Nonlinear model order reduction for structural systems with contact

The design and fabrication of modern engineering systems such as aircraft, automobiles, and naval architectures has benefited tremendously from the adoption of high-fidelity predictive modeling in recent decades. It is difficult to overstate the value that physics-based simulation provides in the understanding of how these complex systems will behave under both nominal and extraordinary circumstances. However, despite its obvious utility, the computational (and hence temporal, and monetary) costs of performing a simulation at the level of detail required to make confident engineering decisions is substantial, to the point where only a few predetermined and carefully considered configurations can be explored through numerical techniques. Projection-based model order reduction is a promising approach to alleviate the massive computational burden associated with the simulation of partial differential equations with complicated domains and boundary conditions. It is an approximation technique defined by the restriction of the solution space to an appropriate subspace that is of low dimension in comparison to the original semi-discrete model. In the physical regimes that are adequately described by linear models, projection-based model reduction has already been demonstrated to increase computational efficiency by orders of magnitude for real-world use cases. It is in these scenarios, which are governed by highly nonlinear operators, where projection-based model reduction has yet to realize its full potential. The current shortcomings of projection-based model order reduction are in two primary areas: robustness of the reduced order model to parameter variations, and efficiently evaluating projected nonlinear terms. The construction of parametrically robust reduced order models is an active area of research and critical to the adoption of projection-based model reduction in industrial settings. This thesis briefly touches on the topic of parametric robustness by presenting a novel state-space clustering approach (known as sparse subspace clustering) designed to identify inherently low-dimensional structure in solution snapshot data. However, the focus of this work is primarily on the latter point, i.e. efficiently evaluating nonlinear terms in the low dimensional subspace, for without significant computational savings, there is no point in achieving parametric robustness. The primary mechanism by which this is accomplished is through the introduction of additional approximation of the projected nonlinear terms, which is known as hyperreduction. Particularly, this thesis presents the energy conserving sampling and weighting method (ECSW), a hyperreduction technique that has provable stability characteristics and tunable accuracy. Hyperreduction methods in general are accompanied by a pre-processing, or training, stage referred to as sampling in which the underlying computational mesh is sampled for the most relevant components to the hyperreduced order model. This is effectively a sparsification of the computational domain. It can be exceptionally expensive to perform for large computational models, hence efficient and scalable sampling algorithms are required for real-world problems. Therefore, a family of scalable, active set methods are presented to compliment the ECSW method of hyperreduction. Lastly, this work is set in the context of structural systems, especially those subject to contact constraints. This thesis extends the applicability of projection-based model order reduction and hyperreduction to models that are subject to large-deformation contact mechanics. This is achieved by leveraging the theoretical development and physical interpretation of the mortar method of constraint enforcement. Crucially, this method is characterized by a one-sided integration domain that naturally lends itself to the definition of a consistent dual snapshot collection procedure. In addition, a novel technique for the construction of a dual reduced order basis from sparse data is introduced based on the principle of nonnegative matrix completion. The techniques developed here are demonstrated on three challenging finite element structural models in order to showcase the improvement to the current state-of-the-art in model reduction technology.

Keywords: Aeronautics and Astronautics.
Advisor: Farhat Charbel 
Alonso Juan José
Kochenderfer Mykel J.