The direct parametrisation method for invariant manifold is a model-order reduction technique that can be directly applied to finite element problems in order to derive efficient and converged reduced-order models (ROMs) for non-linear structures. In the field of nonlinear vibrations, it has already been applied to autonomous and non-autonomous problems in order to propose ROMs that can compute backbones and frequency-response curves of structures with geometric nonlinearity. While previous developments used a first-order development in order to cope with the non-autonomous term, this assumption is here relaxed by proposing a completely different treatment. The key idea is to enlarge the dimension of the dynamical system to make it autonomous and treat the added coordinates related to the forcing as already being written with normal coordinates. The parametrisation method 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, one is able to compute ROMs for superharmonic resonance. 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 3:1 and 2:1 superharmonic resonances, as well as converged results for problems where the first-order truncation on the non-autonomous terms used in previous developments 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，可直接应用于有限元问题，为非线性结构导出高效且收敛的降阶模型。在非线性振动领域，该方法已被应用于自治与非自治问题，以构建几何非线性结构主干曲线与频响曲线的降阶模型。以往研究采用一阶展开处理非自治项，本文则通过提出完全不同的处理方式突破这一假设。核心思路是将动力系统维度扩展以使其变为自治系统，并将与强迫相关的附加坐标视为已用正则坐标表达。基于此初始假设推导参数化方法，其关键结果是同调方程中出现的共振关系包含强制频率的多次出现，表明新方法能够计算超谐共振的降阶模型。该方法的实现通过梁与拱的学术测试案例验证，数值实验证明该方法能有效构建3:1和2:1超谐共振的降阶模型，并在以往方法中非自治项一阶截断显露出局限性的问题上获得收敛结果。