Sensitivity Analysis of Beam Cross-Section Stiffness Using Adjoint Method

The design optimization of rotorcraft through multidisciplinary aeroelastic models with hundreds of thousands of degrees of freedom requires a computationally efficient sensitivity analysis to obtain the objective function gradient. A fundamental part of rotorcraft analysis is the flexible multibody dynamics solver, which in the current work relies on an accurate three-dimensional representation of the beams. This paper presents the theoretical adjoint sensitivity analysis of the first structural analysis step, namely the computation of cross-sectional properties of the beams in the form of six-dimensional stiffness matrices. The adjoint equations are carefully derived, as are the derivatives of the objective function with respect to the design parameters. The method is then validated by comparing certain design sensitivities of a three-ply, composite cross-section with those obtained through real-step and complex-step numerical differentiation. The presented analysis allows the user to quantify the effect of basic structural parameters on fundamental sectional properties that can later be used in the full dynamic simulation.