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Disk Harmonics for Analyzing Self-affine Rough Surfaces and Topological Reconstruction


Core Concepts
The author employs disk harmonics based on Fourier-Bessel basis functions to decompose self-affine rough surfaces, providing a practical alternative for characterizing surface morphology.
Abstract
Disk harmonics based on Fourier-Bessel basis functions are used to analyze self-affine rough surfaces. The method provides an efficient way to characterize surface morphology without introducing biases from boundary conditions or surface discretization. The approach is demonstrated to successfully measure fractal dimensions and Hurst exponents, paving the way for contact mechanics studies based on the Fourier-Bessel spectral representation of curved and rough surface morphologies. When two bodies come into contact, only a small portion of the apparent area is involved in producing contact and friction forces due to surface roughness. Accurately describing the morphology of rough surfaces is crucial, involving extracting fractal dimensions and Hurst exponents. Harmonic decomposition using disk harmonics simplifies this task for open single-edge genus-0 surfaces by employing Fourier-Bessel basis functions. Analytical relationships between power spectrum density decay and Hurst exponent are derived through an extension of the Wiener-Khinchin theorem. The method successfully measures fractal dimensions and Hurst exponents without introducing biases from boundary conditions or curvature of surface patches. Disk harmonics have been used in various engineering applications but not previously employed for studying surface morphology. The method offers stability for any input, solving issues encountered with other harmonic analyses methods like spherical harmonics. Surface morphology analysis using Fourier-Bessel basis functions provides insights into understanding frictional behavior exerted by non-closed surfaces in mechanical engineering applications. The method allows for accurate measurement of fractal dimensions and Hurst exponents without being influenced by boundary conditions or curvature of the surfaces.
Stats
For flat isotropic self-affine surfaces, the decay in amplitudes obtained with the Fourier-Bessel PSD is proportional to the Hurst coefficient. Generated self-affine fractal surfaces were analyzed with an average slope close to -3.1, corresponding to a Hurst exponent of 0.8. Circular patches sampled from square fractal surfaces led to good estimates of the Hurst coefficient with errors less than 0.20%. Neumann boundary conditions imposed at the edge of parameterization domain did not significantly influence results in estimating the Hurst exponent. Reconstruction error maps showed consistent errors across reconstructed surfaces, indicating no effect from Neumann boundary conditions. Projection onto spherical caps allowed analysis of curved rough surfaces while maintaining accuracy in measuring fractal dimensions and Hurst exponents.
Quotes
"The method opens paths for contact mechanics studies based on Fourier-Bessel spectral representation." "Disk harmonics simplify characterizing self-affine rough surfaces without bias from boundary conditions." "The relationship between PSD decay and Hurst exponent enables estimation directly from Fourier-Bessel PSD."

Deeper Inquiries

How can disk harmonics be applied beyond analyzing self-affine rough surfaces?

Disk harmonics can be applied beyond analyzing self-affine rough surfaces by providing a fast and practical alternative for characterizing surface morphology. They offer a stable basis for decomposing open single-edge genus-0 surfaces, making them suitable for studying various types of surfaces, not just self-affine ones. Additionally, the Fourier-Bessel basis functions used in disk harmonics allow for the analysis of curved and non-periodic surfaces, expanding their applicability to a wide range of surface shapes and properties. This versatility makes disk harmonics valuable in contact mechanics studies based on spectral representations of surface morphologies.

What counterarguments exist against using Fourier-Bessel basis functions for surface analysis?

One counterargument against using Fourier-Bessel basis functions for surface analysis is that they may introduce noise or artifacts in the results due to harmonic disagreement between different bases used during generation and analysis. The discretization used in parameterization domains with Fourier-Bessel bases may not always align perfectly with the original data distribution on irregularly shaped surfaces, leading to inaccuracies in the computed power spectral density (PSD). Additionally, the choice of boundary conditions imposed when defining these basis functions could impact the accuracy of results if they are not compatible with certain surface profiles or shapes.

How might understanding harmonic decomposition impact advancements in material science?

Understanding harmonic decomposition can have significant impacts on advancements in material science by providing insights into the structural properties and behavior of materials at different scales. By decomposing complex signals or data into simpler components represented by harmonic functions, researchers can analyze material characteristics such as roughness, fractal dimensions, curvature effects, and other morphological features more effectively. This detailed analysis enables better predictions of material interactions like frictional behavior between surfaces or mechanical responses under varying conditions. Ultimately, this knowledge contributes to developing improved materials with enhanced performance and durability across various applications within material science research and development.
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