Galerkin v. least-squares Petrov-Galerkin projection in nonlinear model reduction
This work proposes a model-reduction methodology that both preserves Lagrangian structure and leads to computationally inexpensive models, even in the presence of high-order nonlinearities. We focus on parameterized simple mechanical systems under Rayleigh damping and external forces, as structural-dynamics models often fit this description. The proposed model-reduction methodology directly approximates the quantities that define the problem’s Lagrangian structure: the Riemannian metric, the potential-energy function, the dissipation function, and the external force. These approximations preserve salient properties (e.g., positive definiteness), behave similarly to the functions they approximate, and ensure computational efficiency. Results applied to a simple parameterized trussstructure problem demonstrate the importance of preserving Lagrangian structure and illustrate the method’s ability to generate speedups while maintaining observed stability, in contrast with other model-reduction techniques that do not preserve structure.