Computational Materials Science

The materials science length scale hierarchy: electrons atoms microstructure macroscopic properties as a multiscale computational challenge; Crystal structures as periodic data structures: lattice vectors, basis atoms, unit cells, and the Bravais lattice classification; Crystallographic information files (CIF) as the standard schema for storing atomic structure data; Reciprocal space and the Brillouin zone as the Fourier dual of the crystal lattice: intuition for why periodicity enables tractable computation; Point defects, dislocations, and grain boundaries as structural irregularities that dominate material behavior: representing them as perturbations to a reference periodic system; Introduction to open materials databases: the Materials Project, AFLOW, and OQMD as large-scale, queryable repositories of computed material properties.

Classical Molecular Dynamics (MD) as a numerical ODE integration problem: the Verlet and velocity-Verlet algorithms for propagating Newton's equations of motion; Interatomic potentials as the force-field approximation: Lennard-Jones, embedded atom method (EAM), and Tersoff potentials as parameterized energy functions replacing quantum mechanics; Periodic boundary conditions as the computational trick that simulates bulk material with a finite simulation box; Thermostats and barostats: the Nosé-Hoover and Parrinello-Rahman algorithms for sampling the NVT and NPT ensembles; Monte Carlo methods for equilibrium sampling: the Metropolis-Hastings algorithm applied to atomic configuration space; Structural analysis of MD trajectories: radial distribution functions, mean squared displacement, and Voronoi tessellation as the post-processing toolkit.

The quantum mechanical many-body problem: the electronic Schrödinger equation and why it is computationally intractable for systems beyond a few electrons; The Hohenberg-Kohn theorems as the theoretical foundation of DFT: ground-state energy as a functional of electron density; The Kohn-Sham equations: mapping the interacting many-electron problem onto a tractable non-interacting system with an exchange-correlation functional; Plane-wave basis sets and pseudopotentials as the computational representation enabling periodic DFT calculations; The self-consistent field (SCF) iteration as a fixed-point algorithm for solving the Kohn-Sham equations; Outputs of a DFT calculation: total energy, atomic forces, electronic band structure, and density of states as the primary computed quantities linked to observable material properties.

The microstructure as the mesoscale link between atomic structure and macroscopic performance: grain size, phase distribution, porosity, and their effect on mechanical and transport properties; Phase-field method as a continuum PDE framework for simulating microstructure evolution: order parameters, free energy functionals, and the Cahn-Hilliard and Allen-Cahn equations; Finite Difference and Finite Element discretization of the phase-field equations: stability, convergence, and the computational cost of 3D microstructure simulations; Voxel-based microstructure representation: 3D arrays as the discrete data structure for storing phase, orientation, and composition fields; CALPHAD (CALculation of PHAse Diagrams) as a thermodynamic database approach to predicting equilibrium phase stability; Synthetic microstructure generation: random grain growth algorithms and their use as training data for downstream ML models.

The computational cost landscape: DFT scales as O(N^3), classical MD as O(N N) with neighbor lists, and phase-field as O(N_ grid); Parallelization strategies for materials codes: domain decomposition in MD (spatial decomposition of the simulation box), k-point parallelism in DFT, and OpenMP threading for shared-memory acceleration; GPU acceleration for MD: neighbor list construction and force evaluation as massively parallel operations; VASP, Quantum ESPRESSO, LAMMPS, and MOOSE as the canonical open-source codes for DFT, MD, and FEM respectively: their input file formats as domain-specific configuration languages; Workflow automation for high-throughput computation: FireWorks and AiiDA as job management frameworks for running thousands of DFT calculations programmatically; Data provenance and reproducibility: tracking simulation inputs, software versions, and pseudopotential choices as a scientific software engineering discipline.