Chapter43 Model Order Reduction for Geometric Nonlinear Structures with Variable State-Dependent Basis Johannes B. Rutzmoser and Daniel J. Rixen Abstract Nonlinear model order reduction based on subspace projection is driven by the exploitation of the structure of the nonlinearity or by analyzing data generated from the nonlinear model. In the proposed approach the one way coupling of bending and stretching in geometrically nonlinear beam or shell elements is considered for a nonlinear projectional framework that is able to consider inertia effects of both, bending and stretching motion. The method increases accuracy in comparison to static condensation approaches, however at the cost of higher computational efforts. Keywords Model reduction • Geometric nonlinearity • Static condensation • Variable basis • Structural dynamics 43.1 Introduction The trend to more advanced, more detailed and more comprehensive simulation models drives the demand on efficient and model order reduction (MOR) techniques. While the properties of reduction methods for linear time invariant systems are well known and established [1], efficient nonlinear MOR is still an open question and thus an active research topic. Basically two approaches are used in nonlinear MOR: the data-driven branch and the branch exploiting the properties of the mathematical structure of the nonlinearity. In the former approach, the strategy is to first generate representative data of the states of the system by a simulation of the full degrees of freedom (dof) model, second to extract the principal components of the states utilizing a singular value decomposition and third to project the dynamic equations on the subspace spanned by the most important principal component vectors. This method, known as Proper Orthogonal Decomposition (POD), is applied successfully in many fields of applications [4]. However the drawback is the necessity to simulate the full-dof system and the need on generating representative data. In the latter approach, the structure of the nonlinearity or the information like linear modes or modal derivatives is exploited and used for either projection or condensation methods. In the field of structural dynamics with large deformations, the structure of geometric nonlinearities is well known and can be used for different approaches like e.g. [2, 3, 5, 7]. As these reduction methods are tailored to geometric nonlinearity, they are not characterized by generality as the POD is but have the advantage to avoid full scale and thus expensive computations. In this paper a reduction method is proposed pertaining to the second category. A projectional framework is described exploiting the structure of the governing equations of geometrically nonlinear beam or shell elements using a nonlinear transformation which can be interpreted as a projection on a nonlinear subspace, or better, a manifold. The results outperform the static condensation approach in all cases in terms of accuracy, but on the cost of higher computational efforts. This paper is organized as follows: After this introduction the general projection framework commonly used in MOR is shortly recalled (Sect. 43.2). As alternative strategy, the static condensation method is explained (Sect. 43.3) and subsequently transformed to the projectional framework as a projection approach with variable basis (Sect. 43.4). Numerical experiments are performed on three differently hinged beams showing the practicability of the projectional method (Sect. 43.5). J.B. Rutzmoser ( ) • D.J. Rixen Technische Universität München, Boltzmannstraße 15 D-85748 Garching, Germany e-mail: johannes.rutzmoser@tum.de; rixen@tum.de M. Allen et al. (eds.), Dynamics of Coupled Structures, Volume 1: Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-04501-6__43, © The Society for Experimental Mechanics, Inc. 2014 455
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