标题： 谐波平衡-自动微分方法：一种开箱即用且高效的通用非线性动力学仿真求解器 显示英文标题

标题： Harmonic balance-automatic differentiation method: an out-of-the-box and efficient solver for general nonlinear dynamics simulation

摘要： 谐波平衡-交替频域-时域（HB-AFT）方法被广泛用于非线性系统的动态响应分析。 然而，由于在牛顿-拉夫森迭代过程中需要手动推导雅可比矩阵，其在高维复杂系统中的应用受到限制，对于复杂的非线性，这些雅可比矩阵变得计算不可行或容易出错。 提出了一种谐波平衡-自动微分（HB-AD）方法，以解决这一限制，其中自动微分与谐波平衡框架相结合。 该方法通过利用自动微分来数值计算精确的雅可比矩阵，消除了所有手动推导，从而实现了对高维复杂非线性系统的通用且高效的分析。 实现中利用了先进的深度学习框架进行原生并行计算和CUDA加速，并将自动微分与弧长延续相结合，建立了一个开箱即用且高效计算架构。 用户只需提供系统的动力学方程，HB-AD便可自主追踪周期响应的完整图景——包括稳定/不稳定解分支。 对带有挤压油膜阻尼器（SFD）的转子系统的计算实验展示了HB-AD在处理具有自动雅可比计算的复杂非线性表达式方面的能力。 对于具有复杂轴承非线性的高维航空发动机转子-轴承-机匣系统，HB-AD的效率是传统HB-AFT的17倍，比Newmark方法快144倍。 HB-AD方法是计算力学和机器学习基本要素的协同融合，为高维工程系统的高保真动态表征提供了一个易于使用、通用且高效的平台。 显示英文摘要

摘要： The Harmonic Balance-Alternating Frequency-Time domain (HB-AFT) method is extensively employed for dynamic response analysis of nonlinear systems. However, its application to high-dimensional complex systems is constrained by the manual derivation of Jacobian matrices during Newton-Raphson iterations, which become computationally intractable or error-prone for intricate nonlinearities. The Harmonic Balance-Automatic Differentiation (HB-AD) method is proposed to address this limitation, in which AD is integrated with the harmonic balance framework. This approach eliminates all manual derivations by leveraging AD to compute exact Jacobians numerically, enabling generic and efficient analysis of high-dimensional complex nonlinear systems. The implementation utilizes advanced deep learning frameworks for native parallel computing and CUDA acceleration, and combines AD with arc-length continuation, establishing an out-of-the-box and high efficiency computational architecture. Users need only supply the system's dynamic equations, HB-AD then autonomously trace the complete panorama of periodic responses -- including stable/unstable solution branches. Computational experiments on a rotor system with squeeze-film damper (SFD) demonstrate HB-AD's capability in handling complex nonlinear expressions with automated Jacobian calculations. For a high-dimensional aero-engine rotor-bearing-casing system with complex bearing nonlinearities, HB-AD achieves 17-fold higher efficiency than traditional HB-AFT and 144-fold acceleration over the Newmark method. The HB-AD method is a synergistic merger of computational mechanics and machine learning primitives, delivers an easy to use, general-purpose, high efficiency platform for high-fidelity dynamic characterization of high-dimensional engineering systems.