Invariant-manifold-based model reduction for state transition equations of geometrically exact beam dynamics

This paper presents a computationally efficient invariant-manifold-based model reduction framework for the nonlinear dynamics of free beams and beam assemblies. Although geometrically exact formulations provide high fidelity, they become prohibitively expensive for high-dimensional problems. To address this issue, a “temporal discretization followed by reduction” procedure is proposed. This methodology represents a significant advancement over the “reduction followed by temporal discretization” sequence used in the authors’ previous research. By fundamentally restructuring the problem to eliminate the computational bottlenecks of repetitive time integration and Newton iterations, this approach provides a substantially more efficient and streamlined framework for reduced-order modeling. The full-order dynamics are first discretized by the discontinuous Galerkin scheme, which provides unconditional stability and suppresses high-frequency components of the response. Invariant-manifold-based reduction is then applied to the resulting state-transition equations, yielding a very low-dimensional reduced state-transition equation and thereby avoiding repetitive time integration and Newton iterations. Besides this primary innovation, the work provides several key advancements: (1) A reformulated state-space governing equation for geometrically exact beams within the floating frame of reference formulation, which eliminates the Lagrange multipliers and explicit frame motion equations required in previous studies. (2) An implicit construction of the state-transition relationship in the full-order equations, which preserves the sparse pattern of the large-scale matrices resulting from the temporal discretization and allows the cohomological equations to leverage this sparsity for enhanced performance. (3) A diagonal decomposition of the coefficient matrix of the reduced-order dynamics is introduced. This strategy uncouples column-wise higher-order cohomological equations into independent linear systems, leading to efficient computation for manifold construction. Numerical examples demonstrate that the proposed approach achieves the expected convergence rates with respect to the time-step size and the interpolation order of the test and trial functions. By leveraging an explicit state-transition mapping for time marching, the method achieves faster online computation, thereby enhancing its practical applicability for rapid simulation of free-flying beam structures.