Hyper-reduction over nonlinear manifolds for large nonlinear mechanical systems

Common trends in model order reduction of large nonlinear finite-element-discretized systems involve the introduction of a linear mapping into a reduced set of unknowns, followed by Galerkin projection of the governing equations onto a constant reduction basis. Though this reduces the number of unknowns in the system, the computational cost for obtaining the solution could still be high due to the prohibitive computational costs involved in the evaluation of nonlinear terms. Hyper-reduction methods are then seen as a fast way of approximating the nonlinearity in the system of equations. In the finite element context, the energy conserving sampling and weighing (ECSW) method has emerged as a stability and structure-preserving method for hyper-reduction. Classical hyper-reduction techniques, however, are applicable only in the context of linear mappings into the reduction subspace. In this work, we extend the concept of hyper-reduction using ECSW to general nonlinear mappings, while retaining its desirable stability and structure-preserving properties. As a proof of concept, the proposed hyper-reduction technique is demonstrated over models of a flat plate and a realistic wing structure, whose dynamics has been shown to evolve over a nonlinear (quadratic) manifold. An online speed-up of over one thousand times relative to the full system has been obtained for the wing structure using the proposed method, which is higher than its linear counterpart using the ECSW.