Reduced order methods for nonlinear parametric problems with branching solutions

The aim of this work is to show the applicability of the reduced basis model reduction in nonlinear systems undergoing bifurcations. Bifurcation analysis, i.e., following the different bifurcating branches, as well as determining the bifurcation point itself, is a complex computational task. Reduced Order Models (ROM) can potentially reduce the computational burden by several orders of magnitude, in particular in conjunction with sampling techniques. In the first task we focus on nonlinear structural mechanics, and we deal with an application of ROM to Von Kármán plate equations, where the buckling effect arises, adopting reduced basis method. Moreover, in the search of the bifurcation points, it is crucial to supplement the full problem with a reduced generalized parametric eigenvalue problem, properly paired with state equations and also a reduced order error analysis. These studies are carried out in view of vibroacoustic applications (in collaboration with A.T. Patera at MIT). As second task we consider the incompressible Navier-Stokes equations, discretized with the spectral element method, in a channel and a cavity. Both system undergo bifurcations with increasing Reynolds - and Grashof - number, respectively. Applications of this model are contraction-expansion channels, found in many biological systems, such as the human heart, for instance, or crystal growth in cavities, used in semiconductor production processes. This last task is in collaboration with A. Alla and M. Gunzburger (Florida State University).