## Theoretical and Numerical aspects for incompressible fluids

### M2, master Mathematiques et Applications, ANEDPSorbonne Université

Course instructors:
Class: wednesday 10:00-12:00, room 15-25 104 ### 1. Syllabus

This class is an introduction to mathematical and computational aspects of incompressible fluid flow simulations. It is presented to the point of view that the students are (going to be) applied mathematicians, physicists or engineers.

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
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. Shape optimization for fluids
6. Appendix
• variational approximation
• error estimates

### 2. Material

• the slides of the course in PDF (Part I P. Frey)
• the slides of the course in PDF (Part II Y. Privat)
• the classroom notes in PDF (part 2)
• the numerical experiments notes in PDF
• the FreeFem++ documentation in PDF

### 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).

### 4. Research papers

Méthode de porosité
• 1. A penalization method to take into account obstacles in incompressible viscous flows
• 2. The Limits of Porous Materials in the Topology Optimization of Stokes Flows
Optim. de forme appliquée au design de bypass ou à la forme de l'arbre bronchique
• 4. Shape minimization of the dissipated energy in dyadic trees
Algorithmes numériques en optim. de forme
• 5. Discrete Gradient Flows for Shape Optimization and Applications
• 5b. Chenais D., Finite element approximation of 2D elliptic optimal design
Contrôle optimal sous contrainte NS et influence des Riblets sur un conduit
• 6. State-constrained optimal control of the three-dimensional stationary Navier-Stokes equations
• 7. On the problems of riblets as a drag reduction device
Modélisation du poumon/arbre bronchique
• 8. Outlet dissipative conditions for air flow in the bronchial tree
• 9. An optimal bronchial tree may be dangerous
Méthodes de type level set et applications
• 10. A fast level set method for propagating interfaces
• 11. Level set methods for fluid interfaces
Optimalité des maillages
• 12. Optimally adapted finite element meshes
• 13. Adaptive finite elements with large aspect ratio based on an anisotropic error estimator
Méthodes des caractéristiques pour NS
• 14. Finite element modified method of characteristics for the Navier-Stokes equations
• 15. A high-order characteristics finite element method for the incompressible Navier-Stokes equations
Méthodes de projection pour NS
• 16. A projection FEM for variable density incompressible flows
• 17. A second order accurate projection method for the incompressible Navier-Stokes equations on non graded adaptive grids
Optimisation de formes, applications biomédicales
• 18. Agoshkov V., Shape design in aorto-coronaric bypass anastomoses using perturbation theory
• 19. Grandmont C., A viscoelastic model with non local damping application to the human lungs
• 20. Schulz A., the optimal shape of a pipe
Optimisation de formes appliquée
• 21. Giraldi L., Optimal Design for Purcell Three-link Swimmer
• 22. Scheid J.F., Shape optimization for a uid-elasticity system

All papers can be accessed in PDF format following this link-> list