Improvements to the linear transform technique for generating randomly rough surfaces with symmetrical autocorrelation functions

Simulating surfaces with defined surface roughness parameters is a common task in tribology. This task is likely to become more relevant as modelling rough surface contact becomes less costly as it enables parametric studies linking behaviour to roughness parameters independent of a particular surface profile. The linear transform method allows the specification of the autocorrelation function (ACF) and gives some control over the height function, however these methods only fit the linear transformation matrix to half of the ACF. Behaviour outside of this half is unspecified and can lead to large errors.

In this work we show that this problem can be overcome by using a symmetrical linear transformation matrix. This ensures that the resulting ACF is symmetrical. The method given by Liao et al. (2018) is extended to include this constraint, including the analytical gradient formula. Additionally, an improvement allowing for the generation of periodic surfaces is given.

The use of a symmetric filter reduced errors in the unfitted region of the ACF, to the same levels as within the fitted region, in one example, this was a reduction from 50% to 3% error. The surface realisations produced by this technique show fewer unphysical effects than those produced by nonsymmetric filters. Particularly, high frequency noise in line with the coordinate axes is removed.