Leitz T, Leyendecker S (2018)
Publication Type: Journal article, Review article
Publication year: 2018
Book Volume: 338
Pages Range: 333-361
DOI: 10.1016/j.cma.2018.04.022
Lie-group variational integrators of arbitrary order are developed using
the Galerkin method, based on unit quaternion interpolation. To our
knowledge, quaternions have not been used before for this purpose,
though they allow a very simple and efficient way to perform the
interpolation. The resulting integrators are symplectic and structure
preserving, in the sense that certain symmetries in the Lagrangian of
the mechanical system are carried over to the discrete setting, which
leads to the preservation of the corresponding momentum maps. The
integrators furthermore exhibit a very good long time energy behavior,
i.e. energy is neither dissipated nor gained artificially. At the same
time, the Lie-group structure is preserved by carefully defining the
variations, the interpolation method and by solving the non-linear
system of equations directly on the manifold, rather than constraining
it in a surrounding space using Lagrange multipliers. As a consequence,
we are able to show that Lie-group variational integrators based on the
special orthogonal group, are equivalent to the variational integrators
for constrained systems using the discrete null-space method employed
e.g. in DMOCC (discrete mechanics and optimal control of constrained
systems). We show new numerical results on the convergence rates, which
are substantially higher than the known theoretical bounds, and on the
relation between accuracy and computational cost.
APA:
Leitz, T., & Leyendecker, S. (2018). Galerkin Lie-group variational integrators based on unit quaternion interpolation. Computer Methods in Applied Mechanics and Engineering, 338, 333-361. https://doi.org/10.1016/j.cma.2018.04.022
MLA:
Leitz, Thomas, and Sigrid Leyendecker. "Galerkin Lie-group variational integrators based on unit quaternion interpolation." Computer Methods in Applied Mechanics and Engineering 338 (2018): 333-361.
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