Mathematical Modeling of Microstructure & Microtexture Evolution¶

Author: Dr. Harsh Kumar Narula

Degree: Ph.D. in Mechanical Engineering, Indian Institute of Technology Bombay (Jan 2024)

Advisors: Prof. Asim Tewari & Prof. Sushil Mishra

Executive Summary¶

Polycrystalline material properties are heavily dictated by two interconnected microstructural features: morphological texture (grain shape/size) and crystallographic texture (preferred lattice orientation). Because generating all possible microstructures experimentally under controlled conditions is practically impossible, this research bridges spatial tessellation models and first-principles physics through novel mathematical formulations and open-source simulation tools.

This PhD work introduces EVOSIM (a unified 2D/3D microstructure evolution engine), REVOSIM (a 3D microstructure reconstruction framework for DCT data), and groundbreaking theoretical extensions to Centroidal Voronoi Tessellations (CVT) and Generalized Balanced Power Diagrams (GBPD).

Key Research Highlights & Contributions¶

1. The EVOSIM Simulation Engine (2D & 3D)¶

Unified Voxel Domain Growth: Developed generic algorithms to simulate space-filling multi-phase microstructure evolution under arbitrary, user-defined spatio-temporal nucleation rates \(\dot{N}(\vec{r}, t)\) and growth rate functions \(G(\vec{r}, t, \Phi)\).
Triaxial & Superellipsoidal Geometry: Built parametric formulations for growing triaxial nuclei shapes including spheres, cuboids, cylinders, octahedrons, hexagonal cells, and superellipsoids.
Software Architecture: Engineered a high-performance cross-platform application utilizing C++, OpenMP parallel processing, Qt5 GUI, VTK visualization, ITK segmentation, PyBind11 Python wrappers, and custom .kin XML kinetics configuration.

2. Microstructure Reconstruction from Experimental Data (REVOSIM)¶

Inverse Problem Solver: Developed the REVOSIM recursive algorithm to reconstruct complete 3D microstructures using limited experimental descriptors (grain centroids, volumes, and orientations) obtained from Diffraction Contrast Tomography (DCT).
Anisotropic Reconstruction: Formulated a non-recursive reconstruction algorithm leveraging grain covariance matrices to estimate ellipsoid orientations and growth rates under site-saturated conditions.

3. Advances in Crystallographic Texture & Probability¶

xyz and \(\hat{n}xyz\) Sampling Algorithms: Proposed two intuitive, fast algorithms for sampling uniform random 3D orientations that achieve high angular uniformity comparable to Mackenzie distributions.
Generalized Fiber & Sheet Textures: Extended classical sheet and fiber texture definitions by introducing disorientation angle distributions \(P(\gamma)\) and fiber axis deviation bounds \(\omega\), establishing complete families of constrained random textures.

4. Novel Tessellation Theories & Physics-Bridging Models¶

Recursive Centroid Tessellations (RCT): Extended Lloyd's CVT algorithm to arbitrary nucleation-growth kinetics, proving that recursive centroid refinement generates regular asymptotic microstructures with controllable multimodal grain size and morphological distributions.
GBPD Physical Significance: Derived a unified tessellation model proving that fitting parameters (\(M_i, w_i\)) in Generalized Balanced Power Diagrams directly correspond to continuous nucleation and size-dependent growth velocities of ellipsoids.
3D-to-2D Kinetics Stereology: Mathematically derived the transformation of 3D nucleation and growth rates onto 2D planar sections, proving that 2D sections of 3D Voronoi structures form Laguerre tessellations and establishing the non-uniqueness of kinetics for spatial tessellations.
Bounded Domain Surface Effects: Discovered and quantified through-thickness variations in grain centroid density and grain size, revealing a characteristic depth-of-peak phenomenon near free surfaces.

Technical Stack & Software Assets¶

Framework / Tool	Core Technologies	Primary Function
EVOSIM 2D	Python, PyQt5, Cython, OpenCV	Fast 2D nucleation & morphological growth sandbox
EVOSIM 3D	C++10.3, OpenMP 4.5, Qt5, VTK 8.2, ITK 5.3	3D voxel space evolution & interactive visualization
REVOSIM	C++ / Python, PyBind11	Reconstruction of 3D microstructures from DCT data
Texture Library	C++, MTEX MATLAB Toolbox	Uniform random, sheet, and fiber texture generation

Thesis Reference¶

Narula, Harsh Kumar. Mathematical Modeling of Microstructure and Microtexture Evolution via Nucleation-Growth Kinetics. Ph.D. Dissertation, Department of Mechanical Engineering, Indian Institute of Technology Bombay, January 2024. Supervised by Prof. Asim Tewari and Prof. Sushil Mishra.