Skip to main content
SpringerLink
Log in

An Extended Finite Element Method for the Analysis of Submicron Heat Transfer Phenomena

  • Chapter
  • First Online:
Multiscale Methods in Computational Mechanics

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 55))

Abstract

Manipulating the spatial layout of heterogeneous materials at submicron scales allows for the design of novel nano-engineered material with unique thermal properties. To analyze heat conduction at submicron scales of geometrically complex nano-structured materials, an extended finite element method (XFEM) is presented. Appropriate for both diffusive and ballistic domains, heat conduction is described by the phonon Boltzmann transport equations. Specifically, the gray phonon model is used along with a diffusive scattering model describing the transmission and reflection of phonons at material interfaces. The geometry of the material interfaces is described by a level-set approach. The phonon distribution is discretized in the velocity space by a discrete ordinate approach and in the spatial domain by a stabilized Galerkin finite element method. Discontinuities of the phonon distribution across material interfaces are captured via enriched shape functions. To enforce interface scattering conditions and boundary conditions, a stabilized Lagrange multiplier method is presented. The proposed method is verified through comparison with benchmark results. The utility of the XFEM approach is demonstrated through the thermal analysis of experimentally characterized material samples and computer-designed nano-composites.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Allaire, F. Jouve, and A.M. Toader. A level-set method for shape optimization. C.R. Math., 334:1125–1130, 2002.

    MATH  MathSciNet  Google Scholar 

  2. E. Bechet, N. Moes, and B. Wohlmuth. A stable Lagrange multiplier space for stiff interface conditions within the extended finite element method. Int. J. Numer. Meth. Engng., 78:931–954, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  3. T. Belytschko and T. Black. Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Methods Eng., 45:601–620, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  4. P.L. Bhatnagar, E.P. Gross, and M. Krook. A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev., 94(3):511–525, 1954.

    Article  MATH  Google Scholar 

  5. F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer-Verlag, 1991.

    Google Scholar 

  6. M.O. Bristeau, O. Pironneau, and R. Glowinski. On the numerical solution of nonlinear problems in fluid dynamics by least squares and finite element methods. (I) Least square formulations and conjugate gradient solution of the continuous problems. Comput. Methods Appl. Mech. Engrg., 17:619–657, 1979.

    Article  MathSciNet  Google Scholar 

  7. A.N. Brooks and T.J.R. Hughes. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier- Stokes equations. Comput. Methods Appl. Mech. Engrg., 32:199–259, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  8. G. Chen. Nonlocal and nonequilibrium heat conduction in the vicinity of nanoparticles. J. Heat Transfer, 118:539–545, 1996.

    Article  Google Scholar 

  9. G. Chen, M.S. Dresselhaus, G. Dresselhaus, J.-P. Fleurial, and T. Cailat. Recent developments in thermoelectric materials. Int. Mater. Rev., 48(1):45–66, 2003.

    Article  Google Scholar 

  10. J. Chessa, P. Smolinski, and T. Belytschko. The extended finite element method (XFEM) for solidification problems. Int. J. Numer. Methods Eng., 53:1959–1977, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  11. S.V. Criekingen. A 2-D/3-D Cartesian geometry non-comforming spherical harmonic neutron transport solver. Ann. Nucl. Energy, 34:177–187, 2007.

    Article  Google Scholar 

  12. J.E. Dolbow. An extended finite element method with discontinuous enrichment for applied mechanics. PhD Thesis, Northwestern University, 1999.

    Google Scholar 

  13. A. Evgrafov, K. Maute, R.G. Yang, and M.L. Dunn. Topology optimization for nano-scale heat transfer. Int. J. Numer. Meth. Engng, 77(285–300), 2009.

    Google Scholar 

  14. K. Fushinobu, A. Majumdar, and K. Hijikata. Heat generation and transport in submicron semiconductor devices. J. Heat Transfer, 117:25–31, February 1995.

    Article  Google Scholar 

  15. A. Gerstenberger and W.A. Wall. An extended finite element method/Lagrange multiplier based approach for fluid-structure interaction. Comput.Methods Appl.Mech. Engrg., 197:1699–1714, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  16. R. Glowinski, T.-W. Pan, and T.I. Hesla. A distributed Lagrange multiplier/fictitious domain method for partiulate flows. Int. J. Multiphase Flow, 25:755–794, 1999.

    Article  MATH  Google Scholar 

  17. K.E. Goodson. Thermal conduction in nonhomogeneous CVD diamond layers in electronic microstructures. J. Heat Transfer, 118:279–286, 1996.

    Article  Google Scholar 

  18. T.C. Harman, P.J. Taylor, M.P. Walsh, and B.E. LaForge. Quantum dot superlattice thermoelectric materials and devices. Science, 297:2229–2232, 2002.

    Article  Google Scholar 

  19. M.-S. Jeng, R. Yang, D.W. Song, and G. Chen. Modeling the thermal conductivity and phonon transport in nanoparticle composites using Monte Carlo simulation. ASME J. Heat Transfer, 130(042410):1–11, 2008.

    Google Scholar 

  20. H. Ji and J.E. Dolbow. On strategies for enforcing interfacial constraints and evaluating jump conditions with the extended finite element method. Int. J. Numer. Meth. Engng., 61:2508–2535, 2004.

    Article  MATH  Google Scholar 

  21. G. Joshi, H. Lee, Y. Lan, X. Wang, G. Zhu, D. Wang, R. W. Gould, D. C. Cuff, M. Y. Tang, M. S. Dresselhaus, G. Chen, and Z. Ren. Enhanced thermoelectric figure-of-merit in nanostructured p-type silicon germanium bulk alloys. Nano Lett., 8(12):4670–4674, 2008.

    Article  Google Scholar 

  22. Y.S. Ju and K.E. Goodson. Phonon scattering in silicon films with thickness of order 100 nm. Appl. Phys. Lett., 74(20):3005–3007, 1999.

    Article  Google Scholar 

  23. S.R. Mathur and J.Y. Murthy. Radiative heat transfer in periodic geometries using a finite volume scheme. J. Heat Transfer, 121:357–364, May 1999.

    Article  Google Scholar 

  24. M.F. Modest. Radiative Heat Transfer. McGraw-Hill, 1993.

    Google Scholar 

  25. N. Moes, E. Bechet, and M. Tourbier. Imposing Dirichlet boundary conditions in the extended finite element method. Int. J. Numer. Meth. Engng., 67:1641–1669, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  26. N. Moes, J. Dolbow, and T. Belytschko. A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng., 46:131–150, 1999.

    Article  MATH  Google Scholar 

  27. J.Y. Murthy and S.R. Mathur. Computation of sub-micron thermal transport using an unstructured finite volume method. Trans. ASME, 124:1176–1181, 2002.

    Article  Google Scholar 

  28. S.V.J. Narumanchi, J.Y. Murthy, and C.H. Amon. Boltzmann transport equation-based thermal modeling approaches for hotspots in microelectronics. Heat Mass Transfer, 42:478–491, 2006.

    Article  Google Scholar 

  29. S. Osher and N. Paragios (Eds.), Geometric Level Set Methods in Imaging, Vision, and Graphics. Springer, 2003.

    Google Scholar 

  30. S.J. Osher and R. P. Fedkiw. Level Set Methods and Dynamic Implicit Surfaces. Springer, 2002.

    Google Scholar 

  31. S.J. Osther and J.A. Sethian. Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comp. Phys., 79:12–49, 1988.

    Article  Google Scholar 

  32. C.C. Pain, M.D. Eaton, R.P. Smedley-Stevenson, A.J.H. Goddard, M.D. Piggott, and C.R.E. de Oliveira. Streamline upwind Petrov-Galerkin methods for the steady-state Boltzmann transport equation. Comput. Methods Appl. Mech. Engrg., 195:4448–4472, 2006.

    Article  MATH  Google Scholar 

  33. C.S. Peskin. The immersed boundary method. Acta Numer., 2:479–517, 2002.

    Article  MathSciNet  Google Scholar 

  34. J.A. Sethian. Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge University Press, 1999.

    Google Scholar 

  35. M. Sussman, P. Smereka, and S. Osher. A level set approach for computing solutions to incompressible two-phase flow. J. Comp. Phys., 114:146–159, 1994.

    Article  MATH  Google Scholar 

  36. E.T. Swartz and R.O. Pohl. Thermal boundary resistance. Rev. Mod. Phys., 61(3):606–668, 1989.

    Article  Google Scholar 

  37. W. Tian and R. Yang. Phonon transport and thermal conductivity percolation in random nanoparticle composites. Comput. Model Eng. Sci., 24:123–141, 2008.

    Google Scholar 

  38. G.J. Wagner, N. Moes, W.K. Liu, and T. Belytschko. The extended finite element method for rigid particles in Stokes flow. Int. J. Numer. Methods Eng., 51:293–313, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  39. R. Yang and G. Chen. Thermal conductivity modeling of periodic two-dimensional nanocmposites. Phys. Rev. B, 69(195316):1–10, 2004.

    Google Scholar 

  40. R. Yang, G. Chen, M. Laroche, and Y. Taur. Multidimensional transient heat conduction at nanoscale using the ballistic-diffusive equations and the Boltzmann equation. ASME J. Heat Transfer, 127:298–306, 2005.

    Article  Google Scholar 

  41. O.C. Zienkiewicz, R.L. Taylor, S.J. Sherwin, and J. Peiro. On discontinuous Galerkin methods. Int. J. Numer. Meth. Engng., 58:1119–1148, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  42. J.M. Ziman. Electrons and Phonons. Oxford University Press, 1960.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, UCB 427, University of Colorado, Boulder, CO, 80309-0427, USA

    Pilhwa Lee, Ronggui Yang & Kurt Maute

  2. Center for Aerospace Structures, Department of Aerospace Engineering Sciences, UCB 429, University of Colorado, Boulder, CO, 80309-0429, USA

    Kurt Maute

Authors
  1. Pilhwa Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ronggui Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kurt Maute
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pilhwa Lee .

Editor information

Editors and Affiliations

  1. Dept. Mechanical Engineering, Eindhoven University of Technology, Eindhoven, Netherlands

    René de Borst

  2. Inst. Mechanik, Lehrstuhl II, Univ. Stuttgart, Pfaffenwaldring 7, Stuttgart, 70569, Germany

    Ekkehard Ramm

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media B.V.

About this chapter

Cite this chapter

Lee, P., Yang, R., Maute, K. (2011). An Extended Finite Element Method for the Analysis of Submicron Heat Transfer Phenomena. In: de Borst, R., Ramm, E. (eds) Multiscale Methods in Computational Mechanics. Lecture Notes in Applied and Computational Mechanics, vol 55. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9809-2_11

Download citation

  • DOI: https://doi.org/10.1007/978-90-481-9809-2_11

  • Published:

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-9808-5

  • Online ISBN: 978-90-481-9809-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation