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Structural-algebraic regularization for coupled systems of DAEs

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Abstract

In the automated modeling of multi-physics dynamical systems, frequently different subsystems are coupled together via interface or coupling conditions. This approach often results in large-scale high-index differential-algebraic equations (DAEs). Since the direct numerical simulation of these kinds of systems leads to instabilities and possibly non-convergence of numerical methods, a regularization or remodeling of such systems is required. In many simulation environments, a structural method that analyzes the system based on its sparsity pattern is used to determine the index and an index-reduced system model. However, this approach is not reliable for certain problem classes, and in particular not suited for coupled systems of DAEs. We present a new approach for the regularization of coupled dynamical systems that combines the Signature method (\({\varSigma }\)-method) for the structural analysis with algebraic regularization techniques. This allows to handle structurally singular systems and also enables a proper treatment of redundancies or inconsistencies in the system.

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Notes

  1. The term structurally singular is also used slightly differently in the literature: on the one hand, it is used for systems that do not allow an assignment of the highest occurring derivative of each variables to a specific equation (i.e., systems that do not have a transversal with finite value) [14], and on the other hand it is used for systems with singular \({\varSigma }\)-Jacobian [12].

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Acknowledgments

We would like to thank two anonymous referees for their careful reading and for thoughtful suggestions for the improvement of the paper. We also thank Jakob Schneck for performing the computations and comparisons of the different simulation environments.

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Authors and Affiliations

  1. Institut für Mathematik, Ma 4-5, TU Berlin, Straße des 17. Juni 136, 10623, Berlin, Federal Republic of Germany

    Lena Scholz & Andreas Steinbrecher

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  1. Lena Scholz
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  2. Andreas Steinbrecher
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Correspondence to Lena Scholz.

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Communicated by Hans-Petter Langtangen.

Research supported by European Research Council, through ERC Advanced Grant MODSIMCONMP.

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Scholz, L., Steinbrecher, A. Structural-algebraic regularization for coupled systems of DAEs. Bit Numer Math 56, 777–804 (2016). https://doi.org/10.1007/s10543-015-0572-y

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  • DOI: https://doi.org/10.1007/s10543-015-0572-y

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