Proposal of texture shape optimization algorithm under constant load condition and considerations on new shape update equation (Texture shape optimization for minimization of friction coefficient)

An algorithm to optimize texture shape under constant load conditions and a shape update equation of the design variables are proposed. The tribological properties are improved by machining grooves and holes, termed“ texture”, on frictional surfaces that are lubricated by fluid. Improvement of tribological properties, such as the friction coefficient, is likely to lead to a reduction in energy loss and extension of machine life, resulting in major economic benefits. Because tribological properties depend on the shape of the texture, the focus of this study was on the dimensional shape of the texture. Most countermeasures to this problem involve size optimization rather than shape optimization. Conventionally, when the effect of the texture shape on the friction coefficient is evaluated experimentally, the load is kept constant. However, when the texture shape is changed as part of the analysis, the pressure field changes. The load, which is the integrated value of the pressure, also changes. Therefore, it is difficult to evaluate the friction coefficient accurately. At this time, the load can be kept constant by adjusting the basic oil film thickness, which is the distance between the frictional surfaces. This occurs naturally in real-world situations. In general, when the adjoint variable method is applied to determine the texture shape, constraint conditions are included in the Lagrange function. But, in this study, the constant load condition, i.e., the constraint condition, was simply added to keep the initial load, because it is difficult to calculate the gradient of the constraint condition with respect to the design variable. Considering the above, the purpose of this study was to find an appropriate oil film thickness for a texture by shape optimization and to reduce the friction coefficient by adding an algorithm that keeps the load constant by varying the basic oil film thickness. In addition, the shape update equation for the design variable was improved, and results based on the present method were compared with those based on the steepest descent and the conjugate gradient methods. This was achieved by replacing the interpretation of the update equation using the steepest descent method with a differential equation and by applying the differential to the step length of the design variable in the Taylor expansion equation of the design variable. By improving the shape update equation, a lower performance function was obtained. Texture shape optimization was performed by the adjoint variable method using the Reynolds equation as the governing equation. The performance function is defined by the frictional force, and the friction coefficient is optimized at the same time by keeping the load constant. FreeFEM++ was used to calculate the optimal shape.