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Electric vehicles offer cleaner transportation with lower emissions, thus their increased popularity. Although, electric powertrains contribute to quieter vehicles, the shift from internal combustion engines to electric powertrains presents new Noise, Vibration, and Harshness challenges. Unlike traditional engines, electric powertrains produce distinctive tonal noise, notably from motor whistles and gear whine. These tonal components have frequency content, sometimes above 10 kHz. Furthermore, the housing of the powertrain is the interface between the excitation from the driveline via the bearings and the radiated noise (NVH). Acoustic features of the radiated noise can be predicted by utilising the transmitted forces from the bearings. Due to tonal components at higher frequencies and dense modal content, full flexible multibody dynamics simulations are computationally expensive.

The aim of this paper is to present the adjoint equations for shape optimization derived from steady incompressible Navier-Stokes (N-S) equations and an objective functional. These adjoint Navier-Stokes equations have a similar form as the N-S equations, while the source terms and the boundary conditions depend on the chosen objective. Additionally, the gradient of the targeted objective with respect to the design variables is calculated. Based on this, a modification of the geometry is computed to arrive at an improved objective value. In order to find out, whether a more sophisticated approach is needed, the adjoint equations are derived by using two different approaches. The first approach is based on the frozen turbulence assumption and the second approach, which is advanced in this paper, is derived from the near wall k − ζ − f turbulence model.