Parametric-ROM of Structures with Varying Geometry using Direct Parameterization of Invariant Manifolds

TL;DR

This work addresses parametric reduction in FEM when geometry varies with a parameter by expanding the inverse of the geometry determinant as a power series in $\mu$ and embedding forcing and the parameter into an enlarged autonomous system. It then applies direct parameterization of invariant manifolds, solving a sequence of order-$p$ homological equations to construct the reduced-order model while preserving the original DoFs of the FOM. Key contributions include explicit zeroth- and first-order terms for $\det(\nabla_{0} \mathbf{x})^{-1}$, a structured ROM framework for forcing and parameters, and a monomial-aware procedure to derive reduced dynamics efficiently. The approach enables efficient parametric studies of geometry-driven structural dynamics without compromising fidelity or incurring excessive computational costs, thereby advancing parametric MOR for complex solid mechanics problems.

Abstract

This work presents a framework for parametric reduction in FEM, where geometry is controlled by a parameter without altering material properties or stress states. The inverse determinant in the weak form is expanded as a power series, with explicit expressions for the zeroth and first-order terms. External forcing and parameter dependence are incorporated into an enlarged autonomous system, reduced via the direct parameterization of invariant manifolds method and homological equations. The parameter is treated as an additional variable with trivial dynamics, isolated for inclusion in the ROM. This approach enables efficient parametric studies and advances reduced-order modeling in structural dynamics.

Theorems & Definitions (1)

• proof