Computational Physics Group

Karel Matous










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.


    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(∆t3) energy evolution while asynchronous time stepping is locally bounded by the O((∆tk)2) estimate. The global energy balance across the interface between domains still holds with O(∆t3). 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, ~∆t1/∆t2= 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.


   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