Model Order Reduction for Geometric Nonlinear Structures with Variable State-Dependent Basis

A quadratic manifold for model order reduction of nonlinear structural dynamics

“…When the deviation from the linearization point increases, the response of (7) can no longer be considered as a good approximation for the original nonlinear counterpart. One might still think of using the VMs obtained from the linearized model to form a reduction basis for the reduction of the nonlinear set of equations.…”

“…The above system can be considered as a 2-DOF-variant of the FE discretized equations of the von karman beam (see e.g. [7] ), where the solution variables w(t), v(t) ∈ R are analogous to the transverse and axial displacements of the beam, respectively, m 1 , m 2 , k 1 , k 2 , a, b, c ∈ R are physical parameters, and g(t) correspond to the externally applied load in the transverse direction.…”

“…As such, the modal amplitudes associated to the MDs are enslaved, in a quadratic fashion, to those of the VMs, and hence do not require independent reduced unknowns for their description. This approach can often be supported by a sound theoretical justification in examples which are characterized by a special dichotomy in time scales, and corresponds to neglecting the inertial forces associated to the fast dynamics of the system at hand [7]. More specifically, the static MDs provide a second order approximation to the underlying critical manifold in such examples [8].…”

On polynomial hyperreduction for nonlinear structural mechanics

“…In contrary to the already discussed methods, the design space for the reduced model must be known in advance. In several previous works, 15,18 it is shown that the POD basis can be a powerful choice for reduction of the nonlinear equations (6). In this paper, we use the POD projection because it appears superior to the mentioned alternatives and, furthermore, naturally matches our approach to hyperreduction, as discussed in Section 5.…”

“…It has been shown 14 that A 1 are the projected eigenmodes and A 2 are the projected modal derivatives, whereas A 3 can be derived as a finite difference. 15,18 As already mentioned, for large deformations in connection with nonlinear material laws, this ansatz works only in special cases. The expansion around zero seems to be a limiting assumption for general applications.…”