The direct parametrisation method for invariant manifold is a model-order reduction technique that can be applied to nonlinear systems described by PDEs and discretised e.g. with a finite element procedure in order to derive efficient reduced-order models (ROMs). In nonlinear vibrations, it has already been applied to autonomous and non-autonomous problems to propose ROMs that can compute backbone and frequency-response curves of structures with geometric nonlinearity. While previous developments used a first-order expansion to cope with the non-autonomous term, this assumption is here relaxed by proposing a different treatment. The key idea is to enlarge the dimension of the parametrising coordinates with additional entries related to the forcing. A new algorithm is derived with this starting assumption and, as a key consequence, the resonance relationships appearing through the homological equations involve multiple occurrences of the forcing frequency, showing that with this new development, ROMs for systems exhibiting a superharmonic resonance, can be derived. The method is implemented and validated on academic test cases involving beams and arches. It is numerically demonstrated that the method generates efficient ROMs for problems involving 3:1 and 2:1 superharmonic resonances, as well as converged results for systems where the first-order truncation on the non-autonomous term showed a clear limitation.

翻译：不变流形的直接参数化方法是一种模型降阶技术，可应用于偏微分方程描述且经有限元等数值方法离散的非线性系统，以构建高效降阶模型。在非线性振动领域，该方法已成功应用于自治与非自治问题，为具有几何非线性的结构生成可计算骨干曲线与频率响应曲线的降阶模型。以往研究采用一阶展开处理非自治项，本文通过提出新的处理方式突破这一假设。核心思想是引入与激励相关的附加坐标，扩展参数化坐标的维度。基于此假设推导出新算法，其关键结果在于通过同调方程出现的共振关系中包含了激励频率的多重作用，表明该新方法可推导出具有超谐共振系统的降阶模型。该方法在含梁与拱的学术算例中得到实现与验证。数值实验证明，该方法能为涉及3:1和2:1超谐共振的问题生成高效降阶模型，且对于非自治项一阶截断存在明显局限的系统，该方法可获得收敛结果。