Advances in Operator Theory

On an elasto-acoustic transmission problem in anisotropic‎, ‎inhomogeneous media

Rainer Picard

Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

‎We consider a coupled system describing the interaction between acoustic and elastic regions‎, ‎where the coupling occurs not via material properties but through an interaction on an interface separating the two regimes‎. ‎Evolutionary well-posedness in the sense of Hadamard well-posedness supplemented by causal dependence is shown for a natural choice of generalized interface conditions‎. ‎The results are obtained in a real Hilbert space setting incurring no regularity constraints on the boundary and almost none on the interface of the underlying regions‎.

Article information

Source
Adv. Oper. Theory, Advance publication (2018), 13 pages.

Dates
Received: 1 March 2018
Accepted: 12 June 2018
First available in Project Euclid: 7 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.aot/1530928840

Digital Object Identifier
doi:10.15352/aot.1803-1323

Subjects
Primary: 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47)
Secondary: 46E40‎ ‎35L50‎ ‎74F10

Keywords
Hilbert space methods‎ evolutionary systems‎‎ ‎‎operator coefficient‎s transmission problems ‎aeroelasticity

Citation

Picard, Rainer. On an elasto-acoustic transmission problem in anisotropic‎, ‎inhomogeneous media. Adv. Oper. Theory, advance publication, 7 July 2018. doi:10.15352/aot.1803-1323. https://projecteuclid.org/euclid.aot/1530928840


Export citation

References

  • B. Flemisch, M. Kaltenbacher, and B. I. Wohlmuth, Elasto-acoustic and acoustic-acoustic coupling on non-matching grids, Internat. J. Numer. Methods Engrg. 67 (2006), no. 13, 1791–1810.
  • K. O. Friedrichs,Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math. 7 (1954), 345–392.
  • K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958), no. 3, 333–418.
  • Y. Gao, P. Li, and B. Zhang, Analysis of transient acoustic-elastic interaction in an unbounded structure, SIAM J. Math. Anal. 49 (2017), no. 5, 3951–3972.
  • G. C. Hsiao, R. E. Kleinman, and G. F. Roach, Weak solutions of fluid-solid interaction problems, Math. Nachr. 218 (2000), 139–163.
  • F. Kang and X. Jiang, Variational approach to shape derivatives for elasto-acoustic coupled scattering fields and an application with random interfaces, J. Math. Anal. Appl. 456 (2017), no. 1, 686–704.
  • H. Lamb, On the vibrations of an elastic plate in contact with water, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. Series A 98 (1920), 205–216.
  • M. Lax, The effect of radiation on the vibrations of a circular diaphragm, J. Acoust. Soc. America 16 (1944), no. 1, 5–13.
  • C. J. Luke and P. A. Martin, Fluid-solid interaction: Acoustic scattering by a smooth elastic obstacle, SIAM J. Appl. Math. 55 (1995), no. 4, 904–922.
  • S. M önk ölä, Numerical simulation of fluid-structure interaction between acoustic and elastic waves, Jyväskylä Stud. Comput. 133, 2011.
  • A. J. Mulholland, R. Picard, S. Trostorff, and M. Waurick, On well-posedness for some thermo-piezoelectric coupling models, Math. Methods Appl. Sci. 39 (2016), no. 15, 4375–4384.
  • D. Natroshvili, D. Sadunishvili, I. Sigua, and Z. Tediashvili, Fluid-solid interaction: Acoustic scattering by an elastic obstacle with Lipschitz boundary, Mem. Differ. Equ. Math. Phys. 35 (2005), 91–127.
  • W. Nowacki, Some theorems of asymmetric thermoelasticity, J. Math. Phys. Sci. 2 (1968), 111–122.
  • W. Nowacki, Dynamische probleme der unsymmetrischen elastizität, Prikl. Mekh. 6 (1970), no. 4, 31–50.
  • W. Nowacki, Theory of asymmetric elasticity, Oxford etc.: Pergamon Press; Warszawa: PWN-Polish Scientific Publishers, 1986.
  • R. Picard, A structural observation for linear material laws in classical mathematical physics, Math. Methods Appl. Sci. 32 (2009), no. 14, 1768–1803.
  • R. Picard. Mother Operators and their Descendants, Technical report, TU Dresden, arXiv:1203.6762v2.
  • R. Picard, Mother operators and their descendants, J. Math. Anal. Appl. 403 (2013), no. 1, 54–62.
  • R. Picard and D. F. McGhee, Partial differential equations: A unified Hilbert space approach, De Gruyter Expositions in Mathematics, 55. Walter de Gruyter GmbH & Co. KG, Berlin, 2011.
  • R. Picard, St. Seidler, S. Trostorff, and M. Waurick, On abstract grad-div systems, J. Differential Equations 260 (2016), no. 6, 4888 – 4917.
  • R. Picard, S. Trostorff, and M. Waurick, On some models for elastic solids with micro-structure, ZAMM Z. Angew. Math. Mech. 95 (2015), no. 7, 664–689.
  • R. Picard, S. Trostorff, and M. Waurick, Well-posedness via Monotonicity – an overview, Operator semigroups meet complex analysis, harmonic analysis and mathematical physics, 397–452, Oper. Theory Adv. Appl., 250, Birkhäuser/Springer, Cham, 2015.
  • A. F. Seybert, T. W. Wu, and X. F. Wu, Radiation and scattering of acoustic waves from elastic solids and shells using the boundary element method, J. Acoust. Soc. America 84 (1988), 1906–1912.
  • L. C. Wilcox, G. Stadler, C. Burstedde, and O. Ghattas, A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media, J. Comput. Phys. 229 (2010), no. 24, 9373–9396.