StokesMFS3D

StokesMFS3D provides a minimal working implementation of the Method of Fundamental Solutions (MFS) for solving Stokes resistance, mobility, and prescribed-slip mobility problems for spherical or ellipsoidal rigid bodies.

Two discretisations are available:

• Basic discretisation (proxy sources only): used by solve_resistance.m and solve_mobility.m, and hence by demo_spheres.m, demo_slip.m and ellipsoid_mobility_run.m. It supports spheres and ellipsoids, FMM acceleration and prescribed slip.
• Image-enhanced discretisation (spheres, resistance and mobility): used by solve_resistance_with_images.m and solve_mobility_with_images.m, and hence by demo_resistance_with_images.m, demo_cluster_resistance_with_images.m, demo_mobility_with_images.m and demo_cluster_mobility_with_images.m. Image lines and extra collocation caps are added at each close contact, following Broms et al. (2025). See image resistance and image mobility.

Note With images, the discretisation of a sphere depends on its neighbours (gaps, contact directions and image counts). In the basic discretisation all identical spheres share one one-body matrix, which is factorised once. With images, the one-body matrix must be recomputed and refactorised for every unique discretisation, i.e. typically for every sphere that has close neighbours. Cross-sphere interactions are also summed directly, without FMM. The image solver is therefore considerably more expensive per sphere and is intended for close interactions. See images/README_resistance.md.

Accuracy note In the current image-enhanced examples, one-body preconditioned mobility can have a larger independent surface residual than resistance. Dense SVD mobility has generally given higher surface accuracy, at substantially greater memory cost. A small GMRES residual alone does not establish surface accuracy; see image mobility.

Note: Without images, it is recommended to use a sufficiently large separation parameter delta between particles to ensure numerical accuracy.

Features

• Resistance problem: prescribe rigid-body velocities and compute forces and torques.
• Mobility problem: prescribe forces and torques and compute rigid-body velocities, optionally with prescribed surface slip.
  • Large scale suspensions of spheres or ellipsoidals simulated at low cost.
  • One-body preconditioning enables close-to-linear scaling via FMM acceleration.
  • On particle i, the generalized boundary condition is

$$\mathbf{u}(\mathbf{x}) = \mathbf{v}^{(i)} + \boldsymbol{\omega}^{(i)} \times (\mathbf{x}-\mathbf{c}^{(i)}) + \mathbf{u}_{\mathrm{slip}}^{(i)}(\mathbf{x}).$$

Dependencies

To run simulations, ensure the following dependencies are available:

• FMM3D
  Recommended for accelerating calculations with many particles (typically >70 with default settings)

• Stokes_Direct
  Used for direct evaluation of fundamental solutions.
  Only SE0P_Stokeslet_direct.c needs to be compiled in the standard setting. A precompiled binary is included in the Stokes_Direct repository.

To use your own method for accelerated or direct evaluation of Stokeslets, just modify the function getStokesletFlow.m.

• Spherical Design Nodes
  Used to construct the source/target geometry. See geometry/spheres/spherical_designs/README.md for details.

• Memorygraph Used for profiling and checks on memory usage.

Example

In the demo folder, we demonstrate the method using minimal input:

• Run demo_spheres.m to reproduce a basic simulation for spherical particles using the MFS.
• Run demo_slip.m for force- and torque-free spherical particles with prescribed surface slip.
• Run ellipsoid_mobility_run.m with no arguments for a mobility solve, followed by a resistance solve for a cluster of ellipsoidal particles.
• Run demo_resistance_with_images.m for a two-sphere resistance problem with lubrication-adapted image systems (see images/README_resistance.md).
• Run demo_cluster_resistance_with_images.m for a resistance problem with a cluster of close spheres using image systems.
• Run demo_mobility_with_images.m for the matching two-sphere resistance-to-mobility check.
• Run demo_cluster_mobility_with_images.m for the matching five-sphere cluster check.

Several functions, including solve_mobility.m and solve_resistance.m, contain self-tests. When run without arguments, they execute an example simulation.

Publications

This work is based on the following research papers:

1. Accurate close interactions of Stokes spheres using lubrication-adapted image systems
   A. Broms, A.H. Barnett, A.-K. Tornberg, J. Comput. Phys. (2024)
   https://doi.org/10.1016/j.jcp.2024.113636

2. A Method of Fundamental Solutions for Large-Scale 3D Elastance and Mobility Problems
   A. Broms, A.H. Barnett, A.-K. Tornberg, Adv. Comput. Math. (2025)
   https://doi.org/10.1007/s10444-025-10258-4

Roadmap

Planned additions include:

• A symmetrized version of the MFS solver to enable Brownian motion (in progress, see dev_DLP folders)