Coupling of mixed finite element and stabilized boundary element methods for a fluid-solid interaction problem in 3D
We introduce and analyze the coupling of a mixed finite element and a boundary element for a three‐dimensional time‐harmonic fluid–solid interaction problem. We consider a formulation in which the Cauchy stress tensor and the rotation are the main variables in the elastic structure and use the usual...
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| Vydáno v: | Numerical methods for partial differential equations Ročník 30; číslo 4; s. 1211 - 1233 |
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01.07.2014
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| Abstract | We introduce and analyze the coupling of a mixed finite element and a boundary element for a three‐dimensional time‐harmonic fluid–solid interaction problem. We consider a formulation in which the Cauchy stress tensor and the rotation are the main variables in the elastic structure and use the usual pressure formulation in the acoustic fluid. The mixed variational formulation in the solid is completed with boundary integral equations relating the Cauchy data of the acoustic problem on the coupling interface. A crucial point in our formulation is the stabilization technique introduced by Hiptmair and coworkers to avoid the well‐known instability issue appearing in the boundary element method treatment of the exterior Helmholtz problem. The main novelty of this formulation, with respect to a previous approach, consists in reducing the computational domain to the solid media and providing a more accurate treatment of the far field effect. We show that the continuous problem is well‐posed and propose a conforming Galerkin method based on the lowest‐order Arnold–Falk–Winther mixed finite element. Finally, we prove that the numerical scheme is convergent with optimal order.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1211–1233, 2014 |
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| AbstractList | We introduce and analyze the coupling of a mixed finite element and a boundary element for a three‐dimensional time‐harmonic fluid–solid interaction problem. We consider a formulation in which the Cauchy stress tensor and the rotation are the main variables in the elastic structure and use the usual pressure formulation in the acoustic fluid. The mixed variational formulation in the solid is completed with boundary integral equations relating the Cauchy data of the acoustic problem on the coupling interface. A crucial point in our formulation is the stabilization technique introduced by Hiptmair and coworkers to avoid the well‐known instability issue appearing in the boundary element method treatment of the exterior Helmholtz problem. The main novelty of this formulation, with respect to a previous approach, consists in reducing the computational domain to the solid media and providing a more accurate treatment of the far field effect. We show that the continuous problem is well‐posed and propose a conforming Galerkin method based on the lowest‐order Arnold–Falk–Winther mixed finite element. Finally, we prove that the numerical scheme is convergent with optimal order.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1211–1233, 2014 We introduce and analyze the coupling of a mixed finite element and a boundary element for a three-dimensional time-harmonic fluid-solid interaction problem. We consider a formulation in which the Cauchy stress tensor and the rotation are the main variables in the elastic structure and use the usual pressure formulation in the acoustic fluid. The mixed variational formulation in the solid is completed with boundary integral equations relating the Cauchy data of the acoustic problem on the coupling interface. A crucial point in our formulation is the stabilization technique introduced by Hiptmair and coworkers to avoid the well-known instability issue appearing in the boundary element method treatment of the exterior Helmholtz problem. The main novelty of this formulation, with respect to a previous approach, consists in reducing the computational domain to the solid media and providing a more accurate treatment of the far field effect. We show that the continuous problem is well-posed and propose a conforming Galerkin method based on the lowest-order Arnold-Falk-Winther mixed finite element. Finally, we prove that the numerical scheme is convergent with optimal order.Copyright copyright 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1211-1233, 2014 We introduce and analyze the coupling of a mixed finite element and a boundary element for a three-dimensional time-harmonic fluid-solid interaction problem. We consider a formulation in which the Cauchy stress tensor and the rotation are the main variables in the elastic structure and use the usual pressure formulation in the acoustic fluid. The mixed variational formulation in the solid is completed with boundary integral equations relating the Cauchy data of the acoustic problem on the coupling interface. A crucial point in our formulation is the stabilization technique introduced by Hiptmair and coworkers to avoid the well-known instability issue appearing in the boundary element method treatment of the exterior Helmholtz problem. The main novelty of this formulation, with respect to a previous approach, consists in reducing the computational domain to the solid media and providing a more accurate treatment of the far field effect. We show that the continuous problem is well-posed and propose a conforming Galerkin method based on the lowest-order Arnold-Falk-Winther mixed finite element. Finally, we prove that the numerical scheme is convergent with optimal order.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1211-1233, 2014 [PUBLICATION ABSTRACT] |
| Author | Gatica, Gabriel N. Meddahi, Salim Heuer, Norbert |
| Author_xml | – sequence: 1 givenname: Gabriel N. surname: Gatica fullname: Gatica, Gabriel N. organization: CI2MA and Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Concepción, 160-C, Casilla, Chile – sequence: 2 givenname: Norbert surname: Heuer fullname: Heuer, Norbert organization: Departamento de Matemática, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile – sequence: 3 givenname: Salim surname: Meddahi fullname: Meddahi, Salim email: salim@uniovi.es organization: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Oviedo, Calvo Sotelo s/n, España, Oviedo |
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| Cites_doi | 10.1137/110836705 10.1007/978-3-642-97146-4 10.1002/num.20074 10.1137/060660370 10.1016/B978-0-444-87272-2.50054-3 10.1137/0519043 10.1007/978-1-4757-4338-8 10.1002/1522-2616(200010)218:1<139::AID-MANA139>3.0.CO;2-S 10.1007/978-3-662-03537-5 10.1007/s00211-004-0579-9 10.1137/050639958 10.1007/b98828 10.1016/j.apnum.2008.12.025 10.1007/978-1-4612-3172-1 10.1090/S0025-5718-07-01998-9 10.1017/S0962492902000041 10.1017/S0962492906210018 10.1137/S0036139993259027 10.1090/qam/1096235 10.1016/j.jcp.2004.02.005 |
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| References_xml | – reference: S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, Springer-Verlag, New York Inc., New York, 1994. – reference: S. A. Sauter and C. Schwab, Boundary element methods, Springer Series in Computational Mathematics 39, Springer-Verlag, Berlin 2011. – reference: S. Meddahi and Sayas F.-J., Analysis of a new BEM-FEM coupling for two dimensional fluid-solid interaction Numer Meth Partial Differen Equ 21 (2005), 1017-1042. – reference: J. Bielak and R. C. MacCamy, Symmetric finite element and boundary integral coupling methods for fluid-solid interaction Q Appl Math 49 (1991) 107-119. – reference: G. N. Gatica, A. Márquez, and S. Meddahi, Analysis of the coupling of BEM, FEM, and mixed-FEM for a two-dimensional fluid-solid interaction problem, Appl Numer Math 59 (2009) 2735-2750. – reference: A. Buffa and R. Hiptmair, Regularized combined field integral equations, Numer Math 100 (2005) 1-19. – reference: G. Hsiao, The coupling of BEM and FEM-A brief review, Boundary elements X, Vol. 1, Springer-Verlag, New York, 1988 pp. 431-445. – reference: D. Boffi, F. Brezzi, and M. Fortin, Reduced symmetry elements in linear elasticity, Commun Pure Appl Anal 8 (2009), 1-28. – reference: A. Márquez, S. Meddahi, and V. Selgas A new BEM-FEM coupling strategy for two-dimensional fluid-solid interaction problems, J Comput Phy 199 (2004) 205-220. – reference: W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. – reference: D. N. Arnold, R. S. Falk, and R. Winther, Mixed finite element methods for linear elasticity with weakly imposed symmetry, Math Comput 76 (2007), 1699-1723. – reference: C. J. Luke and P. A. Martin, Fluid-solid interaction: acoustic scattering by a smooth elastic obstacle, SIAM J Appl Math 55 (1995) 904-922. – reference: G. C. Hsiao, R. E. Kleinman, and G. F Roach, Weak solutions of fluid-solid interaction problems, Mathe Nachrich 218 (2000) 139-163. – reference: A. Ern and J.-L. Guermond, Elément finis: théorie, applications mise en euvre Vol. 36 SMAI Mathématiques et Applications, Springer, Heidelberg, 2002. – reference: R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer 11 (2002), 237-339. – reference: G. N. Gatica, A. Márquez, and S. Meddahi, Analysis of the coupling of Lagrange and Arnold-Falk-Winther finite elements for a fluid-solid interaction problem in 3D, SIAM J Numer Anal 50 (2012), 1648-1674. – reference: F. Ihlenburg, Finite element analysis of acoustic scattering, Springer-Verlag, New York, 1998. – reference: D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, 2nd Ed., Springer-Verlag, Berlin, 1998. – reference: R. Kress, Linear integral equations, Springer-Verlag, Berlin, 1989. – reference: M. Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J Numer Anal 19 (1988) 613-626. – reference: R. Hiptmair and P. Meury, Stabilized FEM-BEM coupling for Helmholtz transmission problems, SIAM J Numer Analy 44 (2006) 2107-2130. – reference: M. Costabel, Symmetric methods for the coupling of finite elements and boundary elements The mathematics of finite elements and applications IV, Academic Press, London, 1988. – reference: G. N. Gatica, A. Márquez, and S. Meddahi, Analysis of the coupling of primal and dual-mixed finite element methods for a two-dimensional fluid-solid interaction problem, SIAM J Numer Anal 45 (2007) 2072-2097. – reference: D. N. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numerica 15 (2006) 1-155. – reference: F. Brezzi and M. 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| Snippet | We introduce and analyze the coupling of a mixed finite element and a boundary element for a three‐dimensional time‐harmonic fluid–solid interaction problem.... We introduce and analyze the coupling of a mixed finite element and a boundary element for a three-dimensional time-harmonic fluid-solid interaction problem.... |
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| SubjectTerms | Acoustics Boundary element method elastodynamic equation Finite element method Galerkin methods Helmholtz equation Joining Mathematical analysis Mathematical models mixed finite elements Three dimensional |
| Title | Coupling of mixed finite element and stabilized boundary element methods for a fluid-solid interaction problem in 3D |
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