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The reduction of acoustic excitation due to piston slap as well as friction loss power and seizure are main issues when simulating the oil film lubricated piston - cylinder contacts of internal combustion engines. For a correct representation of the contact conditions between a piston skirt and a cylinder liner surface both the dynamics of the contacting flexible bodies, the shape of the contacting surfaces, the amount of available oil and the properties of the lubricant itself play important roles. Besides an appropriate representation of the hydrodynamic load carrying capacity using an averaged Reynolds equation with laminar flow conditions, the simulation has to use an appropriate asperity model to consider the mixed lubrication condition. The lubricant properties are in particular influenced by its thermal conditions.

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.