Skip to main content
SpringerLink
Log in

Multi-scale homogenization of moving interface problems with flux jumps: application to solidification

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

In this paper, a multi-scale analysis scheme for solidification based on two-scale computational homogenization is discussed. Solidification problems involve evolution of surfaces coupled with flux jump boundary conditions across interfaces. We provide consistent macro-micro transition and averaging rules based on Hill’s macro- homogeneity condition. The overall macro-scale behavior is analyzed with solidification at the micro-scale modeled using an enthalpy formulation. The method is versatile in the sense that two different models can be employed at the macro- and micro-scales. The micro-scale model can incorporate all the physics associated with solidification including moving interfaces and flux discontinuities, while the macro-scale model needs to only model thermal conduction using continuous (homogenized) fields. The convergence behavior of the tightly coupled macro-micro finite element scheme with respect to decreasing element size is analyzed by comparing with a known analytical solution of the Stefan problem.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Hashin Z (1983) Analysis of composite materials. J Appl Mech 50: 481–505

    Article  MATH  Google Scholar 

  2. Rosen BW, Hashin Z (1970) Effective thermal expansion coefficients and specific heats of composite materials. Int J Eng Sci 8: 157–173

    Article  Google Scholar 

  3. Noor AK, Shah RS (1993) Effective thermoelastic and thermal properties of unidirectional fiber-reinforced composites and their sensitivity coefficients. Comput Struct 26: 7–23

    Article  Google Scholar 

  4. Auriault JL (1983) Effective macroscopic description of heat conduction in periodic composites. Int J Heat Mass Transf 26(6): 861–869

    Article  MATH  Google Scholar 

  5. Boutin C (1995) Microstructural influence on heat conduction. Int J Heat Mass Transf 38(17): 3181–3195

    Article  MATH  Google Scholar 

  6. Guedes JM, Kikuchi N (1990) Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods. Comput Methods Appl Mech Eng 83: 143–198

    Article  MATH  MathSciNet  Google Scholar 

  7. Jiang M, Jasiuk I, Ostoja-Starzewski M (2002) Apparent thermal conductivity of periodic two-dimensional composites. Comput Mater Sci 25: 329–338

    Article  Google Scholar 

  8. Ostoja-Starzewski M, Schulte J (1996) Bounding of effective thermal conductivities of multiscale materials by essential and natural boundary conditions. Phys Rev B 54(1): 278–285

    Article  Google Scholar 

  9. Sigmund O, Torquato S (1997) Design of materials with extreme thermal expansion using a three-phase topology optimization. J Mech Phys Solids 45: 1037–1067

    Article  MathSciNet  Google Scholar 

  10. Wang CY, Beckermann C (1995) Equiaxed dendritic solidification with convection: part I. Multiscale/Multiphase modeling. Metallurgical Mater Trans A 25: 2754–2764

    Google Scholar 

  11. Eck C, Knabner P, Korotov S (2002) A two-scale method for the computation of solid-liquid phase transitions with dendritic microstructure. J Comput Phys 178: 58–80

    Article  MATH  MathSciNet  Google Scholar 

  12. Lee PD, Chirazi A, Atwood RC, Wang W (2004) Multiscale modelling of solidification microstructures, including microsegregation and microporosity, in an Al-Si-Cu alloy. Mater Sci Eng A 365: 57–65

    Article  Google Scholar 

  13. Tan L, Zabaras N (2007) Multiscale modeling of alloy solidification using a database approach. J Comput Phys 227: 728–754

    Article  MATH  MathSciNet  Google Scholar 

  14. Rafii-Tabar H, Chirazi A (2002) Multiscale computational modelling of solidification phenomena. Phys Rep 365: 145–249

    Article  MATH  Google Scholar 

  15. Gravemeier V, Lenz S, Wall WA (2008) Towards a taxonomy for multiscale methods in computational mechanics: building blocks of existing methods. Comput Mech 41: 279–291

    Article  MathSciNet  MATH  Google Scholar 

  16. Smit RJM, Brekelmans WAM, Meijer HEH (1998) Prediction of the mechanical behaviour of nonlinear heterogeneous systems by multi-level finite element modeling. Comput Methods Appl Mech Eng 155: 181–192

    Article  MATH  Google Scholar 

  17. Miehe C, Schroeder J, Schotte J (1999) Computational homogenization analysis in finite plasticity. Simulation of texture development in polycrystalline materials. Comput Methods Appl Mech Eng 171: 387–418

    Article  MATH  Google Scholar 

  18. Kouznetsova VG, Brekelmans WAM, Baaijens FPT (2001) An approach to micromacro modeling of heterogeneous materials. Comput Mech 27: 37–48

    Article  MATH  Google Scholar 

  19. Sundararaghavan V, Zabaras N (2006) Design of microstructure-sensitive properties in elasto-viscoplastic polycrystals using multi-scale homogenization. Int J Plast 22: 1799–1824

    Article  MATH  Google Scholar 

  20. Ozdemir I, Brekelmans WAM, Geers MGD (2008) FE 2 computational homogenization for the thermo-mechanical analysis of heterogeneous solids. Comput Methods Appl Mech Eng 198(3–4): 602–613

    Article  MathSciNet  Google Scholar 

  21. Ozdemir I, Brekelmans WAM, Geers MGD (2008) Computational homogenization for heat conduction in heterogeneous solids. Int J Numer Meth Eng 73: 185–204

    Article  MathSciNet  Google Scholar 

  22. Ostoja-Starzewski M (2002) Towards stochastic continuum thermodynamics. J Non Equilib Thermodyn 27: 335–348

    Article  MATH  Google Scholar 

  23. Miehe C (1996) Numerical computation of algorithmic (consistent) tangent moduli in large-strain computational inelasticity. Comput Meth Appl Mech Eng 134(3–4): 223–240

    Article  MATH  MathSciNet  Google Scholar 

  24. Balay S, Buschelman K, Eijkhout V, Gropp WD, Kaushik D, Knepley MG, McInnes LC, Smith BF, Zhang H (2004) PETSc users manual ANL-95/11—revision 2.1.5. Argonne National Laboratory

  25. Rubenstein LI (1971) The Stefan problem. Trans Math Monographs. American Mathematical Society (AMS), Providence, p 27

  26. Crank J (1984) Free and moving bounday problems. Clarendon Press, Oxford

    Google Scholar 

  27. Shyy W, Udayhumar HS, Rao MM, Smith RW (2007) Computational fluid dynamics with moving boundaries, 1st edn. Dover, Mineola

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI, 48109, USA

    Sangmin Lee & Veera Sundararaghavan

Authors
  1. Sangmin Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Veera Sundararaghavan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Veera Sundararaghavan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lee, S., Sundararaghavan, V. Multi-scale homogenization of moving interface problems with flux jumps: application to solidification. Comput Mech 44, 297–307 (2009). https://doi.org/10.1007/s00466-009-0371-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-009-0371-x

Keywords

Search

Navigation