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Tornberg, Anna-Karin
Publications (10 of 15) Show all publications
Sorgentone, C., Tornberg, A.-K. & Vlahovska, P. M. (2019). A 3D boundary integral method for the electrohydrodynamics of surfactant-covered drops. Journal of Computational Physics, 389, 111-127
Open this publication in new window or tab >>A 3D boundary integral method for the electrohydrodynamics of surfactant-covered drops
2019 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 389, p. 111-127Article in journal (Refereed) Published
Abstract [en]

We present a highly accurate numerical method based on a boundary integral formulation and the leaky dielectric model to study the dynamics of surfactant-covered drops in the presence of an applied electric field. The method can simulate interacting 3D drops (no axisymmetric simplification) in close proximity, can consider different viscosities, is adaptive in time and able to handle substantial drop deformation. For each drop global representations of the variables based on spherical harmonics expansions are used and the spectral accuracy is achieved by designing specific numerical tools: a specialized quadrature method for the singular and nearly singular integrals that appear in the formulation, a general preconditioner for the implicit treatment of the surfactant diffusion and a reparametrization procedure able to ensure a high-quality representation of the drops also under deformation. Our numerical method is validated against theoretical, numerical and experimental results available in the literature, as well as a new second-order theory developed for a surfactant-laden drop placed in a quadrupole electric field.

Place, publisher, year, edition, pages
ACADEMIC PRESS INC ELSEVIER SCIENCE, 2019
Keywords
Boundary integral method, Spherical harmonics, Stokes flow, Surfactant, Electric field, Small deformation theory
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-252584 (URN)10.1016/j.jcp.2019.03.041 (DOI)000467918600006 ()2-s2.0-85064312878 (Scopus ID)
Note

QC 20190611

Available from: 2019-06-11 Created: 2019-06-11 Last updated: 2019-06-11Bibliographically approved
Pålsson, S., Siegel, M. & Tornberg, A.-K. (2019). Simulation and validation of surfactant-laden drops in two-dimensional Stokes flow. Journal of Computational Physics, 386, 218-247
Open this publication in new window or tab >>Simulation and validation of surfactant-laden drops in two-dimensional Stokes flow
2019 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 386, p. 218-247Article in journal (Refereed) Published
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. 

Place, publisher, year, edition, pages
Academic Press, 2019
Keywords
Insoluble surfactants, Stokes flow, Validation, Integral equations, Two-phase flow, Drop deformation, Special quadrature
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-251466 (URN)10.1016/j.jcp.2018.12.044 (DOI)000464675600011 ()2-s2.0-85062883114 (Scopus ID)
Note

QC 20190515

Available from: 2019-05-15 Created: 2019-05-15 Last updated: 2019-11-26Bibliographically approved
Sorgentone, C. & Tornberg, A.-K. (2018). A highly accurate boundary integral equation method for surfactant-laden drops in 3D. Journal of Computational Physics, 360, 167-191
Open this publication in new window or tab >>A highly accurate boundary integral equation method for surfactant-laden drops in 3D
2018 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 360, p. 167-191Article in journal (Refereed) Published
Abstract [en]

The presence of surfactants alters the dynamics of viscous drops immersed in an ambient viscous fluid. This is specifically true at small scales, such as in applications of droplet based microfluidics, where the interface dynamics become of increased importance. At such small scales, viscous forces dominate and inertial effects are often negligible. Considering Stokes flow, a numerical method based on a boundary integral formulation is presented for simulating 3D drops covered by an insoluble surfactant. The method is able to simulate drops with different viscosities and close interactions, automatically controlling the time step size and maintaining high accuracy also when substantial drop deformation appears. To achieve this, the drop surfaces as well as the surfactant concentration on each surface are represented by spherical harmonics expansions. A novel reparameterization method is introduced to ensure a high-quality representation of the drops also under deformation, specialized quadrature methods for singular and nearly singular integrals that appear in the formulation are evoked and the adaptive time stepping scheme for the coupled drop and surfactant evolution is designed with a preconditioned implicit treatment of the surfactant diffusion.

Place, publisher, year, edition, pages
Academic Press, 2018
Keywords
Boundary integral method, Spherical harmonics, Stokes flow, Surfactant
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-227585 (URN)10.1016/j.jcp.2018.01.033 (DOI)000428966300010 ()2-s2.0-85041629805 (Scopus ID)
Funder
Knut and Alice Wallenberg Foundation, KAW 2013.0339Swedish e‐Science Research Center
Note

QC 20180515

Available from: 2018-05-15 Created: 2018-05-15 Last updated: 2018-05-15Bibliographically approved
Siegel, M. & Tornberg, A.-K. (2018). A local target specific quadrature by expansion method for evaluation of layer potentials in 3D. Journal of Computational Physics, 364, 365-392
Open this publication in new window or tab >>A local target specific quadrature by expansion method for evaluation of layer potentials in 3D
2018 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 364, p. 365-392Article in journal (Refereed) Published
Abstract [en]

Accurate evaluation of layer potentials is crucial when boundary integral equation methods are used to solve partial differential equations. Quadrature by expansion (QBX) is a recently introduced method that can offer high accuracy for singular and nearly singular integrals, using truncated expansions to locally represent the potential. The QBX method is typically based on a spherical harmonics expansion which when truncated at order p has O(p2) terms. This expansion can equivalently be written with p terms, however paying the price that the expansion coefficients will depend on the evaluation/target point. Based on this observation, we develop a target specific QBX method, and apply it to Laplace's equation on multiply-connected domains. The method is local in that the QBX expansions only involve information from a neighborhood of the target point. An analysis of the truncation error in the QBX expansions is presented, practical parameter choices are discussed and the method is validated and tested on various problems.

Place, publisher, year, edition, pages
Academic Press, 2018
Keywords
Exterior Dirichlet problem, Integral equations, Layer potentials, Multiply-connected domain, Quadrature by expansion, Spherical harmonics expansions
National Category
Other Mathematics
Identifiers
urn:nbn:se:kth:diva-227521 (URN)10.1016/j.jcp.2018.03.006 (DOI)000432481000017 ()2-s2.0-85044166752 (Scopus ID)
Funder
Knut and Alice Wallenberg Foundation, KAW2014.0338Göran Gustafsson Foundation for Research in Natural Sciences and MedicineSwedish e‐Science Research Center
Note

QC 20180515

Available from: 2018-05-15 Created: 2018-05-15 Last updated: 2018-06-13Bibliographically approved
Srinivasan, S. & Tornberg, A.-K. (2018). Fast Ewald summation for Green's functions of Stokes flow in a half-space. RESEARCH IN THE MATHEMATICAL SCIENCES, 5, Article ID 35.
Open this publication in new window or tab >>Fast Ewald summation for Green's functions of Stokes flow in a half-space
2018 (English)In: RESEARCH IN THE MATHEMATICAL SCIENCES, ISSN 2197-9847, Vol. 5, article id 35Article in journal (Refereed) Published
Abstract [en]

Recently, Gimbutas et al. (J Fluid Mech, 2015. https://doi.org/10.1017/jfm.2015.302) derived an elegant representation for the Green's functions of Stokes flow in a half-space. We present a fast summation method for sums involving these half-space Green's functions (stokeslets, stresslets and rotlets) that consolidates and builds on the work by Klinteberg et al. (Res Math Sci 4(1): 1, 2017. https://doi.org/10.1186/s40687-016-0092-7) for the corresponding free-space Green's functions. The fast method is based on two main ingredients: The Ewald decomposition and subsequent use of FFTs. The Ewald decomposition recasts the sum into a sum of two exponentially decaying series: one in real space (short-range interactions) and one in Fourier space (long-range interactions) with the convergence of each series controlled by a common parameter. The evaluation of short-range interactions is accelerated by restricting computations to neighbours within a specified distance, while the use of FFTs accelerates the computations in Fourier space thus accelerating the overall sum. We demonstrate that while the method incurs extra costs for the half-space in comparison with the free-space evaluation, greater computational savings is also achieved when compared to their respective direct sums.

Place, publisher, year, edition, pages
Springer, 2018
Keywords
Ewald summation, Stokes flow, Green's function, Stokeslet, Rotlet, Stresslet, Half-space
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-234613 (URN)10.1007/s40687-018-0153-1 (DOI)000442336600002 ()
Note

QC 20180914

Available from: 2018-09-14 Created: 2018-09-14 Last updated: 2018-09-14Bibliographically approved
Fryklund, F., Lehto, E. & Tornberg, A.-K. (2018). Partition of unity extension of functions on complex domains. Journal of Computational Physics, 375, 57-79
Open this publication in new window or tab >>Partition of unity extension of functions on complex domains
2018 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 375, p. 57-79Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Academic Press Inc., 2018
Keywords
Boundary integral method, Embedded domain, Function extension, Linear elliptic partial differential equation, Partition of unity, Radial basis function
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-236548 (URN)10.1016/j.jcp.2018.08.012 (DOI)000450907600003 ()2-s2.0-85052310129 (Scopus ID)
Funder
Swedish Research Council, 2015-04998
Note

Funding text: This work has been supported by the Swedish Research Council under Grant No. 2015-04998 and by the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine and is gratefully acknowledged. QC 20181127

Available from: 2018-11-27 Created: 2018-11-27 Last updated: 2018-12-11Bibliographically approved
af Klinteberg, L. & Tornberg, A.-K. (2017). Error estimation for quadrature by expansion in layer potential evaluation. Advances in Computational Mathematics, 43(1), 195-234
Open this publication in new window or tab >>Error estimation for quadrature by expansion in layer potential evaluation
2017 (English)In: Advances in Computational Mathematics, ISSN 1019-7168, E-ISSN 1572-9044, Vol. 43, no 1, p. 195-234Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Springer, 2017
Keywords
Error estimate, Layer potential, Nearly singular, Quadrature
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-201960 (URN)10.1007/s10444-016-9484-x (DOI)000392330500010 ()2-s2.0-84991109241 (Scopus ID)
Funder
Göran Gustafsson Foundation for Research in Natural Sciences and MedicineSwedish Research Council, 2011-3178
Note

QC 20170303

Available from: 2017-03-03 Created: 2017-03-03 Last updated: 2017-11-29Bibliographically approved
af Klinteberg, L., Saffar Shamshirgar, D. & Tornberg, A.-K. (2017). Fast Ewald summation for free-space Stokes potentials. Research in the Mathematical Sciences, 4(1)
Open this publication in new window or tab >>Fast Ewald summation for free-space Stokes potentials
2017 (English)In: Research in the Mathematical Sciences, ISSN 2197-9847, Vol. 4, no 1Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
Springer, 2017
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-203922 (URN)10.1186/s40687-016-0092-7 (DOI)000412664600001 ()
Funder
Göran Gustafsson Foundation for Research in Natural Sciences and MedicineSwedish Research Council, 2011-3178Swedish e‐Science Research Center
Note

QC 20170411

Available from: 2017-03-20 Created: 2017-03-20 Last updated: 2017-12-12Bibliographically approved
Saffar Shamshirgar, D., Hess, B. & Tornberg, A.-K.A comparison of the Spectral Ewald and Smooth Particle Mesh Ewald methods in GROMACS.
Open this publication in new window or tab >>A comparison of the Spectral Ewald and Smooth Particle Mesh Ewald methods in GROMACS
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The smooth particle mesh Ewald (SPME) method is an FFT based methodfor the fast evaluation of electrostatic interactions under periodic boundaryconditions. A highly optimized implementation of this method is availablein GROMACS, a widely used software for molecular dynamics simulations.In this article, we compare a more recent method from the same family ofmethods, the spectral Ewald (SE) method, to the SPME method in termsof performance and efficiency. We consider serial and parallel implementa-tions of both methods for single and multiple core computations on a desktopmachine as well as the Beskow supercomputer at KTH Royal Institute ofTechnology. The implementation of the SE method has been well optimized,however not yet comparable to the level of the SPME implementation thathas been improved upon for many years. We show that the SE method isvery efficient whenever used to achieve high accuracy and that it already atthis level of optimization can be competitive for low accuracy demands.

Keywords
Fast Ewald summation, Fast Fourier transform, Coulomb potentials, SE, SPME, GROMACS
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-219771 (URN)
Funder
Swedish e‐Science Research CenterSwedish Research Council for Environment, Agricultural Sciences and Spatial Planning, 2011-3178
Note

QC 20171213

Available from: 2017-12-12 Created: 2017-12-12 Last updated: 2017-12-18Bibliographically approved
Saffar Shamshirgar, D. & Tornberg, A.-K.A fast multipole method for evaluating exponential integral type kernels.
Open this publication in new window or tab >>A fast multipole method for evaluating exponential integral type kernels
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We present a fast multipole method for evaluation of sums with exponential  integral type kernels. These sums appear while solving free space Poisson problems in two dimensions and in the derivation of 1d-periodic Ewald sums. The presented method uses recurrence relations to derive multipole expansions for computing interactions between particles and far clusters.

Keywords
Fast multipole method, Exponential integral, recurrence relation, Spectral Ewald
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-219774 (URN)
Funder
Swedish e‐Science Research CenterSwedish Research Council for Environment, Agricultural Sciences and Spatial Planning, 2011-3178
Note

QC 20171213

Available from: 2017-12-12 Created: 2017-12-12 Last updated: 2017-12-13Bibliographically approved
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