Computational Materials Track
With five computational faculty (Balakrishna, Begley, Beyerlein, Van der Ven, Van de Walle), the Materials Department offers outstanding opportunities for education and training of students who are attracted to the growing area of Computational Materials Research. Developing mathematical methods to analyze materials at atomic and continuum scales, specifically to understand the relation between material behavior across different scales is a key research topic. A strong set of courses cover ever-more-powerful computational techniques. They include advanced mathematical, numerical and theoretical topics, as well as multiscale and multiphysics approaches and the theoretical principles and mathematical underpinnings of modern machine learning tools as applied to problems in materials science. Cross-departmental interactions (for courses as well as research opportunities; e.g., Fredrickson, Shell in Chemical Engineering) are encouraged. A coordinated course sequence that addresses the educational needs of computational materials scientists can be tailored to specific student interest.
Selected Courses
● MATRL 228: Computational Materials (Van de Walle)
Basic computational techniques and their application to simulating the behavior of materials. Techniques include: finite difference methods, Monte Carlo, molecular dynamics, cellular automata, and simulated annealing.
● MATRL 207: Mechanics of Materials (Begley, McMeeking)
The mathematical foundations of the continuum analysis of materials, including tensors and tensor calculus, conservation laws, and constitutive descriptions spanning anisotropic thermoelasticity, plasticity and viscoelasticty. Variational calculus and the principle of virtual work as foundations of numerical frameworks to solving field equations.
● MATRL 230: Elasticity and Plasticity (Beyerlein)
Review field equations of elasticity and plasticity. Energy principles and uniqueness theorems. Elementary problems in one and two dimensions, stress functions, and complex variable methods. Plastic stress-strain laws; flow potentials. Torsion and bending of plastic flow, slip line theory. Bounding theorems.
● MATRL 232: Advanced Plasticity (Beyerlein)
Plastic, creep, and relaxation behavior of solids. Mechanics of inelastically strained bodies, plastic stress-strain laws; flow potentials. Torsion and bending of prismatic bars, expansion of thick shells, plane plastic flow, slip line theory. Variational formulations, approximate methods.
● MATRL 240: Finite Element Structural Analysis (Beyerlein, Begley)
Definitions and basic element operations displacement approach in linear elasticity. Element formulation: direct methods and variational methods. Global analysis procedures: assemblage and solution. Plane stress and plane strain. Solids of revolution and general solids. Isoparametric representation and numerical integration. Computer implementation.
● MATRL 279: First-Principles Calculations for Materials (Van de Walle)
Basic theory and methods of electronic structure, illustrated with examples of practical computational methods and real-world applications. Topics: Band structure; Uniform electron gas; Density functional theory; Exchange and correlation; Kohn-Sham equations; Pseudopotentials; Basis sets; Predicting materials properties: bulk, surfaces, interfaces, defects.
● MATRL 289Q: Micromechanics (Balakrishna)
Overview of micromechanics, emphasizing the microstructure of materials, its connection to atomic structure, and consequences on macroscopic properties. A key feature of the course is hands-on coding sessions integrated into each lecture. Topics include phase transformations in crystalline solids; energy minimization; Clausius Clapeyron relation; small strain theories; kinetics; variational methods; Applications to structural (e.g., shape-memory alloys) and functional materials (e.g., ferroelectrics, battery materials, and soft magnets).
● MATRL 289H: Statistical Mechanics of Crystalline Solids (Van der Ven)
This course covers first-principles methods to predict thermodynamic and kinetic properties of multi-component crystalline solids. Topics include: (i) a review of thermodynamics and statistical mechanics of multi-component crystals; (ii) effective Hamiltonians for configurational and vibrational degrees of freedom with a focus on the Ising model, cluster expansions, harmonic and anharmonic lattice dynamical Hamiltonians and generalized Heisenberg models; (iii) Monte Carlo methods, low temperature expansions, free energy integration methods; (iv) First-principles methods to calculate multi-component phase diagrams; (v) Order-disorder and structural phase transitions and (vi) Atomic diffusion, transition state theory, kinetic Monte Carlo simulations, Kubo-Green.
