POD approach for unsteady aerodynamic model updating

Abstract

A method for aerodynamic model updating is proposed in this paper. The approach is based upon a correction of the eigenvalues of the reduced-order unsteady aerodynamic matrix through an optimization with objective function defined through the difference in the generalized aerodynamic forces or on the aeroelastic poles. The high-fidelity model in reduced-order form is obtained by the proper orthogonal decomposition (POD) technique applied to the computational fluid dynamics Euler-based formulation. Many of the methods that have been developed in the past years for simpler aeroelastic models that use, for example, doublet-lattice method aerodynamics, can be adopted for this purpose as well. However, this model is not able to capture shocks and flow separation in transonic flow. The proposed approach performs the updating of the aerodynamic model by imposing the minimization of a global error between target aerodynamic performances, namely experimental performances, and an aerodynamic model in reduced-order form via POD approach. After a general presentation of the application of the POD method to the linearized Euler equations, the optimization strategy is presented. First, a simple test on a 2D wing section with theoretical biased data is performed, and then, the performances of different optimization strategies are tested on a 3D model updated by wind tunnel data.

Appendix: some details on POD basis evaluation

The starting point of the POD-based ROM procedure is calculation of the small-disturbance solution response of the fluid dynamic system at N different combinations of excitation and frequency. These solutions, also known as snapshots, are denoted by $U = \left\{ {u^{n} ,n = 1,..,N} \right\}$. The Euler equations, solved for the special case of an harmonic excitation of type $\left( {M_{s} ,t} \right) = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{d} \left( {M_{s} } \right)e^{i\omega t}$, lead us to search for a solution of the form $U = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{U} e^{i\omega t}$, where $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{d}$ is a prescribed structural displacement field, $\omega$ is the frequency, and $i = \sqrt[2]{ - 1}$ is the imaginary number. Thus, the snapshots are the estimate of the complex unsteady field at the centre of the j th cell of the computational grid for a varying frequency ω. The POD technique is next used to find the smallest and best subspace of finite dimension M ≪ N which contains the dominant unsteady characteristics of the flow. The identified subspace $\left\{ {\psi^{i} ,i = 1,..,M} \right\}$ represent the dominate “directions” of the full original solution. Each snapshot can be approximated by a POM (proper orthogonal mode, also known as POD vectors) linear combination:

$$u^{n} =\bar{u} + \sum\nolimits_{i = 1}^{M} {\eta_{i} \psi^{i} }$$

(31)

where $\psi^{i}$ are POMs and $\eta_{i}$ are the unknown coefficients of POD expansions. The POD modes are obtained from the maximization problem:

$$\max_{\psi } \left\langle {\left\| {\left( {u\left( t \right),\frac{\psi }{\psi }} \right)\frac{\psi }{\psi }} \right\|^{2} } \right\rangle .$$

(32)

We maximize the norm of the u projection on the right vectorial direction of $\psi$ on average on T, where T is a discrete continuous set. Note that, in general, the following equivalent formulation is preferred:

$$\max_{\psi } \left\langle {\frac{{\left( {u\left( t \right),\psi } \right)^{2} }}{{\left( {\psi ,\psi } \right)}}} \right\rangle .$$

(33)

The previous maximum problem for a definite-positive function is equivalent to the resolution of the following eigenvalues problem

$${\text{SV}} = V\sigma^{2}$$

(34)

where S is the real snapshot correlation matrix

$$S = {\text{RR}}^{T}$$

(35)

with,

$$R = \left( {Re\left( U \right) Im\left( U \right)} \right)$$

(36)

and each column of U contains a complex valued snapshot. V is an eigenvector. Note that the corresponding eigenvalues are expected to be positive because of the positiveness of the matrix S, and therefore, the positive quantity $\sigma^{2}$ is directly introduced in Eq. 34.

The choice of the eigenvector to build the POD basis is made according to the following criteria:

• 1. Snapshots that are not decorrelated: the modes obtained from decorrelated snapshots are the snapshots themselves and they all have the same eigenvalues.

• 2. Elimination of the eigenvectors associated with eigenvalues that are zero or too small.

• 3. There is a little difference between the partial and total energy.

As shown in Ref. [15], the POMs are simply a linear rearrangement of the original snapshot:

$$\psi^{i} = \sum\nolimits_{n = 1}^{n} {a_{m}^{i} u^{n} }$$

(37)

After the eigenvalues problem (Eq. 34) has been solved, the POMs are computed by the Eq. (37) where

$$a_{m}^{i} = V_{M} \sigma_{M}^{ - 1} .$$

(38)