Computational reduction strategies for bifurcation and stability analysis in fluid-dynamics

Mapping Intimacies ◽

10.14293/p2199-8442.1.sop-math.pt5xwg.v1 ◽

2015 ◽

Author(s):

Giuseppe Pitton ◽

Annalisa Quaini ◽

Gianluigi Rozza

Keyword(s):

Spectral Element ◽

Reduced Basis ◽

Regurgitant Flow ◽

Reduction Strategies ◽

First Results ◽

Uniqueness Of The Solution ◽

Parametrized Partial Differential Equations ◽

Basis Method ◽

Computational Reduction ◽

2D And 3D

We focus on reduced order modelling for nonlinear parametrized Partial Differential Equations, frequently used in the mathematical modelling of physical systems. A common issue in this kind of problems is the possible loss of uniqueness of the solution as the parameters are varied and a singular point is encountered. In the present work, the numerical detection of singular points is performed online through a Reduced Basis Method, coupled with a Spectral Element Method for the numerically intensive offline computations. Numerical results for laminar fluid mechanics problems will be presented, where pitchfork, hysteresis, and Hopf bifurcation points are detected by an inexpensive reduced model. Some of the presented 2D and 3D flow results deal with the study of instabilities in a simplified model of a mitral regurgitant flow in order to understand the onset of the Coanda effect. The first results are in good agreement with the reference.

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Reduced order methods for nonlinear parametric problems with branching solutions

10.14293/p2199-8442.1.sop-math.ezptcy.v1 ◽

2018 ◽

Author(s):

Federico Pichi ◽

Martin Wilfried Hess ◽

Annalisa Quaini ◽

Gianluigi Rozza

Keyword(s):

Stokes Equations ◽

Spectral Element ◽

Navier Stokes ◽

Reduced Basis ◽

State Equations ◽

Order Error ◽

Navier Stokes Equations ◽

Reduced Order ◽

Plate Equations ◽

Basis Method

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).

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