[Paper Review] A Locally Corrected Multiblob Method with Hydrodynamically Matched Grids for the Stokes Mobility Problem

Anna Broms, Mattias Sandberg+1 more|arXiv (Cornell University)|Jul 22, 2022

Lattice Boltzmann Simulation Studies4 citations

TL;DR

This paper presents a locally corrected multiblob method with hydrodynamically matched grids to improve accuracy in solving the 3D Stokes mobility problem for rigid particles. By optimizing blob grid configurations to match the self-interaction of ideal particles and applying local pair-corrections inspired by Stokesian dynamics, the method reduces both self-interaction and near-field errors, enabling high-accuracy simulations even at coarse resolutions for spheres and axisymmetric rods.

ABSTRACT

Inexpensive numerical methods are key to enable simulations of systems of a large number of particles of different shapes in Stokes flow. 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. Optimisation strategies to determine a grid with a certain number of blobs are presented with the aim of matching the hydrodynamic response of a single accurately described ideal particle, alone in the fluid. Small errors in this self-interaction are essential as they determine the basic error level in a system of well-separated particles. With a good match, 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 blob geometry can lead to errors several orders of magnitude larger in size. The pair-correction is local and cheap to apply, and reduces on the error for more closely interacting particles. 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).

Motivation & Objective

• To reduce the dominant self-interaction error in multiblob methods for Stokes flow by optimizing blob grid configurations to match the hydrodynamic response of ideal particles.
• To improve accuracy for moderately separated and close-proximity particles using a low-cost, local pair-correction technique inspired by Stokesian dynamics.
• To enable accurate simulations of rigid particles with coarse blob discretizations by minimizing baseline error from self-interaction.
• To establish a framework for controllable accuracy in multiblob methods applicable to heterogeneous suspensions and complex particle shapes.
• To support future large-scale simulations by ensuring the corrected mobility matrix remains positive definite and suitable for stochastic dynamics.

Proposed method

• Optimizing blob grid geometries (e.g., rt-grid, r- and t-grids) to match the translational and rotational mobility of ideal particles, minimizing self-interaction error.
• Using a combined solve strategy that separately optimizes grids for translation and rotation, then combining solutions to reduce self-interaction error.
• Applying a local, inexpensive pair-correction based on Stokesian dynamics to correct for near-field hydrodynamic interactions between particles.
• Employing a second-kind boundary integral equation (BIE) formulation with quadrature by expansion (QBX) as a high-accuracy reference for error quantification.
• Using numerical experiments with few-particle systems (spheres and rods) to evaluate error across varying inter-particle distances and configurations.
• Validating the corrected method against reference solutions to quantify improvements in velocity accuracy for both far-field and near-field regimes.

Experimental results

Research questions

• RQ1 How can blob grid configurations be optimized to minimize the self-interaction error in the multiblob method for Stokes mobility?
• RQ2 To what extent does a pair-correction based on Stokesian dynamics reduce velocity errors for particles at moderate to close separations?
• RQ3 Can a combined solve strategy—using separate grids for translation and rotation—significantly reduce self-interaction error compared to standard multiblob grids?
• RQ4 What is the relationship between self-interaction error and the effectiveness of pair-corrections in the multiblob method?
• RQ5 How does the accuracy of the multiblob method depend on particle shape, specifically for axisymmetric rods with smooth caps?

Key findings

• Optimizing the blob grid, particularly the rt-grid for rods, reduces the self-interaction error to a level that becomes the dominant error floor in far-field simulations.
• The combined solve strategy reduces self-interaction error significantly, especially for large particle gaps, by separately optimizing grids for translation and rotation.
• Pair-corrections are effective only when near-field interaction errors exceed the self-interaction error floor; thus, they are most beneficial when self-interaction is already well-matched.
• For coarse discretizations, applying pair-corrections to the combined solve yields substantial accuracy improvements, as the lower self-interaction error allows corrections to have a measurable impact.
• The corrected mobility matrix remains positive definite in all numerical tests, as confirmed by explicit eigenvalue computation, supporting its use in stochastic simulations.
• The method enables accurate solutions to the Stokes mobility problem even with coarse blob resolutions, provided the grid is hydrodynamically matched and pair-corrections are applied appropriately.