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Meccanica

, Volume 53, Issue 6, pp 1357–1401 | Cite as

A discontinuous Galerkin method for non-linear electro-thermo-mechanical problems: application to shape memory composite materials

  • Lina Homsi
  • Ludovic NoelsEmail author
Novel Computational Approaches to Old and New Problems in Mechanics

Abstract

A coupled electro-thermo-mechanical discontinuous Galerkin (DG) method is developed considering the non-linear interactions of electrical, thermal, and mechanical fields. In order to develop a stable discontinuous Galerkin formulation the governing equations are expressed in terms of energetically conjugated fields gradients and fluxes. Moreover, the DG method is formulated in finite deformations and finite fields variations. The multi-physics DG formulation is shown to satisfy the consistency condition, and the uniqueness and optimal convergence rate properties are derived under the assumption of small deformation. First the numerical properties are verified on a simple numerical example, and then the framework is applied to simulate the response of smart composite materials in which the shape memory effect of the matrix is triggered by the Joule effect.

Keywords

Discontinuous Galerkin method Electro-thermo-mechanics Non-linear elliptic problems Smart composites 

Notes

Acknowledgements

This project has been funded with support of the European Commission under the grant number 2012-2624/001-001-EM. This publication reflects the view only of the author, and the Commission cannot be held responsible for any use which may be made of the information contained therein. Computational resources have been provided by the supercomputing facilities of the “Consortium des Équipements de Calcul Intensif en Fédération Wallonie Bruxelles (CÉCI)” funded by the “Fond de la Recherche Scientifique de Belgique (FRS-FNRS)”.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Ainsworth M, Kay D (1999) The approximation theory for the p-version finite element method and application to non-linear elliptic pdes. Numer Math 82(3):351–388MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ainsworth M, Kay D (2000) Approximation theory for the hp-version finite element method and application to the non-linear laplacian. Applied numerical mathematics 34(4):329–344MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Amestoy P, Duff I, L’Excellent JY (2000) Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput Methods Appl Mech Eng 184(2):501–520. doi: 10.1016/S0045-7825(99)00242-X ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Anand L, On H (1979) Hencky’s approximate strain-energy function for moderate deformations. J Appl Mech 46:78–82. doi: 10.1115/1.3424532 CrossRefzbMATHGoogle Scholar
  5. 5.
    Arnold DN, Brezzi F, Cockburn B, Marini LD (2002) Unified analysis of discontinuous galerkin methods for elliptic problems. SIAM J Numer Anal 39(5):1749–1779MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Arnold DN, Brezzi F, Marini LD (2005) A family of discontinuous galerkin finite elements for the reissner-mindlin plate. J Sci Comput 22–23:25–45MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Babuška I, Suri M (1987) The \(hp\) version of the finite element method with quasiuniform meshes. RAIRO-Modélisation mathématique et analyse numérique 21(2):199–238MathSciNetzbMATHGoogle Scholar
  8. 8.
    Becker G, Noels L (2013) A full-discontinuous galerkin formulation of nonlinear kirchhofflove shells: elasto-plastic finite deformations, parallel computation, and fracture applications. Int J Numer Meth Eng 93(1):80–117. doi: 10.1002/nme.4381 CrossRefzbMATHGoogle Scholar
  9. 9.
    Behl M, Lendlein A (2007) Shape-memory polymers. Mater Today 10(4):20–28CrossRefGoogle Scholar
  10. 10.
    Bonet J, Burton A (1998) A simple orthotropic, transversely isotropic hyperelastic constitutive equation for large strain computations. Comput Methods Appl Mech Eng 162(1):151–164ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    Chung DD (1994) CHAPTER 4—properties of carbon fibers. In: Chung DD (ed) Carbon fiber composites. Butterworth-Heinemann, Boston, pp. 65–78. doi: 10.1016/B978-0-08-050073-7.50008-7. URL www.sciencedirect.com/science/article/pii/B9780080500737500087
  12. 12.
    Ciarlet P (2002) Conforming finite element methods for second-order problems, chapter 3, pp. 110–173. SIAM. doi: 10.1137/1.9780898719208.ch3
  13. 13.
    Cockburn B, Karniadakis GE, Shu CW (2000) The development of discontinuous Galerkin methods. Springer, New YorkCrossRefzbMATHGoogle Scholar
  14. 14.
    Culebras M, Gómez CM, Cantarero A (2014) Review on polymers for thermoelectric applications. Materials 7(9):6701–6732ADSCrossRefGoogle Scholar
  15. 15.
    Douglas J, Dupont T (1976) Interior penalty procedures for elliptic and parabolic Galerkin methods. Springer, Berlin, pp 207–216. doi: 10.1007/BFb0120591 Google Scholar
  16. 16.
    Engel G, Garikipati K, Hughes TJR, Larson MG, Mazzei L, Taylor RL (2002) Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput Methods Appl Mech Eng 191(34):3669–3750ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fabré M, Binst J, Bocsan I, De Smet C, Ivens J (2012) Heating shape memory polymers with alternative ways: microwave and direct electrical heating. In: 15th European Conference on composite materials. University of Padova, pp. 1–8Google Scholar
  18. 18.
    Ferreira ADBL, Nóvoa PRO, Torres Marques A (2016) Multifunctional material systems: a state-of-the-art review. Compos Struct 151:3–35. doi: 10.1016/j.compstruct.2016.01.028. Smart composites and composite structures In honour of the 70th anniversary of Professor Carlos Alberto Mota Soares
  19. 19.
    Georgoulis EH (2003) Discontinuous Galerkin methods on shape-regular and anisotropic meshes. University of Oxford D. Phil, ThesisGoogle Scholar
  20. 20.
    Geuzaine C, Remacle JF (2009) Gmsh: a 3-d finite element mesh generator with built-in pre- and post-processing facilities. Int J Numer Meth Eng 79(11):1309–1331. doi: 10.1002/nme.2579 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gilbarg D, Trudinger NS (2015) Elliptic partial differential equations of second order. Springer, New YorkzbMATHGoogle Scholar
  22. 22.
    Gudi T (2006) Discontinuous Galerkin methods for nonlinear elliptic problems. Ph.D. thesis, Indian Institute of Technology, BombayGoogle Scholar
  23. 23.
    Gudi T, Nataraj N, Pani AK (2008) hp-Discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems. Numer Math 109(2):233–268MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hansbo P, Larson MG (2002) A discontinuous Galerkin method for the plate equation. Calcolo 39(1):41–59MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hansbo P, Larson MG (2002) Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method. Comput Methods Appl Mech Eng 191(17–18):1895–1908ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Homsi L (2017) Development of non-linear electro-thermo-mechanical discontinuous Galerkin formulations. Ph.D. thesis, University of Liège, BelgiumGoogle Scholar
  27. 27.
    Homsi L, Geuzaine C, Noels L (2017) A coupled electro-thermal discontinuous Galerkin method. J Comput Phys 348:231–258ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Houston P, Robson J, Süli E (2005) Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: the scalar case. IMA J Numer Anal 25(4):726–749MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Huang W, Yang B, An L, Li C, Chan Y (2005) Water-driven programmable polyurethane shape memory polymer: demonstration and mechanism. Appl Phys Lett 86(11):114,105CrossRefGoogle Scholar
  30. 30.
    Issi JP (2003) Electronic and thermal properties of carbon fibers. World of Carbon. CRC Press, Boca Raton, pp 207–216. doi: 10.1201/9780203166789.ch3 Google Scholar
  31. 31.
    Karypis G, Kumar V (1998) A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J Sci Comput 20(1):359–392MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kaufmann P, Martin S, Botsch M, Gross M (2009) Flexible simulation of deformable models using discontinuous galerkin fem. Graphical Models 71(4):153–167. doi: 10.1016/j.gmod.2009.02.002. http://www.sciencedirect.com/science/article/pii/S15240703090%00125. Special Issue of ACM SIGGRAPH / Eurographics Symposium on Computer Animation 2008
  33. 33.
    Keith JM, Janda NB, King JA, Perger WF, Oxby TJ (2005) Shielding effectiveness density theory for carbon fiber/nylon 6, 6 composites. Polym Compos 26(5):671–678CrossRefGoogle Scholar
  34. 34.
    Langer R, Tirrell DA (2004) Designing materials for biology and medicine. Nature 428(6982):487–492ADSCrossRefGoogle Scholar
  35. 35.
    Lendlein A, Jiang H, Jünger O, Langer R (2005) Light-induced shape-memory polymers. Nature 434(7035):879–882ADSCrossRefGoogle Scholar
  36. 36.
    Leng J, Lan X, Liu Y, Du S (2011) Shape-memory polymers and their composites: stimulus methods and applications. Prog Mater Sci 56(7):1077–1135CrossRefGoogle Scholar
  37. 37.
    Liang C, Rogers C, Malafeew E (1997) Investigation of shape memory polymers and their hybrid composites. J Intell Mater Syst Struct 8(4):380–386CrossRefGoogle Scholar
  38. 38.
    Liu L (2012) A continuum theory of thermoelectric bodies and effective properties of thermoelectric composites. Int J Eng Sci 55:35–53MathSciNetCrossRefGoogle Scholar
  39. 39.
    Liu R, Wheeler M, Dawson C (2009) A three-dimensional nodal-based implementation of a family of discontinuous galerkin methods for elasticity problems. Comput Struct 87(3–4):141–150. doi: 10.1016/j.compstruc.2008.11.009 CrossRefGoogle Scholar
  40. 40.
    Liu R, Wheeler M, Dawson C, Dean R (2009) Modeling of convection-dominated thermoporomechanics problems using incomplete interior penalty Galerkin method. Comput Methods Appl Mech Eng 198(9–12):912–919. doi: 10.1016/j.cma.2008.11.012 ADSCrossRefzbMATHGoogle Scholar
  41. 41.
    Lu H, Liu Y, Leng J, Du S (2010) Qualitative separation of the physical swelling effect on the recovery behavior of shape memory polymer. Eur Polymer J 46(9):1908–1914CrossRefGoogle Scholar
  42. 42.
    Mahan GD (2000) Density variations in thermoelectrics. J Appl Phys 87(10):7326–7332ADSCrossRefGoogle Scholar
  43. 43.
    McBride A, Reddy B (2009) A discontinuous Galerkin formulation of a model of gradient plasticity at finite strains. Comput Methods Appl Mech Eng 198(21–26):1805–1820. doi: 10.1016/j.cma.2008.12.034. Advances in Simulation-Based Engineering Sciences Honoring J. Tinsley Oden
  44. 44.
    Meng H, Li G (2013) A review of stimuli-responsive shape memory polymer composites. Polymer 54(9):2199–2221CrossRefGoogle Scholar
  45. 45.
    Meng Q, Hu J (2009) A review of shape memory polymer composites and blends. Compos A Appl Sci Manuf 40(11):1661–1672CrossRefGoogle Scholar
  46. 46.
    Miehe C (1994) Aspects of the formulation and finite element implementation of large strain isotropic elasticity. Int J Numer Meth Eng 37(12):1981–2004MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Muliana A, Li KA (2010) Time-dependent response of active composites with thermal, electrical, and mechanical coupling effect. Int J Eng Sci 48(11):1481–1497CrossRefGoogle Scholar
  48. 48.
    Noels L (2009) A discontinuous galerkin formulation of non-linear Kirchhoff–Love shells. Int J Numer Meth Eng 78(3):296–323MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Noels L, Radovitzky R (2006) A general discontinuous Galerkin method for finite hyperelasticity. Formulation and numerical applications. Int J Numer Meth Eng 68(1):64–97MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Noels L, Radovitzky R (2008) An explicit discontinuous Galerkin method for non-linear solid dynamics: formulation, parallel implementation and scalability properties. Int J Numer Meth Eng 74(9):1393–1420MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Pilate F, Toncheva A, Dubois P, Raquez JM (2016) Shape-memory polymers for multiple applications in the materials world. Eur Polymer J 80:268–294CrossRefGoogle Scholar
  52. 52.
    Prudhomme S, Pascal F, Oden J, Romkes A (2000) Review of a priori error estimation for discontinuous Galerkin methods. Technical report, TICAM, UTexasGoogle Scholar
  53. 53.
    Reed W, Hill T (1973) Triangular mesh methods for the neutron transport equation. Technical report LA-UR-73-479, Los Alamos Scientific Laboratory. http://www.osti.gov/scitech/servlets/purl/4491151
  54. 54.
    Romkes A, Prudhomme S, Oden J (2003) A priori error analyses of a stabilized discontinuous Galerkin method. Comput Math Appl 46(8):1289–1311MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Rothe S, Schmidt JH, Hartmann S (2015) Analytical and numerical treatment of electro-thermo-mechanical coupling. Arch Appl Mech 85(9–10):1245–1264CrossRefzbMATHGoogle Scholar
  56. 56.
    Schmidt AM (2006) Electromagnetic activation of shape memory polymer networks containing magnetic nanoparticles. Macromol Rapid Commun 27(14):1168–1172CrossRefGoogle Scholar
  57. 57.
    Spencer A (1982) The formulation of constitutive equations for anisotropic solids. Nijhoff, Amsterdam, pp. 3–26zbMATHGoogle Scholar
  58. 58.
    Spencer AJM (1986) Modelling of finite deformations of anisotropic materials. In: John G, Joseph Z, Siavouche N-N (eds) Large deformations of solids: physical basis and mathematical modelling. Springer, Dordrecht, pp 41–52Google Scholar
  59. 59.
    Srivastava V, Chester SA, Anand L (2010) Thermally actuated shape-memory polymers: experiments, theory, and numerical simulations. J Mech Phys Solids 58(8):1100–1124ADSCrossRefzbMATHGoogle Scholar
  60. 60.
    Sun S, Wheeler MF (2005) Discontinuous Galerkin methods for coupled flow and reactive transport problems. Appl Numer Math 52(2):273–298MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Ten Eyck A, Celiker F, Lew A (2008) Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity: motivation, formulation, and numerical examples. Comput Methods Appl Mech Eng 197(45–48):3605–3622. doi: 10.1016/j.cma.2008.02.020 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Ten Eyck A, Lew A (2006) Discontinuous Galerkin methods for non-linear elasticity. Int J Numer Meth Eng 67(9):1204–1243. doi: 10.1002/nme.1667 MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Truster TJ, Chen P, Masud A (2015) Finite strain primal interface formulation with consistently evolving stabilization. Int J Numer Meth Eng 102(3–4):278–315. doi: 10.1002/nme.4763 MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Truster TJ, Chen P, Masud A (2015) On the algorithmic and implementational aspects of a discontinuous Galerkin method at finite strains. Comput Math Appl 70(6):1266–1289. doi: 10.1016/j.camwa.2015.06.035. http://www.sciencedirect.com/science/article/pii/S08981221150%03211
  65. 65.
    Vilčáková J, Sáha P, Quadrat O (2002) Electrical conductivity of carbon fibres/polyester resin composites in the percolation threshold region. Eur Polymer J 38(12):2343–2347CrossRefGoogle Scholar
  66. 66.
    Wells GN, Dung NT (2007) AC 0 discontinuous Galerkin formulation for Kirchhoff plates. Comput Methods Appl Mech Eng 196(35–36):3370–3380ADSCrossRefzbMATHGoogle Scholar
  67. 67.
    Wells GN, Garikipati K, Molari L (2004) A discontinuous Galerkin formulation for a strain gradient-dependent damage model. Comput Methods Appl Mech Eng 193(33–35):3633–3645ADSCrossRefzbMATHGoogle Scholar
  68. 68.
    Wen S, Chung D (1999) Seebeck effect in carbon fiber-reinforced cement. Cem Concr Res 29(12):1989–1993CrossRefGoogle Scholar
  69. 69.
    Wheeler MF (1978) An elliptic collocation-finite element method with interior penalties. SIAM J Numer Anal 15(1):152–161MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Wu L, Tjahjanto D, Becker G, Makradi A, Jérusalem A, Noels L (2013) A micro-meso-model of intra-laminar fracture in fiber-reinforced composites based on a discontinuous galerkin/cohesive zone method. Eng Fract Mech 104:162–183CrossRefGoogle Scholar
  71. 71.
    Yadav S, Pani A, Park EJ (2013) Superconvergent discontinuous Galerkin methods for nonlinear elliptic equations. Math Comput 82(283):1297–1335MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Yang Y, Xie S, Ma F, Li J (2012) On the effective thermoelectric properties of layered heterogeneous medium. J Appl Phys 111(1):013,510CrossRefGoogle Scholar
  73. 73.
    Zheng XP, Liu DH, Liu Y (2011) Thermoelastic coupling problems caused by thermal contact resistance: a discontinuous Galerkin finite element approach. Sci China Phys Mech Astron 54(4):666–674ADSCrossRefGoogle Scholar
  74. 74.
    Zhupanska OI, Sierakowski RL (2011) Electro-thermo-mechanical coupling in carbon fiber polymer matrix composites. Acta Mech 218(3–4):319–332CrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Aerospace and Mechanical Engineering, Computational and Multiscale Mechanics of Materials (CM3)University of Liège, Quartier Polytech 1LiègeBelgium

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