Artificial conditions for the linear elasticity equations
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- by Virginie Bonnaillie-Noël, Marc Dambrine, Frédéric Hérau and Grégory Vial PDF
- Math. Comp. 84 (2015), 1599-1632 Request permission
Abstract:
In this paper, we consider the equations of linear elasticity in an exterior domain. We exhibit artificial boundary conditions on a circle, which lead to a non-coercive second order boundary value problem. In the particular case of an axisymmetric geometry, explicit computations can be performed in Fourier series proving the well-posedness except for a countable set of parameters. A perturbation argument allows us to consider near-circular domains. We complete the analysis by some numerical simulations.References
- Yves Achdou, O. Pironneau, and F. Valentin, Effective boundary conditions for laminar flows over periodic rough boundaries, J. Comput. Phys. 147 (1998), no. 1, 187–218. MR 1657773, DOI 10.1006/jcph.1998.6088
- Y. Amirat, G. A. Chechkin, and R. R. Gadyl′shin, Asymptotics of simple eigenvalues and eigenfunctions for the Laplace operator in a domain with oscillating boundary, Zh. Vychisl. Mat. Mat. Fiz. 46 (2006), no. 1, 102–115 (English, with Russian summary); English transl., Comput. Math. Math. Phys. 46 (2006), no. 1, 97–110. MR 2239730, DOI 10.1134/S0965542506010118
- H. Ammari, E. Bonnetier, Y. Capdeboscq, M. Tanter, and M. Fink, Electrical impedance tomography by elastic deformation, SIAM J. Appl. Math. 68 (2008), no. 6, 1557–1573. MR 2424952, DOI 10.1137/070686408
- Habib Ammari, Hyeonbae Kang, Mikyoung Lim, and Habib Zribi, Layer potential techniques in spectral analysis. Part I: Complete asymptotic expansions for eigenvalues of the Laplacian in domains with small inclusions, Trans. Amer. Math. Soc. 362 (2010), no. 6, 2901–2922. MR 2592941, DOI 10.1090/S0002-9947-10-04695-7
- M. F. Ben Hassen and E. Bonnetier, An asymptotic formula for the voltage potential in a perturbed $\epsilon$-periodic composite medium containing misplaced inclusions of size $\epsilon$, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), no. 4, 669–700. MR 2250439, DOI 10.1017/S0308210500004650
- A. Bendali and K. Lemrabet, The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation, SIAM J. Appl. Math. 56 (1996), no. 6, 1664–1693. MR 1417476, DOI 10.1137/S0036139995281822
- P. Bettess, Infinite Elements, Penshaw Press, Paris, 1992.
- V. Bonnaillie-Noël, D. Brancherie, M. Dambrine, F. Hérau, S. Tordeux, and G. Vial, Multiscale expansion and numerical approximation for surface defects, CANUM 2010, $40^\textrm {e}$ Congrès National d’Analyse Numérique, ESAIM Proc., vol. 33, EDP Sci., Les Ulis, 2011, pp. 22–35 (English, with English and French summaries). MR 2863307, DOI 10.1051/proc/201133003
- V. Bonnaillie-Noël, M. Dambrine, F. Hérau, and G. Vial, On generalized Ventcel’s type boundary conditions for Laplace operator in a bounded domain, SIAM J. Math. Anal. 42 (2010), no. 2, 931–945. MR 2644364, DOI 10.1137/090756521
- Virginie Bonnaillie-Noël, Marc Dambrine, Sébastien Tordeux, and Grégory Vial, Interactions between moderately close inclusions for the Laplace equation, Math. Models Methods Appl. Sci. 19 (2009), no. 10, 1853–1882. MR 2573145, DOI 10.1142/S021820250900398X
- Blaise Bourdin, Gilles A. Francfort, and Jean-Jacques Marigo, The variational approach to fracture, Springer, New York, 2008. Reprinted from J. Elasticity 91 (2008), no. 1-3 [MR2390547]; With a foreword by Roger Fosdick. MR 2473620, DOI 10.1007/978-1-4020-6395-4
- Didier Bresch and Vuk Milisic, High order multi-scale wall-laws, Part I: the periodic case, Quart. Appl. Math. 68 (2010), no. 2, 229–253. MR 2663000, DOI 10.1090/S0033-569X-10-01135-0
- Monique Dauge, Elliptic boundary value problems on corner domains, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions. MR 961439, DOI 10.1007/BFb0086682
- B. Delourme, Modèles et asymptotiques des interfaces fines et périodiques, Ph.D. thesis, Université Pierre et Marie Curie, Paris VI, 2010.
- Bérangère Delourme, Houssem Haddar, and Patrick Joly, Approximate models for wave propagation across thin periodic interfaces, J. Math. Pures Appl. (9) 98 (2012), no. 1, 28–71 (English, with English and French summaries). MR 2935369, DOI 10.1016/j.matpur.2012.01.003
- Bjorn Engquist and Andrew Majda, Absorbing boundary conditions for the numerical simulation of waves, Math. Comp. 31 (1977), no. 139, 629–651. MR 436612, DOI 10.1090/S0025-5718-1977-0436612-4
- B. Engquist and J. C. Nedelec, Effective boundary conditions for electromagnetic scattering in thin layers, Rapport interne 278, CMAP, 1993.
- Dan Givoli, Nonreflecting boundary conditions, J. Comput. Phys. 94 (1991), no. 1, 1–29. MR 1103713, DOI 10.1016/0021-9991(91)90135-8
- P. Grisvard, Boundary Value Problems in Non-Smooth Domains, Pitman, London, 1985.
- P. Grisvard, Problèmes aux limites dans les polygones. Mode d’emploi, EDF Bull. Direction Études Rech. Sér. C Math. Inform. 1 (1986), 3, 21–59 (French). MR 840970
- P. Grisvard, Singularités en elasticité, Arch. Rational Mech. Anal. 107 (1989), no. 2, 157–180 (French, with English summary). MR 996909, DOI 10.1007/BF00286498
- H. Haddar and P. Joly, Effective boundary conditions for thin ferromagnetic coatings. Asymptotic analysis of the 1D model, Asymptot. Anal. 27 (2001), no. 2, 127–160. MR 1852003
- L. Halpern and J. Rauch, Absorbing boundary conditions for diffusion equations, Numer. Math. 71 (1995), no. 2, 185–224 (English, with English and French summaries). MR 1347164, DOI 10.1007/s002110050141
- Willi Jäger and Andro Mikelić, On the roughness-induced effective boundary conditions for an incompressible viscous flow, J. Differential Equations 170 (2001), no. 1, 96–122. MR 1813101, DOI 10.1006/jdeq.2000.3814
- P. Joly and C. Tsogka, Numerical methods for treating unbounded media, Effective computational methods for wave propagation, Numer. Insights, vol. 5, Chapman & Hall/CRC, Boca Raton, FL, 2008, pp. 425–472. MR 2404885, DOI 10.1201/9781420010879.ch14
- D. Leguillon and E. Sánchez-Palencia, Computation of singular solutions in elliptic problems and elasticity, John Wiley & Sons, Ltd., Chichester; Masson, Paris, 1987. MR 995254
- M. Lenoir, M. Vullierme-Ledard, and C. Hazard, Variational formulations for the determination of resonant states in scattering problems, SIAM J. Math. Anal. 23 (1992), no. 3, 579–608. MR 1158823, DOI 10.1137/0523030
- J.-J. Marigo and C. Pideri, The effective behavior of elastic bodies containing microcracks or microholes localized on a surface, International Journal of Damage Mechanics 20 (2011), no. 8, 1151–1177.
- D. Martin, Mélina, Bibliothèque de calculs éléments finis,http://anum-maths.univ-rennes1.fr/melina/danielmartin/melina/.
- V. Maz′ya, S. Nazarov, and B. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. I, Operator Theory: Advances and Applications, vol. 111, Birkhäuser Verlag, Basel, 2000. MR 1779977 (2001e:35044a)
- —, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. II, Operator Theory: Advances and Applications, vol. 112, Birkhäuser Verlag, Basel, 2000. MR 1779978 (2001e:35044b)
- V. G. Maz′ja and B. A. Plamenevskiĭ, Weighted spaces with inhomogeneous norms, and boundary value problems in domains with conical points, Elliptische Differentialgleichungen (Meeting, Rostock, 1977) Wilhelm-Pieck-Univ., Rostock, 1978, pp. 161–190 (Russian). MR 540196
- Gen Nakamura and Gunther Uhlmann, Global uniqueness for an inverse boundary problem arising in elasticity, Invent. Math. 118 (1994), no. 3, 457–474. MR 1296354, DOI 10.1007/BF01231541
- Gen Nakamura and Gunther Uhlmann, Inverse problems at the boundary for an elastic medium, SIAM J. Math. Anal. 26 (1995), no. 2, 263–279. MR 1320220, DOI 10.1137/S0036141093247494
- Daniel Rabinovich, Dan Givoli, Jacobo Bielak, and Thomas Hagstrom, A finite element scheme with a high order absorbing boundary condition for elastodynamics, Comput. Methods Appl. Mech. Engrg. 200 (2011), no. 23-24, 2048–2066. MR 2795161, DOI 10.1016/j.cma.2011.03.006
- V. Bonnaillie-Noël; D. Brancherie; M. Dambrine; S. Tordeux and G. Vial, Effect of micro-defects on structure failure. Coupling asymptotic analysis and strong discontinuity, Eur. J. Comput. Mech. 19 (2010), no. 1-3, 165–175 (English).
- D. Brancherie; M. Dambrine; G. Vial and P. Villon, Effect of surface defects on structure failure. A two-scale approach, Rev. Eur. Méc. Numér. 17 (2008), no. 5-7, 613–624 (English).
- Michael S. Vogelius and Darko Volkov, Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter, M2AN Math. Model. Numer. Anal. 34 (2000), no. 4, 723–748. MR 1784483, DOI 10.1051/m2an:2000101
- O. C. Zienkiewicz and P. Bettess, Infinite elements in the study of fluid-structure interaction problems, Computing methods in applied sciences (Second Internat. Sympos., Versailles, 1975) Lecture Notes in Phys., Vol. 58, Springer, Berlin, 1976, pp. 133–172. MR 0443580
Additional Information
- Virginie Bonnaillie-Noël
- Affiliation: IRMAR - UMR6625, ENS Rennes, Univ. Rennes 1, CNRS, UEB, av. Robert Schuman, 35170 Bruz, France
- Email: bonnaillie@math.cnrs.fr
- Marc Dambrine
- Affiliation: LMAP - UMR5142, Université de Pau et des Pays de l’Adour, av. de l’Université, BP 1155, 64013 Pau Cedex, France
- Email: marc.dambrine@univ-pau.fr
- Frédéric Hérau
- Affiliation: LMJL - UMR6629, Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France
- Email: frederic.herau@univ-nantes.fr
- Grégory Vial
- Affiliation: Université de Lyon, CNRS UMR 5208, École Centrale de Lyon, Institut Camille Jordan, 36 avenue Guy de Collongue, 69134 Écully cedex, France
- Email: gregory.vial@ec-lyon.fr
- Received by editor(s): December 10, 2012
- Received by editor(s) in revised form: September 4, 2013, and October 7, 2013
- Published electronically: November 18, 2014
- © Copyright 2014 American Mathematical Society
- Journal: Math. Comp. 84 (2015), 1599-1632
- MSC (2010): Primary 35J47, 35J57, 35P10, 35S15, 47A10, 47G30, 65N20
- DOI: https://doi.org/10.1090/S0025-5718-2014-02901-3
- MathSciNet review: 3335885