Indirect reduced-order modelling: using nonlinear manifolds to conserve kinetic energy

Abstract

Nonlinear dynamic analysis of complex engineering structures modelled using commercial finite element (FE) software is computationally expensive. Indirect reduced-order modelling strategies alleviate this cost by constructing low-dimensional models using a static solution dataset from the FE model. The applicability of such methods is typically limited to structures in which (a) the main source of nonlinearity is the quasi-static coupling between transverse and in-plane modes (i.e. membrane stretching); and (b) the amount of in-plane displacement is limited. We show that the second requirement arises from the fact that, in existing methods, in-plane kinetic energy is assumed to be negligible. For structures such as thin plates and slender beams with fixed/pinned boundary conditions, this is often reasonable, but in structures with free boundary conditions (e.g. cantilever beams), this assumption is violated. Here, we exploit the concept of nonlinear manifolds to show how the in-plane kinetic energy can be accounted for in the reduced dynamics, without requiring any additional information from the FE model. This new insight enables indirect reduction methods to be applied to a far wider range of structures while maintaining accuracy to higher deflection amplitudes. The accuracy of the proposed method is validated using an FE model of a cantilever beam.

Model Order Reduction based on Direct Normal Form: Application to Large Finite Element MEMS Structures Featuring Internal Resonance

“…3 In order to better highlight the quality of the results obtained with the DNF method, we compare with another reduction strategy using implicit condensation (IC) as proposed in [39,41]. As shown for example in [50,42] with the case of a cantilever beam, IC method fails at reproducing correctly the nonlinear dynamics of structures when inertia nonlinearities are importantly excited. In particular, the IC method condenses statically the contribution of slave modes on the master mode trajectory, hence neglecting velocity dependent terms [49].…”

“…The stiffness evaluation procedure (STEP) uses the linear modes as projection basis [37,38], but can be simply used as a technique to compute non-intrusively linear and nonlinear characteristics. The implicit condensation and expansion (ICE) method relies on a set of applied forces to keep trace of non-resonant couplings among modes [39,40,41,42]. Finally modal derivatives (MD) have been introduced with the aim of taking the amplitude dependence of modes into account, thus pursuing the same goal as the NNMs [43,44,45,46].…”

Detecting internal resonances during model reduction

“…The linear properties of the ROM, Λ r , Φ r and Φ s , can be computed directly using the linear mass and stiffness matrices of the FE model. The reduced nonlinear stiffness functions,f r (r), and the quasi-static coupling functions, g(r), are approximated indirectly in a least-squares manner using a force-displacement dataset for a series of nonlinear static solutions extracted from the FE model, as detailed in [32,35]. These are approximated as Kth-order polynomials of the reduced coordinates.…”

“…The second class, denoted by the subscript ∙s (static), is composed of S ≪ N modes, which may contain a substantial fraction of the energy of the full-order system, yet their response can be approximated as being quasi-statically coupled to the reduced modes. These modes need not be included as independent DOFs in the reduction basis, as their effects can be incorporated implicitly in the reduced dynamics [30,31,35]. Finally, the third class, denoted by the subscript ∙u (unmodelled), contains the remaining U = ( N − R − S ) modes, which are very weakly coupled to the reduced modes, such that they always contain a negligible amount of energy under the operating conditions of interest.…”

“…This suggests that, for the response range considered here, a higher truncation order is not necessary 6. Here, the condensed basis was chosen based on the relative amplitude of each mode evaluated at the static solutions, as discussed in[35]. For more complex structures, more intricate methods of selecting good candidate modes for the condensed…”