Materials Modelling II
FSI-WMQAcad. year: 2019/2020
Computational modelling of materials is an indispensable tool to understand the relationship between microstructure and physical properties of materials. Atomic models based on empirical and semiempirical potentials represent essential and frequently used tools for computer simulations of nanostructures such as nanotubes, epitaxial films or graphene, studies of radiation damage and the motion of dislocations under stress. Spin-based models investigated using the Monte Carlo method and continuum mesoscopic models are standard approaches to study the ordering of solid solutions, phase transitions in multiferroics and their changes caused by crystal lattice defects. Macroscopic studies employing the Finite Element Method, which are often enriched by the results of atomistic and mesoscopic studies, represent an essential tool for the prediction of macroscopic behavior of real-world structures. This course provides a broad overview of the basic theoretical methods used in computational modelling of materials from the level of interacting atoms to the continuum macroscopic description, including postprocessing and visualizations of results. An important part of the course is to gain practical experience with these approaches through a series of exercises (implementation, solution and analysis of each model problem), and through individual problem assignments.
Learning outcomes of the course unit
Within this course, the students will acquire knowledge of a broad range of computational methods used to study the relationships between microstructure and physical properties of materials. It will provide basic theoretical and practical skills for the studies of nanostructures, interacting many-body systems and for simulations of mesoscopic and macroscopic systems based on their continuum descriptions.
Knowledge of mathematics at the level of the 2nd year of FME (differentiations of the functions of many variables, basic probability theory, numerical methods), and basic knowledge of programming.
Recommended optional programme components
Recommended or required reading
M. P. Allen, D. J. Tildesley: Computer simulation of liquids. Clarendon Press (1987).
D. Frenkel, B. Smith: Understanding molecular simulation. Academic Press (2002).
J. P. Sethna: Statistical mechanics: Entropy, order parameters, and complexity. Oxford University Press
Planned learning activities and teaching methods
The objective of the lectures is to introduce fundamental theoretical descriptions of individual methods, their analyses, and in some cases also analytical solutions. The main focus of the exercises is on understanding the implementation of each method and their use to solve model problems and students‘ individual assignments.
Assesment methods and criteria linked to learning outcomes
At the end of the semester, each student will be assigned a problem that will be tightly linked to some of the methods explained in the lectures and more deeply studied in the exercises. The output of each such assignment will be the formulation (or modification of already existing) simulation code, its application to study the given problem and writing a report that summarizes these developments and the principal results. The exam will then consist of an oral defense of this report.
Language of instruction
This course will provide a broad overview of the most frequently used methods for computational simulations of materials from the atomic level, via a range of mesoscopic descriptions to the continuum simulations of macroscopic bodies. In the series of exercises, the students will get acquianted with computer implementations of the individual algorithms, which will make it possible to understand the inputs into, methods used and results obtained from standard commercial and open-source packages for computer simulations of materials.
Specification of controlled education, way of implementation and compensation for absences
The attendance at exercises is mandatory and each absence must be propertly justified. The absence will be accepted upon the student submitting a written report from the missed exercise which proves that the student understood the method explained.
Type of course unit
26 hours, optionally
Teacher / Lecturer
1. Modelling of relationships between microstructure and physical properties, history and presence.
2. Equilibrium statistical mechanics, spin models and their mean field solutions.
3. Phase space, phase trajectory, ergodic theorem, entropy.
4. Numerical methods for the minimizations of functions of N variables.
5. Crystallography and symmetry in the real and reciprocal spaces.
6. Molecular statics, atomic-level forces, energies and stresses in many-body systems.
7. Molecular dynamics, stability of numerically integrated equations of motions, thermostats, barostats.
8. More advanced interaction potentials and their physical origins.
9. Mesoscopic phase field models.
10. Phase field crystal model.
11. Methods for finding the minimum energy paths of systems.
12. Finite Element Method, shape functions and elasticity.
13. Modern trends in computational studies of materials.
26 hours, compulsory
Teacher / Lecturer
1. Investigation of the Fermi-Pasta-Ulam problem.
2. Monte Carlo studies of the 1D-3D Ising models and calculations of their phase diagrams.
3. Calculation of the density of states of the 2D Ising model using the Wang-Landau method.
4. Implementation of numerical methods for the minimizations of functions of N variables.
5. Construction of an arbitrary Bravais lattice and introduction to visualizations.
6. Ground state of crystalline argon in 2D a 3D using the Lennard-Jones potential.
7. Crystallization of inert gas in the Lennard-Jones potential.
8. Calculation of the energies of point defects and surfaces in an fcc material.
9. Study of twinning in ferroelastic materials.
10. Evolution of microstructure in the phase field crystal model.
11. Obtaining the transition pathway of a model system using the Nudged Elastic Band method.
12. Distribution of stresses and strains in a deformed elastic body using the Finite Element Method.
13. Discussions on the assigned problems.