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Modern High Performance Computing (HPC) machines are distributed memoryclusters, consisting of multi-core compute nodes. Engineering simulation and analysis tools must employ efficient parallel algorithms in order to fully utilize the compute capability of modern HPC machines. The trend in Computational Fluid Dynamcis (CFD) has been to construct parallel solution algorithms based on some form of spatial domain decomposition. This approach has been shown to be a success for many practical applications. However, as one attempts to utlize more compute cores, limitations in strong scalability are inevitably reached due to a diminishing compute workload per compute core and either fixed or increasing communication cost. Furthermore, spatial domain decomposition approaches cannot be easily applied to mid-fidelity structural dynamics or rigid body dynamics models. A significant majority of industrial fluid and structural dynamic models utilize some form of time marching. Thus, if the domain decomposition strategy may be extended to include the temporal dimension, additional opportunity for increased parallelism may be realized. A new form of periodic multiple shooting is proposed that ismatrix-free and may be applied to high-fidelity multiphysics models or other high dimensional systems. The proposed methodology is formulated entirely in the time domain. Therefore, existing time-domain simulation tools may utilize the proposed approach to achieve a high degree of distributed memory parallelism without requiring any reformulation. Furthermore, the proposed methodology may be combined with conventional space domain decomposition techniques and other forms of data parallelism to achieve maximal performance on modern HPC architectures. The proposed algorithm retains the iterative shoot-correct approach of conventational periodic shooting methods. However, the correction stage is formulated using a hierarchical evaluation strategy combined with an Arnoldi subspace approximation to eliminate the need for explicit formulation of Jacobian matricies. The local convergence of the proposed method is formally proven for the case of an asyptotically stable dynamical system. The proposed method is numerically tested for a 2D limit cycle problem, a rigid blade helicoper rotor model with quasi-steady aerodynamics and autopilot trim, and an OVERSET CFD model of a helicopter rotor with prescribed elastic blade motions. The method is observed to be convergent in all test cases and found to exhibit good scalability. The proposed periodic multiple shooting method is a practical means of reducingtime-to-solution for numerical simulations of asymptotically stable periodic systems on distributed memory parallel computers. Furthermore, the proposed method may be used to enhance the parallel scalability of OVERSET CFD models of helicopter rotors in steady periodic flight. |
