Generalization of Quadratic Manifolds for Reduced Order Modeling of Nonlinear Structural Dynamics

TL;DR

This paper introduces a generalized quadratic manifold approach for reduced order modeling in nonlinear structural dynamics, shifting the quadratic behavior to the manifold for more efficient simulations.

Contribution

It presents a novel generalization that incorporates static derivatives and Krylov vectors, improving the efficiency and applicability of reduced order models for nonlinear problems.

Findings

The approach reduces the number of coordinates needed compared to linear methods.

Numerical studies demonstrate the method's effectiveness across various examples.

The generalized framework extends modal derivatives to broader displacement fields.

Abstract

In this paper, a generalization of a quadratic manifold approach for the reduction of geometrically nonlinear structural dynamics problems is presented. This generalization is constructed by a linearization of the static force with respect to the generalized coordinates, resulting in a shift of the quadratic behavior from the force to the manifold. In this framework, static derivatives emerge as natural extensions to modal derivatives for displacement fields other than the vibration modes, such as the Krylov subspace vectors. Here the dynamic problem is projected onto the tangent space of the quadratic manifold, allowing for a much less number of generalized coordinates compared to linear basis methods. The potential of the quadratic manifold approach is investigated in a numerical study, where several variations of the approach are compared on different examples, indicating a clear pattern where the proposed approach is applicable.