Nonlinear Model Reduction Techniques with Application to Geometrically Nonlinear Structures

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.