Comptes Rendus
On the space-time separated representation of integral linear viscoelastic models
Comptes Rendus. Mécanique, Volume 343 (2015) no. 4, pp. 247-263.

The analysis of materials mechanical behavior involves many computational challenges. In this work, we are addressing the transient simulation of the mechanical behavior when the time of interest is much larger than the characteristic time of the mechanical response. This situation is encountered in many applications, as for example in the simulation of materials aging, or in structural analysis when small-amplitude oscillatory loads are applied during a long period, as it occurs for example when characterizing viscoelastic behaviors by calculating the complex modulus or when addressing fatigue simulations. Moreover, in the case of viscoelastic behaviors, the constitutive equation is many times expressed in an integral form avoiding the necessity of using internal variables, fact that results in an integro-differential model. In order to efficiently simulate such a model, we explore in this work the use of a space-time separated representation.

L'analyse du comportement mécanique des matériaux entraîne de nombreuses difficultés du point de vue numérique. Dans ce travail, nous allons nous focaliser sur l'une d'entre elles, celle associée à la simulation transitoire du comportement mécanique quand l'intervalle temporel d'intérêt est substantiellement plus long que le temps caractéristique associé à la réponse mécanique. Cette situation est fréquemment retrouvée dans la caractérisation rhéologique des matériaux viscoélastiques (pour la détermination du module complexe) ainsi que quand on s'attaque à la simulation de la fatigue. De plus, dans le cas des matriaux viscoélastiques, le comportement est généralement décrit par une loi de comportement intégrale qui évite le besoin d'utiliser des variables internes, donnant lieu à un modèle mécanique integro-différentiel. Pour une résolution efficace, nous analysons ici l'utilisation d'une représentation séparée en espace-temps.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2015.02.002
Keywords: PGD, Viscoelasticity, Integro-differential models, Fatigue
Mot clés : PGD, Viscoélasticité, Modèle integro-differentiel, Fatigue

Amine Ammar 1, 2; Ali Zghal 1; Franck Morel 2; Francisco Chinesta 3

1 UMSSDT, ENSIT, Université de Tunis, 5, avenue Taha-Hussien, Montfleury 1008, Tunis, Tunisia
2 Arts et Métiers ParisTech, 2, bd du Ronceray, BP 93525, 49035 Angers cedex 01, France
3 GEM, UMR CNRS–Centrale Nantes, 1, rue de la Noe, BP 92101, 44321 Nantes cedex 3, France
@article{CRMECA_2015__343_4_247_0,
     author = {Amine Ammar and Ali Zghal and Franck Morel and Francisco Chinesta},
     title = {On the space-time separated representation of integral linear viscoelastic models},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {247--263},
     publisher = {Elsevier},
     volume = {343},
     number = {4},
     year = {2015},
     doi = {10.1016/j.crme.2015.02.002},
     language = {en},
}
TY  - JOUR
AU  - Amine Ammar
AU  - Ali Zghal
AU  - Franck Morel
AU  - Francisco Chinesta
TI  - On the space-time separated representation of integral linear viscoelastic models
JO  - Comptes Rendus. Mécanique
PY  - 2015
SP  - 247
EP  - 263
VL  - 343
IS  - 4
PB  - Elsevier
DO  - 10.1016/j.crme.2015.02.002
LA  - en
ID  - CRMECA_2015__343_4_247_0
ER  - 
%0 Journal Article
%A Amine Ammar
%A Ali Zghal
%A Franck Morel
%A Francisco Chinesta
%T On the space-time separated representation of integral linear viscoelastic models
%J Comptes Rendus. Mécanique
%D 2015
%P 247-263
%V 343
%N 4
%I Elsevier
%R 10.1016/j.crme.2015.02.002
%G en
%F CRMECA_2015__343_4_247_0
Amine Ammar; Ali Zghal; Franck Morel; Francisco Chinesta. On the space-time separated representation of integral linear viscoelastic models. Comptes Rendus. Mécanique, Volume 343 (2015) no. 4, pp. 247-263. doi : 10.1016/j.crme.2015.02.002. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2015.02.002/

[1] H.M. Park; D.H. Cho The use of the Karhunen–Loève decomposition for the modelling of distributed parameter systems, Chem. Eng. Sci., Volume 51 (1996), pp. 81-98

[2] Y. Maday; E.M. Ronquist The reduced basis element method: application to a thermal fin problem, SIAM J. Sci. Comput., Volume 26 (2004) no. 1, pp. 240-258

[3] R.A. Bialecki; A.J. Kassab; A. Fic Proper orthogonal decomposition and modal analysis for acceleration of transient FEM thermal analysis, Int. J. Numer. Methods Eng., Volume 62 (2005), pp. 774-797

[4] J. Burkardt; M. Gunzburger; H.-Ch. Lee POD and CVT-based reduced-order modeling of Navier–Stokes flows, Comput. Methods Appl. Mech. Eng., Volume 196 (2006), pp. 337-355

[5] M.D. Gunzburger; J.S. Peterson; J.N. Shadid Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data, Comput. Methods Appl. Mech. Eng., Volume 196 (2007), pp. 1030-1047

[6] A. Ammar; D. Ryckelynck; F. Chinesta; R. Keunings On the reduction of kinetic theory models related to finitely extensible dumbbells, J. Non-Newton. Fluid Mech., Volume 134 (2006), pp. 136-147

[7] S. Niroomandi; I. Alfaro; E. Cueto; F. Chinesta Real-time deformable models of non-linear tissues by model reduction techniques, Comput. Methods Programs Biomed., Volume 91 (2008), pp. 223-231

[8] S. Niroomandi; I. Alfaro; E. Cueto; F. Chinesta Model order reduction for hyperelastic materials, Int. J. Numer. Methods Biomed. Eng., Volume 81 (2010) no. 9, pp. 1180-1206

[9] S. Niroomandi; I. Alfaro; E. Cueto; F. Chinesta Accounting for large deformations in real-time simulations of soft tissues based on reduced order models, Comput. Methods Programs Biomed., Volume 105 (2012), pp. 1-12

[10] S. Niroomandi; I. Alfaro; D. Gonzalez; E. Cueto; F. Chinesta Real time simulation of surgery by reduced order modelling and X-FEM techniques, Int. J. Numer. Methods Biomed. Eng., Volume 28 (2012) no. 5, pp. 574-588

[11] A. Ammar; E. Pruliere; F. Chinesta; M. Laso Reduced numerical modeling of flows involving liquid–crystalline polymeres, J. Non-Newton. Fluid Mech., Volume 160 (2009), pp. 140-156

[12] F. Schmidt; N. Pirc; M. Mongeau; F. Chinesta Efficient mould cooling optimization by using model reduction, Int. J. Material Form., Volume 4 (2011) no. 1, pp. 71-82

[13] D. Ryckelynck; L. Hermanns; F. Chinesta; E. Alarcon An efficient a priori model reduction for boundary element models, Eng. Anal. Bound. Elem., Volume 29 (2005), pp. 796-801

[14] D. Ryckelynck; F. Chinesta; E. Cueto; A. Ammar On the a priori model reduction: overview and recent developments, Arch. Comput. Methods Eng., Volume 13 (2006) no. 1, pp. 91-128

[15] Y. Maday; E.M. Ronquist A reduced-basis element method, C. R. Acad. Sci. Paris, Ser. I, Volume 335 (2002), pp. 195-200

[16] Y. Maday; A.T. Patera; G. Turinici A priori convergence theory for reduced-basis approximations of single-parametric elliptic partial differential equations, J. Sci. Comput., Volume 17 (2002) no. 1–4, pp. 437-446

[17] K. Veroy; A. Patera Certified real-time solution of the parametrized steady incompressible Navier–Stokes equations: rigorous reduced-basis a posteriori error bounds, Int. J. Numer. Methods Fluids, Volume 47 (2005), pp. 773-788

[18] G. Rozza; D.B.P. Huynh; A.T. Patera Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations—application to transport and continuum mechanics, Arch. Comput. Methods Eng., Volume 15 (2008) no. 3, pp. 229-275

[19] P. Ladevèze The large time increment method for the analyze of structures with nonlinear constitutive relation described by internal variables, C. R. Acad. Sci. Paris, Ser. I, Volume 309 (1989), pp. 1095-1099

[20] P. Ladevèze; A. Nouy A multiscale computational method with time and space homogenization, C. R. Mecanique, Volume 330 (2002) no. 10, pp. 683-689

[21] P. Ladevèze; A. Nouy; O. Loiseau A multiscale computational approach for contact problems, Comput. Methods Appl. Mech. Eng., Volume 191 (2002) no. 43, pp. 4869-4891

[22] P. Ladevèze; A. Nouy On a multiscale computational strategy with time and space homogenization for structural mechanics, Comput. Methods Appl. Mech. Eng., Volume 192 (2003) no. 28–30, pp. 3061-3087

[23] P. Ladevèze; D. Néron; P. Gosselet On a mixed and multiscale domain decomposition method, Comput. Methods Appl. Mech. Eng., Volume 96 (2007), pp. 1526-1540

[24] P. Ladevèze; J.-C. Passieux; D. Néron The Latin multiscale computational method and the proper generalized decomposition, Comput. Methods Appl. Mech. Eng., Volume 199 (2010) no. 21–22, pp. 1287-1296

[25] 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, J. Non-Newton. Fluid Mech., Volume 139 (2006), pp. 153-176

[26] 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: Transient simulation using space-time separated representation, J. Non-Newton. Fluid Mech., Volume 144 (2007), pp. 98-121

[27] 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., Volume 6 (2008) no. 3, pp. 191-213

[28] A. Ammar; E. Cueto; F. Chinesta Reduction of the chemical master equation for gene regulatory networks using proper generalized decompositions, Int. J. Numer. Methods Biomed. Eng., Volume 28 (2012) no. 9, pp. 960-973

[29] H. Lamari; A. Ammar; A. Leygue; F. Chinesta On the solution of the multidimensional Langer's equation by using the proper generalized decomposition method for modeling phase transitions, Model. Simul. Mater. Sci. Eng., Volume 20 (2012) no. 1, p. 015007

[30] A. Ammar; M. Normandin; F. Chinesta Solving parametric complex fluids models in rheometric flows, J. Non-Newton. Fluid Mech., Volume 165 (2010), pp. 1588-1601

[31] E. Pruliere; F. Chinesta; A. Ammar On the deterministic solution of multidimensional parametric models by using the proper generalized decomposition, Math. Comput. Simul., Volume 81 (2010), pp. 791-810

[32] Ch. Ghnatios; F. Chinesta; E. Cueto; A. Leygue; P. Breitkopf; P. Villon Methodological approach to efficient modeling and optimization of thermal processes taking place in a die: application to pultrusion, Composites, Part A, Appl. Sci. Manuf., Volume 42 (2011), pp. 1169-1178

[33] Ch. Ghnatios; F. Masson; A. Huerta; E. Cueto; A. Leygue; F. Chinesta Proper generalized decomposition based dynamic data-driven control of thermal processes, Comput. Methods Appl. Mech. Eng., Volume 213 (2012), pp. 29-41

[34] D. Gonzalez; F. Masson; F. Poulhaon; A. Leygue; E. Cueto; F. Chinesta Proper generalized decomposition based dynamic data-driven inverse identification, Math. Comput. Simul., Volume 82 (2012) no. 9, pp. 1677-1695

[35] A. Ammar; E. Cueto; F. Chinesta Non-incremental PGD solution of parametric uncoupled models defined in evolving domains, Int. J. Numer. Methods Eng., Volume 93 (2013) no. 8, pp. 887-904

[36] F. Chinesta; A. Ammar; E. Cueto Recent advances and new challenges in the use of the proper generalized decomposition for solving multidimensional models, Arch. Comput. Methods Eng., Volume 17 (2010) no. 4, pp. 327-350

[37] F. Chinesta; A. Ammar; A. Leygue; R. Keunings An overview of the proper generalized decomposition with applications in computational rheology, J. Non-Newton. Fluid Mech., Volume 166 (2011), pp. 578-592

[38] F. Chinesta; P. Ladeveze; E. Cueto A short review in model order reduction based on proper generalized decomposition, Arch. Comput. Methods Eng., Volume 18 (2011), pp. 395-404

[39] F. Chinesta; A. Leygue; F. Bordeu; J.V. Aguado; E. Cueto; D. Gonzalez; I. Alfaro; A. Ammar; A. Huerta Parametric PGD based computational vademecum for efficient design, optimization and control, Arch. Comput. Methods Eng., Volume 20 (2013) no. 1, pp. 31-59

[40] A. Ammar; M. Normandin; F. Daim; D. Gonzalez; E. Cueto; F. Chinesta Non-incremental strategies based on separated representations: applications in computational rheology, Commun. Math. Sci., Volume 8 (2010) no. 3, pp. 671-695

[41] F. Chinesta; A. Ammar; E. Cueto Proper generalized decomposition of multiscale models, Int. J. Numer. Methods Biomed. Eng., Volume 83 (2010) no. 8–9, pp. 1114-1132

[42] A. Ammar; F. Chinesta; E. Cueto; M. Doblare Proper generalized decomposition of time-multiscale models, Int. J. Numer. Methods Biomed. Eng., Volume 90 (2012) no. 5, pp. 569-596

[43] H. Lamari; A. Ammar; P. Cartraud; G. Legrain; F. Jacquemin; F. Chinesta Routes for efficient computational homogenization of non-linear materials using the proper generalized decomposition, Arch. Comput. Methods Eng., Volume 17 (2010) no. 4, pp. 373-391

[44] B. Bognet; A. Leygue; F. Chinesta; A. Poitou; F. Bordeu Advanced simulation of models defined in plate geometries: 3D solutions with 2D computational complexity, Comput. Methods Appl. Mech. Eng., Volume 201 (2012), pp. 1-12

[45] F. Chinesta, A. Leygue, B. Bognet, Ch. Ghnatios, F. Poulhaon, F. Bordeu, A. Barasinski, A. Poitou, S. Chatel, S. Maison-Le-Poec, First steps towards an advanced simulation of composites manufacturing by automated tape placement, Int. J. Material Form., . | DOI

[46] A. Leygue; F. Chinesta; M. Beringhier; T.L. Nguyen; J.C. Grandidier; F. Pasavento; B. Schrefler Towards a framework for non-linear thermal models in shell domains, Int. J. Numer. Methods Heat Fluid Flow, Volume 23 (2013) no. 1, pp. 55-73

[47] B. Bognet; A. Leygue; F. Chinesta Separated representations of 3D elastic solutions in shell geometries, Adv. Model. Simul. Eng. Sci., Volume 1 (2014), p. 4 www.amses-journal.com/content/1/1/4

[48] D. Gonzalez; A. Ammar; F. Chinesta; E. Cueto Recent advances in the use of separated representations, Int. J. Numer. Methods Biomed. Eng., Volume 81 (2010) no. 5, pp. 637-659

[49] C. Ghnatios, G. Xu, M. Visonneau, A. Leygue, F. Chinesta, On the space separated representation when addressing the solution of PDE in complex domains, AIMS J., submitted for publication.

[50] F. Chinesta; R. Keunings; A. Leygue The proper generalized decomposition for advanced numerical simulations. A primer, Springer briefs, Springer, 2014

[51] A. Ammar; F. Chinesta; P. Diez; A. Huerta An error estimator for separated representations of highly multidimensional models, Comput. Methods Appl. Mech. Eng., Volume 199 (2010), pp. 1872-1880

[52] P. Ladevèze; L. Chamoin On the verification of model reduction methods based on the proper generalized decomposition, Comput. Methods Appl. Mech. Eng., Volume 200 (2011), pp. 2032-2047

Cited by Sources:

Comments - Policy