AI 中文总结
研究针对谐波平衡法在高维复杂系统应用受限问题，提出开源框架pyHB。它利用局部非线性和自动微分，结合多种策略，集成完整HB工作流程。通过实例验证其能追踪多种响应，加速效果显著，为非线性动力学分析提供通用基准平台。
AI 中文摘要
谐波平衡（HB）方法被广泛用于计算和分析非线性系统的周期响应。然而，其在高维复杂系统中的应用受到处理非线性项偏导数的负担限制。本文提出了pyHB，一个开源的、自动微分增强的半解析框架，集成了针对一般用户定义非线性系统的完整HB工作流程。该公式利用局部非线性，仅将基于PyTorch的自动微分应用于简化的非线性力，避免了用户提供非线性力的导数，并保持了可控的GPU内存使用。加权弧长延续、稀疏矩阵组装、增强延续方程的分块求解策略以及基于弗洛凯的稳定性分析被纳入一个模块化架构中，该架构将模型定义与可重复使用的数值程序分开。因此，pyHB仅基于用户定义的动力学方程就能提供非线性系统周期响应的完整图景。四个例子展示了pyHB追踪稳定和不稳定解分支以及捕捉次谐波共振、组合共振和混合阶机电响应的能力。特别是，在具有202000个HB未知数的伯努利梁例子中，AD增强求解器每个延续点大约需要0.44秒，与Newmark-β方法相比实现了数百倍的加速，并且还剩余637.8MB的额外RAM和243.5MB的GPU内存。所提出的pyHB为基于HB的非线性动力学分析提供了一个通用的一站式基准平台。
英文摘要
The Harmonic Balance (HB) method is widely used to compute and analyze the periodic responses of nonlinear systems. However, its application to high-dimensional complex systems is limited by the burden of handling the partial derivatives of the nonlinearities. This work presents pyHB, an open-source, automatic-differentiation-enhanced semi-analytical framework that integrates the complete HB workflow for general user-defined nonlinear systems. The proposed formulation exploits localized nonlinearities and applies PyTorch-based automatic differentiation (AD) only to the reduced nonlinear force, thereby avoiding the need for user-supplied derivatives of the nonlinear force and maintaining controllable GPU memory usage. Weighted arc-length continuation, sparse matrix assembly, a blocked solution strategy for the augmented continuation equations, and Floquet-based stability analysis are incorporated within a modular architecture that separates model definition from reusable numerical procedures. Hence, pyHB can provide a complete landscape of the nonlinear system's periodic response based solely on the user-defined dynamical equations. Four examples, including a quasi-zero-stiffness isolator, a nonlinear piezoelectric energy harvester, a 284 degrees of freedom (DOFs) aeroengine model, and a 2000 DOFs Bernoulli beam, demonstrate the ability of pyHB to trace stable and unstable solution branches and capture subharmonic resonance, combination resonance, and mixed-order electromechanical responses. Notably, in the Bernoulli beam example with 202000 HB unknowns, the AD-enhanced solver requires approximately 0.44s per continuation point, achieving several-hundred-fold speedup compared to the Newmark-$β$ method and remaining 637.8MB of additional RAM and 243.5MB of GPU memory. The proposed pyHB provides a general, one-stop benchmark platform for HB-based nonlinear dynamics analysis.
Comments32 pages, 8 figures, 1 table
