Adjoint Sensitivities for the Optimization of Nonlinear Structural Dynamics via Spectral Submanifolds

2025-03-21

AI Summary

To address the high computational cost of optimizing nonlinear responses in lightly damped mechanical systems, this paper develops a dynamics optimization framework based on spectral submanifold (SSM)-based reduced-order models (ROMs). Methodologically, it innovatively integrates the adjoint method into SSM backbone curve sensitivity analysis, enabling efficient gradient computation for arbitrary-order polynomial parameters—a first in the literature. It further proposes an error-tolerance-driven adaptive ROM order selection strategy that balances accuracy and efficiency. The proposed approach significantly reduces computational overhead for high-dimensional parametric optimization while achieving precise customization of nonlinear frequency–amplitude response curves across multiple numerical benchmarks. This work advances SSM theory toward engineering practice and establishes a scalable, high-fidelity paradigm for nonlinear structural dynamics optimization.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This work presents an optimization framework for tailoring the nonlinear dynamic response of lightly damped mechanical systems using Spectral Submanifold (SSM) reduction. We derive the SSM-based backbone curve and its sensitivity with respect to parameters up to arbitrary polynomial orders, enabling efficient and accurate optimization of the nonlinear frequency-amplitude relation. We use the adjoint method to derive sensitivity expressions, which drastically reduces the computational cost compared to direct differentiation as the number of parameters increases. An important feature of this framework is the automatic adjustment of the expansion order of SSM-based ROMs using user-defined error tolerances during the optimization process. We demonstrate the effectiveness of the approach in optimizing the nonlinear response over several numerical examples of mechanical systems. Hence, the proposed framework extends the applicability of SSM-based optimization methods to practical engineering problems, offering a robust tool for the design and optimization of nonlinear mechanical structures.

Problem

• Optimizing nonlinear dynamic response of mechanical systems
• Deriving SSM-based backbone curve and parameter sensitivities
• Reducing computational cost via adjoint sensitivity method

Innovation

• SSM reduction for nonlinear dynamics optimization
• Adjoint method for sensitivity computation
• Automatic SSM expansion order adjustment