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An integrated method for the transient solution of reduced order models of geometrically nonlinear structures

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Abstract

For repeated transient solutions of geometrically nonlinear structures, the numerical effort often poses a major obstacle. Thus it may become necessary to introduce a reduced order model which accelerates the calculations considerably while taking into account the nonlinear effects of the full order model in order to maintain accuracy. This work yields an integrated method that allows for rapid, accurate and parameterisable transient solutions. It is applicable if the structure is discretised in time and in space and its dynamic equilibrium described by a matrix equation. The projection on a reduced basis is introduced to obtain the reduced order model. Three approaches, each responding to one of the requirements of rapidity, accuracy and parameterisation, are united to form the integrated method. The polynomial formulation of the nonlinear terms renders the solution of the reduced order model autonomous from the finite element formulation and ensures a rapid solution. The update and augmentation of the reduced basis ensures the accuracy, because the simple introduction of a constant basis seems to be insufficient to account for the nonlinear behaviour. The interpolation of the reduced basis allows adapting the reduced order model to different external parameters. A Newmark-type algorithm provides the backbone of the integrated method. The application of the integrated method on test-cases with geometrically nonlinear finite elements confirms that this method enables a rapid, accurate and parameterisable transient solution.

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Acknowledgments

The fourth author gratefully acknowledges the support of Alexander von Humboldt Foundation through a Gay Lussac Research Award received in 2013.

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Author notes

  1. Fritz Adrian Lülf

    Present address: Institute of Continuum Mechanics, Leibniz Universität Hannover, Appelstr. 11, 30167, Hannover, Germany

Authors and Affiliations

  1. ONERA, The French Aerospace Lab, 92322, Châtillon, France

    Fritz Adrian Lülf & Duc-Minh Tran

  2. Institut für Wissenschaftliches Rechnen, Technische Universität Braunschweig, 38092, Braunschweig, Germany

    Hermann G. Matthies & Roger Ohayon

  3. Structural Mechanics and Coupled Systems Laboratory, Conservatoire National des Arts et Métiers, 2 rue Conté, 75003, Paris, France

    Roger Ohayon

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  1. Fritz Adrian Lülf
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  2. Duc-Minh Tran
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  3. Hermann G. Matthies
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  4. Roger Ohayon
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Correspondence to Fritz Adrian Lülf.

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Lülf, F.A., Tran, DM., Matthies, H.G. et al. An integrated method for the transient solution of reduced order models of geometrically nonlinear structures. Comput Mech 55, 327–344 (2015). https://doi.org/10.1007/s00466-014-1103-4

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  • DOI: https://doi.org/10.1007/s00466-014-1103-4

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