Generalized flutter reliability analysis with adjoint and direct approaches for aeroelastic eigen-pair derivatives computation

Abstract

The article presents physics based time invariant generalized flutter reliability approach for a wing in detail. For carrying flutter reliability analysis, a generalized first order reliability method (FORM) and a generalized second order reliability method (SORM) algorithms are developed. The FORM algorithm requires first derivative and the SORM algorithm requires both the first and second derivatives of a limit state function; and for these derivatives, an adjoint and a direct approaches for computing eigen-pair derivatives are proposed by ensuring uniqueness in eigenvector and its derivative. The stability parameter, damping ratio (real part of an eigenvalue), is considered as implicit type limit state function. To show occurrence of the flutter phenomenon, the limit state function is defined in conditional sense by imposing a condition on flow velocity. The aerodynamic parameter: slope of the lift coefficient curve ($C_{L}$) and structural parameters: bending rigidity (EI) and torsional rigidity (GJ) of an aeroelastic system are considered as independent Gaussian random variables, and also the structural parameters are modeled as second-order constant mean stationary Gaussian random fields having exponential type covariance structures. To represent the random fields in finite dimensions, the fields are discretized using Karhunen–Loeve expansion. The analysis shows that the derivatives of an eigenvalue obtained from both the adjoint and direct approaches are the same. So the cumulative distribution functions (CDFs) of flutter velocity will be the same, irrespective of the approach chosen, and it is also reflected in CDFs obtained using various reliability methods based on adjoint and direct approaches: first order second moment method, generalized FORM, and generalized SORM.

Appendices

Appendix A: Detailed expression of symbolic notation appeared in both adjoint and direct approaches

For the random variable/random field case, the right hand side of Eqs. (40) and (54) can be written as:

$$\begin{aligned}{} & {} Re({\textbf{y}}_{2})+iIm({\textbf{y}}_{2})\nonumber \\{} & {} \quad = - \left( \lambda _j^{2} \frac{\partial {\textbf{A}}}{\partial {r_{\beta }}} + \lambda _j \frac{\partial {\textbf{B}}}{\partial {r_{\beta }}} + \frac{\partial {{\textbf{C}}}}{\partial {r_{\beta }}}\right) \frac{\partial {\bar{{\textbf{q}}}_j}}{\partial r_{\alpha }} \nonumber \\{} & {} \qquad -\frac{\partial {\lambda _j}}{\partial r_{\beta }}(2\lambda _j{\textbf{A}}+{\textbf{B}}) \frac{\partial {\bar{{\textbf{q}}}_j}}{\partial r_{\alpha }} - \frac{\partial Im(\lambda _j)}{\partial r_{\beta }}\bigg (\lambda _{j}\frac{b}{U}\frac{\partial {{\textbf{B}}}}{\partial C(k_{j})}\frac{\partial C(k_j)}{\partial k_j} \nonumber \\{} & {} \qquad +\frac{b}{U}\frac{\partial {{\textbf{C}}}}{\partial C(k_{j})}\frac{\partial C(k_j)}{\partial k_j} \bigg ) \frac{\partial {\bar{{\textbf{q}}}_j}}{\partial r_{\alpha }}-\frac{\partial \lambda _{j}}{\partial {r_{\alpha }}}\bigg (2\lambda _{j}\frac{\partial {{\textbf{A}}}}{\partial {r_{\beta }}} + \frac{\partial {{\textbf{B}}}}{\partial {r_{\beta }}} + 2\frac{\partial \lambda _{j}}{\partial r_{\beta }} {\textbf{A}} \nonumber \\{} & {} \qquad +\frac{Im(\lambda _{j})}{\partial {r_{\beta }}}\frac{b}{U}\frac{\partial {\textbf{B}}}{\partial {C(k_j)}}\frac{\partial {C(k_j)}}{\partial k_j}\bigg )\bar{{\textbf{q}}}_j - \frac{\partial {\lambda _{j}}}{\partial {r_{\alpha }}}\left( 2\lambda _{j}{\textbf{A}}+{\textbf{B}}\right) \frac{\partial {\bar{{\textbf{q}}}_j}}{\partial r_{\beta }}\nonumber \\{} & {} \qquad -\frac{\partial {Im(\lambda _{j})}}{\partial {r_{\alpha }}}\bigg (\frac{b}{U}\frac{\partial {\lambda _{j}}}{\partial {r_{\beta }}}\frac{\partial {{\textbf{B}}}}{\partial {C(k_j)}}\frac{\partial {C(k_{j})}}{\partial {k_{j}}} + \lambda _{j}\frac{b^2}{U^2}\frac{\partial {Im(\lambda _{j})}}{\partial {r_{\beta }}}\nonumber \\{} & {} \qquad \bigg (\left( \frac{\partial {C(k_j)}}{\partial {k_{j}}}\right) ^2 \frac{\partial ^2{{\textbf{B}}}}{\partial {C(k_{j})^2}} +\frac{\partial ^2C(k_j)}{\partial k_j^2}\frac{\partial {{\textbf{B}}}}{\partial {C(k_j)}} \bigg ) \nonumber \\{} & {} \qquad + \frac{b}{U}\lambda _{j}\frac{\partial ^2{{\textbf{B}}}}{\partial {r_{\beta }}\partial {C(k_j)}}\frac{\partial C(k_j)}{\partial {k_j}}\bigg )\bar{{\textbf{q}}}_j -\frac{\partial {Im(\lambda _{j})}}{\partial {r_{\alpha }}}\bigg (\frac{b^2}{U^2}\frac{\partial {Im(\lambda _{j})}}{\partial {r_{\beta }}} \nonumber \\{} & {} \qquad \bigg (\left( \frac{\partial {C(k_j)}}{\partial k_j}\right) ^2 \frac{\partial ^2{{\textbf{C}}}}{\partial C(k_j)^2} + \frac{\partial {{\textbf{C}}}}{\partial {C(k_j)}}\frac{\partial ^2C(k_{j})}{\partial {k_j^2}}\bigg )\nonumber \\{} & {} \qquad +\frac{b}{U}\frac{\partial {C(k_{j})} }{\partial {k_j}}\frac{\partial ^{2 }{\textbf{C}}}{\partial {r_{\beta }}\partial {C(k_{j})}}\bigg )\bar{{\textbf{q}}}_j - \frac{\partial {Im(\lambda _{j})}}{\partial {r_{\alpha }}}\bigg (\frac{b}{U}\lambda _{j}\frac{\partial {{\textbf{B}}}}{\partial {C(k_j)}}\nonumber \\{} & {} \qquad \frac{\partial {C(k_{j})}}{\partial k_{j}} +\frac{b}{U}\frac{\partial {{\textbf{C}}}}{\partial {C(k_{j})}}\frac{\partial {C(k_{j})}}{\partial {k_{j}}} \bigg )\frac{\partial {\bar{{\textbf{q}}}_j}}{\partial r_{\beta }} - \bigg (\lambda _{j}^{2}\frac{\partial ^{2}{\textbf{A}}}{\partial r_{\beta }\partial r_{\alpha }} + \lambda _{j}\frac{\partial ^{2}{\textbf{B}}}{\partial r_{\beta }\partial r_{\alpha }} \nonumber \\{} & {} \qquad + \frac{\partial ^{2}{\textbf{C}}}{\partial r_{\beta }\partial r_{\alpha }} + 2\lambda _{j}\frac{\partial \lambda _{j}}{\partial r_{\beta }}\frac{\partial {{\textbf{A}}}}{\partial {r_{\alpha }}}+\frac{\partial \lambda _{j}}{\partial r_{\beta }}\frac{\partial {{\textbf{B}}}}{\partial {r_{\alpha }}}+\frac{b}{U}\frac{\partial {Im(\lambda _{j})}}{\partial {r_{\beta }}}\frac{C(k_{j})}{\partial {k_j}}\lambda _{j}\nonumber \\{} & {} \qquad \frac{\partial ^2{\textbf{B}}}{\partial C(k_j)\partial {r_{\alpha }}} +\frac{b}{U}\frac{\partial {Im(\lambda _{j})}}{\partial {r_{\beta }}}\frac{\partial C(k_{j})}{\partial {k_j}}\frac{\partial ^{2}{\textbf{C}}}{\partial C(k_j) \partial {r_{\alpha }}}\bigg )\bar{{\textbf{q}}}_j \nonumber \\{} & {} \qquad - \bigg (\lambda _{j}^{2}\frac{\partial {{\textbf{A}}}}{\partial {r_{\alpha }}}+\lambda _{j} \frac{\partial {{\textbf{B}}}}{\partial {r_{\alpha }}}+\frac{\partial {{\textbf{C}}}}{\partial {r_{\alpha }}}\bigg ) \frac{\partial {\bar{{\textbf{q}}}_j}}{\partial r_{\beta }} \end{aligned}$$

(A.1)

Appendix B: Derivatives of Theodorsen’s function

First derivatives of Theodorsen’s function with respect to k is written as:

$$\begin{aligned} \frac{dC(k)}{dk} =\frac{i\left( \frac{dH_{1}^{2}(k)}{dk}H_{o}^{2}(k)-H_{1}^{2}(k)\frac{dH_{o}^{2}(k)}{dk}\right) }{\left( H_{1}^{2}(k)+iH_{o}^{2}(k)\right) ^2} \end{aligned}$$

(B.1)

And the second derivatives of Theodorsen’s function with respect to k can be written as:

$$\begin{aligned} \frac{d^2C(k)}{dk^2}= & {} \frac{i\bigg (\frac{d^2H_{1}^{2}(k)}{dk^2}H_{o}^{2}(k)-H_{1}^{2}(k)\frac{d^2H_{o}^{2}(k)}{dk^2}\bigg )}{(H_{1}^{2}(k)+iH_{o}^{2}(k))^2}\nonumber \\{} & {} - \dfrac{ 2i\bigg (\left( \frac{dH_{1}^{2}(k)}{dk}\right) ^{2}H_{o}^{2}(k) - H_{1}^{2}(k)\frac{dH_{o}^{2}(k)}{dk}\frac{dH_{1}^{2}(k)}{dk} \bigg )}{\left( H_{1}^{2}(k)+iH_{o}^{2}(k)\right) ^3} \nonumber \\{} & {} +\dfrac{2\bigg ( { \frac{dH_{1}^{2}(k)}{dk}\frac{dH_{o}^{2}(k)}{dk}H_{o}^{2}(k)-H_{1}^{2}(k)\left( \frac{dH_{o}^{2}(k)}{dk}\right) ^2\bigg )}}{\left( H_{1}^{2}(k)+iH_{o}^{2}(k)\right) ^3} \end{aligned}$$

(B.2)

The derivative terms of Hankel’s function given in Eqs. (B.1) and (B.2) are obtained from Bessel’s function properties as:

$$\begin{aligned} \frac{dH_{o}^{2}(k)}{dk}= & {} \frac{dJ_{o}(k)}{dk}-i\frac{dY_{o}(k)}{dk}=-J_{1}(k) + i Y_{1}(k) \end{aligned}$$

(B.3)

$$\begin{aligned} \frac{d^2H_{o}(k)}{dk^2}= & {} -\frac{\left( J_{o}(k)-J_{2}(k)\right) }{2}+i\frac{\left( Y_{o}(k)-Y_{2}(k)\right) }{2} \end{aligned}$$

(B.4)

$$\begin{aligned} \frac{dH_{1}^{2}(k)}{dk}= & {} \frac{J_{o}(k)-J_{2}(k)}{2} - \frac{i\left( Y_{o}(k)-Y_{2}(k)\right) }{2} \end{aligned}$$

(B.5)

$$\begin{aligned} \frac{d^2H_{1}^{2}(k)}{dk^2}= & {} \frac{1}{4}\left( -3J_{1}(k)+J_{3}(k)\right) -\frac{i}{4}\left( -3Y_{1}(k)+Y_{3}(k)\right) \end{aligned}$$

(B.6)

Appendix C: Derivatives of eigenvalues: adjoint and direct approaches

1.1 C.1. Random variables

Tables 10 and 11 show the first derivative of eigenvalues with respect to structural parameters EI and GJ using adjoint and direct approaches and central difference scheme. Table 12 shows the first and second derivatives of eigenvalues, obtained from both the adjoint and direct approaches, with respect to aerodynamic parameter $C_{L}$.

1.2 C.2. Random fields

Tables 13 and 14 show the derivatives of eigenvalues with respect to discretized field random variables $\xi _{n}$ $(n = 1, 2, 3, 4)$ obtained using adjoint and direct approaches, when the structural parameters EI and GJ are modeled as random fields.

Table 10 Derivative of eigenvalues with respect to EI at various flow velocities

Table 11 Derivative of eigenvalues with respect to GJ at various flow velocities

Table 12 First and second derivatives of eigenvalues with respect to $C_{L}$ at various flow velocities

Table 13 Derivative of eigenvalues with respect to $\xi _{n}$ at various flow velocities (EI modeled as random field)

Table 14 Derivative of eigenvalues with respect to $\xi _{n}$ at various flow velocities (GJ modeled as random field)