PX91215 Multiscale Modelling Methods & Applications II
Introductory description
N/A.
Module aims
To provide students with a firm foundation in the macroscopic modelling of fluid flows and solid deformation in order to describe a phenomenal range of natural and technological processes.
Classical approaches to formulating and solving the equations of continuum mechanics will be covered, but our primary focus will be to explore and understand recent approaches to developing classes of macroscopic models that are connected to an underlying microscopic description via multiscale techniques.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
The following syllabus is a superset of what can be covered in this module. The majority of Sections 1,2,4 will be covered whilst the topics for Sections 3 and 5 are intended vary yeartoyear (Exemplars are provided).

Foundations of Continuum Mechanics (4 lectures)
(a) physical properties of materials
(b) continuum approximation
(c) Eulerian/Lagrangian descriptions
(d) vectors and tensors
(e) theory of deformation
(f) stress and deformation measure
(g) conservation of mass
(h) equations of motion/linear momentum balance
(i) principle of virtual work 
Classical Fluid Mechanics (4 lectures)
(a) Viscous incompressible Newtonian fluids
(b) interfacial dynamics
(c) boundary conditions, initial conditions
(d) dimensionless parameters
(e) general concepts of thermodynamics
(f) energy equations
(g) computational fluid dynamics 
Recent Developments in Multiscale Fluid Mechanics  Topic Flexible (4 lectures)
Example 1: From Kinetic to Continuum for Gas Microflows
(a) introduction to the Boltzmann equation
(b) Grad’s method of moments
(c) ChapmanEnskog expansions
(d) connection to and derivation of macroscopic models
Exemplar 2: Secondary Continuum Limit for Particulate Flows
(a) motion of single particles within Stokes flow, fundamental solutions
(b) ensemble averages for dilute suspensions
(c) emergence of nonNewtonian behaviour
(d) method of fundamental solutions for microflows
Exemplar 3: Thermal Fluctuations in Fluid Flow
(a) introduction to fluctuating hydrodynamics
(b) LandauLifshitz NavierStokes
(c) simplified configurations
(d) computation of SPDEs 
Classical Solid Mechanics (4 lectures)
(a) general theory
(b) linear elasticity, displacement formulation, derivation of NavierLame equation, boundary conditions (Neumann, Dirichlet, mixed)
(c) simple configurations
(d) weak form of BVP
(e) discretisation and FE solution 
Recent Developments in Multiscale Solid Mechanics  Topic Flexible (4 lectures)
Exemplar 1: Incorporating Microstructure
(a) meanfield approaches
(b) heterogeneous multiscale methods
(c) separation of scales, localisation, homogenisation
(d) Representative Volume Element (RVE), periodic boundary conditions
(e) nested approach
Learning outcomes
By the end of the module, students should be able to:
 It is expected that by the end of this module students will be able to understand the foundations of continuum mechanics and derive the classical governing equations of fluid and solid mechanics.
 It is expected that by the end of this module students will be able to appreciate the relationships between different modelling frameworks.
 It is expected that by the end of this module students will be able to engage with cuttingedge topics in the multiscale modelling of materials.
 It is expected that by the end of this module students will be able to develop and apply appropriate computational methods to solve real world phenomena.
Indicative reading list
D.J. Acheson, Elementary Fluid Dynamics
G. K. Batchelor, An Introduction to Fluid Dynamics
P.A. Kelly, Mechanics Lecture Notes: An introduction to Solid Mechanics. University of Auckland, 2019 (online).
G.A. Holzapfel, Nonlinear Solid Mechanics: a continuum approach for engineering, Wiley, 2010.
Subject specific skills
It is expected that by the end of this module students will be able to:
understand the foundations of continuum mechanics and derive the classical governing equations of fluid and solid mechanics
appreciate the relationships between different modelling frameworks
engage with cuttingedge topics in the multiscale modelling of materials
develop and apply appropriate computational methods to solve real world phenomena
Transferable skills
Coding, problem solving, writing
Study time
Type  Required 

Lectures  10 sessions of 2 hours (50%) 
Supervised practical classes  10 sessions of 2 hours (50%) 
Total  40 hours 
Private study description
Reading set material, preparation for projects.
Costs
No further costs have been identified for this module.
You must pass all assessment components to pass the module.
Assessment group D
Weighting  Study time  

Fluids Project  30%  30 hours 
On Section 3 (Multiscale Fluids). Each of which will contain a computational component. 

Solids Project  30%  30 hours 
On Section 5 (Multiscale Solids). Each of which will contain a computational component. 

Viva Voce Exam  40%  10 hours 
On core material. 30 minutes long. 
Feedback on assessment
\tWritten annotations on submitted projects\r\n\tVerbal discussion during viva voce exam\r\n\tWritten summary of viva performance
Courses
This module is Core for:
 Year 1 of TPXAF344 Postgraduate Taught Modelling of Heterogeneous Systems
 Year 1 of TPXAF345 Postgraduate Taught Modelling of Heterogeneous Systems (PGDip)