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Various constrained problem formulations for the optimization of an electro-thermal wing anti-icing system in both running-wet and evaporative regimes are presented. The numerical simulation of the system is performed by solving the conjugate heat transfer problem between the fluid and solid domains. The optimization goal is to reduce the energy use and power demand of the anti-icing system while ensuring a safe protection. The formulations are carefully proposed from the physical and mathematical viewpoints; their performance is assessed by means of several numerical test cases to discern the most promising for each regime. The design optimization is conducted using the mesh adaptive direct search algorithm using quadratic and statistical surrogate models in the search step. The influence of the models on the convergence speed and the quality of the obtained design solutions is investigated.

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The numerical solution of an inverse airfoil design problem requires computational tools for grid generation, flow solution and the geometry update to provide the desirable shape corresponding to a given surface target pressure. Un-coupled as well as fully and semi-coupled shape design strategies have already been developed and employed. The target pressure is directly imposed in the fully coupled approach in contrast to un-coupled methods. Available semi-coupled methods use additional computational tools to couple the flow solver and the shape updater to directly impose the target pressure. In this article, a semicoupled numerical shape design strategy is introduced, in the context of an ideal flow model, in which the target tangential velocity is directly imposed while no additional computational tool is employed. Numerical test cases, implemented on both structured and unstructured grids, are carried out to compare the performance of the proposed method with an adjoint-based optimization technique. Results show that the semi-coupled method has a two to three times faster convergence rate as compared to the adjoint approach. However, the semicoupled solution convergence rate usually stalls after three orders of magnitude reduction of the objective function, while the un-coupled optimization method provides a more accurate solution. Therefore, a hybrid solution methodology is also advised, which provides faster convergence as well as lower computational errors as compared to both semi and un-coupled methods.