Abstract
The method of regularized stokeslets is extensively used in biological fluid dynamics due to its conceptual simplicity and meshlessness. This simplicity carries a degree of cost in computational expense and accuracy because the number of degrees of freedom used to discretise the unknown surface traction is generally significantly higher than that required by boundary element methods. We describe a meshless method based on nearest-neighbour interpolation that significantly reduces the number of degrees of freedom required to discretise the unknown traction, increasing the range of problems that can be practically solved, without excessively complicating the task of the modeller. The nearest-neighbour technique is tested against the classical problem of rigid body motion of a sphere immersed in very viscous fluid, then applied to the more complex biophysical problem of calculating the rotational diffusion timescales of a macromolecular structure modelled by three closely-spaced non-slender rods. A heuristic for finding the required density of force and quadrature points by numerical refinement is suggested. Matlab/GNU Octave code for the key steps of the algorithm is provided, which predominantly use basic linear algebra operations, with a full implementation being provided on github. Compared with the standard Nyström discretisation, more accurate and substantially more efficient results can be obtained by de-refining the force discretisation relative to the quadrature discretisation: a cost reduction of over 10 times with improved accuracy is observed. This improvement comes at minimal additional technical complexity. Future avenues to develop the algorithm are then discussed.
Highlights
When attempting to formulate and solve mathematical models of microscopic biological flow systems, for example involving macromolecular structures, swimming cells and cilia, a significant challenge to overcome is that the flow domain is typically bounded by curved, moving surfaces
It should be noted that there have been major algorithmic developments in numerical methods for Stokes flow in the intervening period, including the completed double-layer boundary integral equation [10,11], hybrid boundary integral-multipole methods [12], spectral discretisation combined with the fast multipole method [13,14], quadrature by expansion [11], and slender body theory combined with these techniques [15]
The issue of the computational cost of the method of regularized stokeslets was discussed in an earlier paper [22], in which we suggested employing a boundary element discretisation of the regularized stokeslet boundary integral equation
Summary
When attempting to formulate and solve mathematical models of microscopic biological flow systems, for example involving macromolecular structures, swimming cells and cilia, a significant challenge to overcome is that the flow domain is typically bounded by curved, moving surfaces. This reduction in dimensionality both removes the need to mesh and re-mesh the evolving flow domain, and vastly reduces the size of the linear algebra problem resulting from discretisation In certain respects these methods were anticipated by the computational slender body theory work of Higdon [6] and [7]; relatively early examples of the ‘fully-fledged’ boundary element method for Stokes flow was developed by Phan-Thien and colleagues [8,9]. It should be noted that there have been major algorithmic developments in numerical methods for Stokes flow in the intervening period, including the completed double-layer boundary integral equation [10,11], hybrid boundary integral-multipole methods [12], spectral discretisation combined with the fast multipole method [13,14], quadrature by expansion [11], and slender body theory combined with these techniques [15] These approaches are generally employed by numerical experts to solve problems at the limits of computational feasibility, involving very large numbers of interacting bodies. An implementation in Matlab®/GNU Octave will be given, and applied to a simple test problem of the drag and moment on a sphere or prolate spheroid undergoing rigid body motion, followed by a more complex problem of calculating the rotational diffusion timescale of a biological macromolecule
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