variational integrators for nonlinear hyperelastic solids
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.
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.
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
) energy evolution
while asynchronous time stepping is locally bounded by the O
estimate. The global energy balance across the interface between
domains still holds with O
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, ~∆t1
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
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.