![]() ![]() A different fitting method, again relying on optimization, is described by Šedivý et al. (2015), in which a fitting procedure based on a (very high-dimensional) linear optimization is also proposed. ![]() A quite general tessellation model, the so-called generalized balanced power diagram (GBPD), is introduced in Alpers et al. Heuristics for fitting some of these tessellation models are described in Altendorf et al. In this case, other tessellation models-often generalizations of the Voronoi/Laguerre tessellations-have been proposed we refer to Altendorf et al. This is unacceptable when it comes to the investigation of curvature-related phenomena like grain growth. A major drawback of the Laguerre tessellation, however, is the fact that its facets are planar and therefore apply only to grains having nearly flat boundaries. Additional details regarding the method proposed in Spettl et al. Of particular interest is the description of 3D image data, e.g., from 3D electron backscatter diffraction (EBSD) or 3D x-ray diffraction (3DXRD) microscopy, which was studied in Liebscher (2015) Quey and Renversade (2018) Spettl et al. ![]() (2019) Quey and Renversade (2018) the problem of finding good representations for statistical data, such as grain volumes and centroids, is discussed. It is therefore not surprising that the fitting of Laguerre tessellations to experimental data has already received much attention. A prominent tessellation type in materials science is the Laguerre tessellation ( Lautensack and Zuyev, 2008), which is a generalization of the well-known Voronoi tessellation ( Møller, 1994 Okabe et al., 2000). For the latter, realistic “virtual polycrystals” generated by parametric stochastic models for these tessellations are particularly helpful (see, e.g., Allen et al., 2021). For example, the representation of a material’s microstructure by means of tessellations can be utilized for the analysis of microstructure-property relationships ( Raabe, 1998 Westhoff et al., 2018). For this purpose, tessellations have proven to be a powerful tool, as they provide a partitioning of space into disjoint subsets called cells. In many such cases, the investigation and modeling of grain boundaries presupposes that their locations can be represented precisely. The grain boundaries of polycrystalline materials play an important role in many different phenomena, ranging from fundamental processes like grain growth and extending to applied scenarios like the degradation of electrodes in lithium-ion batteries. Furthermore, we investigate the effect of noisy image data and whether the smoothing of image data prior to the fitting step is advantageous. ![]() We demonstrate this on a three-dimensional x-ray diffraction (3DXRD) mapping of an AlCu alloy, but we also evaluate the modeling errors for simulated data. The resulting reduction in runtime makes it feasible to find approximations to real experimental datasets. We therefore propose a modification of the traditional definition of GBDPs that allows gradient-based optimization methods to be employed. With as many as ten parameters for each cell, it is computationally demanding to fit GBPDs to three-dimensional image data containing hundreds of grains. For this reason, we consider generalizations of Laguerre tessellations-variations of so-called generalized balanced power diagrams (GBPDs)-that exhibit non-convex cells. However, most traditional tessellation models that are used for modeling the microstructure morphology of these materials, e.g., Voronoi or Laguerre tessellations, have flat faces and thus fail to incorporate the curvature of the latter. The curvature of grain boundaries in polycrystalline materials is an important characteristic, since it plays a key role in phenomena like grain growth. 2Institute of Functional Nanosystems, Faculty of Engineering, Computer Science and Psychology, Ulm University, Ulm, Germany.1Institute of Stochastics, Faculty of Mathematics and Economics, Ulm University, Ulm, Germany.Lukas Petrich 1*, Orkun Furat 1, Mingyan Wang 2, Carl E. ![]()
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