Reduced Order Models

Owing to the large complexity and size of modern mathematical models nowadays employed in numerical simulations, the study of relevant engineering problems often requires too much time and too many computational resources. Model order reduction is an effective strategy to overcome these difficulties, by reducing the state space dimension or degrees of freedom of the system. An approximation to the original model is therefore evaluated, generally with lower accuracy but also in significantly less time.

Coupled Electromechanical Systems:

Model order reduction of coupled electromechanical systems is highly desirable when aiming to perform optimization studies of electromechanical devices, such as piezoelectric energy harvesters. For instance, a multidisciplinary optimization of the full coupled system requires an immense computational effort. An approximation to the high-fidelity model may be obtained by introducing several simplification assumptions. The resulting low-order model is described simply by a pair of coupled ordinary differential equations, which is much more amenable for use with advanced optimization algorithms.



Aerodynamics of Flapping Airfoils:

Massive savings in the time required by parametric studies for the aerodynamic design of flapping airfoils may be obtained via the evaluation of low-order models rather than using high-fidelity simulations. The latter generally comprise obtaining solutions of the unsteady Navier-Stokes equations, in a framework accounting for body motion, involving tenths or hundreds of thousands of degrees of freedom. Vortex methods provide a means to obtain approximate solutions of the flow, as well as describing the non-linear behavior of the aerodynamic forces generated by flapping airfoils, employing only a minute fraction of the resources demanded by a high-fidelity model. In addition, achieving real-time simulations represents a key issue for flow control in micro-aerial vehicles.



Proper Orthogonal Decomposition:

Reduced order models of large-scale fluid flow problems may be obtained via Proper Orthogonal Decomposition (POD), which may be included within the set of techniques used for principal component analysis. The method is data-based in the sense that a suitable orthonormal basis is determined from computational (e.g., from CFD) or experimental data (e.g., from PIV), aiming to capture and hierarchize relevant information regarding the spatial dynamics of the system. This approach may, to a certain extent, be associated with machine learning.



N-factor Transition Prediction:

In flows where laminar-to-turbulent transition is dominated by the linear phase of breakdown, small-disturbance (linear) stability theory coupled with the semi-empirical N-factor method can provide accurate enough estimates of the transition location, thereby keeping the computational burden within reasonable bounds. This methodology is particulary useful in the analysis of transonic aircraft wings designed to operate with large extensions of laminar flow, so that skin-friction drag may be reduced.



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