Resolution of linear viscoelastic equations in the frequency domain using real Helmholtz boundary integral equations

Stéphanie Chaillat; Huy Duong Bui

doi:10.1016/j.crme.2007.10.005

Resolution of linear viscoelastic equations in the frequency domain using real Helmholtz boundary integral equations

Presented by: Huy Duong Bui

Stéphanie Chaillat ^1,² ; Huy Duong Bui ^1,³

¹ LMS, CNRS UMR 7649, École polytechnique, 91128 Palaiseau cedex, France
² LCPC, 58, boulevard Lefebvre, 75732 Paris cedex 15, France
³ Lamsid/EDF/R&D, 1, avenue du Général de Gaulle, 92141 Clamart, France

Comptes Rendus. Mécanique, Volume 335 (2007) no. 12, pp. 746-750.

Abstract
Résumé

Boundary integral equations are well suitable for the analysis of seismic waves propagation in unbounded domains. Formulations in elastodynamics are well developed. In contrast, for the dynamic analysis of viscoelastic media, there are very seldom formulations by boundary integral equations. In this Note, we propose a new and simple formulation of time harmonic viscoelasticity with the Zener model, which reduces to classical elastodynamics if a compatibility condition is satisfied by boundary conditions. Intermediate variables which satisfy the classical elastodynamic equations are introduced. It makes it possible to utilize existing numerical tools of time harmonic elastodynamics.

Les équations intégrales de frontière sont bien adaptées à l'étude de la propagation des ondes sismiques notamment pour les domaines non bornés. Les formulations pour l'élastodynamique sont très développées. Par contre, pour la dynamique des corps viscoélastiques modélisant le sol, peu de formulations par équations intégrales de frontière existent. On propose dans cette Note une nouvelle formulation simple de la viscoélasticité en domaine fréquentiel par un modèle de Zener qui se ramène à la formulation classique de l'élastodynamique, si les conditions à la frontière respectent une condition de compatibilité. Des variables intermédiaires qui respectent les équations de l'élastodynamique classique sont introduites. Il est alors possible de réutiliser les outils numériques existant pour l'élastodynamique harmonique.

Received: 2007-07-17
Accepted: 2007-10-09
Published online: 2007-11-19

DOI: 10.1016/j.crme.2007.10.005

Keywords: Computational solid mechanics, Viscoelasticity, Boundary integral equations
Mot clés : Mécanique des solides numérique, Viscoélasticité, Équations intégrales de frontière

Author's affiliations:

Stéphanie Chaillat ^{1, 2}; Huy Duong Bui ^{1, 3}

¹ LMS, CNRS UMR 7649, École polytechnique, 91128 Palaiseau cedex, France
² LCPC, 58, boulevard Lefebvre, 75732 Paris cedex 15, France
³ Lamsid/EDF/R&D, 1, avenue du Général de Gaulle, 92141 Clamart, France

@article{CRMECA_2007__335_12_746_0,
     author = {St\'ephanie Chaillat and Huy Duong Bui},
     title = {Resolution of linear viscoelastic equations in the frequency domain using real {Helmholtz} boundary integral equations},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {746--750},
     publisher = {Elsevier},
     volume = {335},
     number = {12},
     year = {2007},
     doi = {10.1016/j.crme.2007.10.005},
     language = {en},
}

TY  - JOUR
AU  - Stéphanie Chaillat
AU  - Huy Duong Bui
TI  - Resolution of linear viscoelastic equations in the frequency domain using real Helmholtz boundary integral equations
JO  - Comptes Rendus. Mécanique
PY  - 2007
SP  - 746
EP  - 750
VL  - 335
IS  - 12
PB  - Elsevier
DO  - 10.1016/j.crme.2007.10.005
LA  - en
ID  - CRMECA_2007__335_12_746_0
ER  -

%0 Journal Article
%A Stéphanie Chaillat
%A Huy Duong Bui
%T Resolution of linear viscoelastic equations in the frequency domain using real Helmholtz boundary integral equations
%J Comptes Rendus. Mécanique
%D 2007
%P 746-750
%V 335
%N 12
%I Elsevier
%R 10.1016/j.crme.2007.10.005
%G en
%F CRMECA_2007__335_12_746_0

Stéphanie Chaillat; Huy Duong Bui. Resolution of linear viscoelastic equations in the frequency domain using real Helmholtz boundary integral equations. Comptes Rendus. Mécanique, Volume 335 (2007) no. 12, pp. 746-750. doi : 10.1016/j.crme.2007.10.005. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2007.10.005/

References
Cited by

[1] J.F. Semblat; A.M. Duval; P. Dangla Seismic site effects in deep alluvial basin: numerical analysis by boundary element method, Comput. Geotech., Volume 29 (2002), pp. 573-585

[2] M. Schanz Wave Propagation in Viscoelastic and Poroelastic Continua, Springer, 2001

[3] F.J. Rizzo; D.J. Shippy An application of the correspondence principle of linear viscoelasticity theory, SIAM J. Appl. Math., Volume 21 (1971), pp. 321-330

[4] T. Kusama; Y. Mitsui Boundary element method applied to linear viscoelastic analysis, Appl. Math. Model., Volume 6 (1982), pp. 285-290

[5] L. Gaul; M. Schanz Dynamics of viscoelastic solids treated by boundary element approaches in time domain, Eur. J. Mech. A/Solids, Volume 13 (1994), pp. 43-59

[6] C. Lubich Convolution quadrature and discretized operational calculus, I, Numer. Math., Volume 52 (1988), pp. 129-145

[7] M. Schanz; H. Antes A new visco- and elastodynamic time domain boundary element formulation, Comput. Mech., Volume 20 (1997), pp. 452-459

[8] A.D. Mesquita; H.B. Coda A simple Kelvin and Boltzmann viscoelastic analysis of three-dimensional solids by boundary element method, Eng. Anal. Bound. Elements, Volume 27 (2003), pp. 885-895

[9] J. Dominguez Boundary Elements in Dynamics, International Series on Computational Engineering, Computational Mechanics/Elsevier Applied Science, Southampton, London UK, 1993

[10] W. Flügge Viscoelasticity, Springer-Verlag, 1975

[11] I.G. Goriacheva Contact problem of rolling of a viscoelastic cylinder on a base of the same material, PMM, Volume 37 (1973), pp. 925-933

[12] M. Bonnet Boundary Integral Equation Method for Solids and Fluids, Wiley, 1999

[13] S. Chaillat; M. Bonnet; J.F. Semblat A Fast Multipole Method formulation for 3D elastodynamics in the frequency domain, C. R. Mecanique, Volume 335 (2007) no. 11, pp. 714-719

Cited by Sources:

Comments - Policy