Lubrication Correction

One of the innovation done by Stokesian dynamics [4,6,5] is the model of lubrication into the system.

$$\bm{R} \simeq \left( \bm{M} \right)^{-1} + \bm{L} .$$ (3.32)

In a straightforward implementation of this lubrication correction, even in F version, the mobility problem needs twice matrix-inversions.

Here's some trick to prevent the extra matrix-inversion, which is usually the bottleneck of the calculation (especially for large systems). Substituting the resistance matrix (3.32) into the resistance equation and multiplying $\bm{M}$ from the left, we have the inverse-free matrix equation

$$\Bigl( \bm{I} + \bm{M} \cdot \bm{L} \Bigr) \cdot \bm{U} = \bm{M} \cdot \bm{F} ,$$ (3.33)

where $\bm{U}$ and $\bm{F}$ contains all velocity and force moments for simplicity. Writing the equation in this form, we can always apply the generalized linear set of equation as in Eq. (3.30) for any case.