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Scalar boundary value problems on junctions of thin rods andplates

I. Asymptotic analysis and error estimates

Published online by Cambridge University Press:  13 August 2014

R. Bunoiu ,
G. Cardone  and
S. A. Nazarov
R. Bunoiu
Affiliation:
Universitéde Lorraine, Institut Elie Cartan de Lorraine, 7502 UMR, 57045 Metz, France.. [email protected]
G. Cardone
Affiliation:
University of Sannio − Department of Engineering, Piazza Roma, 21, 84100 Benevento, Italy.; [email protected]
S. A. Nazarov
Affiliation:
Mathematics and Mechanics Faculty, St. Petersburg State University 198504, Universitetsky pr., 28, Stary Peterhof, Russia.; [email protected]
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Abstract

We derive asymptotic formulas for the solutions of the mixed boundary value problem forthe Poisson equation on the union of a thin cylindrical plate and several thin cylindricalrods. One of the ends of each rod is set into a hole in the plate and the other one issupplied with the Dirichlet condition. The Neumann conditions are imposed on the wholeremaining part of the boundary. Elements of the junction are assumed to have contrastingproperties so that the small parameter, i.e. the relative thickness,appears in the differential equation, too, while the asymptotic structures cruciallydepend on the contrastness ratio. Asymptotic error estimates are derived in anisotropicweighted Sobolev norms.

Keywords

Junction of thin plate and rodsasymptotic analysisdimension reductionboundary layerserror estimates
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 5 , September 2014 , pp. 1495 - 1528
Copyright
© EDP Sciences, SMAI 2014

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References

Arsen’ev, A.A., The existence of resonance poles and resonances under scattering in the case of boundary conditions of the second and third kind.Ž. Vyčisl. Mat. i Mat. Fiz. 16 (1976) 718–724. Google Scholar
Beale, J., Thomas Scattering frequencies of reasonators. Commun. Pure Appl. Math. 26 (1973) 549–563. Google Scholar
Berlyand, L., Cardone, G., Gorb, Y. and Panasenko, G.P., Asymptotic analysis of an array of closely spaced absolutely conductive inclusions. Netw. Heterog. Media 1 (2006) 353–377. Google Scholar
Blanchard, D., Gaudiello, A. and Griso, G., Junction of a periodic family of elastic rods with a 3d plate. I. J. Math. Pures Appl. 88 (2007) 1–33. Google Scholar
Blanchard, D., Gaudiello, A. and Griso, G., Junction of a periodic family of elastic rods with a thin plate. II. J. Math. Pures Appl. 88 (2007) 149–190. Google Scholar
Blanchard, D. and Griso, G., Microscopic effects in the homogenization of the junction of rods and a thin plate. Asymptot. Anal. 56 (2008) 1–36. Google Scholar
Blanchard, D. and Griso, G., Asymptotic behavior of a structure made by a plate and a straight rod. Chin. Annal. Math. Ser. B 34 (2013) 399–434. Google Scholar
Borisov, D., Bunoiu, R. and Cardone, G., On a waveguide with frequently alternating boundary conditions: homogenized Neumann condition. Annal. Henri Poincaré 11 (2010) 1591–1627. Google Scholar
Borisov, D., Bunoiu, R. and Cardone, G., Homogenization and asymptotics for a waveguide with an infinite number of closely located small windows. J. Math. Sci. 176 (2011) 774–785. Google Scholar
Borisov, D. and Bunoiu, R., Cardone, G., On a waveguide with an infinite number of small windows. C. R. Math. Acad. Sci. Paris, Ser. I 349 (2011) 53–56. Google Scholar
Borisov, D., Bunoiu, R. and Cardone, G., Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics. Z. Angew. Math. Phys. 64 (2013) 439–472. Google Scholar
Borisov, D. and Cardone, G., Homogenization of the planar waveguide with frequently alternating boundary conditions. J. Phys. A: Math. Theor. 42 (2009) 365–205. Google Scholar
Borisov, D. and Cardone, G., Complete asymptotic expansions for the eigenvalues of the Dirichlet Laplacian in thin three-dimensional rods. ESAIM: COCV 17 (2011) 887–908. Google Scholar
Borisov, D. and Cardone, G., Planar Waveguide with “Twisted” Boundary Conditions: Small Width. J. Math. Phys. 53 (2012) 023–503. Google Scholar
Borisov, D., Cardone, G., Faella, L. and Perugia, C., Uniform resolvent convergence for strip with fast oscillating boundary. J. Differ. Eqs. 255 (2013) 4378–4402. Google Scholar
Cardone, G., Corbo Esposito, A. and Panasenko, G.P., Asymptotic partial decomposition for diffusion with sorption in thin structures. Nonlinear Anal. 65 (2006) 79–106. Google Scholar
Cardone, G., Corbo Esposito, A. and Pastukhova, S.E., Homogenization of a scalar problem for a combined structure with singular or thin reinforcement. Z. Anal. Anwend. 26 (2007) 277–301. Google Scholar
Cardone, G., Fares, R. and Panasenko, G.P., Asymptotic expansion of the solution of the steady Stokes equation with variable viscosity in a two-dimensional tube structure. J. Math. Phys. 53 (2012) 103–702. Google Scholar
Cardone, G., Panasenko, G.P. and Sirakov, Y., Asymptotic analysis and numerical modeling of mass transport in tubular structures. Math. Models Methods Appl. Sci. 20 (2010) 397–421. Google Scholar
Cardone, G., Nazarov, S.A. and Piatnitski, A.L., On the rate of convergence for perforated plates with a small interior Dirichlet zone. Z. Angew. Math. Phys. 62 (2011) 439–468.Google Scholar
Ciarlet, P.G., Mathematical elasticity. Vol. II. Theory of plates. Studies Math. Appl. 27 (1997). Google Scholar
Cioranescu, D., Oleĭnik, O.A. and Tronel, G., Korn’s inequalities for frame type structures and junctions with sharp estimates for the constants. Asymptot. Anal. 8 (1994) 1–14. Google Scholar
Cioranescu, D. and Saint Jean Paulin, J., Homogenization of reticulated structures. Appl. Math. Sci. 136 (1999). Google Scholar
Gadyl’shin, R.R., On the eigenvalues of a dumbbell with a thin handle. Izv. Ross. Akad. Nauk Ser. Mat. 69 (2005) 45–110; Izv. Math. 69 (2005) 265–329. Google Scholar
Gaudiello, A., Monneau, R., Mossino, J., Murat, F. and Sili, A., Junction of elastic plates and beams. ESAIM: COCV 13 (2007) 419–457. Google Scholar
Gaudiello, A. and Sili, A., Asymptotic analysis of the eigenvalues of a Laplacian problem in a thin multidomain. Indiana Univ. Math. J. 56 (2007) 1675–1710. Google Scholar
Gaudiello, A. and Sili, A., Asymptotic analysis of the eigenvalues of an elliptic problem in an anisotropic thin multidomain. Proc. Roy. Soc. Edinburgh Sect. A 141 (2011) 739–754. Google Scholar
Gruais, I., Modélisation de la jonction entre une plaque et une poutre en élasticité linéarisée. RAIRO Modél. Math. Anal. Numér. 27 (1993) 77–105. Google Scholar
Gruais, I., Modeling of the junction between a plate and a rod in nonlinear elasticity. Asymptot. Anal. 7 (1993) 179–194. Google Scholar
Il’in, A.M., A boundary value problem for an elliptic equation of second order in a domain with a narrow slit. I. The two-dimensional case. Mat. Sb. 99 (1976) 514–537. Google Scholar
Il’in A.M., Matching of asymptotic expansions of solutions of boundary value problems. Moscow, Nauka (1989); Translations: Math. Monogr., vol. 102. AMS, Providence (1992).
Joly, P. and Tordeux, S.. Matching of asymptotic expansions for waves propagation in media with thin slots II: The error estimates. ESAIM: M2AN 42 (2008) 193–221. Google Scholar
Kondratiev, V.A., Boundary problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obshch. 16 (1967) 209−292; Trans. Moscow Math. Soc. 16 (1967) 227−313. Google Scholar
V. Kozlov, V. Maz’ya and A. Movchan, Asymptotic analysis of fields in multi-structures. Oxford Math. Monogr. Oxford University Press (1999).
Kozlov, V.A., Maz’ya, V.G. and Movchan, A.B., Asymptotic analysis of a mixed boundary value problem in a multi-structure. Asymptot. Anal. 8 (1994) 105–143. Google Scholar
Kozlov, V.A., Maz’ya, V.G. and Movchan, A.B., Asymptotic representation of elastic fields in a multi-structure. Asymptot. Anal. 11 (1995) 343–415. Google Scholar
Kozlov, V.A., Maz’ya, V.G. and Movchan, A.B., Fields in non-degenerate 1D-3D elastic multi-structures. Quart. J. Mech. Appl. Math. 54 (2001) 177–212. Google Scholar
O.A. Ladyzhenskaya, The boundary value problems of mathematical physics. Moscow, Nauka (1973); Appl. Math. Sci., vol. 49. Springer-Verlag, New York (1985).
N.S. Landkof, Foundations of modern potential theory. Die Grundlehren der mathematischen Wissenschaften, vol. 180. Springer-Verlag, New York-Heidelberg (1972).
H. Le Dret, Problèmes variationnels dans le multi-domaines: modélisation des jonctions et applications. Res. Appl. Math., vol. 19. Masson, Paris (1991).
Leguillon, D. and Sanchez-Palencia, E., Approximation of a two-dimensional problem of junction. Comput. Mech. 6 (1990) 435–455. Google Scholar
J.L. Lions, Magenes E., Non-homogeneous boundary value problems and applications. Springer-Verlag, New York-Heidelberg (1972).
J.-L. Lions, Some more remarks on boundary value problems and junctions. Proc. of Asymptotic methods for elastic structures, Lisbon 1993. De Gruyter, Berlin (1995) 103–118.
V.G. Maz’ya, S.A. Nazarov and B.A. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, Tbilisi Univ. 1981; Operator Theory. Adv. Appl., vol. 112. Birkhäuser, Basel (2000).
S.A. Nazarov, Asymptotic Theory of Thin Plates and Rods. Dimension Reduction and Integral Estimates, vol. 1. Nauchnaya Kniga, Novosibirsk (2001).
Nazarov, S.A., Selfadjoint extensions of the operator of the Dirichlet problem in weighted function spaces. Mat. Sb. 137 (1988) 224–241; Math. USSR-Sb. 65 (1990) 229–247. Google Scholar
Nazarov, S.A., Asymptotic behavior of the solution of a boundary value problem in a thin cylinder with a nonsmooth lateral surface. Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993) 202–239; Russian Acad. Sci. Izv. Math. 42 (1994) 183–217. Google Scholar
Nazarov, S.A., Junctions of singularly degenerating domains with different limit dimensions. I. Tr. Semin. im. I. G. Petrovskogo 18 (1995) 3–78; J. Math. Sci. 80 (1996) 1989–2034. Google Scholar
Nazarov, S.A., Korn’s inequalities for junctions of bodies and thin rods. Math. Meth. Appl. Sci. 20 (1997) 219–243. Google Scholar
S.A. Nazarov, Asymptotic conditions at a point, selfadjoint extensions of operators, and the method of matched asymptotic expansions. Proc. St. Petersburg Math. Society, V, 77–125; Amer. Math. Soc. Transl. Ser. 2, 193, Amer. Math. Soc., Providence (1999).
Nazarov, S.A., Asymptotic expansions at infinity of solutions of a problem in the theory of elasticity in a layer. Tr. Mosk. Mat. Obs. 60 (1999) 3–97; Trans. Moscow Math. Soc. (1999) 1–85. Google Scholar
Nazarov, S.A., Junctions of singularly degenerating domains with different limit dimensions. II. Tr. Semin. im. I. G. Petrovskogo 20 (2000) 155–195; 312–313; J. Math. Sci. 97 (1999) 155–195. Google Scholar
Nazarov, S.A., Asymptotic analysis and modeling of the junction of a massive body and thin rods. Tr. Semin. im. I. G. Petrovskogo 24 (2004) 95–214, 342–343; J. Math. Sci. 127 (2005) 2192–2262. Google Scholar
Nazarov, S.A., Estimates for the accuracy of modeling boundary value problems on the junction of domains with different limit dimensions. Izv. Ross. Akad. Nauk Ser. Mat. 68 (2004) 119–156; Izv. Math. 68 (2004) 1179–1215. Google Scholar
Nazarov, S.A., Elliptic boundary value problems on hybrid domains. Funktsional. Anal. i Prilozhen 38 (2004) 55–72; Funct. Anal. Appl. 38 (2004) 283–297. Google Scholar
Nazarov, S.A., Korn’s inequalities for elastic joints of massive bodies, thin plates, and rods. Uspekhi Mat. Nauk 63 (2008) 379, 37–110; Russian Math. Surveys 63 (2008) 35–107. Google Scholar
Nazarov, S.A., Asymptotic behavior of the solutions of the spectral problem of the theory of elasticity for a three-dimensional body with a thin coupler. Sibirsk. Mat. Zh. 53 (2012) 345–364; Sib. Math. J. 53 (2012) 274–290. Google Scholar
S.A. Nazarov and B.A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries. Moscow: Nauka. (1991); de Gruyter Expositions Math., vol. 13. Walter de Gruyter & Co., Berlin (1994).
G.P. Panasenko, Multi-scale Modeling for Structures and Composites. Springer, Dordrecht (2005).
G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annal. Math. Studies, vol. 27, Princeton University Press, Princeton (1951).
J. Sanchez-Hubert, Sanchez-Palencia E., Coques élastiques minces. Propriétés asymptotiques. Recherches en Mathématiques Appliquées. Paris, Masson (1997).
V.I. Smirnov, A course of higher mathematics. Advanced calculus, vol. II. Sneddon Pergamon Press, London (1964).
V.I. Smirnov, A course of higher mathematics. Integral equations and partial differential equations, vol. IV. Sneddon Pergamon Press, London (1964).
M. Van Dyke, Perturbation methods in fluid mechanics. Appl. Math. Mech., vol. 8 Academic Press, New York, London (1964).
V.S. Vladimirov, Generalized Functions in Mathematical Physics, Mir Moscow (1979).