SIMULATION-FREE HYPER-REDUCTION FOR GEOMETRICALLY NONLINEAR STRUCTURES BASED ON STOCHASTIC KRYLOV TRAINING SETS

Abstract eng:
Many advanced structural systems like MEMS or aerospace structures undergo large deformations. Finite element analysis of the dynamics of such structures typically leads to a system of nonlinear equations of large size. Thereby, the nonlinear terms, which are represented by an internal force vector, arise from the geometric nonlinearity due to large displacements of the underlying system. Although modern computers allow the solution of large-sized systems, dynamic analysis, such as time integration of the nonlinear equations, is still very cost-intensive. These high computational costs often make analyses that are highly desired for structural design, like large parametric studies or optimization, not feasible. To overcome this issue, model order reduction (MOR) techniques can be applied to reduce the problem size. Similar to reduction methods for linear systems, the displacement field is approximated by a reduction basis which transforms a small set of generalized coordinates to the physical domain. Thus, the system only has to be solved for this smaller set of generalized coordinates. However, the nonlinear force vector still has to be evaluated in the full element domain for each iteration step, so that despite the reduction degrees of freedom no substantial reduction of computational cost can be reached. In order to reduce the evaluation costs of the nonlinear term, hyper-reduction techniques are employed. Prominent examples for those techniques are the Discrete Empirical Interpolation Method (DEIM) [1] and the Energy Conserving Sampling and Weighting (ECSW) method [2]. Instead of evaluating all element forces at each iteration step, these methods only evaluate a subset of all finite elements to approximate the nonlinear force vector with much less computational effort. However, both methods require a training data set, which is usually generated by simulation-based methods, such as proper orthogonal decomposition. The main drawback of these methods is their requirement for simulations of the full unreduced system. It is highly desired to avoid such simulations to avoid their computational effort. In this contribution we propose a simulation-free hyper-reduction method by using a nonlinear stochastic Krylov training set. It is shown that this training set can be used for common hyper-reduction methods, such as ECSW, to save the high computational effort of generating simulation based training data while accuracy of the methods is still preserved.

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