Generalized Finite Element
Method for Modeling Nearly Incompressible Bimaterial Hyperelastic Solids
K.R. Srinivasan1, K. Matous1,2 and
1Department of Aerospace Engineering
2Computational Science and Engineering
University of Illinois at Urbana-Champaign
Urbana, IL 61801, USA.
An extension of the generalized finite element method to the class of
mixed finite element methods is presented to tackle heterogeneous
systems with nearly-incompressible nonlinear hyperelastic material
behavior. In particular, heterogeneous systems with large modulus
mismatch across the material interface undergoing large strains are
investigated using two formulations, one based on a continuous
deformation map, the other on a discontinuous one. A bimaterial patch
test is formulated to assess the ability of the two formulations to
reproduce constant stress fields, while a mesh convergence study is
used to examine the consistency of the formulations. Finally,
compression of a model heterogeneous propellant pack is simulated to
demonstrate the robustness of the discontinous deformation map
The present work provides a numerical framework that combines the
generalized finite element method with the classical mixed finite
element method. Two formulations, based on a continuous and
discontinuous deformation map, were derived and discretized for the
motion of a bimaterial nearly-incompressible hyperelastic solid. The
two formulations were assessed numerically on the low-order Q1
element, which is very popular in engineering practice, using a
bimaterial patch test and mesh convergence studies were carried out to
evaluate the consistencies of the formulations. It was observed that
both the continuous and discontinuous deformation maps yield convergent
schemes for moderate modulus mismatches, while the continuous
deformation map appears to be non-convergent for large mismatches.
Finally, an idealized heterogeneous solid propellant pack is chosen as
an example to demonstrate the capability and robustness of the
discontinuous deformation map formulation.
This work was supported by the Center for Simulation of Advanced
Rockets (CSAR) under contract number B341494 by the U.S. Department of
Energy. K. Matous also acknowledges support from ATK/Thiokol,
ATK-21316 (Program Managers, J. Thompson and Dr. I. L. Davis). The
authors would also like to thank Prof. C. A. Duarte for helpful
discussions and comments.
© 2009 Notre Dame and Dr.