, edition, pages
Springer Nature, 2021
Keywords
Boundary integral equations, Creeping flow, Multiphase flow, Numerical analysis
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-285362 (URN)10.1007/s10543-020-00816-1 (DOI)000557133500001 ()2-s2.0-85089183171 (Scopus ID)
Note

QC 20250318

Available from: 2020-12-01 Created: 2020-12-01 Last updated: 2025-03-18Bibliographically approved
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ORCID iD: ORCID iD iconorcid.org/0000-0002-4290-1670

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