The least-squares finite element method is used to solve the compressible Euler equations around airfoils in transonic regime. The symbolic analysis method is used to generate the element stiffness and force matrices. The equations of the element matrices are derived symbolically based on the flow primitive variables and the position of the element nodes. The symbolic analysis is also used to compute the exact derivatives of the residuals with respect to both design variables (e.g. the airfoil geometry) and the state variables (e.g. the flow velocity). The symbolic analysis allows to compute the exact Jacobian of the governing equations in a computationally efficient way, which is used for Newton iteration. Besides, using the symbolic analysis the sensitivities of the outputs, such as the airfoil drag, with respect to the design variables, such as the airfoil geometry, are computed using the discrete adjoint method without the need for automatic differentiation. This makes the analysis and optimization computationally more efficient.
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