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

Homsi, Lina; Noels, Ludovic

doi:10.1007/s11012-017-0743-9

Article (Scientific journals)

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

Homsi, Lina; Noels, Ludovic

2018 • In Meccanica, 53 (6), p. 1357-1401

Peer reviewed

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https://hdl.handle.net/2268/214128

DOI
10.1007/s11012-017-0743-9

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Keywords :

Discontinuous Galerkin method; Electro-thermo-mechanics; Nonlinear elliptic problems; Smart composites; CECI; LIMARC

Abstract :

[en] 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.

Research center :

A&M - Aérospatiale et Mécanique - ULiège

Disciplines :

Mechanical engineering
Materials science & engineering

Author, co-author :

Homsi, Lina ; Université de Liège - ULiège > Form. doct. sc. ingé. & techno. (aéro. & mécan. - Paysage)

Noels, Ludovic ; Université de Liège > Département d'aérospatiale et mécanique > Computational & Multiscale Mechanics of Materials (CM3)

Language :

English

Title :

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

Publication date :

April 2018

Journal title :

Meccanica

ISSN :

0025-6455

eISSN :

1572-9648

Publisher :

Kluwer Academic Publishers, Netherlands

Special issue title :

Novel computational approaches to old and new problems in mechanics

Volume :

Issue :

Pages :

1357-1401

Peer reviewed :

Peer reviewed

Tags :

CÉCI : Consortium des Équipements de Calcul Intensif

Additional URL :

http://dx.doi.org/10.1007/s11012-017-0743-9

Name of the research project :

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 Equipements de Calcul Intensif en Federation Wallonie Bruxelles (CECI) funded by the Fond de la Recherche Scientifique de Belgique (FRS-FNRS).

Funders :

UE - Union Européenne [BE]
F.R.S.-FNRS - Fonds de la Recherche Scientifique [BE]
CÉCI - Consortium des Équipements de Calcul Intensif [BE]

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since 11 September 2017

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Bibliography

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):351–388
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):329–344
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
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
Arnold DN, Brezzi F, Cockburn B, Marini LD (2002) Unified analysis of discontinuous galerkin methods for elliptic problems. SIAM J Numer Anal 39(5):1749–1779
Arnold DN, Brezzi F, Marini LD (2005) A family of discontinuous galerkin finite elements for the reissner-mindlin plate. J Sci Comput 22–23:25–45
Babuška I, Suri M (1987) The hp version of the finite element method with quasiuniform meshes. RAIRO-Modélisation mathématique et analyse numérique 21(2):199–238
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
Behl M, Lendlein A (2007) Shape-memory polymers. Mater Today 10(4):20–28
Bonet J, Burton A (1998) A simple orthotropic, transversely isotropic hyperelastic constitutive equation for large strain computations. Comput Methods Appl Mech Eng 162(1):151–164
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
Ciarlet P (2002) Conforming finite element methods for second-order problems, chapter 3, pp. 110–173. SIAM. doi:10.1137/1.9780898719208.ch3
Cockburn B, Karniadakis GE, Shu CW (2000) The development of discontinuous Galerkin methods. Springer, New York
Culebras M, Gómez CM, Cantarero A (2014) Review on polymers for thermoelectric applications. Materials 7(9):6701–6732
Douglas J, Dupont T (1976) Interior penalty procedures for elliptic and parabolic Galerkin methods. Springer, Berlin, pp 207–216. doi:10.1007/BFb0120591
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):3669–3750
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. 1–8
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
Georgoulis EH (2003) Discontinuous Galerkin methods on shape-regular and anisotropic meshes. University of Oxford D. Phil, Thesis
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
Gilbarg D, Trudinger NS (2015) Elliptic partial differential equations of second order. Springer, New York
Gudi T (2006) Discontinuous Galerkin methods for nonlinear elliptic problems. Ph.D. thesis, Indian Institute of Technology, Bombay
Gudi T, Nataraj N, Pani AK (2008) hp-Discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems. Numer Math 109(2):233–268
Hansbo P, Larson MG (2002) A discontinuous Galerkin method for the plate equation. Calcolo 39(1):41–59
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):1895–1908
Homsi L (2017) Development of non-linear electro-thermo-mechanical discontinuous Galerkin formulations. Ph.D. thesis, University of Liège, Belgium
Homsi L, Geuzaine C, Noels L (2017) A coupled electro-thermal discontinuous Galerkin method. J Comput Phys 348:231–258
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):726–749
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):114,105
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
Karypis G, Kumar V (1998) A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J Sci Comput 20(1):359–392
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
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):671–678
Langer R, Tirrell DA (2004) Designing materials for biology and medicine. Nature 428(6982):487–492
Lendlein A, Jiang H, Jünger O, Langer R (2005) Light-induced shape-memory polymers. Nature 434(7035):879–882
Leng J, Lan X, Liu Y, Du S (2011) Shape-memory polymers and their composites: stimulus methods and applications. Prog Mater Sci 56(7):1077–1135
Liang C, Rogers C, Malafeew E (1997) Investigation of shape memory polymers and their hybrid composites. J Intell Mater Syst Struct 8(4):380–386
Liu L (2012) A continuum theory of thermoelectric bodies and effective properties of thermoelectric composites. Int J Eng Sci 55:35–53
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
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
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):1908–1914
Mahan GD (2000) Density variations in thermoelectrics. J Appl Phys 87(10):7326–7332
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
Meng H, Li G (2013) A review of stimuli-responsive shape memory polymer composites. Polymer 54(9):2199–2221
Meng Q, Hu J (2009) A review of shape memory polymer composites and blends. Compos A Appl Sci Manuf 40(11):1661–1672
Miehe C (1994) Aspects of the formulation and finite element implementation of large strain isotropic elasticity. Int J Numer Meth Eng 37(12):1981–2004
Muliana A, Li KA (2010) Time-dependent response of active composites with thermal, electrical, and mechanical coupling effect. Int J Eng Sci 48(11):1481–1497
Noels L (2009) A discontinuous galerkin formulation of non-linear Kirchhoff–Love shells. Int J Numer Meth Eng 78(3):296–323
Noels L, Radovitzky R (2006) A general discontinuous Galerkin method for finite hyperelasticity. Formulation and numerical applications. Int J Numer Meth Eng 68(1):64–97
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):1393–1420
Pilate F, Toncheva A, Dubois P, Raquez JM (2016) Shape-memory polymers for multiple applications in the materials world. Eur Polymer J 80:268–294
Prudhomme S, Pascal F, Oden J, Romkes A (2000) Review of a priori error estimation for discontinuous Galerkin methods. Technical report, TICAM, UTexas
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
Romkes A, Prudhomme S, Oden J (2003) A priori error analyses of a stabilized discontinuous Galerkin method. Comput Math Appl 46(8):1289–1311
Rothe S, Schmidt JH, Hartmann S (2015) Analytical and numerical treatment of electro-thermo-mechanical coupling. Arch Appl Mech 85(9–10):1245–1264
Schmidt AM (2006) Electromagnetic activation of shape memory polymer networks containing magnetic nanoparticles. Macromol Rapid Commun 27(14):1168–1172
Spencer A (1982) The formulation of constitutive equations for anisotropic solids. Nijhoff, Amsterdam, pp. 3–26
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 41–52
Srivastava V, Chester SA, Anand L (2010) Thermally actuated shape-memory polymers: experiments, theory, and numerical simulations. J Mech Phys Solids 58(8):1100–1124
Sun S, Wheeler MF (2005) Discontinuous Galerkin methods for coupled flow and reactive transport problems. Appl Numer Math 52(2):273–298
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
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
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
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
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):2343–2347
Wells GN, Dung NT (2007) AC 0 discontinuous Galerkin formulation for Kirchhoff plates. Comput Methods Appl Mech Eng 196(35–36):3370–3380
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):3633–3645
Wen S, Chung D (1999) Seebeck effect in carbon fiber-reinforced cement. Cem Concr Res 29(12):1989–1993
Wheeler MF (1978) An elliptic collocation-finite element method with interior penalties. SIAM J Numer Anal 15(1):152–161
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 104:162–183
Yadav S, Pani A, Park EJ (2013) Superconvergent discontinuous Galerkin methods for nonlinear elliptic equations. Math Comput 82(283):1297–1335
Yang Y, Xie S, Ma F, Li J (2012) On the effective thermoelectric properties of layered heterogeneous medium. J Appl Phys 111(1):013,510
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):666–674
Zhupanska OI, Sierakowski RL (2011) Electro-thermo-mechanical coupling in carbon fiber polymer matrix composites. Acta Mech 218(3–4):319–332