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An Improved Continuous Sensitivity Equation Method for Optimal Shape Design in Mixed Convection

This article presents an optimal shape design methodology for mixed-convection problems. The Navier-Stokes equations and the continuous sensitivity equations (CSEs) are solved using an adaptive finite-element method to obtain flow and sensitivity fields. A new procedure is presented to extract accurate values of the flow derivatives at the boundary, appearing in the CSE boundary conditions for shape parameters. Flow and sensitivity information are then employed to calculate the value and gradient of a design objective function. A Broyden-Fletcher-Goldfarb-Shanno (BFGS) optimization algorithm is used to find optimal shape parameter values. The proposed approach is first verified on a problem with a closed-form solution, obtained by the method of manufactured solutions. The method is then applied to determine the optimal shape of a model cooling system.