Improved identification of stiffness coefficients of non intrusive nonlinear geometric reduced order models of structures

Significant, successful efforts have been devoted in recent years to the non intrusive construction from finite element software of reduced order models of the nonlinear geometric response of structures. A key assumption in these models is that the elastic restoring forces can be represented as cubic polynomials of their generalized coordinates. However, there has been only very limited discussion of whether this representation is appropriate, how deviations from this polynomial form affect the identification of its coefficients and ultimately the prediction of the structural response, and how current identification strategies can be improved. These questions are the focus of the present investigation which considers first a simple structural model that exhibits the potential for snap-through and thus represents an interesting validation case with strongly nonlinear geometric behavior. It is shown that the force-displacement relationship which can be derived in closed form is not exactly a cubic polynomial but can be very closely modeled by one. However, the match between exact curve and polynomial fit is very sensitive to small changes in the linear, quadratic, and cubic stiffness coefficients and thus the prediction of the response may range from excellent to significantly in error depending on how the identification of the coefficients is performed. In particular, it is found that a cubic approximation obtained by Taylor series around the undeformed configuration is not very accurate post snap-through. Based on these findings, two strategies are proposed to improve the identification of the stiffness coefficients of these reduced order models. The first one is a tuning procedure in which a limited number of key coefficients of the model are adjusted to fit additional static response data. The second approach performs the identification of the stiffness coefficients at a series of different levels, as opposed to one as currently performed, and seeks the most appropriate value of each coefficient based on a "small change"criterion. The applicability of these two novel strategies is demonstrated on several challenging structural models: a clamped-clamped curved beam undergoing snap-through as well as a curved panel and a hat stiffened one. It is found that the reduced order models identified via the tuning or multi-level strategies not only lead to more accurate predictions of the structural response as compared to the original finite element model but also are more computationally stable than their counterparts identified with the current single-level strategy.