Accurate modeling of the aeroelastic response due to gust loads is becoming a design necessity for many novel concepts. Both the high aspect ratio wings of high altitude long endurance (HALE) aircraft and flexible membrane wings of micro aerial vehicles (MAVs) are susceptible to large deformations, geometric nonlinearities, and rigid body motion when gust loads are encountered. Using gradient based optimization for aeroelastic shape design of such configurations can be computationally expensive. Furthermore, multidisciplinary design optimization frameworks that use gradient-based algorithms are becoming increasingly popular. A major challenge in developing such frameworks is implementing accurate and efficient sensitivity analyses for the various disciplines. This becomes especially challenging when a "black box" analysis tool is used and access to the source code is not possible. Presented in this paper is a continuum shape sensitivity method which does not require access to or knowledge of source code formulation. A static nonlinear beam bending problem and a static linear plate bending problem are presented as benchmarks. In addition, continuum shape sensitivities are computed for an aeroelastic gust response of both a one-and two-dimensional structure. In all cases, the analysis tool is treated as a "black box".
Nomenclature
ߙ= Incident angle of attack ߙ = Gust angle ߙ = Effective angle of attack ܣ = General time-space differential operator ܣ ̅ = General time-space differential operator of local sensitivity system ܣ ̅ ்௧ = General time-space differential operator of total sensitivity system ܾ = Design variable (shape parameter) ܤ = General boundary operator ܤ ത = General boundary operator of local sensitivity system ܥ ഀ = Lift curve slope ߜ = Perturbation ∇ = Gradient operator ∆ = Essential boundary displacement ݂ = Generalized loads ܨ = Applied load ܨ = Quasi-steady lift force ߁ = Spatial boundary *The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Research Laboratory or the U.S. Government. 2 ݃ = Loads applied at the boundary ℎ = Plunge degree of freedom ܯ = Internal bending moment ܰ = Internal axial force Ω = Spatial domain ߮ = Rotation ݍ ஶ = Dynamic pressure ܳ = Generalized element force ߩ ஶ = Air density ݏ = Effective lifting area ݐ = Time variable ߠ = Rotation degree of freedom ݑ = General response variable and horizontal displacement ܷ = Gust velocity ܷ ஶ = Free stream velocity Ѵ = Design velocity ܸ = Internal shear force ݓ = Transverse displacement ݔ = Point in spatial domain