Boundary velocity method for continuum shape sensitivity of nonlinear fluid–structure interaction problems

A Continuum Sensitivity Equation (CSE) method was developed in local derivative form for fluid–structure shape design problems. The boundary velocity method was used to derive the continuum sensitivity equations and sensitivity boundary conditions in local derivative form for a built-up joined beam structure under transient aerodynamic loads. For nonlinear problems, when the Newton–Raphson method is used, the tangent stiffness matrix yields the desired sensitivity coefficient matrix for solving the linear sensitivity equations in the Galerkin finite element formulation. For built-up structures with strain discontinuity, the local sensitivity variables are not continuous at the joints, requiring special treatment to assemble the elemental local sensitivities and the generalized force vector. The coupled fluid–structure physics and continuum sensitivity equations for gust response of a nonlinear joined beam with an airfoil model were posed and solved. The results were compared to the results obtained by finite difference (FD) method.