Finite Element Methods for the Temperature in Composite Media with Contact Resistance

详细信息    查看全文

• 作者：Faker Ben Belgacem (1)
  Christine Bernardi (2) (3)
  Faten Jelassi (4) (5)
  Maimouna Mint Brahim (4)
  
  1. UTC ; EA 2222 ; LMAC ; Sorbonne Universit茅s ; 60205 ; Compi猫gne ; France
  2. CNRS ; UMR 7598 ; Laboratoire Jacques-Louis Lions ; 75005 ; Paris ; France
  3. UPMC Univ Paris 06 ; UMR 7598 ; LJLI ; Sorbonne Universit茅s ; 75005 ; Paris ; France
  4. UB1-I2M UMR CNRS 5295 ; Universit茅 de Bordeaux ; 33405 ; Talence ; France
  5. ENIT-LAMSIN ; BP 37 ; 1002 ; Tunis ; Tunisia
  
• 关键词：Composite medium ; Temperature contact resistance ; Finite elements
• 刊名：Journal of Scientific Computing
• 出版年：2015
• 出版时间：May 2015
• 年：2015
• 卷：63
• 期：2
• 页码：478-501
• 全文大小：2,611 KB
• 参考文献：1. Adams, RA, Fournier, J (2003) Sobolev Spaces. Academic Press, London
  2. Arnold, DN, Brezzi, F, Cackburn, B, Marini, LD (2002) Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39: pp. 1749-1779 CrossRef
  3. Bernard, J-M (2011) Density results in Sobolev spaces whose elements vanish on a part of the boundary. Chin. Ann. Math. Ser. B 32: pp. 823-846 CrossRef
  4. Bernardi, C, Maday, Y, Patera, AT A new nonconforming approach to domain decomposition: the mortar element method. In: Brezis, H, Lions, J-L eds. (1990) Coll猫ge de France Seminar. Pitman, London
  5. Bernardi, C., Maday, Y., Rapetti, F.: Discr茅tisations variationnelles de probl猫mes aux limites elliptiques. Collection 鈥淢ath茅matiques et Applications鈥?45. Springer, Berlin (2004)
  6. Brenner, S, Scott, LR (2008) The Mathematical Theory of Finite Element Method, Texts in Applied Mathematics 15. Springer, Berlin CrossRef
  7. Brezzi, F, Fortin, M (1991) Mixed and Hybrid Finite Element Methods. Springer, Berlin CrossRef
  8. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978)
  9. Fletcher, L.S.: Conduction in solids, imperfect metal-to-metal contacts: thermal contact resistance, Section 502.5, Heat Transfer and Fluid Mechanics Data Books, Genium Publishing Company, Schenectady, New York (1991)
  10. Girault, V, Raviart, P-A (1986) Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Springer, Berlin CrossRef
  11. Grisvard, P (1985) Elliptic Problems in Nonsmooth Domains. Pitman, London
  12. Hecht, F.: Freefem \(_{++}\) . Third Edition, Version 3.30, http://www.freefem.org/ff++
  13. Hecht, F (2012) New development in freefem++. J. Numer. Math. 20: pp. 251-265 CrossRef
  14. Jelassi, F, Aza茂ez, M, Palomo Del Barrio, E (2013) A substructuring method for phase change modelling in hybrid media. Comput. Fluids 88: pp. 81-92 CrossRef
  15. Jerison, D, Kenig, CE (1981) The Neumann problem on Lipschitz domains. Bull. Am. Math. Soc. 4: pp. 203-207 CrossRef
  16. Lions, J-L, Magenes, E (1968) Probl猫mes aux limites non homog猫nes et applications. Dunod, Paris
  17. Raviart, P.-A., Thomas, J.-M.: A mixed finite element method for second order elliptic problems. In: Mathematical Aspects of Finite Element Methods, Lecture Notes in Math., Vol. 606. Springer, Berlin, pp. 292鈥?15 (1977)
  18. Roberts, J.E., Thomas, J.-M.: Mixed and Hybrid Methods, Handbook of Numerical Analysis. In: Ciarlet, P.G., Lions, J.-L. (eds.) Finite Element Methods (Part I), vol. II, pp. 523鈥?39. Elsevier Science Publishers, Amsterdam (1991)
  19. Safa, Y.: Simulation num茅rique des ph茅nom猫nes thermiques et magn茅tohydrodynamiques dans une cellule de Hall-H茅roult. Ph. D, Ecole Polytechnique F茅d茅rale de Lauzanne (2005)
  20. Swartz, ET, Pohl, RO (1989) Thermal boundary resistance. Rev. Mod. Phys. 61: pp. 605 CrossRef
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algorithms
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Mathematical and Computational Physics
  
• 出版者：Springer Netherlands
• ISSN：1573-7691

文摘

We consider a heat diffusion problem inside a composite medium. The contact resistance at the interface of constitutive materials allows for jumps of the temperature field. The transmission conditions need to be handled carefully and efficiently. The main concerns are accuracy and feasibility. Hybrid dual formulations are recommended here as the most popular mixed finite elements well adapted to account for the discontinuity of the temperature field. We therefore write the discretization of the heat problem by mixed finite elements and perform its numerical analysis. Of course, applying Lagrangian finite elements is possible in simple composite media but it turns out to be problematic for complex geometries. Nevertheless, we study the convergence of this finite element method to highlight some particularities related to the model under consideration and point out the effect of the contact resistance on the accuracy. Illustrative numerical experiments are finally provided to assess the theoretical findings.