Computational Physics Group
Asynchronous multi-domain variational integrators for nonlinear problems
M. Gates, K. Matous and M.T. Heath
Computational Science and Engineering
University of Illinois at Urbana-Champaign
Urbana, IL 61801, USA.
We develop an asynchronous time integration and coupling method with domain decomposition for linear and nonlinear problems in mechanics. To ensure stability in the time integration and in coupling between domains, we use variational integrators with local Lagrange multipliers to enforce continuity at the domain interfaces. The asynchronous integrator lets each domain step with its own time step, using a smaller time step where required by stability and accuracy constraints, and a larger time step where allowed. We show that in practice the time step is limited by accuracy requirements rather than stability requirements.
In this paper we have shown how to derive both synchronous
and asynchronous integrators with domain decomposition in
both Lagrangian and Hamiltonian frameworks. For the
Lagrangian integrator we enforce continuity of position at
the interface, while for the Hamiltonian integrator we
enforce continuity of both position and velocity at the
interface. Each domain can be computed in parallel, making
the method suitable for parallel computing. The similarity
of implementation to the Newmark method makes our
integrators easy to incorporate into existing codes.
As we show by various examples, these integrators have stability properties superior to traditional integrators for long term time integration. Experiments on a nonlinear problem have shown that our methods are symplectic and approximately preserve energy, such that the energy oscillates about a fixed value. The stability limit becomes independent of the time step ratio, and in practice the system time step is limited by the accuracy requirement rather than the stability requirement.
There are a number of directions for future work. We are interested in further investigating the asynchronous scheme to improve its order of accuracy and stability, particularly the loss of unconditional stability. One potential direction is to investigate other interpolations of the system interface than the linear interpolation we used here. Another improvement for the Hamiltonian integrator would be to constrain the velocity at every substep, rather than at just the system time step. Composition methods  also afford a means of generating variational integrators with higher order accuracy, which may benefit the asynchronous integrators.
We plan to add external forces and dissipation, whose effects can be accurately computed by variational integrators . We intend to extend the methods developed here to partial differential equations, and investigate how to solve the resulting systems efficiently in parallel. We are also interested in extending this work to domains with non-matching meshes at the interface. To handle non-matching meshes, we believe the common interface between domains that we have included in this work will be of vital importance . Ultimately, we want to extend the proposed method to multi-physics problems.
The authors gratefully acknowledge support from the Center for Simulation of Advanced Rockets (CSAR) under contract number B523819 by the U.S. Department of Energy as a part of its Advanced Simulation and Computing program (ASC).
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© 2009 Notre Dame and Dr. Karel Matous