Efficient Optimal Surface Texture Design Using Linearization

Surface textures reduce friction in lubricated sliding contact. This behavior can be modeled using the Reynolds equation, a single partial differential equation (PDE) that relates the hydrodynamic pressure to the gap height. In a previous study, a free-form texture design optimization problem was solved based on this model and two competing design objectives. A pseudo-spectral method was used for PDE solution, which was treated as a black box in the optimization problem. This optimization implementation did not exploit model structure to improve numerical efficiency, so design representation fidelity was limited. Here a new strategy is introduced where design representation resolution and computational efficiency are improved simultaneously. This is achieved by introducing a new optimization variable involving both pressure gradient and the cube of gap height at each mesh node location, and simultaneously solving the flow and texture design problems. This transformation supports linearization of the governing equations and design objectives. Sequential Linear Programming (SLP) is used with the epsilon-constraint method to generate Pareto-optimal texture designs with high resolution and low computational expense. An adaptive trust region is used, based on solution improvement, to manage linearization error. Comparing to the non-linear programming implementation, the solution converged to a set of slightly suboptimal points (maximum 25% objective function degradation when normalized apparent viscosity is 0.5, and moderately better when normalized apparent viscosity is 0.2), but results in significant improvement in computational speed (8.4 times faster).