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PGD-Based Computational Homogenization

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PGD-Based Modeling of Materials, Structures and Processes

Part of the book series: ESAFORM Bookseries on Material Forming ((EBMF))

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Abstract

Homogenization approaches are now intensively used in numerous engineering applications.

Calculemus! (Let us calculate!)

—Gottfried Wilhelm Leibniz.

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References

  1. J. Fish, Bridging the scales in nano engineering and science. J. Nanopart. Res. 8(5), 577–594 (2006)

    Article  Google Scholar 

  2. R. De Borst, Challenges in computational materials science: multiple scales, multi-physics and evolving discontinuities. Comput. Mater. Sci. 43, 1–15 (2008)

    Google Scholar 

  3. C. McVeigh, W.K. Liu, Linking microstructure and properties through a predictive multiresolution continuum. Comput. Methods Appl. Mech. Eng. 197(41–42), 3268–3290 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. T. Zohdi, P. Wriggers, An Introduction to Computational Micromechanics (Springer, Berlin, 2005)

    Book  Google Scholar 

  5. H.J. Bőhm, A Short Introduction to Basic Aspects of Continuum Micromechanics, ISLB Report 208 (TU Wien, Vienna, 2009)

    Google Scholar 

  6. T.J.R. Hughes, Multiscale phenomena: green’s functions, the dirichlet-to-neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Eng. 127(1–4), 387–401 (1995)

    Article  MATH  Google Scholar 

  7. J. Fish, Z. Yuan, Multiscale enrichment based on partition of unity. Int. J. Numer. Meth. Eng. 62(10), 1341–1359 (2005)

    Article  MATH  Google Scholar 

  8. T. Strouboulis, L. Zhang, I. Babuška, P-version of the generalized fem using mesh-based handbooks with applications to multiscale problems. Int. J. Numer. Meth. Eng. 60(10), 1639–1672 (2004)

    Google Scholar 

  9. F. Feyel, Multiscale fe2 elastoviscoplastic analysis of composite structures. Comput. Mater. Sci. 16(1–4), 344–354 (1999)

    Article  Google Scholar 

  10. M.G.D. Geers, V.G. Kouznetsova, W.A.M. Brekelmans, Multi-scale computational homogenization: trends and challenges. J. Comput. Appl. Math. (2009 in press). doi:10.1016/j.cam.2009.08.077

  11. J.C. Michel, P. Suquet, Computational analysis of nonlinear composite structures using the nonuniform transformation field analysis. Comput. Methods Appl. Mech. Eng. 193(48–51), 5477–5502 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. I. Temizer, P. Wriggers, An adaptive method for homogenization in orthotropic nonlinear elasticity. Comput. Methods Appl. Mech. Eng. 196(35–36), 3409–3423 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. I. Temizer, T. Zohdi, A numerical method for homogenization in non-linear elasticity. Comput. Mech. 40(2), 281–298 (2007)

    Article  MATH  Google Scholar 

  14. J. Yvonnet, D. Gonzalez, Q.-C. He, Numerically explicit potentials for the homogenization of nonlinear elastic heterogeneous materials. Comput. Methods Appl. Mech. Eng. 198(33–36), 2723–2737 (2009)

    Article  MATH  Google Scholar 

  15. T. Kanit, S. Forest, I. Galliet, V. Mounoury, D. Jeulin, Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Solids Struct. 40(13–14), 3647–3679 (2003)

    Article  MATH  Google Scholar 

  16. T. Kanit, F. N’Guyen, S. Forest, D. Jeulin, M. Reed, S. Singleton, Apparent and effective physical properties of heterogeneous materials: representativity of samples of two materials from food industry. Comput. Methods Appl. Mech. Eng. 195(33–36), 3960–3982 (2006)

    Article  MATH  Google Scholar 

  17. M. Ostoja-Starzewski, Material spatial randomness: from statistical to representative volume element. Probab. Eng. Mech. 21(2), 112–132 (2006)

    Article  Google Scholar 

  18. K. Sab, On the homogenization and the simulation of random materials. Eur. J. Mech. A. Solids 11(5), 585–607 (1992)

    MATH  MathSciNet  Google Scholar 

  19. M. Jiang, I. Jasiuk, M. Ostoja-Starzewski, Apparent thermal conductivity of periodic two-dimensional composites. Comput. Mater. Sci. 25(3), 329–338 (2002)

    Article  Google Scholar 

  20. I. Ozdemir, W.A.M. Brekelmans, M.G.D. Geers, Computational homogenization for heat conduction in heterogeneous solids. Int. J. Numer. Meth. Eng. 73(2), 185–204 (2008)

    Article  MathSciNet  Google Scholar 

  21. P. Arbenz, G.H. van Lenthe, U. Mennel, R. Műller, M. Sala, A scalable multi-level preconditioner for matrix-free \(\mu \)-finite element analysis of human bone structures. Int. J. Numer. Meth. Eng. 73(7), 927–947 (2008)

    Article  MATH  Google Scholar 

  22. T.G. Kolda, B.W. Bader, Tensor decompositions and applications. Technical Report (SAND2007-6702), SANDIA National Laboratories, Nov 2007

    Google Scholar 

  23. A. Ammar, F. Chinesta, P. Joyot, The nanometric and micrometric scales of the structure and mechanics of materials revisited: an introduction to the challenges of fully deterministic numerical descriptions. Int. J. Multiscale Comput. Eng. 6(3), 191–213 (2008)

    Article  Google Scholar 

  24. F. Chinesta, G. Chaidron, On the steady solution of linear advection problems in steady recirculating flows. J. NonNewton. Fluid Mech. 98, 65–80 (2001)

    Article  MATH  Google Scholar 

  25. G. Chaidron, F. Chinesta, On the steady solution of non-linear advection problems in steady recirculating flows. Comput. Methods Appl. Mech. Eng. 191, 1159–1172 (2002)

    Article  MATH  Google Scholar 

  26. F. Chinesta, G. Chaidron, A. Poitou, On the solution of the fokker-planck equations in steady recirculating flows involving short fiber suspensions. J. NonNewton. Fluid Mech. 113(2–3), 97–125 (2003)

    Article  MATH  Google Scholar 

  27. A. Ammar, B. Mokdad, F. Chinesta, R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. part ii. J. NonNewton. Fluid Mech. 144, 98–121 (2007)

    Article  MATH  Google Scholar 

  28. S. Tokarzewski, I. Andrianov, Effective coefficients for real non-linear and fictitious linear temperature-dependent periodic composites. Int. J. Non-Linear Mech. 36(1), 187–195 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  29. J.J. Telega, S. Tokarzewski, A. Galka, Effective conductivity of nonlinear two-phase media: homogenization and two-point pade approximants. Acta Appl. Math. 61(1), 295–315 (2000)

    Article  MathSciNet  Google Scholar 

  30. G. Laschet, Homogenization of the thermal properties of transpiration cooled multi-layer plates. Comput. Methods Appl. Mech. Eng. 191(41–42), 4535–4554 (2002)

    Article  MATH  Google Scholar 

  31. A. Alzina, E. Toussaint, A. Beakou, B. Skoczen, Multiscale modelling of thermal conductivity in composite materials for cryogenic structures. Compos. Struct. 74(2), 175–185 (2006)

    Article  Google Scholar 

  32. A. Mobasher-Amini, D. Dureisseix, P. Cartraud, Multi-scale domain decomposition method for large-scale structural analysis with a zooming technique: application to plate assembly. Int. J. Numer. Meth. Eng. 79(4), 417–443 (2009)

    Article  Google Scholar 

  33. P. Kanouté, D.P. Boso, J.L. Chaboche, B.A. Schrefler, Multiscale methods for composites: a review. Arch. Computat. Methods Eng. 16(1), 31–75 (2009)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. UMR CNRS, Ecole Centrale de Nantes, Nantes, France

    Francisco Chinesta

  2. Aragon Institute of Engineering Research, Universidad de Zaragoza, Zaragoza, Spain

    Elías Cueto

Authors
  1. Francisco Chinesta
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  2. Elías Cueto
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Correspondence to Francisco Chinesta .

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Chinesta, F., Cueto, E. (2014). PGD-Based Computational Homogenization. In: PGD-Based Modeling of Materials, Structures and Processes. ESAFORM Bookseries on Material Forming. Springer, Cham. https://doi.org/10.1007/978-3-319-06182-5_3

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  • DOI: https://doi.org/10.1007/978-3-319-06182-5_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-06181-8

  • Online ISBN: 978-3-319-06182-5

  • eBook Packages: EngineeringEngineering (R0)

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