Structural dynamics models with nonlinear stiffness appear, for example, when analyzing systems with nonlinear material behavior or undergoing large deformations. For complex systems, these models become too large for real-time applications or multi-query workflows. Hence, model reduction is needed. However, the mathematical operators of these models are often not available since, as is common in industry practice, the models are constructed using commercial simulation software. In this work, we propose an operator inference-based approach aimed at inferring, from data generated by the simulation model, reduced-order models (ROMs) of structural dynamics systems with stiffness terms represented by polynomials of arbitrary degree. To ensure physically meaningful models, we impose constraints on the inference such that the model is guaranteed to exhibit stability properties. Convexity of the optimization problem associated with the inference is maintained by applying a sum-of-squares relaxation to the polynomial term. To further reduce the size of the ROM and improve numerical conditioning of the inference, we also propose a novel clustering-based sparsification of the polynomial term. We validate the proposed method on several numerical examples, including a representative 3D Finite Element Model (FEM) of a steel piston rod.

翻译：具有非线性刚度的结构动力学模型常见于分析非线性材料行为或经历大变形的系统。对于复杂系统，这些模型规模过大，难以应用于实时场景或多查询工作流程，因此需要进行模型降阶。然而，这些模型的数学算子通常难以直接获取，因为工业实践中普遍采用商业仿真软件构建模型。本研究提出一种基于算子推断的方法，旨在利用仿真模型生成的数据，推断刚度项由任意阶多项式表示的结构动力学系统的降阶模型。为确保模型的物理意义，我们在推断过程中施加约束，以保证模型具备稳定性。通过对多项式项应用平方和松弛，保持了与推断相关的优化问题的凸性。为进一步减小降阶模型规模并改善推断的数值条件，我们还提出了一种基于聚类的新型多项式项稀疏化方法。我们在多个数值算例中验证了所提方法的有效性，包括一个具有代表性的钢制活塞杆三维有限元模型。