Hostname: page-component-77c78cf97d-xcx4r Total loading time: 0 Render date: 2026-04-23T16:27:50.166Z Has data issue: false hasContentIssue false
  • English
  • Français

A multiplicative Schwarz method and its applicationto nonlinear acoustic-structure interaction

Published online by Cambridge University Press:  08 April 2009

Roland Ernst ,
Bernd Flemisch  and
Barbara Wohlmuth
Roland Ernst
Affiliation:
Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Germany.
Bernd Flemisch
Affiliation:
Institute of Hydraulic Engineering, University of Stuttgart, Germany. bernd@iws.uni-stuttgart.de
Barbara Wohlmuth
Affiliation:
Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Germany.
Get access

Abstract

A new Schwarz method for nonlinear systems is presented, constituting the multiplicative variant of a straightforward additive scheme. Local convergence can be guaranteed under suitable assumptions.The scheme is applied to nonlinear acoustic-structure interaction problems. Numerical examples validate the theoretical results. Further improvements are discussed by means of introducing overlapping subdomains and employing an inexact strategy for the local solvers.

Keywords

Schwarz methodfluid-structure interactioncoupled problemsnonlinear elasticitynonlinear acousticselasto-acoustic.

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 3 , May 2009 , pp. 487 - 506
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

On, H.-B. An convergence of the additive Schwarz preconditioned inexact Newton method. SIAM J. Numer. Anal. 43 (2005) 1850–1871.
Bermúdez, A., Rodríguez, R. and Santamarina, D., Finite element approximation of a displacement formulation for time-domain elastoacoustic vibrations. J. Comput. Appl. Math. 152 (2003) 17–34. CrossRef
C. Bernardi, Y. Maday and A.T. Patera, Domain decomposition by the mortar element method, in Asymptotic and numerical methods for partial differential equations with critical parameters (Beaune, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 384, Kluwer Acad. Publ., Dordrecht (1993) 269–286.
C. Bernardi, Y. Maday and A.T. Patera, A new nonconforming approach to domain decomposition: the mortar element method, in Nonlinear partial differential equations and their applications, Collège de France Seminar, Vol. XI (Paris, 1989–1991), Pitman Res. Notes Math. Ser. 299, Longman Sci. Tech., Harlow (1994) 13–51.
Cai, X.-C. and Keyes, D.E., Nonlinearly preconditioned inexact Newton algorithms. SIAM J. Sci. Comput. 24 (2002) 183–200. CrossRef
P.G. Ciarlet, Mathematical elasticity, Vol. I: Three-dimensional elasticity, Studies in Mathematics and its Applications 20. North-Holland Publishing Co., Amsterdam (1988).
Dryja, M. and Hackbusch, W., On the nonlinear domain decomposition method. BIT 37 (1997) 296–311. CrossRef
Flemisch, B., Kaltenbacher, M. and Wohlmuth, B.I., Elasto-acoustic and acoustic-acoustic coupling on non-matching grids. Int. J. Numer. Meth. Engng. 67 (2006) 1791–1810. CrossRef
M.F. Hammilton and D.T. Blackstock, Nonlinear Acoustics. Academic Press (1998).
T. Hughes, The Finite Element Method. Prentice-Hall, New Jersey (1987).
M. Kaltenbacher. Numerical Simulation of Mechatronic Sensors and Actuators. Springer, Berlin-Heidelberg-New York (2007).
Kuhl, D. and Crisfield, M.A., Energy-conserving and decaying algorithms in non-linear structural dynamics. Int. J. Numer. Meth. Engng. 45 (1999) 569–599. 3.0.CO;2-A>CrossRef
Kuznetsov, V.I., Equations of nonlinear acoustics. Soviet Phys.-Acoust. 16 (1971) 467–470.
N.M. Newmark, A method of computation for structural dynamics. J. Engng. Mech. Div., Proc. ASCE 85 (EM3) (1959) 67–94.
A. Quarteroni and A. Valli, Domain decomposition methods for partial differential equations, Numerical Mathematics and Scientific Computation. Oxford University Press, New York (1999).
A.-M. Sändig, Nichtlineare Funktionalanalysis mit Anwendungen auf partielle Differentialgleichungen. Vorlesung im Sommersemester 2006, IANS preprint 2006/012, Technical report, University of Stuttgart, Germany (2006).
B.F. Smith, P.E. Bjørstad and W.D. Gropp, Domain decomposition, Parallel multilevel methods for elliptic partial differential equations. Cambridge University Press, Cambridge (1996).
A. Toselli and O. Widlund, Domain decomposition methods – algorithms and theory, Springer Series in Computational Mathematics 34. Springer-Verlag, Berlin (2005).