在非线性模型降阶中保持拉格朗日结构及其在结构动力学中的应用
Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics
AI总结：
本文提出一种保持拉格朗日结构的降阶方法，用于高阶非线性和参数依赖性场景，通过保留能量守恒和辛时间演化映射等关键特性，提升计算效率。
摘要：
本文提出了一种降阶方法，旨在保持拉格朗日结构（等价于哈密顿结构）并提高在高阶非线性和任意参数依赖性下的计算效率。所得到的降阶模型保留了关键属性，如能量守恒和辛时间演化映射。我们专注于受拉格朗日阻尼和外部力作用的参数化简单机械系统，并考虑其在非线性结构动力学中的应用。为保持结构，该方法首先近似系统的'拉格朗日成分'——黎曼度量、势能函数、耗散函数和外部力，然后通过应用带有这些量的强制欧拉-拉格朗日方程推导降阶运动方程。从代数角度看，关键贡献包括两种高效的近似参数化降阶矩阵的技术：矩阵空洞POD和降阶基集稀疏化（RBS）。参数化桁架结构问题的结果展示了保持拉格朗日结构的重要性，并展示了所提方法的优点：在保持高精度和稳定性的同时减少了计算时间，与不保持结构的现有非线性降阶技术形成对比。
英文摘要：
This work proposes a model-reduction methodology that preserves Lagrangian structure (equivalently Hamiltonian structure) and achieves computational efficiency in the presence of high-order nonlinearities and arbitrary parameter dependence. As such, the resulting reduced-order model retains key properties such as energy conservation and symplectic time-evolution maps. We focus on parameterized simple mechanical systems subjected to Rayleigh damping and external forces, and consider an application to nonlinear structural dynamics. To preserve structure, the method first approximates the system's `Lagrangian ingredients'---the Riemannian metric, the potential-energy function, the dissipation function, and the external force---and subsequently derives reduced-order equations of motion by applying the (forced) Euler--Lagrange equation with these quantities. From the algebraic perspective, key contributions include two efficient techniques for approximating parameterized reduced matrices while preserving symmetry and positive definiteness: matrix gappy POD and reduced-basis sparsification (RBS). Results for a parameterized truss-structure problem demonstrate the importance of preserving Lagrangian structure and illustrate the proposed method's merits: it reduces computation time while maintaining high accuracy and stability, in contrast to existing nonlinear model-reduction techniques that do not preserve structure.
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