The motion equations for the rigid bodies are integrated using a predictor-corrector scheme.
For the translation of the center-of-mass the predictor step reads
\begin{align} \begin{split} \tilde{\mv{a}}^{t} &= \mv{a}^{t-1}, \\ \tilde{\mv{u}}^{t} &= \mv{u}^{t-1} + \frac12 \Delta t (\tilde{\mv{a}}^{t} + \mv{a}^{t-1}), \\ \tilde{\mv{x}}^{t} &= \mv{x}^{t-1} + \Delta t \mv{u}^{t-1} + \frac14 \Delta t (\tilde{\mv{a}}^{t} + \mv{a}^{t-1}). \end{split} \label{eq:rb_trans_pred} \end{align}
The obtained information ( \(\tilde{\mv{a}}^{t}, \tilde{\mv{v}}^{t}, \tilde{\mv{x}}^{t}\)) is then used to:
The corrector step is given by
\begin{align} \begin{split} \mv{a}^{t} &= \frac{\mv{F}^{t}}{m} + \mv{g}(1-\frac{1}{\rho_B / \rho_F}), \\ \mv{u}^{t} &= \mv{u}^{t-1} + \frac12 \Delta t (\tilde{\mv{a}}^{t} + \mv{a}^{t-1}), \\ \mv{x}^{t} &= \mv{x}^{t-1} + \Delta t \mv{u}^{t-1} + \frac14 \Delta t (\tilde{\mv{a}}^{t} + \mv{a}^{t-1}). \end{split} \label{eq:rb_trans_corr} \end{align}
For the rotational part, please be referred to Seelen2016
The parallelization algorithm for the rigid bodies solver is based on the space filling curve approach chosen for the hierarchical Cartesian grid.
For a given domain, there are three types of bodies:
Local bodies These bodies are local to the current domain, where all data describing these bodies is stored. The current domain is their home domain. The integration of the motion equation is only performed for local bodies. A body is local to a domain, if the center-of-mass is located inside the domain, i.e. is contained by one of its partition cells (see Parallelization).
Remote bodies These bodies are local to another domain, but their domain of influence intersects with the current domain. The current domain is (one of) their remote domains. Remote bodies are integrated on their respective home domain and all relevant data is communicated to their remote domains. In turn, remote bodies can still have external forces acting on them from the current domain, which have to be communicated back to the home domain.
Example: Body 0 is local to domain 0, body 1 is local to domain 1. Body 1 is a remote body to domain 0 and, in turn, domain 0 is a remote domain of body 1. The same goes for body 0 and domain 1 (see figure above).
Periodic dummy bodies If a domain has a periodic connection with itself, there might be bodies which are local and remote to this domain at the same time, thus have to exist as two individual entities in the data structure. The remote version of this body is called periodic dummy body.
There are two major types of data structures in this solver:
Body collector: This collector, holds all body data in a specific order:
Connectivity mappings: There are two mappings for the communication of body data:
Initially, each domain determines its local and remote bodies.
In the figure above the procedure using the partitioned Cartesian grid is shown: Bodies are local as long as their center-of-mass is contained by a partition cell of a given domain. Local bodies become relevant to another domain as soon as their domain of influence intersects the partition level halo cells associated to the other domain; This body is a remote body to this domain.
During the simulation, bodies change their position and therefore their association with specific domains.
1.9.5