Abstract
Three-dimensional mathematical problems of the interaction between thermoelastic and scalar oscillation fields in a general physically anisotropic case are studied by the boundary integral equation methods. Uniqueness and existence theorems are proved by the reduction of the original interface problems to equivalent systems of boundary pseudodifferential equations. In the non-resonance case the invertibility of the corresponding matrix pseudodifferential operators in appropriate functional spaces is shown on the basis of the generalized Sommerfeld-Kupradze type thermoradiation conditions for anisotropic bodies. In the resonance case the co-kernels of the pseudodifferential operators are analysed and the efficient conditions of solvability of the original interface problems are established.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bielak, J. and MacCamy, R.C.,Symmetric finite element and boundary integral coupling methods for fluid-solid interaction, Quart. Appl. Math.,49 (1991), 107–119.
Bielak, J., MacCamy, R.C. and Zeng X.,Stable coupling method for interface scattering problems, Research Report R-91-199, Dept. of Civil Engineering, Carnegie Mellon University, 1991.
Brekhovskikh, L.M.,Waves in Layered Medium, Nauka, Moscow, 1973 (Russian).
Boström, A.,Scattering of stationary acoustic waves by an elastic obstacle immersed in a fluid. J. Acoust. Soc. Amer.,67(1980), 390–398.
Boström, A.Scattering of acoustic waves by a layered elastic obstacle in a fluid — a improved null field approach. J. Acoust. Soc. Amer.,76(1984), 588–593.
Colton, D. and Kress, R.,Integral Equation Methods in Scattering Theory. John Wiley, New York, 1983.
Ershov, N.E.,Solution of a three-dimensional diffraction problem of acoustic waves by potential method. In: Mathematical Problems of Geophysics. Novosibirsk, 1985, 47–58 (Russian).
Ershov, N.E. and Smagin, S.I.,On Solving of a stationary diffraction problem by potential method. Soviet Doklady,311, 2(1990), 339–342 (Russian).
Everstine, G.C. and Au-Yang, M.K. eds.,Advances in Fluid-Structure Interaction — 1984, American Society of Mechanical Engineers New York, 1984.
Everstine, G.C. and Henderson, F.M.,Coupled finite element/boundary element approach for fluid-structure interaction. J. Acoust. Soc. Amer.,87(1990), 1938–1947.
Fichera, G.,Existence Theorems in Elasticity. Handb. der Physik, Bd. 6/2, Springer-Verlag, Heidelberg, 1973.
Goswami, P.P., Rudolphy, T.J., Rizzo, F.J. and Shippy, D.J.,A boundary element method for acoustic-elastic interaction with applications in ultrasonic NDE. J. Nondestruct. Eval.,9(1990), 101–112.
Hargé, T.,Valeurs propres d'un corps élastique, C.R. Acad. Sci. Paris, Sér. I Math.,311(1990), 857–859.
Hörmander, L.,The Analysis of Linear Partial Differential Operators, III, Pseudodifferential Operators, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1985.
Hsiao, G.C.,On the boundary-field equation methods for fluid-structure interactions. Proceedings of the 10.TMP, Teubner Texte zur Mathematik, Bd.134, Stuttgart-Leipzig, 1994, 79–88.
Hsiao, G.C., Kleinman, R.E. and Schuetz, L.S.,On variational formulation of boundary value problems for fluid-solid interactions, In: MacCarthy M.F. and Hayes M.A. (eds): Elastic Wave Propagation. IUTAM Symposium on Elastic Wave Propagation. North-Holland-Amsterdam, 1989, 321–326.
Jentsch, L. and Natroshvili, D.,Non-classical interface problems for piecewise homogeneous anistropic bodies, Math. Methods Appl. Sci.,18(1995), 27–49.
Jentsch, L. and Natroshvili, D.,Thermoelastic oscillations of anisotropic bodies. Preprint 96-1. Technische Universität Cheminitz-Zwickau, Fakultät für Mathematik, 1996. (See also ‘Proceedings of Sommerfeld'96 Workshop, Freudenstadt, Schwarzwald, 30 September–4 October, 1996’).
Jentsch, L. and Natroshvili, D.,Interaction between thermoelastic and scalar oscillation fields (general anisotropic case). Preprint 97-5. Technische Universität Chemnitz-Zwickau, Fakultät für Mathematik, 1997.
Jentsch, L., Natroshvili, D. and Wendland, W.,General transmission problems in the theory of elastic oscillations of anisotropic bodies. DFG-Schwerpunkt Randelementmethoden, Bericht Nr. 95-7, Stuttgart University, 1995. (To appear in ‘Journal of Math. Anal. and Applications’).
Jones, D.S.,Low-frequency scattering by a body in lubricated contact. Quart. J. Mech. Appl. Math.,36(1983), 111–137.
Junger, M.C. and Fiet, D.,Sound, Structures and Their Interaction, MIT Press, Cambridge, MA, 1986.
Kupradze, V.D., Gegelia, T.G., Basheleishvili, M.O. and Burchuladze, T.V.,Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, Nauka, Moscow, 1976 (Russian).
Luke, C.J. and Martin, P.A.,Fluid-solidinteraction: acoustic scattering by a smooth elastic obstacle. SIAM J. Appl. Math.,55, 4(1995), 904–922.
Mikhlin, S.G. and Prössdorf, S.,Singular Integral Operators, Springer-Verlag, Berlin, 1986.
Miranda, C.,Partial Differential Equations of Elliptic Type, 2-nd ed., Springer-Verlag, Berlin-Heidelberg-New York, 1970.
Natroshvili, D.,Investigation of Boundary Value and Initial Boundary Value Problems of the Mathematical Theory of Elasticity and Thermoelasticity for Homogeneous Anisotropic Bodies by Potential Methods, Doct. Thesis, Tbilisi Math. Inst. of Acad. of Sci. Georgian SSR. Tbilisi, 1–325, 1984.
Natroshvili, D., Djagmaidze, A. and Svanadze, M.,Some Linear Problems of the Theory of Elastic Mixtures, Tbilisi University Press, Tbilisi, 1986.
Natroshvili, D.,Boundary integral equation method in the steady state oscillation problems for anisotropic bodies. Math. Methods Appl. Sci. (to appear)
Natroshvili, D. and Sadunishvili, G.,Interaction of elastic and scalar fields. Math. Methods Appl. Sci.19, No.18(1996), 1445–1469.
Nowacki, W.,Dynamic Problems of Thermoelasticity. PWN-Polish Scientific Publishers, Warszawa, Poland, Nordhoff International Publishing, Leyden, The Nederlands, 1975.
Ohayon, R. and Sanchez-Palencia, E.,On the vibration problem for an elastic body surrounded by a slightly compressible fluid, RAIRO Anal. Numér.,17(1983), 311–326.
Sarchez Hubert, J. and Sanchez-Palencia, E.,Vibration and Coupling of Continuous Systems, Springer-Verlag, Berlin, 1989.
Schneck, H.A.,Improved integral formulation for acoustic radiation problems, J. Acoust. Soc. Amer.,44(1968), 41–58.
[35]+Seybert, A.F., Wu, T.W. and Wu, X.F.,Radiation and scattering of acoustic waves from elastic solids and shells using the boundary element method. J. Acoust. Soc. Amer.,84(1988), 1906–1912.
Smagin, S.I.,Potential method in a three-dimensional problem of the diffraction electromagnetic waves, Journal of Computational Mathematics and Mathematical Physics,29, 1(1989), 82–92 (Russian).
Smagin, S.I.,On a new class of integral equations lowering dimensionality of diffraction problems. Soviet Doklady,316, 3(1991), 580–585 (Russian).
Vainberg, B.R.,The radiation, limiting absorption and limiting amplitude principles in the general theory of partial differential equations. Usp. Math. Nauk,21, 3(129)(1966), 115–194 (Russian).
Vekua, I.N.,On metaharmonic functions. Proc. Tbilisi Mathem. Inst. of Acad. Sci. Georgian SSR,12(1943), 105–174 (Russian).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jentsch, L., Natroshvili, D. Interaction between thermoelastic and scalar oscillation fields. Integr equ oper theory 28, 261–288 (1997). https://doi.org/10.1007/BF01294154
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01294154