GALCIT Colloquium

Model reduction is rapidly becoming an indispensable tool for computational mechanics-based design and optimization, statistical analysis, embedded computing, and online optimal control. It is also essential for "what-if" engineering scenarios and parametric studies where real-time or at least dramatically faster simulation responses are desired. This is because in many engineering applications, high-fidelity, time-dependent numerical simulations remain so computationally intensive that they cannot be used routinely, or even as often as needed. This is the case, for example, for turbulent CFD computations at high Reynolds numbers, the simulation of blast-induced fracture of thin shells, and the prediction of large shear deformations in soft tissues. During the last decade, both theoretical and algorithmic aspects of linear model reduction have significantly advanced and matured, and their impact on practical and important applications has been successfully demonstrated in many engineering fields. Unfortunately, this is not yet the case for nonlinear model reduction, where problems with shocks, turbulence, large displacements and rotations, large deformations, material yielding, contact, and evolving domains and discontinuities raise significant barriers. This lecture will begin by briefly overviewing this context and the aforementioned issues. It will then proceed with presenting a general framework for nonlinear model reduction that is based on a recently developed concept of local reduced-order bases where locality focuses on the regimes of the solution manifold rather than spatial or temporal considerations, a nonlinear projection method that preserves numerical stability, and two different hyper-reduction approaches tailored for finite volume and finite element approximations that deliver the sought-after speedups. Significant results recently obtained for the application of this nonlinear model reduction framework to the parametric simulation of turbulent flows in high Reynolds number aerodynamic applications and that of structural failures caused by under-body blasts  will also be presented and discussed. Finally, personal perspectives will also be offered.