Characterizing digital microstructures by the Minkowski-based quadratic normal tensor

TL;DR

This paper introduces a robust computational method based on Minkowski tensors, especially the quadratic normal tensor, for characterizing digital microstructures, aiding material modeling and microstructure synthesis.

Contribution

It presents a modular algorithm for computing the quadratic normal tensor from voxel-based microstructures, demonstrating convergence and robustness for large-scale applications.

Findings

Algorithm remains unaffected by interface area inaccuracies

Demonstrates multigrid convergence on numerical examples

Applicable to diverse microstructure types

Abstract

For material modeling of microstructured media, an accurate characterization of the underlying microstructure is indispensable. Mathematically speaking, the overall goal of microstructure characterization is to find simple functionals which describe the geometric shape as well as the composition of the microstructures under consideration, and enable distinguishing microstructures with distinct effective material behavior. For this purpose, we propose using Minkowski tensors, in general, and the quadratic normal tensor, in particular, and introduce a computational algorithm applicable to voxel-based microstructure representations. Rooted in the mathematical field of integral geometry, Minkowski tensors associate a tensor to rather general geometric shapes, which make them suitable for a wide range of microstructured material classes. Furthermore, they satisfy additivity and continuity properties, which makes them suitable and robust for large-scale applications. We present a modular algorithm for computing the quadratic normal tensor of digital microstructures. We demonstrate multigrid convergence for selected numerical examples and apply our approach to a variety of microstructures. Strikingly, the presented algorithm remains unaffected by inaccurate computation of the interface area. The quadratic normal tensor may be used for engineering purposes, such as mean-field homogenization or as target value for generating synthetic microstructures.