Computational Mechanics

, Volume 55, Issue 2, pp 327–344

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

• Duc-Minh Tran
• Hermann G. Matthies
• Roger Ohayon
Original Paper

## 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.

### Keywords

Structural dynamics Geometric nonlinearities Model reduction Reduced bases Normal modes Tangent modes Basis update Parameters

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## Authors and Affiliations

• 1
• 2
• Duc-Minh Tran
• 1
• Hermann G. Matthies
• 3
• Roger Ohayon
• 3
• 4
1. 1.ONERA, The French Aerospace LabChâtillonFrance
2. 2.Institute of Continuum MechanicsLeibniz Universität HannoverHannoverGermany
3. 3.Institut für Wissenschaftliches RechnenTechnische Universität BraunschweigBraunschweigGermany
4. 4.Structural Mechanics and Coupled Systems LaboratoryConservatoire National des Arts et MétiersParisFrance