A multifidelity approach for the construction of explicit decision boundaries: application to aeroelasticity

Abstract

This paper presents a multifidelity approach for the construction of explicit decision boundaries (constraints or limit-state functions) using support vector machines. A lower fidelity model is used to select specific samples to construct the decision boundary corresponding to a higher fidelity model. This selection is based on two schemes. The first scheme selects samples within an envelope constructed from the lower fidelity model. The second technique is based on the detection of regions of inconsistencies between the lower and the higher fidelity decision boundaries. The approach is applied to analytical examples as well as an aeroelasticity problem for the construction of a nonlinear flutter boundary.

A Stability analysis

In order to assess the stability of a given airfoil configuration, its response is studied in the time domain. This approach is essential in the case of “black-box” codes for which the Jacobian is not available. In addition, the study of the system’s response is the only way to assess the true stability boundary (as opposed to based on a linear assumption) of a nonlinear system in the general case. In this study, the response considered is the mechanical energy defined as the sum of the kinetic and the elastic energies. This approach has the advantage of encompassing all the degrees of freedom of the system in one quantity. For an asymptotically stable system, the energy will converge. For an unstable system the system energy will continue to grow unboundedly. In order to capture the trend, the following function:

$$ y(\tau)=p_{1}e^{p_{2}\tau}$$

(13)

with parameters p ₁, p ₂ approximates in a least square sense the mechanical energy. If p ₂ is negative, the system is classified as stable, otherwise as unstable. Figure 17 provides examples of stable and unstable configurations. The system energy is calculated from the pitch and plunge velocities and the deformation of the springs. For the two DOF system the classification is not based on the system energy E directly, but on the dimensionless system energy $\bar{E}$ defined by:

$$ \bar{E}=\frac{E}{\rho U^{2}b^{2}}$$

(14)

Where U denotes the free stream velocity, ρ denotes the two-dimensional air density and b denotes the airfoil semi-chord. The restoring forces due to the springs are given in terms of plunge and pitch by:

$$ F_{h}\left(\xi\right)=\xi+k_{3h}\xi^{3}+k_{5h}\xi^{5}\label{plunge spring force} $$

(15)

$$ M_{\alpha}\left(\alpha\right)=\alpha+k_{3\alpha}\alpha^{3}+k_{5\alpha}\alpha^{5}\label{pitch spring force} $$

(16)

The energy stored in the spring in plunge is calculated as:

$$ \bar{E}_{spring\,\xi}=\mu\pi\left(\frac{\omega}{U_{R}}\right)^{2}\left(\frac{1}{2}\xi^{2}+\frac{1}{4}k_{3h}\xi^{4}+\frac{1}{6}k_{5h}\xi^{6}\right)\label{elasticenergyplunge}$$

(17)

Similarly for the spring in pitch:

$$ \bar{E}_{spring\,\alpha}=\mu\pi\left(\frac{r_{\alpha}}{U_{R}}\right)^{2}\left(\frac{1}{2}\alpha^{2}+\frac{1}{4}k_{3\alpha}\alpha^{4}+\frac{1}{6}k_{5\alpha}\alpha^{6}\right)\label{elasticenergypitch}$$

(18)

The kinetic energies are calculated as:

$$ \bar{E}_{kinetic\,\xi}=\frac{1}{2}\mu\pi\xi'^{2}+2x_{\alpha}\alpha'\xi'\cos(\alpha)+(x_{\alpha}\alpha')^{2} $$

(19)

$$ \bar{E}_{kinetic\,\alpha}=\frac{1}{2}\mu\pi U_{R}^{2}\alpha'^{2}$$

(20)

Where the prime sign for the degrees of freedom represents the derivative with respect to the non-dimensional time.

Fig. 17

Energy for a stable (a) and an unstable configuration (b). The dashed line represents an exponential least square approximation whose coefficient is either positive (unstable) or negative (stable)