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A barrier method for contact avoiding particles in Stokes flow
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.ORCID iD: 0000-0003-0613-1426
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.ORCID iD: 0000-0002-4290-1670
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. Vol. 497, p. 112648-112648, article id 112648
Keywords [en]
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: urn:nbn:se:kth:diva-340423DOI: 10.1016/j.jcp.2023.112648ISI: 001123740300001Scopus ID: 2-s2.0-85177875562OAI: oai:DiVA.org:kth-340423DiVA, id: diva2:1816997
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
In thesis
1. Accuracy, efficiency and robustness for rigid particle simulations in Stokes flow
Open this publication in new window or tab >>Accuracy, efficiency and robustness for rigid particle simulations in Stokes flow
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
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, 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:nbn:se:kth:diva-344768 (URN)978-91-8040-879-0 (ISBN)
Public defence
2024-04-26, F3, Lindstedtsvägen 26, Stockholm, 10:00 (English)
Opponent
Supervisors
Funder
Swedish Research Council, 2019-05206Swedish Research Council, 2016-06119
Available from: 2024-03-28 Created: 2024-03-27 Last updated: 2024-04-08Bibliographically approved

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Broms, AnnaTornberg, Anna-Karin

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