Simulation-lean training-sets for hyper-reduction of parametric geometric non-linear structures

Parameter studies to design the dynamics of structures that undergo large deformations, such as wind turbine blades or MEMS, can be a tedious task. These studies are usually done with finite element models. The equations of motion provided by the finite element method are nonlinear and high-dimensional. This typically leads to high computation times, especially if the equations must be solved for each parameter variation of the design study. Model reduction can reduce the computation time by approximating the high dimensional parametric models. This is done in two steps: First, a Galerkin projection is done that approximates the solution by a linear combination of basis vectors. Second, the evaluation of the nonlinear force term is accelerated by a hyper-reduction. Both steps are challenging if large parameter variations are present because suitable basis vectors as well as the coefficients for the hyper-reduction are dependent on these parameters. An update of the coefficients for the hyper-reduction is very time consuming and can make the benefit of the model reduction useless. The hyper-reduction method that is used in this contribution is the Energy Conserving Sampling and Weighting method (ECSW). This method requires training sets for computing coefficients for the hyper-reduction. This contribution shows how cost-intensive updates of the reduced order model can be avoided by computing coefficients for the ECSW that are valid for the whole parameter space of interest. We propose to generate training sets for some design parameter samples and compute coefficients that can be used for the whole parameter space. These training sets are computed without the need for expensive time integration of the full order model. Instead, Nonlinear Stochastic Krylov Training Sets (NSKTS) are used that only require static solutions to some applied training forces. A simple case study of a shape parameterized cantilever beam illustrates the performance of the method. It is shown that computation time can be reduced for large numbers of design changes, but shows worse accuracy compared to local methods.