Boundary Integral Methods for Stokes Equations
A course intended for PhD and master-level students offered during Spring semester 2009, 7.5 hp.
The course started on Thursday, March 19, 2009. Please see the schedule of the course below.
Micro fluid dynamics or "micro-fluidics" is a rapidly growing research
area within fluid mechanics, that deals with phenomena on the micro
scale. At these small scales, the Reynolds numbers are typically low,
and the Stokes equations are often applicable. This course focuses on
numerical methods based on boundary integral equations for Stokes
flow; in particular for multi-phase flows and flows with suspended
particles of various shapes. The course includes topics ranging from
relevant theory for integral equations to practical implementation
issues, all ingredients that are needed in the design of a numerical
method based on a boundary integral formulation.
After completing the course, you should be able to:
Design a solution algorithm for a collocation scheme to solve the Stokes equations for a simple 3D-geometry, and motivate the choices made.
Identify strengths and weaknesses about boundary integral methods. Argue for if a boundary integral method is advantageous to use for a specific problem, and how it compares to other solution methods, such as finite difference or finite element methods.
As smaller sub-goals, you should specifically be able to:
Give examples of applications for which the Stokes equations are
Formulate the Stokes equations as a boundary integral equation (BIE).
Master a few basic techniques to derive simple analytical solutions to
the Stokes equations using singularity methods and slender body
Explain key concepts of the mathematical theory for integral equations
(e.g. properties of integral equations of the first and second kind,
practical consequences) and of theory specific to Stokes flow (
e.g. Lorentz reciprocal theorem).
Explain what difficulties arise in the design of quadrature formulas
for BIEs, and some techniques that can remedy these difficulties.
Describe the need of so called "fast summation methods". Explain the
underlying ideas of the Fast Multipole Method (FMM) and the
particle-mesh Ewald method.
List of topics:
Topics to be discussed include:
Introduction to Stokes equations. Fundamental solutions of Stokes equations.
Singularity methods for derivation of analytical solutions. Slender body approximations.
Theory for integral equations, and specifically boundary integral equations. Starting with the Laplace equation, then moving on to Stokes equations. Single layer and double layer formulations.
Numerical discretization of boundary integral equations. Collocation and Galerkin methods. Quadrature rules, including singularity treatments.
Fast summation methods. FFT based methods, fast multipole method (FMM).
Periodicity treatment. Theory and practical methods.
There will be two assignments, involving both theoretical questions
and practical implementation, in the first part of the course. These
assignments will be presented orally in class. The course ends with short individual oral exams.
For PhD students, there will also be a larger course project for which
each student is to select his or her own subject. Students can also
choose to work together on this project. The project work will continue after the oral exam. The class will meet again for oral project presentations in late August, and a written project report will be due shortly after.
The project part of the course is not mandatory for master
Format of the course:
The course will run with one two hour lecture each week. The second
hour of each lecture will be used to introduce a new topic, and the
first hour of the lecture the following week will be used for a follow
up discussion of that topic. In between, you will work with different
discussion themes and two specific assignments.
Schedule and lectures:
- Lecture 1, Thursday, March 19, 10-12 in D35
Formalia, outline of the course
Introduction to Stokes equations (KG)
Fundamental solutions (AKT)
Chapter 1-2 in Kim&Karilla
- Lecture 2, Thursday, March 26, 10-12 in 4523
Follow up on lecture 1
Singularity methods (KG)
Chapter 3 in Kim&Karilla
- Lecture 3, Thursday, April 2, 16-18 in D42
Follow up on lecture 2
Theory for boundary integral equations (AKT)
- Lecture 4, Tuesday, April 14, 13-15 in 4523
Follow up on lecture 3
Theory for boundary integral equations, contd. (AKT)
- Lecture 5, Thursday, April 23, 13-15 in 4523
Follow up on lecture 4
Numerical discretization of boundary integral equations (KG)
Chapter 14 and 15 in Kim&Karilla
- Lecture 6, Tuesday, April 28, 13-15 in 4523
Follow up on lecture 5
Numerical discretization of boundary integral equations, contd. (KG)
Chapter 14 and 15 in Kim&Karilla
Chapter 18.5 (Singularity subtraction) in Kim&Karilla
- Lecture 7, Friday, May 8, 13-15 in 1537
Presentations and discussion on assignments
- Lecture 8, Tuesday, May 12, 13-15 in 4523
Follow up on lecture 6 and presentations during lecture 7.
- Lecture 9, Tuesday, May 19, 13-15 in 4523
Follow up on lecture 8
Particle mesh methods & Fast multipole methods (AKT)
- Lecture 10, Friday, May 29, 13-15 in 4523
Presentations and discussion on Assignment 2.
- Oral exams in week 23
At each lecture, copies of the slides presented at that lecture will be given out. Theere will also be additional handouts, as listed below.
Life at Low Reynolds number, E.M. Purcell, Journal of Physics, 45,1, 1977.
Microfluidics: Fluid physics at the nanoliter scale, T.M. Squires, S.R. Quake, Reviews of Modern Physics, 77, 2005
Hydromechanics of low-Reynolds number flow, Part 2. Singularity methods for Stokes flow ,
A.T. Chwang and T. Wu, Journal of Fluid Mechanics, 67,4, 1975.
Simulating particles in Stokes flow,
T. Götz, Journal of Computational and Applied Mathematics, 2005.
Some pages from the book "The numerical solution of integral equations of the second kind", by K. Atkinson was given out in class. In case they inspire to further reading, check out the bibiography from that book.
Some pages from the book "Boundary integral and singularity methods for linearized viscous flow", by C. Pozrikidis was given out in class.
A comparison of integral formulations for the analysis of low Reynolds number flows, M.S. Ingber and A.A.Mammoli, Engineering Analysis with Boundary Elements, 23, 1999.
Computation of periodic Green's functions of
Stokes flow, C. Pozrikidis, Journal of Engineering Mathematics, 1996.
On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres, H. Hasimoto, Journal of Fluid Mechanics, 1959.
A new version of the Fast Multipole Method for the Laplace equation in three dimensions, L. Greengard and V. Rokhlin, Acta Numerica (6), 1997.
From Sean's applied math book (free resource on the web):
A short chapter on Dirac Delta functions
A few sections on Green's functions in 1D
Assignment 1,part I
. Part II of Assignment 1 will be handed out later. The due date for Assignment 1 is May 8, 2009.
NOTE: a new version 29/4-09. There was an error in the potential dipole.
Assignment 1,part II
. The due date for Assignment 1, both part I and II, is May 8, 2009. NOTE: a new version 28/4-09. There was an error in the formula for the spherical coordinates in the earlier version.
. The due date for Assignment 2 is May 29, 2009.
We will use the book Microhydrodynamics - Principles and Selected Applications by S. Kim and S. J. Karilla (Dover Publications, 2005), together with selected pages from Boundary integral and singularity methods for linearized viscous flow, by C. Pozrikidis (Cambridge University Press, 1992). We will complement this with handouts regarding e.g. boundary integral equations for the Laplace equation, and will read several research papers on the topic (e.g. concerning the FMM method and periodicity treatment).
The course will be taught by Anna-Karin Tornberg and Katarina
Anna-Karin Tornberg, email firstname.lastname@example.org.
Office: Osquars backe 2, floor 5, room 4517.
Phone: 08-790 6266.
Katarina Gustavsson, email email@example.com.
Office: Osquars backe 2, floor 5, room 1517.
Phone: 08-790 6696.
Upp till Nadas hemsida.