Abstract
We use spherical cap harmonic (SCH) basis functions to analyse and reconstruct the morphology of scanned genus-0 rough surface patches with open edges. We first develop a novel one-to-one conformal mapping algorithm with minimal area distortion for parameterising a surface onto a polar spherical cap with a prescribed half angle. We then show that as a generalisation of the hemispherical harmonic analysis, the SCH analysis provides the most added value for small half angles, i.e., for nominally flat surfaces where the distortion introduced by the parameterisation algorithm is smaller when the surface is projected onto a spherical cap with a small half angle than onto a hemisphere. From the power spectral analysis of the expanded SCH coefficients, we estimate a direction-independent Hurst exponent. We also estimate the wavelengths associated with the orders of the SCH basis functions from the dimensions of the first degree ellipsoidal cap. By windowing the spectral domain, we limit the bandwidth of wavelengths included in a reconstructed surface geometry. This bandlimiting can be used for modifying surfaces, such as for generating finite or discrete element meshes for contact problems. The codes and data developed in this paper are made available under the GNU LGPLv2.1 license.
Highlights
The mechanics of granular materials, such as soil mechanics, investigates the interaction of particles at various loading conditions where the shape, size and material properties of the particles can vary widely
Two further questions were investigated: (i) How can spherical harmonics analysis (SHA) shape descriptors be linked to the well-known, traditional shape descriptors? (ii) How can the SHA shape descriptors be linked to mechanical parameters, such as friction coefficients, and traditional morphology measures, such as fractal dimensions (FD)? Here, we review studies in mechanics that used SHA for generating particles as well as for simulating particle mechanics
Similar to the Power Spectral Density (PSD) analysis, here we found that the accumulated power at each degree correlates with the Hurst exponent H as follows: By investigating several benchmarks, we observed that the algorithm chooses the θc that best represents the global concavity of the open surface
Summary
The mechanics of granular materials, such as soil mechanics, investigates the interaction of particles at various loading conditions where the shape, size and material properties of the particles can vary widely. Similar problems exist in structural engineering when modelling, for example, the microstructure of concrete aggregates or the various stones in stone masonry elements, as the shape and roughness of the particles affect the matrices of these materials Addressing these problems numerically requires the modelling of randomly shaped closed objects and their interactions. We propose the use of spherical cap harmonic analysis (SCHA) for describing the geometry of open shapes. The proposed SCHA method allows us to study the structure of any arbitrary surface patch with faster convergence than the traditional SHA as it scales to any level of detail with reasonable computational complexity It can be used for reconstructing surfaces and modifying the morphology of digital twins of real surfaces, which is useful when numerically studying contact problems such as friction. We summarise the main results of the paper and discuss possible future works in Sections 10 and 11, respectively
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