On the Nonnormality of Subiteration for a Fluid-Structure-Interaction Problem

Subiteration forms the basic iterative method for solving the aggregated equations in fluid-structure-interaction problems, in which the fluid and structure equations are solved alternatingly subject to complementary partitions of the interface conditions. In the present work we establish for a prototypical model problem that the subiteration method can be characterized by recursion of a nonnormal operator. This implies that the method typically converges nonmonotonously. Despite formal stability, divergence can occur before asymptotic convergence sets in. It is shown that the transient divergence can amplify the initial error by many orders of magnitude, thus inducing a severe degradation in the robustness and efficiency of the subiteration method. Auxiliary results concern the dependence of the stability and convergence of the subiteration method on the physical parameters in the problem and on the computational time step.

[1]  Hans J. Stetter,et al.  The Defect Correction Approach , 1984 .

[2]  Charbel Farhat,et al.  Partitioned analysis of coupled mechanical systems , 2001 .

[3]  Lloyd N. Trefethen,et al.  Pseudospectra of Linear Operators , 1997, SIAM Rev..

[4]  M. Kreĭn,et al.  Introduction to the theory of linear nonselfadjoint operators , 1969 .

[5]  Eberhard Zeidler,et al.  Applied Functional Analysis: Applications to Mathematical Physics , 1995 .

[6]  M. Heil.  Stokes flow in an elastic tube—a large-displacement fluid-structure interaction problem , 1998 .

[7]  Charbel Farhat,et al.  The discrete geometric conservation law and the nonlinear stability of ALE schemes for the solution of flow problems on moving grids , 2001 .

[8]  Charbel Farhat,et al.  Partitioned procedures for the transient solution of coupled aeroelastic problems , 2001 .

[9]  Eberhard Zeidler,et al.  Applied Functional Analysis , 1995 .

[10]  R. Kamm,et al.  A fluid--structure interaction finite element analysis of pulsatile blood flow through a compliant stenotic artery. , 1999, Journal of biomechanical engineering.

[11]  Vimal Singh,et al.  Perturbation methods , 1991 .

[12]  Juan J. Alonso,et al.  Fully-implicit time-marching aeroelastic solutions , 1994 .

[13]  R. de Borst,et al.  Interface-GMRES(R) Acceleration of Subiteration for Fluid-Structure-Interaction Problems , 2005 .

[14]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[15]  Gregory W. Brown,et al.  Application of a three-field nonlinear fluid–structure formulation to the prediction of the aeroelastic parameters of an F-16 fighter , 2003 .

[16]  Scott A. Morton,et al.  Fully implicit aeroelasticity on overset grid systems , 1998 .

[17]  Gene H. Golub,et al.  Matrix computations , 1983 .

[18]  Anne Greenbaum,et al.  Iterative methods for solving linear systems , 1997, Frontiers in applied mathematics.

[19]  Anne E. Trefethen,et al.  Hydrodynamic Stability Without Eigenvalues , 1993, Science.

[20]  H. Weitzner,et al.  Perturbation Methods in Applied Mathematics , 1969 .

[21]  Dan S. Henningson,et al.  Pseudospectra of the Orr-Sommerfeld Operator , 1993, SIAM J. Appl. Math..

[22]  P. Tallec,et al.  Fluid structure interaction with large structural displacements , 2001 .

[23]  van Eh Harald Brummelen,et al.  Energy conservation under incompatibility for fluid-structure interaction problems , 2003 .

[24]  John W. Leonard,et al.  NONLINEAR RESPONSE OF MEMBRANES TO OCEAN WAVES USING BOUNDARY AND FINITE ELEMENTS , 1995 .

[25]  Serge Piperno,et al.  Explicit/implicit fluid/structure staggered procedures with a structural predictor and fluid subcycling for 2D inviscid aeroelastic simulations , 1997 .

[26]  S. Piperno.  Simulation numérique de phénomènes d'interaction fluide-structure , 1995 .

[27]  Roland Glowinski,et al.  An introduction to the mathematical theory of finite elements , 1976 .

[28]  C. Farhat,et al.  Partitioned procedures for the transient solution of coupled aroelastic problems Part I: Model problem, theory and two-dimensional application , 1995 .