Asynchronous
multi-domain variational integrators for nonlinear
hyperelastic solids
M. Benes2 and K. Matous1
1Department of Aerospace and Mechanical
Engineering
University of Notre Dame
Notre Dame, IN 46556, USA.
2Computational Science and Engineering
University of Illinois at Urbana-Champaign
Urbana, IL 61801, USA.
Abstract
We present the asynchronous
multi-domain variational time integrators with a dual
domain decomposition method for the initial hyperbolic
boundary value problem in hyperelasticity. Variational
time integration schemes, based on the principle of
minimal action within the Lagrangian framework, are
constructed for the equation of motion and implemented
into a variational finite element framework, which is
systematically derived from the three-field de
Veubeke-Hu-Washizu variational principle to accommodate
the incompressibility constraint present in an analysis of
nearly-incompressible materials. For efficient parallel
computing, we use the dual domain decomposition method
with local Lagrange multipliers to ensure the continuity
of the displacement field at the interface between
subdomains. The alpha-method for time discretization and
the multi-domain spatial decomposition enable us to use
different types of integrators (explicit vs. implicit) and
different time steps on different parts of a computational
domain, and thus efficiently capture the underlying
physics with less computational effort. The energy
conservation of our nonlinear, midpoint, asynchronous
integration scheme is investigated using the Energy
method, and both local and the global energy error
estimates are derived. We illustrate the performance of
proposed variational multi-domain time integrators by
means of three examples. First, the method of manufactured
solutions is used to examine the consistency of the
formulation. In the second example, we investigate energy
conservation and stability. Finally, we apply the method
to the motion of a heterogeneous plane domain, where
different integrators and time discretization steps are
used accordingly with disparate material data of
individual parts.
Conclusions
In this paper, we have proposed
asynchronous multi-domain variational integrators for
nonlinear hyperelastic solids. Variational time
integration schemes have been derived in the context of
the Lagrangian variational framework, and the three-field
de Veubeke-Hu-Washizu variational principle has been used
for the spatial discretization to accommodate the
incompressibility constraint. The subdomain coupling has
been achieved by the Lagrange multiplier method to ensure
the continuity of the displacement field at the interface
between subdomains. In particular, the dual domain
decomposition method has been exploited in order to
preserve the efficient parallelization of the algorithm.
The Energy method has been employed to
assess the energy conservation of the proposed midpoint
asynchronous integrator. The local and the global energy
conservation criteria have been derived. For synchronous
time integration, we retain the
O(∆t
3) energy evolution while
asynchronous time stepping is locally bounded by the
O((∆t
k)
2)
estimate. The global energy balance across the interface
between domains still holds with
O(∆t
3). Based on the
numerical observations, the investigated integrators are
conditionally stable only. However, we have not
investigated the nonlinear stability of our integrators
and their observed conditional stability in detail, and
more rigorous study is required.
Several examples have been solved to
show the consistency and robustness of the method. We have
adopted the method of manufactured solutions to show the
optimal convergence rates as well as to investigate the
energy evolution and the stability criteria. The mixed
time integration, implicit versus explicit, has been
presented in order to illustrate the solution error
control, and the applicability of the method in
engineering applications. Only moderate time
asynchronicity has been investigated in this work, ~∆t
1/∆t
2=
1/8; future investigation is necessary to extend this
approach to large time step differences. Moreover, a
moderate number of time steps has been studied (~10,000),
and further study is required to assess the long time
performance of our integrators.
The emphasis of this work has been on
the development of asynchronous multi-domain time
integrators and their energy conservation. However,
realsize applications are likely to necessitate solution
strategy improvements, when a system of algebraic
equations is solved, requiring an efficient parallel
implementation of the computational scheme presented
above. The extension to three-dimensions is also of
importance. Ultimately, we want to extend the proposed
method to examples with non-matching discretizations and
multiphysics problems.
Acknowledgment
The authors gratefully acknowledge support
from the Center for Simulation of Advanced Rockets (CSAR)
under contract number B523819 the U.S. Department of
Energy as a part of its Advanced Simulation and Computing
program (ASC). The authors also thank Prof. Joseph M.
Powers from University of Notre Dame for numerous
suggestions that improved the presentation of this paper.
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© 2010 Notre Dame and Dr.
Karel Matous