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Multi-objective robust design optimization (MORDO) of an aeroelastic high-aspect-ratio wing

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Abstract

In this paper, a new approach for multi-objective robust optimization of flutter velocity and maximum displacement of the wing tip are investigated. The wing is under the influence of bending–torsion coupling and its design variables have different levels of uncertainty. In designing and optimizing wings with a high aspect ratio, the optimization process can be done in such a way to increase the flutter velocity, but this can increase the amplitude of the wing tip displacement to a point that leads to the wings damage and structural failure. Therefore, single-objective design optimization may lead to infeasible designs. Thus, for multi-objective optimization, modeling is based on the Euler–Bernoulli cantilever beam model in quasi-steady aerodynamic condition. Using the Galerkin’s techniques, the aeroelastic equations are converted to ODE equations. After validating the results, the system time response is obtained by the numerical solution of the governing equations using 4th Runge–Kutta method and the flutter velocity of the wing is obtained using the theory of eigenvalues. Subsequently, by choosing bending and torsional rigidity and mass per unit wing length as the optimization variables, using Monte Carlo–Latin hypercube (MC-LH) simulation and 4th polynomial chaos expansion (PCE), the effect of uncertainty on these variables is modeled in modeFRONTIER™ software coupled with MATLAB™ and optimization is performed by genetic algorithm. Finally, by plotting the Pareto front, it is observed that with an acceptable increase in flutter velocity, the maximum wing displacement amplitude is reduced as much as possible. The results of the multi-objective robust optimization show more feasible results compared with deterministic optimization.

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Abbreviations

a :

Non-dimensional length between elastic axis and center of mass

α eff :

Effective angle of attack (deg)

b :

Half chord length (m)

A Bend :

Displacement wing tip due to bending (mm)

A Couple :

Displacement wing tip due to bending and torsion coupling

F Bend(t):

Bending-time response

C Lα :

Lift coefficients of quasi-steady aerodynamics

C Lin :

Non-dimensional linear damping matrix

C mα :

Moment coefficients of quasi-steady aerodynamics

D x :

In-plane torsional stiffness (N m2)

D y :

In-plane bending stiffness (N m2)

EI:

Bending rigidity of wing (N m2)

e :

Cg offset from elastic axis (m)

GJ:

Torsional rigidity of wing (N m2)

Iy:

Mass moment of inertia about elastic axis (kg m2)

K Lin :

Linear stiffness matrix

K N :

Geometric nonlinear matrix

L :

Span length (m)

L QS :

Aerodynamic lift force (kg deg s−2)

m :

Total mass (kg)

M Lin :

Linear mass matrix

(m/L)nom :

Total mass per unit span length (kg m−1)

M QS :

Aerodynamic moment (kg m deg s−2)

Obj :

Objective function

U MEAN F :

Average flutter velocity (m s−1)

S :

Nominal value of optimization variables (N m2, kg m−1)

sL:

Lower bound

su:

Upper bound

A Tor :

Displacement wing end due to torsion (deg)

F Tor(t):

Torsion-time response

V :

Fluid flow velocity (m s−1)

w :

Bending displacement (mm)

X :

Generated random numbers for optimization variables

\(\phi\) :

Torsional displacement (deg)

ρ :

Air density (kg m−3)

μ :

Mass ratio

β Z :

Stiffness ratio

\(\sigma_{{U_{\rm F} }}\) :

Standard deviation of flutter velocity

\(\sigma_{{{\rm MA}_{\rm Cou} }}\) :

Standard deviation of maximum displacement of the wing tip due to the coupled effect of bending and torsion

\(\sigma_{{{\rm MA}_{\rm Ben} }}\) :

Standard deviation of maximum displacement of the wing tip due to the effect of bending

\(\sigma_{{{\rm MA}_{\rm Tor} }}\) :

Standard deviation of maximum displacement of the wing tip due to the effect of torsion

\(\delta m\) :

Variation of mass per unit span length (kg m−1)

\(\delta D_{y}\) :

Variation of in-plane bending stiffness (N m2)

\(\delta D_{x}\) :

Variation of in-plane torsional stiffness (N m2)

\(\tilde{m}\) :

Total mass per unit span length with uncertainty term (kg m−1)

\(\tilde{D}_{y}\) :

In-plane bending stiffness with uncertainty term (N m2)

\(\tilde{D}_{x}\) :

In-plane torsional stiffness with uncertainty term (N m2)

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Authors and Affiliations

  1. Department of Mechanical and Energy Engineering, Shahid Beheshti University, Tehran, Iran

    Meysam Elyasi

  2. Department of Aeronautical Engineering, Sharif University of Technology, 13878-93967, Tehran, Iran

    Alireza Roudbari & Pedram Hajipourzadeh

Authors
  1. Meysam Elyasi
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  2. Alireza Roudbari
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  3. Pedram Hajipourzadeh
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Correspondence to Alireza Roudbari.

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Technical Editor: José Roberto de França Arruda.

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Elyasi, M., Roudbari, A. & Hajipourzadeh, P. Multi-objective robust design optimization (MORDO) of an aeroelastic high-aspect-ratio wing. J Braz. Soc. Mech. Sci. Eng. 42, 560 (2020). https://doi.org/10.1007/s40430-020-02633-7

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  • DOI: https://doi.org/10.1007/s40430-020-02633-7

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