Theoretical and Numerical Methods for Fluid Mechanics

School of Mathematics, Statistics and Computer Sciences, College of Science, University of Tehran Course instructor: Pascal Frey lab. J.L. Lions & institute for computing and data science, Sorbonne Université 4, place Jussieu, 75005 Paris, France Collaborators: Yannick Privat, Charles Dapogny, Dena Kazerani, Thi Thu Cuc Bui, Thi Thanh Mai Ta

Information: summer school

The institute for computing and data science (ISCD) is organizing its 7th summer school "Scientific Trends at the Interfaces".
This year's topic is Computational Chemistry-Mathematical Modelling-High Performance Computing.
Have a look at our web page and read our brochure (in PDF format).
Applications can be submitted and a limited number of grants (accomodation) are available.

1. Syllabus

Computational fluid dynamics is at the crossroad of many disciplines and, like many topics at the interfaces between disciplines, its access may seem a bit harsh for (under)graduate students in mathematics or engineering. Our goal is to cover the main aspects of finite element methods for incompressible flows. We have sought to achieve a right balance between theoretical concepts, numerical analysis, description of schemes and algorithms and engineering applications. Numerical experiments using FreeFem++ will help students to understand these concepts and see advanced numerical methods in action.

1. A fluid mechanics primer
• notations, vectors, tensors
• conservation laws
• flow models and simplifications
2. The Stokes model
• mathematical and numerical analysis
• finite element approximation, resolution
• unsteady Stokes problem
3. The Navier-Stokes model
• analysis of the steady-state problem
• discretization procedures
4. Two-fluid or two-phase flows
• level set formalism
• bifluid simulations
5. Error estimates and mesh adaptation
• residual and geometric estimates
• mesh adaptation
6. Shape optimization for fluids
• framework of shape optimization, examples
• shape sensitivity analysis using shape derivatives
• Céa's method to compute derivatives
• numerical issues
7. Appendix
• variational approximation
• error estimates
• mesh adaptation

2. Material

• introduction (15 pages, 1.6 Mb)
  
• chapter 1: fluid mechanics (45 pages, 1.5 Mb)
  
• chapter 2: the Stokes model (93 pages, 3.4 Mb)
  
• chapter 3: the Navier-Stokes model (60 pages, 3.8 Mb)
  
• chapter 4: two-fluids and two-phase flows (63 pages, 37.1 Mb)
  
• chapter 5: error estimates and mesh adaptation (41 pages, 2.8 Mb)
  
• chapter 6: mesh generation and adaptation (129 pages, 34.4 Mb)
  
• chapter 7: shape optimization (74 pages, 5.4 Mb)
  
• appendix: variational approximation of linear problems (26 pages, 1.7 Mb)
• appendix: the finite element method (44 pages, 1.3 Mb)
• FreeFem++ documentation in PDF

• homework with FreeFem++
  

3. References

• Functional and numerical analysis
1. Allaire G., Numerical analysis and optimizaton, Oxford Science Publishing, (2007)
2. Brezis H., Analisis funcional, Teoria y applicaciones, Allianza Editorial, (1983).
3. Ciarlet P.G., The finite element methods for elliptic problems, SIAM classics, 40, (2002)
4. Ern A., Guermond J.L., Theory and practice of finite elements Applied Mathematical Series, 159, Springer, (2004)
5. Evans L.C., Partial differential equations, AMS, (2002).
6. Frey P., George P.L., Mesh generation, application to finite element methods, Wiley, (2008)
7. Johnson C., Numerical solution of partial differential equations by the finite element method, Cambridge University Press, (1987)
8. Lax P.D., Functional Analysis, Wiley Interscience, (2002)
9. Oden J.T., Applied Functional Analysis, Prentice-Hall, (1979)
10. Quarteroni A., Valli A., Numerical approximation of partial differential equations, 23, Springer Series in Computational Mathematics, (1997)
11. Quarteroni A., Sacco R., Saleri F., Numerical Mathematics Texts in Applied Mathematics, 37, Springer, (1991)
12. Rudin W., Functional Analysis, Mc-Graw Hill, (1973)
13. Saxe K., Beginning Functional Analysis, Springer, New York, (2001)
14. Solin P., Partial differential equations and the finite element method, Wiley Interscience, (2006)
15. Yosida K., Functional Analysis, Springer, (1980)

• Fluid mechanics
1. Chorin A.J., Marsden J.E., A mathematical introduction to fluid mechanics, Springer, (1992)
2. Durst F., Fluid Mechanics, an introduction to the theory of fluid flows, Springer, (2008)
3. Landau L.D., Lifschitz E.M., Fluid mechanics, course in theoretical physics, 6, Pergamon Press, (1987)
4. Pnueli D., Gutfinger C., Fluid Mechanics, Cambridge University Press, (1992)
5. Temam R., Miranville A., Mathematical modeling in continuum mechanics, Cambridge University Press, (2005).

• Computational fluid dynamics
1. Acheson D.J., Elementary Fluid Dynamics, Oxford Applied Mathematics and Computing Science Se- ries, (2005)
2. Batchelor G.K, An introduction to fluid dynamics, Cambridge University Press, (2002)
3. Blazek J., Computational Fluid Dynamics, Principles and Applications, Elsevier, (2005)
4. Donea J., Huerta A., Finite element methods for flow problems, Wiley, (2003)
5. Feistauer M., Mathematical Methods in Fluid Dynamics, Longman Scientific & Technical, Harlow, (1993)
6. Ferziger J.H., Peric M., Computational Methods for Fluid Dynamics, Springer, (1999)
7. Girault V., Raviart P.A., Finite element methods for Navier-Stokes equations. Theory and Algorithms, Springer, (1986)
8. Glowinski R., Finite Element Methods fo Incompressible Viscous Flows, in Handbook of numerical analysis, vol. 9 (part 3), North-Holland, (2003)
9. Gresho P. M., Sani R.L., Incompressible flow and the finite element method, Wiley (1998)
10. Gunzburger M., Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Prac- tice, and Algorithms, Academic Press, (1989)
11. Kwak D., Kiris C.C., Computation of viscous incompressible flows, Scientific Computation series, Springer, (2011)
12. Marion M.,Temam R., Navier-Stokes equations: Theory and approximation,in Handbook of numerical analysis, vol. 6, 503-689, North-Holland, (1998)
13. Peyret R., Taylor T.D., Computational Methods for Fluid Flow, Springer, (1983)
14. Pironneau O., Finite element methods for fluids,Wiley & Sons, (1989)
15. Wesseling P., Principles of Computational Fluid Dynamics, Springer, (2000)

• Numerical programming
1. Hecht F. et al., FreeFem++, UPMC
2. Quarteroni A., Scientific Computing in Matlab and Octave, 2nd ed., Springer, Texts in Computational Science and Engineering, (2006)

• Shape optimization
1. Allaire G., Conception optimale de structures, Mathématiques et Applications 58, Springer, (2006)
2. Bendsoe M.P. and Sigmund O., Topology Optimization, Theory, Methods and Applications, 2nd Edition, Springer, (2003)
3. Henrot A., and Pierre M., Variation et optimisation de formes, une analyse géométrique, Springer, (2005)
4. Mohammadi B. and Pironneau O., Applied shape optimization for fluids, Oxford University Press,28, (2001)
5. Pironneau O., Optimal Shape Design for Elliptic Systems, Springer, (1984)
6. Sethian J.A., Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry,Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, (1999).