Abstract
A cell-mapping approach is implemented and parallelized to analyze three-body problem orbits in the vicinity of icy moons (Europa and Enceladus). The cell-mapping method is developed for studying nonlinear dynamics with periodic motions. The method does not require previously known solutions as inputs, which is an essential requirement of continuation approaches, and does not impose symmetric constraints. As major strengths of the method, multiple-period periodic solutions and bifurcation studies can be easily performed. This method is especially applicable to a systematic periodic orbit search over a region of interest using an integration time of one period. The parallelized cell-mapping method facilitates a rapid understanding of the global dynamics.
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Acknowledgments
The research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. NASA Postdoctoral Program fellowship acknowledged. The High Performance Computing resources used in this investigation were provided by funding from the JPL Office of the Chief Information Officer.
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Appendix: :Parallelization of Unraveling Algorithm
Appendix: :Parallelization of Unraveling Algorithm
In the parallel algorithm all processes run a synchronized sequence of steps. An integer variable ‘step’ keeps track of the step number during the execution. The following local arrays, of length equal to the local number of cells in each process, are defined: ‘target’, ‘next-target’, ‘sum’, and ‘final’. When a cell z is identified as periodic during the unraveling algorithm execution, the corresponding ‘step’ values will be set to positive integers corresponding to the Step Number (S) defined earlier. At each ‘step’, the values ‘target’, ‘next-target’ and ‘sum’ are updated from the cell-map, C, as ‘target(z)=Cstep(z)’, ‘next-target(z)=C(target(z))’ and ‘sum(z)=sum(z)+target(z)’, for each cell ‘z’ that has not been assigned to a group number yet. A vector ‘PO’ keeps track of the periodic solutions found in the successive iterations, adding any newly found periodic solutions in each step to the existing ones. Components of ‘final’ are finally updated to track where a cell z is mapped eventually, and this eventually sorts the different Groups defined earlier.
At initialization, all elements of the ‘final’ and ‘sum’ arrays are set to − 1 and 0, respectively, and the ‘PO’ vector is empty. The iterative process stops when all cells have been assigned to a group, that is, when all values in the ‘final’ array are different from − 1, for all processes. The parallelized unraveling algorithm can be split into four stages. Figures 3 – 6 provide a visual representation of these four stages, where each circle 1 to N represents a process (MPI rank) running in parallel and the two-way arrows represent communications between processes:
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1.
Figure 18 summarizes the first step of the parallelized unraveling algorithm. For each cell ‘z’ such that ‘final(z)’ is − 1, obtain ‘target(z)=Cstep(z)’. If the cell given by ‘target(z)’ resides in this process (i.e., the MPI rank of cell ‘target(z)’ is the same as the MPI rank of cell ‘z’), then ‘next-target(z)’ can be obtained as ‘C(target)’ locally (i.e., without communication). Otherwise, if the cell given by ‘target(z)’ is local to another process (i.e., the MPI rank of the cell ‘target(z)’ is different from the MPI rank of the cell ‘z’), then inter-process communication is required to retrieve ‘next-target(z)’, which is temporarily set to − 1. The inter-process communication pattern is set by grouping data requests per process, to minimize the number of communication exchanges, by means of the arrays ‘comm’ and ‘pckg’.
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Inter-process data communication to ‘C(target(z))’ and update ‘next-target(z)’ for those cells with non-local ‘target(z)’. This step uses the ‘comm’ and ‘pckg’ arrays to formulate the send and receive MPI call pairs between processes at once. A broadcast call from each process is needed first to warn all the other processes that should expect to receive data requests. A schematic diagram for this step is presented in Fig. 19.
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Identify periodic cells and update the corresponding ‘final(z)’ by comparing ‘next-target(z)’ to ‘z’ and existing periodic cell groups. Newly found periodic solutions are added in this step to the collection of existing ones in the vector called ‘PO.’ A flow chart for this is shown in Fig. 20.
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Check for the stop condition: for each process if ‘final(z)’ is equal to − 1 for any z set ‘locally_not_done’ to 1, and to 0 otherwise. Sum ‘locally_not_done’ among all processes via a call to ‘MPI_Allreduce’ into ‘globally_not_done’. If ‘globally_not_done’ is 0 then stop, otherwise continue proceeding to stage 1 again. A schematic diagram for this step is presented in Fig. 21.
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Koh, D., Anderson, R.L. & Bermejo-Moreno, I. Cell-mapping orbit search for mission design at ocean worlds using parallel computing. J Astronaut Sci 68, 172–196 (2021). https://doi.org/10.1007/s40295-021-00251-6
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DOI: https://doi.org/10.1007/s40295-021-00251-6