Nonlinear Model Reduction Techniques with Application to Geometrically Nonlinear Structures

随着不断提高性能，结构变得越来越轻，越来越灵活。以波音SUGAR Volt等新飞机概念的大展弦比机翼为例。这种柔性结构的构件通常表现出大的位移和旋转，导致所谓的非线性几何效应。非线性的存在带来了重要的挑战，因为可能出现无法用工业中常用的线性工具处理的新动态现象。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。虽然这些方法是严格的数学和先验广泛适用，与高维系统的分析，如在现实生活中的工业应用中遇到的计算成本仍然棘手。本论文旨在发展一套适用于非线性机械结构的严格模型降阶技术。所提出的工作的独创性将是使用模态导数（MD）的概念，以提出一种基于非线性Galerkin变换的简化技术。具体来说，在论文的第一年，MD将被用来减少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的降阶模型是否可以准确地捕获分叉。分叉代表稳定性边界，在那里系统的动态可能发生剧烈的定性和定量变化，因此，它们通常是理解系统动态的关键。此外，在第一年，将定义一个更复杂，更大规模的结构，灵感来自工业中遇到的结构。这个结构将被用来展示和比较在论文的其余部分研究的不同方法的性能。MD已被限制为相对于模态振荡振幅的振动模式的一阶导数。在第一年之后，本论文将把这个概念延伸到高阶导数，以更精确地捕捉非线性失真。然后将开发一种基于非线性坐标变换的新模型简化技术。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降阶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数。最后，本计画的另一个重要方面是提出一个有效且精确的方法，当全尺寸系统的物理性质被修改时，用来更新降阶模型。到目前为止，降阶模型通常仅对给定的一组物理参数有效，即材料和几何特性。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是非常有益的分叉分析和设计优化。在这篇论文中开发的方法和工具将不限于一个特定的结构-事实上，他们将适用于任何结构，提出高度灵活的组件。

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.