Fixed Particles - Mixed Problem

When we are interested in particle dynamics, the mobility problem is the main target. However, in several situations, we may want to introduce fixed particles into the simulation representing some obstacles or vessel to support other mobile particles. [16] In those cases, it is not a simple mobility problem but the mixed problems, that is, the ``mobility and resistance'' problem:

$$\left[ \begin{array}{c} \bm{U}^{m}\\ \bm{E}^{m}\\ \bm{U}^{f}\\ \b... ...ray}{c} \bm{F}^{m}\\ \bm{S}^{m}\\ \bm{F}^{f}\\ \bm{S}^{f} \end{array} \right] .$$ (3.29)

Here, we denote the mobile and fixed particles by the superscripts $m$ and $f$, respectively. After sorting by the known and unknown variables, we have

$$\bm{B} \cdot \left[ \begin{array}{c} \bm{F}^{m}\\ \bm{E}^{m}\\ \b... ...ray}{c} \bm{U}^{m}\\ \bm{S}^{m}\\ \bm{F}^{f}\\ \bm{S}^{f} \end{array} \right] ,$$ (3.30)

where given parameters are in the left-hand side and the unknowns are in the right-hand side. It is straightforward to obtain the matrices $\bm{A}$ and $\bm{B}$ by $\bm{M}$'s.

$$\left[ \begin{array}{cccc} -\bm{M}_{UF}^{mm} & \bm{0} & \bm{0} & ... ...ray}{c} \bm{U}^{m}\\ \bm{S}^{m}\\ \bm{F}^{f}\\ \bm{S}^{f} \end{array} \right] .$$ (3.31)

Because the matrices are known variables, the generalized form of linear set of equations (3.30) can be solved by the standard iterative method by giving the procedure to calculate $\bm {A}\cdot \bm {x}$ with $\bm{A}$ in (3.30) for a given vector $\bm{b} = \bm{B}\cdot\bm{b}'$ appeared in the left-hand side of Eq. (3.30).

To reduce the debugging cost (and extra complexity of the code), the implementation of the scheme in the library libstokes is in this way.^3.1