Nonlinear Model Reduction

In the context of Multibody Simulation, often flexible bodies need to be taken into account. In general, one relies on a detailed Finite Element Model of the deformable structure, which captures the flexibilities as well as the dynamical effects. Due to the large number of degrees of freedom, FEM models can only be used to compute rather short time spans and can not directly be applied for durability computations.

Commercial MBS tools usually accomplish the inclusion into a full model using linear model reduction techniques like the Craig-Bampton method. The number of dofs of the model is reduced by projection of the equations, which adapts the computational effort to the one of the MBS model.

FE tire model
© Photo ITWM

FE tire model

Proper Orthogonal Decomposition (POD)

However, these methods are based on a linearization step of the FEM model, often yielding wrong results for nonlinear systems. Such equations arise in the case of nonlinear materials (e.g. rubber), geometrical nonlinearities (induced by large deformations) or contact conditions. These effects are crucial when describing tyres or rubber bushings.

We work on methods for nonlinear model reduction. The number of dofs of high dimensional structural models is diminished while maintaining the nonlinearities. Mathematical methods like the Proper Orthogonal Decomposition (POD) are applied to achieve a projection basis for the model equations without linearization.

 

Lookup-Table methods

The nonlinear equations of the reduced model are treated by lookup table methods from stored information of the full FE model. With this procedure, one can handle models created with commercial codes like Abaqus, where the black-box type prohibits the use of analytical reduction methods.

This method requires a training computation of the detailed FEM model. The external loading is chosen to represent a typical excitation for the structure's later use over a short time span. The extracted information is used as foundation for the projection basis as well as for the lookup table of the nonlinearities.

Comparing the reduced model with the full one in Abaqus, effort reductions of two orders of magnitude are achieved. The errors of the nonlinear model reduction method lie in an acceptable range and are significantly smaller than the ones produced by the Craig-Bampton method. In this example, the model of an air spring is used consisting of nearly 3000 dofs and containing geometrical and material nonlinearities. With this treatment, the reduced model can be included into the full MBS model.