Asynchronous Variational Integrators
We describe a new class of asynchronous variational integrators (AVI) for nonlinear elastodynamics. The AVIs are distinguished by the following attributes: (i) The algorithms permit the selection of independent time steps in each element, and the local time steps need not bear an integral relation to each other; (ii) the algorithms derive from a spacetime form of a discrete version of Hamilton's variational principle. As a consequence of this variational structure, the algorithms conserve local momenta and a local discrete multisymplectic structure exactly. To guide the development of the discretizations, a spacetime multisymplectic formulation of elastodynamics is presented. The variational principle used incorporates both configuration and spacetime reference variations. This allows a unified treatment of all the conservation properties of the system.A discrete version of reference configuration is also considered, providing a natural definition of a discrete energy. The possibilities for discrete energy conservation are evaluated. Numerical tests reveal that, even when local energy balance is not enforced exactly, the global and local energy behavior of the AVIs is quite remarkable, a property which can probably be traced to the symplectic nature of the algorithm
© Springer-Verlag (2003). Accepted April 1, 2002. Published online February 28, 2003. Support from NSF/DARPA through the OPAAL grant is gratefully acknowledged. J. Marsden and M. West were partially supported by NSF/KDI grant ATM-9873133 and NSF/ITR grant ACI-0204932 as well.We are grateful to John Ball, Stuart Antman, Tom Hughes, Stefan Müller, Fehmi Ciraki, Steve Shkoller, Couro Kane, Anna Pandolfi, Melvin Leok and Razvan Fetecau for helpful discussions and suggestions.
Accepted Version - LeMaOrWe2003.pdf