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Accuracy, efficiency and robustness for rigid particle simulations in Stokes flow
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.ORCID iD: 0000-0003-0613-1426
2024 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The thesis concerns simulation techniques for systems of nano- to micro-scaled rigid particles immersed in a viscous fluid, ubiquitous in nature and industry. With negligible fluid inertia, the set of PDEs known as the Stokes equations can be used to model the hydrodynamics. For a dynamic study, the PDEs have to be solved at any given instance of time, provided the particle configuration and any non-hydrodynamic interactions. The resulting particle velocities can then be used to update the particle coordinates, and the equations repeatedly solved anew. For any simulation result of a physical system to be reliable, it is crucial to control different error contributions, with two error types here particularly in focus: those related to solving the Stokes equations and those related to the update in time.

The PDEs can be recast as boundary integral equations (BIEs) that hold on the particle surfaces. Hydrodynamic interactions are challenging: they are simultaneously long-ranged and expensive to resolve both in time and space for closely interacting particles. The latter is caused by strong lubrication forces resulting from bodies in relative motion. We approach two alternative and related techniques to BIEs that allow for more cost-effective simulations, namely the rigid multiblob method and the method of fundamental solutions. The former is a regularisation technique that allows for generally shaped particles in large systems, both with and without thermal fluctuations. We make two improvements: the basic error level is tied to the discretisation and set by solving a small optimisation problem off-line for each given particle shape, and the accuracy for closely interacting particles is improved by pair-corrections. With the method of fundamental solutions, we present a technique with linear or close to linear scaling in the number of particles, depending on if a so-called resistance or mobility problem is solved. For circles and spheres, the accuracy can be controlled to a target level independently of the particle separations. This is done by the introduction of a small set of image points for every pair of particles close to contact that well manage to represent lubrication forces.        

In the model, particles can neither touch nor overlap, and our work on time-stepping is tied to the problem of contact avoiding. We develop a new strategy that guarantees contact free simulations in 3D, essential for studying the system of particles over long time spans.   

Controlled accuracy in solutions to the Stokes equations can together with robust timestepping allow for simulations that can complement physical experiments of particle systems for a better understanding of their behaviour, to drive the development in fields such as materials science, biomedical engineering and environmental engineering.

Abstract [sv]

Avhandlingen behandlar simuleringstekniker för system av stela partiklar på nano- till mikroskala i en viskös vätska. Sådana system har en stor spännvidd av tillämpningsområden både i naturen och i industrin. Då vätskans tröghet anses försumbar utgör uppsättningen av partiella differentialekvationer (PDEer) känd som Stokes ekvationer en modell för vätskans fysik. För att studera dynamiska förlopp behöver PDEerna lösas vid varje given tidpunkt, givet partikelkonfigurationen och eventuell extern påverkan mellan partiklarna. De resulterande hastigheterna på partiklarna används för att uppdatera dess positioner och ekvationerna kan sedan lösas på nytt. För att ett simuleringsresultat av ett fysiskt system ska vara tillförlitligt är det viktigt att kontrollera olika felkällor. Vi fokuserar specifikt på de numeriska fel som uppstår när Stokes ekvationer löses approximativt och felet från tidsstegningen, alltså uppdateringen av koordinater över tid.               

Interaktionerna i vätskan är utmanande att hantera: de avtar långsamt med ökande partikelavstånd och är dyra att lösa upp vid nära kontakt. Det sistnämnda är en konsekvens av de starka lubrikationskrafter som relativ rörelse mellan partiklar resulterar i på korta avstånd. PDEerna kan omformuleras som randintegralekvationer på partiklarnas ytor. Vi behandlar två alternativa men relaterade tekniker som möjliggör billigare simuleringar. Den stela multiblob-metoden bygger på regularisering och kan hantera stora system av partiklar med generell geometri. Två förbättringar utvecklas: den basala felnivån relaterar till diskretiseringen av partiklarna och sätts genom att förberäkna lösningen till ett litet optimeringsproblem för varje unik partikeltyp. Noggrannheten för nära interaktion förbättras sedan med hjälp av parkorrektioner. Genom en alternativ metod baserad på fundamentallösningar presenterar vi en ny snabb teknik som skalar linjärt med antalet partiklar. För cirklar och sfärer kan noggrannheten kontrolleras oberoende av partikelavstånd genom att introducera en uppsättning reflektionspunkter för varje par av partiklar nära varandra, som väl kan representera de lubrikationskrafter som uppstår.        

I ett Stokesflöde kan partiklar varken kollidera eller överlappa och vårt arbete relaterat till tidsstegning behandlar kontaktundvikande algoritmer. Vi utvecklar en ny optimeringsbaserad strategi som garanterar att partiklar förblir kontaktfria i 3D. En sådan teknik är nödvändig för att kunna studera partiklar över långa tidsintervall.                

Kontrollerad noggrannhet kan tillsammans med robust tidsstegning möjliggöra att simuleringar kan komplettera fysiska experiment så att en ökad förståelse av partikelsystemen kan leda till utveckling inom exempelvis materialvetenskap, biomedicin och miljövetenskap.

Place, publisher, year, edition, pages
Stockholm, Sweden: KTH Royal Institute of Technology, 2024. , p. 107
Series
TRITA-SCI-FOU ; 2024:17
Keywords [en]
Stokes flow, rigid particles, accuracy, fundamental solutions, method of images, multiblob, contact avoiding, complementarity problem, barrier method, elliptic PDE, grid optimisation, pair-correction, boundary integral equations
Keywords [sv]
Stokesflöde, noggrannhet, stela partiklar, fundamentallösningar, randintegralekvation, reflektionspunkter, multiblob, kontaktundvikande algoritmer, komplementaritetsproblem, barriärmetod, elliptisk PDE, gridoptimering, parkorrektion
National Category
Computational Mathematics
Research subject
Applied and Computational Mathematics, Numerical Analysis
Identifiers
URN: urn:nbn:se:kth:diva-344768ISBN: 978-91-8040-879-0 (print)OAI: oai:DiVA.org:kth-344768DiVA, id: diva2:1847481
Public defence
2024-04-26, F3, Lindstedtsvägen 26, Stockholm, 10:00 (English)
Opponent
Supervisors
Funder
Swedish Research Council, 2019-05206Swedish Research Council, 2016-06119Available from: 2024-03-28 Created: 2024-03-27 Last updated: 2024-04-08Bibliographically approved
List of papers
1. A locally corrected multiblob method with hydrodynamically matched grids for the Stokes mobility problem
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, p. 112172-112172, 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: 2024-03-27Bibliographically approved
2. A reflection enhanced method of fundamental solutions for Laplace and Stokes boundary value problems in 2D
Open this publication in new window or tab >>A reflection enhanced method of fundamental solutions for Laplace and Stokes boundary value problems in 2D
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Two elliptic PDEs with Dirichlet boundary conditions are considered in 2D for a collection of simple objects: the exterior Laplace and Stokes boundary value problems. We present a novel, cost-effective, accurate and singularity-free solution technique based on the method of fundamental solutions. For circular objects, controllable accuracy is obtained for close-to-touching neighbours, with sources on inner proxy-boundaries complemented as needed by a small set of extra singularities in near-contact regions. The locations of the extra sources are deduced from the fractal obtained by repeated inversion of circles in circles, sometimes referred to as Indra’s pearls. For Stokes, results for coarsely resolved closely interacting circular particles, undergoing rigid body motion, are compared to results from a well-resolved boundary integral equation equipped with a special quadrature method. A careful parameter study is made for the locations of the additional sources and their singularity types to reach a target accuracy of 10^{-6} for circles of unequal radii, down to particle separations of a thousandth of the particle radii. For well-separated objects, a one-body preconditioning strategy allows for acceleration with the fast multipole method. 

Keywords
Method of fundamental solutions, method of images, elliptic PDE, Stokes flow, accuracy
National Category
Computational Mathematics Fluid Mechanics and Acoustics
Research subject
Applied and Computational Mathematics, Numerical Analysis
Identifiers
urn:nbn:se:kth:diva-344757 (URN)
Funder
Swedish Research Council, 2019-05206Swedish Research Council, 2016-06119
Available from: 2024-03-27 Created: 2024-03-27 Last updated: 2024-04-02
3. Accurate close interactions of Stokes spheres using lubrication-adapted image systems
Open this publication in new window or tab >>Accurate close interactions of Stokes spheres using lubrication-adapted image systems
(English)Manuscript (preprint) (Other academic)
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 sources on interior spheres, augmented by lines of sources adapted to each near-contact to resolve lubrication. The source strengths are found via a least-squares solve at contact-adapted boundary collocation nodes. Results for coarsely resolved spheres undergoing rigid body motion are compared to reference solutions determined with a well-resolved boundary integral formulation equipped with a special quadrature method. With less than 60 additional sources per particle per contact, we show controlled accuracy to three digits in the relative surface velocities for separations between the particles down to a thousandth of the particle radius. Computed forces and torques are more accurate than surface velocities, by a few orders of magnitude. For fixed spheres in a given background flow, the proxy-surface discretization alone gives controlled accuracy. A one-body preconditioning strategy allows for acceleration with the fast multipole method that combined yield close to linear scaling in the number of particles. This is demonstrated by solving problems of up to 2000 spheres on a workstation using 700 unknown proxy-sources per particle.

Keywords
Method of fundamental solutions, method of images, elliptic PDE, Stokes flow, collocation
National Category
Computational Mathematics Fluid Mechanics and Acoustics
Research subject
Applied and Computational Mathematics, Numerical Analysis
Identifiers
urn:nbn:se:kth:diva-344760 (URN)
Funder
Swedish Research Council, 2019-05206Swedish Research Council, 2016-06119
Note

QC 20240405

Available from: 2024-03-27 Created: 2024-03-27 Last updated: 2024-04-05Bibliographically approved
4. A Method of Fundamental Solutions for Large-Scale 3D Elastance and Mobility Problems
Open this publication in new window or tab >>A Method of Fundamental Solutions for Large-Scale 3D Elastance and Mobility Problems
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The method of fundamental solutions (MFS) is effective for the 3D Laplace and Stokes Dirichlet BVPs in the exterior of a collection of simple objects. Here, we present new formulations for the related elastance and mobility problems in the same geometries. The mobility problem computes rigid body velocities, given net forces and torques on the particles, while the elastance problem is the counterpart for Laplace and solves the problem of conductors with known net charges to determine unknown constant potentials. The new formulations allow for application of one-body preconditioning of the resulting systems, and for large suspensions with moderate lubrication forces, sources on inner proxy-surfaces give accuracy on par with a well-resolved boundary integral formulation. The performance of the well-conditioned mobility solver is demonstrated for a suspension of 10000 nearby ellipsoids.   Using a discretization with $2.59\times 10^7$ total degrees of freedom in the preconditioned system,   GMRES converges to five digits of accuracy in less than two hours on a workstation and the scaling is linear in the number of particles.

Keywords
Elliptic PDE, accuracy, Stokes flow, rigid bodies, completion formulation
National Category
Computational Mathematics Fluid Mechanics and Acoustics
Research subject
Applied and Computational Mathematics, Numerical Analysis
Identifiers
urn:nbn:se:kth:diva-344761 (URN)
Funder
Swedish Research Council, 2019-05206Swedish Research Council, 2016-0611
Note

QC 20240405

Available from: 2024-03-27 Created: 2024-03-27 Last updated: 2024-04-05Bibliographically approved
5. A barrier method for contact avoiding particles in Stokes flow
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, p. 112648-112648, 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 and Acoustics
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: 2024-03-27Bibliographically approved

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