We present an enhanced version of the parametric nonlinear reduced order model for shape imperfections in structural dynamics we studied in a previous work [1]. The model is computed intrusively and with no training using information about the nominal geometry of the structure and some user-defined displacement fields representing shape defects, i.e. small deviations from the nominal geometry parametrized by their respective amplitudes. The linear superposition of these artificial displacements describe the defected geometry and can be embedded in the strain formulation in such a way that, in the end, nonlinear internal elastic forces can be expressed as a polynomial function of both these defect fields and the actual displacement field. This way, a tensorial representation of the internal forces can be obtained and, owning the reduction in size of the model given by a Galerkin projection, high simulation speed-ups can be achieved. We show that by adopting a rigorous deformation framework we are able to achieve better accuracy as compared to the previous work. In particular, exploiting Neumann expansion in the definition of the Green-Lagrange strain tensor, we show that our previous model is a lower order approximation with respect to the one we present now. Two numerical examples of a clamped beam and a MEMS gyroscope finally demonstrate the benefits of the method in terms of speed and increased accuracy.

中文翻译：

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使用Neumann展开的非理想结构的增强参数非线性降阶模型

我们提供了参数非线性降阶模型的增强版本，用于我们在先前的工作中研究过的结构动力学中的形状缺陷[1]。使用关于结构的标称几何形状的信息和一些代表形状缺陷的用户定义的位移场（即与通过其各自的振幅参数化的标称几何形状的微小偏差），无需干预就可以侵入式地计算模型。这些人工位移的线性叠加描述了缺陷的几何形状，并且可以以如下方式嵌入应变公式中：最终，非线性内部弹性力可以表示为这些缺陷场和实际位移场的多项式函数。这样，可以获得内力的张量表示，并且 拥有通过Galerkin投影给出的模型尺寸的减小，可以实现较高的仿真速度。我们表明，通过采用严格的变形框架，与以前的工作相比，我们可以获得更好的精度。特别是，利用格林-拉格朗日应变张量定义中的诺伊曼展开式，我们证明了我们以前的模型相对于现在的模型是低阶近似。夹紧梁和MEMS陀螺仪的两个数值示例最终证明了该方法在速度和提高准确性方面的优势。利用格林-拉格朗日应变张量定义中的诺伊曼展开，我们证明了我们先前的模型相对于现在的模型是低阶近似。夹紧梁和MEMS陀螺仪的两个数值示例最终证明了该方法在速度和提高准确性方面的优势。利用格林-拉格朗日应变张量定义中的诺伊曼展开，我们证明了我们先前的模型相对于现在的模型是低阶近似。夹紧梁和MEMS陀螺仪的两个数值示例最终证明了该方法在速度和提高准确性方面的优势。