The proper orthogonal decomposition (POD) is the prevailing method for basis generation in the model reduction of fluids. A serious limitation of this method, however, is that it is empirical. In other words, this basis accurately represents the flow data used to generate it, but may not be accurate when applied 'off-design'. Thus, the reduced-order model may lose accuracy for flow parameters (e.g. Reynolds number, initial or boundary conditions and forcing parameters) different from those used to generate the POD basis and generally does. This paper investigates the use of sensitivity analysis in the basis selection step to partially address this limitation. We examine two strategies that use the sensitivity of the POD modes with respect to the problem parameters. Numerical experiments performed on the flow past a square cylinder over a range of Reynolds numbers demonstrate the effectiveness of these strategies. The newly derived bases allow for a more accurate representation of the flows when exploring the parameter space. Expanding the POD basis built at one state with its sensitivity leads to low-dimensional dynamical systems having attractors that approximate fairly well the attractor of the full-order Navier-Stokes equations for large parameter changes.
IntroductionA number of practical engineering problems requires the repeated simulation of unsteady fluid flows for a large number of parameter values. These problems include the control, optimization and uncertainty quantification of fluid systems. To make many of these problems tractable, reduced-order modelling has been used to minimize the simulation requirements. The use of reduced-order modelling in control and optimization has led to practical solutions for extremely challenging problems (Ito & Ravindran 1996) Copyright by the Cambridge University Press. Hay, A.; Borggaard, J. T.; Pelletier, D., "Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition,"
Reduced-order models for parameter dependent geometries based on shape sensitivity analysis Hay, A.; Borggaard, J.; Akhtar, I.; Pelletier, D.Contact us / Contactez nous: nparc.cisti@nrc-cnrc.gc.ca. The proper orthogonal decomposition (POD) is widely used to derive low-dimensional models of large and complex systems. One of the main drawback of this method, however, is that it is based on reference data. When they are obtained for one single set of parameter values, the resulting model can reproduce the reference dynamics very accurately but generally lack of robustness away from the reference state. It is therefore crucial to enlarge the validity range of these models beyond the parameter values for which they were derived. This paper presents two strategies based on shape sensitivity analysis to partially address this limitation of the POD for parameters that define the geometry of the problem at hand (design or shape parameters.) We first detail the methodology to compute both the POD modes and their Lagrangian sensitivities with respect to shape parameters. From them, we derive improved reduced-order bases to approximate a class of solutions over a range of parameter values. Secondly, we demonstrate the efficiency and limitations of these approaches on two typical flow problems: (1) the one-dimensional Burgers' equation; (2) the two-dimensional flows past a square cylinder over a range of incidence angles.
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