ME595M: Computational Methods for Nanoscale Thermal Transport

Course Syllabus

Profs. Jayathi Murthy and Tim Fisher*


Course description: This one-credit course will provide an introductory understanding of numerical methods for solving phonon transport problems in spatially confined materials. Methods to be studied include: the atomistic Green’s function (AGF) method, the finite volume (FV) method, and the Monte Carlo (MC) method.


Pre-requisites: ME 597F or permission of instructor


Textbook: None (course notes only)


Computer programming: Experience with Matlab’s matrix tools is helpful. Alternatively, other scientific computing languages can be used. 


Course Dates/Times/Location: normally MTWTh, May 14 to June 7, 8:40 to 9:40 AM, Room ME118


Course grading: The overall course grade will be based entirely on homework.


Tentative Lecture schedule:


Week 1

Assignments (all homework is due the following lecture unless otherwise noted)

May 14    Lattice dynamics (TSF)

-        1D atomic chain

-        Lattice glossary

Reading (Sections 1-5)

May 15    Statistical mechanics (TSF)

-        Statistical ensembles

-        Definition of temperature

Reading, Homework 1, Demo Code

May 16    No lecture


May 17    No lecture


Week 2


May 21    AGF introduction (TSF)

-        General problem description

-        Harmonic matrix

-        Green’s functions

 Homework 2 (due May 23)

May 22    AGF energy transport  (TSF)

-        Energy flux and transmission

-        Introduction to the nanoHUB atomic chain tool


May 23    AGF DOS and multi-dimensionality (TSF)

-        1D chain results

-        Density of states

-        Introduction to multi-dimensional systems

 Homework 3

May 24    No lecture


May 25    Introduction to BTE (JYM)

-        General derivation

-        General form of scattering term

-        U, N processes, impurity and boundary scattering

-        Relaxation time form and single mode relaxation time

-        Thick limit of the BTE

BTE Homework (due June 4)

BTE code

How-to-run instructions

Other files

Week 3


May 28    No lecture


May 29    Gray BTE (JYM)

-        Derivation of gray energy form

-        Thick limit of gray BTE

-        Boundary and initial conditions – diffuse, specular, mixed, Fourier+BTE interfaces


May 30    Introduction to finite volume method (JYM)

-        Space and angle integration

-        Upwinding schemes

-        Time stepping

-        Linear solvers

-        Accuracy and convergence issues


May 31    Introduction to finite volume code (JYM)

-        Discuss FV code data structures and usage

-        Assign problem and discuss


June 1     Extensions and modifications to FV (JYM)

-        Issues with sequential solutions and coupling

-        Accuracy issues – false scattering, false diffusion

-        Point-coupled techniques

-        Ray-tracing techniques


Week 4


June 4      Higher Order BTE Models (JYM)

-        Semi-gray models – limiting behavior and discussion

-        Non-gray models

-        Full BTE simulation


June 5   Monte Carlo method (TSF)

-        Introduction

-        Equation of motion

-        Scattering

Homework (see lecture notes): Due June 7

June 6    Monte Carlo (TSF)

-        Statistical sampling

-        Relation to the BTE


June 7    Course wrap-up (TSF and JYM)



*TSF gratefully acknowledges partial support of this work by the Chemical and Transport Systems division of NSF through its CAREER program