Existence of optimal shape for a system of conservation laws in a free air-porous domain
Author:
Arian Novruzi
Journal:
Quart. Appl. Math. 64 (2006), 641-661
MSC (2000):
Primary 49J20, 93C20, 76D05, 76D06
DOI:
https://doi.org/10.1090/S0033-569X-06-01012-X
Published electronically:
November 15, 2006
MathSciNet review:
2284464
Full-text PDF Free Access
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Abstract: We consider a shape optimization problem related to a nonlinear system of PDE describing the gas dynamics in a free air-porous domain, including gas concentrations, temperature, velocity and pressure. The velocity and pressure are described by the Stokes and Darcy laws, while concentrations and temperature are given by mass and heat conservation laws. The system represents a simplified dry model of gas dynamics in the channel and graphite diffusive layers of hydrogen fuel cells. The model is coupled with the other part of the domain through some mixed boundary conditions, involving nonlinearities, and pressure boundary conditions. Under some assumptions we prove that the system has a solution and that there exists a channel domain in the class of Lipschitz domains minimizing a certain functional measuring the membrane temperature distribution, total current, water vapor transport and channel inlet/outlet pressure drop
ADN-1 S. Douglis, A. Douglis, L. Nirenberg. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary value conditions I, Comm. Pure Appl. Math., 12 (1959).
Beavers.Joseph.1 G. S. Beavers, D. D. Joseph. Boundary condition at a naturally permeable wall. J. Fluid Mech. 30 (1967), 99. 197-207.
- Alain Bensoussan and Lucio Boccardo, Nonlinear systems of elliptic equations with natural growth conditions and sign conditions, Appl. Math. Optim. 46 (2002), no. 2-3, 143–166. Special issue dedicated to the memory of Jacques-Louis Lions. MR 1944757, DOI https://doi.org/10.1007/s00245-002-0753-3
Brezis.1 H. Brézis, J. L. Lions (eds). Nonlinear partial differential equations and their applications. V. 5, Collége de France seminar, in Research Notes in Mathematics, Pitman.
- T. Gallouet and A. Monier, On the regularity of solutions to elliptic equations, Rend. Mat. Appl. (7) 19 (1999), no. 4, 471–488 (2000) (English, with English and Italian summaries). MR 1789483
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
- P. Grisvard, Singularities in boundary value problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 22, Masson, Paris; Springer-Verlag, Berlin, 1992. MR 1173209
- J. Haslinger and R. A. E. Mäkinen, Introduction to shape optimization, Advances in Design and Control, vol. 7, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003. Theory, approximation, and computation. MR 1969772
henrot.pierre.1 A. Henrot, M. Pierre. Variation et optimisation de formes. Une analyse géometrique. Series: Mathématiques et Applications, Vol. 48, 2005, XII.
- Willi Jäger and Andro Mikelić, On the interface boundary condition of Beavers, Joseph, and Saffman, SIAM J. Appl. Math. 60 (2000), no. 4, 1111–1127. MR 1760028, DOI https://doi.org/10.1137/S003613999833678X
- Willi Jäger and Andro Mikelić, On the boundary conditions at the contact interface between a porous medium and a free fluid, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), no. 3, 403–465. MR 1440029
- O. A. Ladyženskaja, Matematicheskie voprosy dinamiki vyazkoĭ neszhimaemoĭ zhidkosti, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1961 (Russian). MR 0155092
- William J. Layton, Friedhelm Schieweck, and Ivan Yotov, Coupling fluid flow with porous media flow, SIAM J. Numer. Anal. 40 (2002), no. 6, 2195–2218 (2003). MR 1974181, DOI https://doi.org/10.1137/S0036142901392766
lions-m-1 J. L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications I, Springer-Verlag, 1972.
lions-m-2 J. L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications II, Springer-Verlag, 1972.
- Béatrice Rivière and Ivan Yotov, Locally conservative coupling of Stokes and Darcy flows, SIAM J. Numer. Anal. 42 (2005), no. 5, 1959–1977. MR 2139232, DOI https://doi.org/10.1137/S0036142903427640
Saffman.1 P. G. Saffman. On the boundary condition at the surface of a porous medium. Studies in applied mathematics. Vol. L, No. 2, June 1971.
- Roger Temam, Navier-Stokes equations. Theory and numerical analysis, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Studies in Mathematics and its Applications, Vol. 2. MR 0609732
ADN-1 S. Douglis, A. Douglis, L. Nirenberg. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary value conditions I, Comm. Pure Appl. Math., 12 (1959).
Beavers.Joseph.1 G. S. Beavers, D. D. Joseph. Boundary condition at a naturally permeable wall. J. Fluid Mech. 30 (1967), 99. 197-207.
Bensoussan.1 A. Bensoussan, L. Boccardo. Nonlinear systems of elliptic equations with natural growth conditions and sign conditions. Appl. Math. Optim., 46:143-166 (2002).
Brezis.1 H. Brézis, J. L. Lions (eds). Nonlinear partial differential equations and their applications. V. 5, Collége de France seminar, in Research Notes in Mathematics, Pitman.
gallouet.1 T. Gallouët, A. Monier. On the regularity of Solutions to Elliptic Equations. Rendiconti di Matematica, Serie VII, Volume 19, Roma (1999), 471-488.
G.T.1 D. Gilbarg and N. S. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin, 1983.
grisvard-1 P. Grisvard. Singularities in boundary value problems. Masson, Springer-Verlag, 1992.
haslinger.1 J. Haslinger, R.A.E. Mäkinen. Introduction to Shape Optimization. Theory, Approximation and Computation. SIAM 2003.
henrot.pierre.1 A. Henrot, M. Pierre. Variation et optimisation de formes. Une analyse géometrique. Series: Mathématiques et Applications, Vol. 48, 2005, XII.
Jager.Mikelic.1 W. Jager, A. Mikelić. On the interface boundary condition of Beavers, Joseph and Saffman. SIAM, J. Appl. Math. Vol. 60, No. 4, pp. 1111-1127.
Jager.Mikelic.2 W. Jager, A. Mikelić. On the boundary conditions at the contact interface between a porous medium and a free fluid. Annali Scuola Norm. Super. Pisa Cl. Sci. (4), 23 (1996), pp. 403-465.
ladyz-1 O. L. Ladyzhenskaya. The Mathematical Theory of Viscous Incompressible Flow. New York: Gordon and Breach Science publishers (1963).
layton.1 W. J. Layton, F. Schieweck, I. Yotov. Coupling fluid flow with porous media flow. SIAM J. Numer. Anal., Vol. 40, No. 6, pp. 2195-2218.
lions-m-1 J. L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications I, Springer-Verlag, 1972.
lions-m-2 J. L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications II, Springer-Verlag, 1972.
riviere-1 B. Rivière, I. Yotov. Locally conservative coupling of Stokes and Darcy flows. SIAM Journal on Numerical Analysis, 42, no. 5, p. 1959-1977, 2005.
Saffman.1 P. G. Saffman. On the boundary condition at the surface of a porous medium. Studies in applied mathematics. Vol. L, No. 2, June 1971.
temam-1 R. Temam. Navier-Stokes Equations. Theory and Numerical Analysis. North-Holland Publishing Company (1977).
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Additional Information
Arian Novruzi
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, Canada
Address at time of publication:
Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave., Ottawa, ON, K1N 6N5, Canada
Email:
novruzi@uottawa.ca
Received by editor(s):
June 21, 2005
Published electronically:
November 15, 2006
Additional Notes:
This work was supported by NSERC Discovery Grants and by Ballard through the MITACS project Mathematical Modeling and Scientific Computation (MMSC)
Article copyright:
© Copyright 2006
Brown University
The copyright for this article reverts to public domain 28 years after publication.
