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).

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.

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.

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.