Sensitivity algorithms for an inverse design problem involving a shock wave

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Abstract
A model inverse design problem is used to investigate the effect of flow discontinuities on the optimization process. The optimization involves finding the cross-sectional area distribution of a duct that produces velocities which closely match a targeted velocity distribution. Quasi-one-dimensional flow theory is used, and the target is chosen to have a shock wave in its distribution. The objective function which quantifies the difference between the targeted and calculated velocity distributions may become non-smooth due to the presence of the shock in the discretized flow field. This paper offers two techniques to resolve the resulting problems for the optimization algorithms. The first, shock fitting, involves careful integration of the objective function through the shock wave. The second, coordinate straining with shock penalty, uses a coordinate transformation to align the calculated shock with the target and then adds a penalty proportional to the square of the distance between the shocks. These techniques are evaluated and tested using several sensitivity methods, including finite-differences, direct and adjoint methods.

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