Hessian-Enhanced Alternating Frequency/Time method for Computing Resonance Backbone Curves

Computing resonance and anti-resonance backbone curves in complex nonlinear mechanical systems is of high engineering relevance but remains computationally challenging, especially for large finite-element (FE) models. Existing manifold-based approaches often rely on polynomial parameterizations, limiting their effectiveness for general smooth, non-polynomial nonlinearities. To overcome these limitations, we develop a direct optimization framework that employs a Lagrange multiplier formulation to determine the resonance backbone curve on the response surface constrained by the harmonic balance governing equations. Crucially, solving this formulation efficiently requires second-order sensitivity information. Therefore, the primary innovation of this work is the derivation of a analytical Hessian Tensor for generic $C^2$-continuous nonlinear elements. This is achieved by combining an extended Alternating Frequency/Time (AFT) method for computing second-order derivatives with local-coordinate tensor transformations. By integrating this analytical Hessian into the solver, the proposed framework ensures robust convergence and significantly reduces runtime, making it practical for large-scale models where numerical differentiation is computationally prohibitive. The method is validated on three benchmarks of increasing complexity: a two-degree-of-freedom (2-DOF) system with cubic nonlinearity, a beam with cubic stiffness or hyperbolic tangent (tanh) friction nonlinearities, and an industrial-scale finite element model of a compressor bladed disk (blisk) with a friction ring damper. Results demonstrate that the proposed framework accurately and efficiently computes both resonance and anti-resonance backbone curves, providing a robust frequency-domain tool for structures with non-polynomial nonlinearities.

💡 Research Summary

This paper presents a novel optimization-based framework for efficiently computing resonance and anti-resonance backbone curves in nonlinear mechanical systems, with a particular focus on enabling analysis of large-scale finite element (FE) models. The core challenge addressed is the computational expense and limitations of existing methods, such as manifold-based techniques (e.g., Spectral Submanifolds) which rely on polynomial expansions and struggle with general smooth, non-polynomial nonlinearities (like friction modeled by hyperbolic tangent functions), or methods requiring exhaustive frequency response sweeps.

The authors reformulate the problem of finding the backbone curve—the locus of response peaks as forcing parameters vary—as a constrained optimization problem. The governing equations are the algebraic equations derived from the Harmonic Balance Method (HBM), which are enforced as constraints. The objective is to maximize a response amplitude metric (the Euclidean norm of Fourier coefficients for a monitored degree of freedom). Solving this optimization problem efficiently and robustly requires second-order sensitivity information (the Hessian). The primary methodological innovation of this work is the derivation and implementation of an analytical Hessian tensor for generic C²-continuous nonlinear elements within the HBM/AFT framework.

This is achieved through two key developments: 1) An extended Alternating Frequency/Time (AFT) method that generalizes the conventional AFT (used for efficient Jacobian calculation) to compute second-order derivatives. The nonlinear forces and their first and second derivatives are evaluated analytically in the time domain for each nonlinear element, then transformed to the frequency domain via FFT. 2) A systematic process for local-to-global coordinate tensor transformations that assembles the element-level Hessians into the global system Hessian required by the optimizer.

The analytical Hessian eliminates the prohibitive cost of numerical differentiation for large models and ensures robust convergence of the optimization solver (a Lagrange multiplier formulation combined with parameter continuation using the COCO package). The framework is inherently general, applicable to any smooth nonlinearity, including displacement-dependent (e.g., cubic stiffness) and velocity-dependent (e.g., regularized friction) types.

The method is validated on three benchmarks of increasing complexity:

1. A 2-DOF system with cubic stiffness, serving as a basic verification.
2. A cantilever beam with either cubic stiffness or a hyperbolic tangent (tanh) friction nonlinearity, demonstrating capability with non-polynomial nonlinearities.
3. An industrial-scale FE model of a compressor bladed disk (blisk) with a friction ring damper, where the contact interface is modeled using a regularized tanh function. Model order reduction via a complex-valued basis is employed here to handle the high dimensionality.

Results show that the proposed Hessian-enhanced framework accurately and efficiently computes both resonance and anti-resonance backbone curves. It significantly reduces runtime compared to approaches relying on numerical derivatives or full response surface sweeps, making it a practical and robust frequency-domain tool for analyzing structures with complex, non-polynomial nonlinearities, especially in large-scale engineering applications like turbomachinery components with friction interfaces.