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Global proper orthogonal decomposition for parametric model reduction of geometrically nonlinear structures
Author(s) -
Meyer Christian H.,
Rixen Daniel
Publication year - 2019
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201900436
Subject(s) - nonlinear system , reduction (mathematics) , subspace topology , parametric statistics , parameterized complexity , computation , basis (linear algebra) , model order reduction , mathematics , equations of motion , concatenation (mathematics) , degrees of freedom (physics and chemistry) , sampling (signal processing) , finite element method , dimensionality reduction , algorithm , mathematical analysis , computer science , projection (relational algebra) , geometry , physics , classical mechanics , statistics , quantum mechanics , filter (signal processing) , combinatorics , artificial intelligence , computer vision , thermodynamics
Finite element simulations of structures that undergo large deformations can imply long simulation times due to the nonlinearity of the resulting equations of motion and a high number of degrees of freedom. Especially for design studies, where the equations of motion must be solved several times, it is highly desired to reduce the simulation time. The simulation time can be reduced by nonlinear model reduction. Nonlinear model reduction is carried out in two steps: First, the dimension of the problem is reduced by projecting the equations of motion onto a low‐order subspace that is defined by a reduction basis. Second, the evaluation of the nonlinear restoring force term, that originates from large deflections, is accelerated through hyperreduction. The first step can be challenging for design studies spanning large parameter intervals. For those studies, the calculated reduction basis either must be updated several times or must provide a subspace that captures the solution vectors for the whole parameter space. The latter option can be performed by sampling the parameter space, building local reduction bases at these sampling points and then applying a proper orthogonal decomposition on their concatenation. This contribution shows how this procedure is used for parametric model reduction of geometrically nonlinear structures. It demonstrates its accuracy and computation time on a shape‐parameterized beam structure that undergoes large deflections. It turns out that the success of the proposed method is highly dependent on the implementation of the shape parameterization.