A nonintrusive reduced order method able to solve a parametric modal analysis is proposed in this work. The main objective is being able to efficiently identify how a variation of user-defined parameters affects the dynamic response of the structure in terms of fundamental natural frequencies and corresponding mode shapes. A parametric version of the inverse power method (IPM) is presented by using the proper generalised decomposition (PGD) rationale. The proposed approach utilises the so-called encapsulated PGD toolbox and includes a new algorithm for computing the square root of a parametric object. With only one offline computation, the proposed PGD-IPM approach provides an analytical parametric expression of the smallest (in magnitude) eigenvalue (or natural frequency) and corresponding eigenvector (mode shape), which contains all the possible solutions for every combination of the parameters within pre-defined ranges. A Lagrange multiplier deflation technique is introduced in order to compute subsequent eigenpairs, which is also valid to overcome the stiffness matrix singularity in the case of a free-free structure. The proposed approach is nonintrusive and it is therefore possible to be integrated with commercial finite element (FE) packages. Two numerical examples are shown to underline the properties of the technique. The first example includes one material and one geometric parameter. The second example shows a more realistic industrial example, where the nonintrusivity of the approach is demonstrated by employing a commercial FE package for assembling the FE matrices. Finally, a multi-objective optimisation study is performed proving that the developed method could significantly assist the decision-making during the preliminary phase of a new design process.

在这项工作中提出了一种能够解决参数模态分析的非侵入式降阶方法。主要目标是能够有效地识别用户定义参数的变化如何影响结构在基本固有频率和相应模式形状方面的动态响应。通过使用适当的广义分解 (PGD) 原理，提出了逆幂方法 (IPM) 的参数版本。所提出的方法利用所谓的封装 PGD 工具箱，并包括一种用于计算参数对象平方根的新算法。仅通过一次离线计算，所提出的 PGD-IPM 方法提供了最小（幅度）特征值（或自然频率）和相应特征向量（模态形状）的解析参数表达式，其中包含预定义范围内每个参数组合的所有可能解决方案。引入了拉格朗日乘数紧缩技术以计算后续特征对，这对于克服自由自由结构情况下的刚度矩阵奇异性也是有效的。所提出的方法是非侵入式的，因此可以与商业有限元 (FE) 包集成。显示了两个数值示例以强调该技术的特性。第一个示例包括一种材料和一个几何参数。第二个示例显示了一个更现实的工业示例，其中通过使用商业 FE 包来组装 FE 矩阵来证明该方法的非侵入性。最后，