Nonlinear Model Reduction Techniques with Application to Geometrically Nonlinear Structures

随着不断提高性能的驱动，结构变得越来越轻和更灵活。以波音Suge Volt等新概念飞机的大展弦比机翼为例。这种柔性结构的构件往往表现出很大的位移和转动，从而导致所谓的非线性几何效应。非线性的存在带来了重大挑战，因为可能会出现工业中常用的线性工具无法处理的新的动态现象。在科学文献中已经提出了许多分析非线性系统动力学的方法，例如，数值延拓技术。虽然这些方法在数学上是严格的，先验的广泛适用，但与高维系统分析相关的计算成本仍然很难处理，例如在现实生活中工业应用中遇到的那些系统。本文的目的是发展一种适用于非线性机械结构的严格的模型降阶技术。本文的创新之处在于引入了模态导数的概念，提出了一种基于非线性Galerkin变换的降阶方法。具体地说，在论文的第一年，MD将被用来简化具有几何非线性的简单学术结构(如二维曲梁)的有限元模型。该项目将确定基于MD的降阶模型是否能够准确地捕捉分叉。分叉代表的是稳定性边界，在这个边界上，系统的动力学可能发生质的和量的剧烈变化，因此，它们往往是理解系统动力学的关键。此外，在第一年，将定义一个更复杂、更大规模的结构，其灵感来自于工业中遇到的结构。这一结构将被用来演示和比较整个论文中所研究的不同方法的性能。MDS仅限于振型相对于振型振幅的一阶导数。在第一年之后，为了更准确地捕捉非线性失真，本论文将把这个概念扩展到高阶导数。在此基础上，提出了一种基于非线性坐标变换的模型降阶方法。到目前为止，结构动力学文献中提出的模型降阶技术考虑了完整模型和简化模型之间的线性变换。非线性变换有可能进一步降低模型的维度。最后，本项目的另一个重要方面是提出了一种有效而准确的方法，用于在全尺度系统的物理属性发生变化时更新降维模型。到目前为止，降阶模型通常只对给定的一组物理参数有效，即材料和几何属性。更新(内插)非线性降阶模型而不需要完全重新计算它们的可能性将对分叉分析和设计优化非常有利。本论文中开发的方法和工具不会局限于特定的结构--事实上，它们将适用于任何提供高度灵活组件的结构。

With the constant drive to improve performance, structures become increasingly lighter and more flexible. Take for example the high aspect ratio wings of new aircraft concepts such as the Boeing SUGAR Volt. The components of such flexible structures often exhibit large displacements and rotations, leading to so-called nonlinear geometric effects. The presence of nonlinearity poses important challenges as novel dynamic phenomena that cannot be treated with the linear tools commonly used in industry can arise. A number of methods to analyse the dynamics of nonlinear systems have been proposed in the scientific literature as, for instance, numerical continuation techniques. Although these methods are rigorous mathematically and a priori broadly applicable, the computational cost associated with the analysis of high-dimensional systems such as those met in real-life industrial applications remains intractable. The aim of this thesis is to develop a rigorous model reduction technique adapted to nonlinear mechanical structures. The originality of the proposed work will be to use the concept of modal derivatives (MDs) in order to propose a reduction technique based on nonlinear Galerkin transformations. Specifically, during the first year of the thesis, MDs will be exploited to reduce the finite element models of simple, academic structures featuring geometric nonlinearities (such as a two-dimensional, curved beam). The project will determine if reduced-order models based on MDs can accurately capture bifurcations. Bifurcations represent stability boundaries where dramatic qualitative and quantitative changes in the dynamics of a system can occur and, as such, they are often key to the understanding of a system's dynamics. In addition, during this first year, a more complex, larger-scale structure inspired from structures met in industry will be defined. This structure will be used to demonstrate and compare the performance of the different methods studied throughout the rest of the thesis. MDs have been restricted to the first-order derivatives of vibration modes with respect to modal oscillation amplitudes. After the first year, the thesis will extend the concept to higher-order derivatives in order to capture more accurately nonlinear distortions. A new model reduction technique based on nonlinear coordinate transformations will then be developed. As of now, model reduction techniques proposed in the structural dynamics literature consider linear transformations between full and reduced models. Nonlinear transformations have the potential to reduce the dimensionality of the model even further. Finally, another important aspect of this project is to propose an effective and accurate method for updating reduced-order models when physical properties of the full-scale system are modified. As of now, reduced-order models are typically valid only for a given set of physical parameters, i.e. material and geometrical properties. The possibility to update (interpolate) nonlinear reduced-order models without the need to fully recompute them would be extremely beneficial for bifurcation analysis and design optimization. The methodologies and tools developed in this thesis will not be limited to a specific structure - in fact, they will be applicable any structure that presents highly-flexible components.