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Spectral Accuracy in Fast Ewald Methods and Topics in Fluid Interface Simulation
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

This work contains two separate but related parts: one on spectrally  accurate and fast Ewald methods for electrostatics and viscous flow,  and one on micro- and complex fluid interface problems.  In Part I we are concerned with fast and spectrally accurate methods  to compute sums of slowly decaying potentials over periodic  lattices. We consider two PDEs: Laplace (electrostatics, the Coulomb  potential) and Stokes (viscous flow, the Stokeslet''  potential). Moreover, we consider both full and planar periodicity,  the latter meaning that periodicity applies in two dimensions and  the third is free''. These are major simulation tasks in current  molecular dynamics simulations and in many areas of computational  fluid mechanics involving e.g. particle suspensions.   For each of the four combinations of PDE and periodic structure, we  give spectrally accurate and O(N log N) fast methods based on  Ewald's or Ewald-like decompositions of the underlying potential  sums. In the plane-periodic cases we derive the decompositions in a  manner that lets us develop fast methods. Associated error estimates  are developed as needed throughout. All four methods can be placed  in the P3M/PME (Particle Mesh Ewald) family. We argue that they  have certain novel and attractive features: first, they are spectral  accurate; secondly, they use the minimal amount of memory possible  within the PME family; third, each has a clear and reliable view of  numerical errors, such that parameters can be chosen  wisely. Analytical and numerical results are given to support these  propositions. We benchmark accuracy and performance versus an  established (S)PME method.  Part II deals with free boundary problems, specifically numerical  methods for multiphase flow. We give an interface tracking method  based on a domain-decomposition idea that lets us split the  interface into overlapping patches. Each patch is discretized on a  uniform grid, and accurate and efficient numerical methods are given  for the equations that govern interface transport. We demonstrate  that the method is accurate and how it's used in immersed boundary,  and interface, Navier-Stokes methods, as well as in a boundary  integral Stokes setting.  Finally, we consider a problem in complex fluidics where there is a  concentration of surfactants \emph{on} the interface and the  interface itself is in contact with a solid boundary (the contact  line problem). We argue that the domain-decomposition framework is  attractive for formulating and treating complex models  (e.g. involving PDEs on a dynamic interface) and proceed with  developing various aspects of such a method.

##### Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology , 2011. , p. xv, 104
##### Series
Trita-CSC-A, ISSN 1653-5723 ; 2011:19
##### National Category
Computational Mathematics
##### Identifiers
ISBN: 978-91-7501-195-0 (print)OAI: oai:DiVA.org:kth-48805DiVA, id: diva2:458575
##### Public defence
2011-12-16, Salongen, KTHB, Osquars backe 25, Stockholm, 10:00 (English)
##### Funder
Swedish e‐Science Research Center
##### Note
QC 20111125Available from: 2011-11-25 Created: 2011-11-23 Last updated: 2012-05-24Bibliographically approved
##### List of papers
1. Spectral accuracy in fast Ewald-based methods for particle simulations
Open this publication in new window or tab >>Spectral accuracy in fast Ewald-based methods for particle simulations
2011 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 230, no 24, p. 8744-8761Article in journal (Refereed) Published
##### 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.

##### Keywords
Ewald summation, FFT, Molecular dynamics, PME, Spectral accuracy, SPME
##### National Category
Computational Mathematics Other Physics Topics
##### Identifiers
urn:nbn:se:kth:diva-48765 (URN)10.1016/j.jcp.2011.08.022 (DOI)000297081700007 ()2-s2.0-80053628839 (Scopus ID)
##### Funder
Knut and Alice Wallenberg FoundationSwedish e‐Science Research Center
##### Note
QC 20111124Available from: 2011-11-23 Created: 2011-11-23 Last updated: 2017-12-08Bibliographically approved
2. Spectrally accurate fast summation for periodic Stokes potentials
Open this publication in new window or tab >>Spectrally accurate fast summation for periodic Stokes potentials
2010 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 229, no 23, p. 8994-9010Article in journal (Refereed) Published
##### 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.

##### Keywords
Viscous flow, Stokes equations, Potential theory, Ewald summation, FFT, Spectral accuracy
##### National Category
Computational Mathematics
##### Identifiers
urn:nbn:se:kth:diva-26264 (URN)10.1016/j.jcp.2010.08.026 (DOI)000283405700019 ()2-s2.0-77956947285 (Scopus ID)
##### Note
QC 20110114Available from: 2011-01-14 Created: 2010-11-21 Last updated: 2017-12-11Bibliographically approved
3. Fast and spectrally accurate Ewald summation for 2-periodic electrostatic systems
Open this publication in new window or tab >>Fast and spectrally accurate Ewald summation for 2-periodic electrostatic systems
2012 (English)In: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 136, no 16, p. 164111-1-164111-16Article in journal (Refereed) Published
##### 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.

##### Keywords
Ewald summations, Fast methods, Fast particle, K-space, Numerical results, Parameter selection, Spectral representations, Two-dimension, Type methods
##### National Category
Physical Sciences Computational Mathematics
##### Identifiers
urn:nbn:se:kth:diva-48766 (URN)10.1063/1.4704177 (DOI)000303602200013 ()2-s2.0-84860487992 (Scopus ID)
##### Funder
Knut and Alice Wallenberg FoundationSwedish e‐Science Research Center
##### Note

Updated from manuscript to article in journal. QC 20120605

Available from: 2011-11-23 Created: 2011-11-23 Last updated: 2017-12-08Bibliographically approved
4. Fast and spectrally accurate summation of 2-periodic Stokes potentials
Open this publication in new window or tab >>Fast and spectrally accurate summation of 2-periodic Stokes potentials
##### 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.

##### National Category
Computational Mathematics
##### Identifiers
urn:nbn:se:kth:diva-48803 (URN)
##### Note
QS 2011Available from: 2011-11-23 Created: 2011-11-23 Last updated: 2011-11-25Bibliographically approved
5. Interface tracking using patches
Open this publication in new window or tab >>Interface tracking using patches
##### National Category
Computational Mathematics
##### Identifiers
urn:nbn:se:kth:diva-48764 (URN)
##### Note
QS 2011Available from: 2011-11-23 Created: 2011-11-23 Last updated: 2011-11-25Bibliographically approved
6. An explicit Eulerian method for multiphase flow with contact line dynamics and insoluble surfactant
Open this publication in new window or tab >>An explicit Eulerian method for multiphase flow with contact line dynamics and insoluble surfactant
2014 (English)In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 101, p. 50-63Article in journal (Refereed) Published
##### 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.

##### Keywords
Multiphase flow, Insoluble surfactant, Marangoni force, Moving contact line, Immersed boundary method
##### National Category
Fluid Mechanics and Acoustics
##### Identifiers
urn:nbn:se:kth:diva-48763 (URN)10.1016/j.compfluid.2014.05.029 (DOI)000340851500005 ()2-s2.0-84903152815 (Scopus ID)
##### Funder
Swedish Research Council, 621-2007-6375
##### Note

QC 20140919. Updated from accepted to published.

Available from: 2011-11-23 Created: 2011-11-23 Last updated: 2017-12-08Bibliographically approved
7. Generic compressed sparse matrix insertion: algorithms and implementations in MTL4 and FEniCS
Open this publication in new window or tab >>Generic compressed sparse matrix insertion: algorithms and implementations in MTL4 and FEniCS
2009 (English)In: Proceedings of the 8th workshop on Parallel/High-Performance Object-Oriented Scientific Computing, 2009, p. 2-1Conference paper, Published paper (Refereed)
##### Abstract [en]

Sparse matrices are indispensable components of most scientific applications. Nevertheless, there is very little general-purpose software support. With the Matrix Template Library 4 (MTL4) we provide a generic library support for dense and compressed sparse matrices. The first challenge in working with compressed matrices is how to set the nonzero entries in an efficient manner. The implementation in MTL4 does not need any pre-allocation or pre-sorting phase, uses a minimal amount of memory and was in all measures as fast or faster than comparable libraries. We demonstrate the performance on well-defined benchmarks.

##### Keywords
FEniCS, generic programming, matrix template library, sparse matrices
##### National Category
Computer Sciences
##### Identifiers
urn:nbn:se:kth:diva-48762 (URN)10.1145/1595655.1595657 (DOI)2-s2.0-70450186197 (Scopus ID)
##### Note
QC 20111125Available from: 2011-11-23 Created: 2011-11-23 Last updated: 2018-01-12Bibliographically approved

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Cite
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