The method of fundamental solutions (MFS) is known to be effective for solving 3D Laplace and Stokes Dirichlet boundary value problems in the exterior of a large collection of simple smooth objects. Here, we present new scalable MFS formulations for the corresponding elastance and mobility problems. The elastance problem computes the potentials of conductors with given net charges, while the mobility problem—crucial to rheology and complex fluid applications—computes rigid body velocities given net forces and torques on the particles. The key idea is orthogonal projection of the net charge (or forces and torques) in a rectangular variant of a “completion flow.” The proposal is compatible with one-body preconditioning, resulting in well-conditioned square linear systems amenable to fast multipole accelerated iterative solution, thus a cost linear in the particle number. For large suspensions with moderate lubrication forces, MFS sources on inner proxy-surfaces give accuracy on par with a well-resolved boundary integral formulation. Our several numerical tests include a suspension of 10,000 nearby ellipsoids, using 2.6 x 10⁷ total preconditioned degrees of freedom, where GMRES converges to five digits of accuracy in under two hours on one workstation.

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.

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.

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.

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.

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).

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.

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. 

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.