This work proposes a model-reduction methodology that preserves Lagrangianstructure (equivalently Hamiltonian structure) and achieves computationalefficiency in the presence of high-order nonlinearities and arbitrary parameterdependence. As such, the resulting reduced-order model retains key propertiessuch as energy conservation and symplectic time-evolution maps. We focus onparameterized simple mechanical systems subjected to Rayleigh damping andexternal forces, and consider an application to nonlinear structural dynamics.To preserve structure, the method first approximates the system's `Lagrangianingredients'---the Riemannian metric, the potential-energy function, thedissipation function, and the external force---and subsequently derivesreduced-order equations of motion by applying the (forced) Euler--Lagrangeequation with these quantities. From the algebraic perspective, keycontributions include two efficient techniques for approximating parameterizedreduced matrices while preserving symmetry and positive definiteness: matrixgappy POD and reduced-basis sparsification (RBS). Results for a parameterizedtruss-structure problem demonstrate the importance of preserving Lagrangianstructure and illustrate the proposed method's merits: it reduces computationtime while maintaining high accuracy and stability, in contrast to existingnonlinear model-reduction techniques that do not preserve structure.
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