Spectral submanifolds (SSMs) have emerged as accurate and predictive model reduction tools for dynamical systems defined either by equations or data sets. While finite-elements (FE) models belong to the equation-based class of problems, their implementations in commercial solvers do not generally provide information on the nonlinearities required for the analytical construction of SSMs. Here, we overcome this limitation by developing a data-driven construction of SSM-reduced models from a small number of unforced FE simulations. We then use these models to predict the forced response of the FE model without performing any costly forced simulation. This approach yields accurate forced response predictions even in the presence of internal resonances or quasi-periodic forcing, as we illustrate on several FE models. Our examples range from simple structures, such as beams and shells, to more complex geometries, such as a micro-resonator model containing more than a million degrees of freedom. In the latter case, our algorithm predicts accurate forced response curves in a small fraction of the time it takes to verify just a few points on those curves by simulating the full forced-response.

翻译：频谱子流形（SSMs）已成为动力系统精确且具有预测能力的降阶建模工具，既可适用于方程定义的系统，也可适用于数据驱动的系统。尽管有限元（FE）模型属于基于方程的问题类别，但商业求解器中的实现通常无法提供解析构建SSM所需的非线性信息。本文通过仅利用少量无受迫有限元模拟数据，开发了一种数据驱动的SSM降阶模型构建方法，从而克服了这一局限。我们利用这些模型预测有限元模型的受迫响应，而无需执行任何昂贵的受迫模拟。即使存在内共振或准周期受迫情况，该方法仍能获得精确的受迫响应预测——我们在多个有限元模型上验证了这一特性。我们的算例涵盖从梁、壳等简单结构到包含超过百万自由度的微谐振器等复杂几何模型。就后者而言，我们的算法能在极短时间内预测出精确的受迫响应曲线，而验证曲线上少数几个点所需的全受迫响应模拟时间占其中极小部分。