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A gradient-based approach for optimal actuator design with morphing wings

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Abstract

In this paper, a design optimization approach is developed to design the placement and stroke of actuators at a conceptual design phase. This method is applied to the actuator design problem of the morphing leading edge of a high-performance sailplane to achieve predetermined optimal shapes. The centerpiece of this work is the derived analytical sensitivity analysis via direct differentiation to efficiently and effectively find the optimal design. This methodology is illustrated by breaking the solving routine into five blocks: meshing module, actuation stroke module, actuated deformation module, geometric deviation module and geometric constraint module. Although applied here to a morphing wing profile, the methodology and derived equations are general and can be applied to a number of applications of form-variable structures in which the optimal placement of actuators is desired.

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Notes

  1. Note that sans serif \({\textsf{x}}\) stands for a design variable, while serif x is a position.

Abbreviations

\({\underline{\left( \cdot \right) }}\) :

Vector

\(\underline{\underline{\left( \cdot \right) }}\) :

Matrix

\(\underline{{\underline{B}}}_{\textrm{control}}\) :

Boolean matrix for control degrees of freedom

\(d_{\min }\) :

Minimum allowable seperation of neighboring actuators

\({\underline{e}}\) :

Unit vector

\({\textsf{f}}\) :

Optimization objective function

\({\textsf{g}}\) :

Optimization constraint function

\(\underline{{\underline{H}}}\) :

Actuator sensitivity matrix

\(\ell _{\text {el}}\) :

Element length

L :

Arc length

\(L_{\textrm{act}i}\) :

Arc length position of ith actuator

\(n_{\textrm{act}}\) :

Number of actuators

\({n}_N\) :

Number of nodes

u :

Displacement

\({u}_0\) :

Prescribed displacement (Neumann boundary condition)

\({u}_{\textrm{act}}\) :

Calculated actuator stroke, displacement at actuator

\({u}_{\textrm{control}}\) :

Control nodal displacement

\({u}_{\textrm{target}}\) :

Target displacement

\({\textsf{x}}\) :

Optimization design variable

x :

x-position

X :

Optimization design domain

y :

y-position

\(\varDelta \) :

Finite perturbation

\(\delta \) :

Geometric deviation

\(\epsilon _{\textrm{MS}}\) :

Mean square geometric deviation

\(\theta \) :

Rotation about z-axis (out of plane)

\(\nabla \) :

Nabla operator denoting total derivatives w.r.t. \({\textsf{x}}\), i.e. \(\frac{\textrm{d}\left( \cdot \right) }{\textrm{d}{\textsf{x}}}\)

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Acknowledgements

This work is supported by the project CRC 2017 TN2091, doloMULTI Design of lightweight optimized structures and systems under multidisciplinary considerations through integration of multibody dynamics in a multiphysics framework, funded by the Free University of Bozen-Bolzano. Further support was provided in the framework MILAN, Morphing w ings for sai lpl anes, funded by the German Federal Ministry for Economic Affairs and Climate Action under the grant of the German Federal Aviation Research Program (Luftfahrtforschungsprogramm, LuFo) V-3.

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Authors and Affiliations

  1. Collins Aerospace, Applied Research & Technology, Multidisciplinary Design Optimization Research Group, Munich, Germany

    Erich Wehrle

  2. Faculty of Engineering, Free University of Bozen-Bolzano, Universitätsplatz 1, 39100, Bozen, South Tyrol, Italy

    Erich Wehrle & Veit Gufler

  3. Chair of Aircraft Design, Technical University of Munich, Boltzmannstr. 15, 85748, Garching, Germany

    Fabian Sturm

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  1. Erich Wehrle
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Correspondence to Erich Wehrle.

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Wehrle, E., Gufler, V. & Sturm, F. A gradient-based approach for optimal actuator design with morphing wings. Optim Eng (2023). https://doi.org/10.1007/s11081-023-09835-7

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