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Mechanical Sciences An open-access journal for theoretical and applied mechanics
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Volume 4, issue 1
Mech. Sci., 4, 243-250, 2013
https://doi.org/10.5194/ms-4-243-2013
© Author(s) 2013. This work is distributed under
the Creative Commons Attribution 3.0 License.

Special issue: Recent advances and current trends in numerical multibody...

Mech. Sci., 4, 243-250, 2013
https://doi.org/10.5194/ms-4-243-2013
© Author(s) 2013. This work is distributed under
the Creative Commons Attribution 3.0 License.

Research article 06 Jun 2013

Research article | 06 Jun 2013

A graph-theoretic approach to sparse matrix inversion for implicit differential algebraic equations

H. Yoshimura H. Yoshimura
  • Department of Applied Mechanics and Aerospace Engineering, Waseda University, Ohkubo, Shinjuku, Tokyo 169-8555, Japan

Abstract. In this paper, we propose an efficient numerical scheme to compute sparse matrix inversions for Implicit Differential Algebraic Equations of large-scale nonlinear mechanical systems. We first formulate mechanical systems with constraints by Dirac structures and associated Lagrangian systems. Second, we show how to allocate input-output relations to the variables in kinematical and dynamical relations appearing in DAEs by introducing an oriented bipartite graph. Then, we also show that the matrix inversion of Jacobian matrix associated to the kinematical and dynamical relations can be carried out by using the input-output relations and we explain solvability of the sparse Jacobian matrix inversion by using the bipartite graph. Finally, we propose an efficient symbolic generation algorithm to compute the sparse matrix inversion of the Jacobian matrix, and we demonstrate the validity in numerical efficiency by an example of the stanford manipulator.

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