Time‐parallel implicit integrators for the near‐real‐time prediction of linear structural dynamic responses

The time-parallel framework for constructing parallel implicit time-integration algorithms (PITA) is revisited in the specific context of linear structural dynamics and near-real-time computing. The concepts of decomposing the time-domain in time-slices whose boundaries define a coarse time-grid, generating iteratively seed values of the solution on this coarse time-grid, and using them to time-advance the solution in each time-slice with embarrassingly parallel time-integrations are maintained. However, the Newton-based corrections of the seed values, which so far have been computed in PITA and related approaches on the coarse time-grid, are eliminated to avoid artificial resonance and numerical instability. Instead, the jumps of the solution on the coarse time-grid are addressed by a projector which makes their propagation on the fine time-grid computationally feasible while avoiding artificial resonance and numerical instability. The new PITA framework is demonstrated for a complex structural dynamics problem from the aircraft industry. Its potential for near-real-time computing is also highlighted with the solution of a relatively small-scale problem on a Linux cluster system. Copyright © 2006 John Wiley & Sons, Ltd.

[1]  Andrea Toselli,et al.  Recent developments in domain decomposition methods , 2002 .

[2]  Stefan Vandewalle,et al.  Efficient Parallel Algorithms for Solving Initial-Boundary Value and Time-Periodic Parabolic Partial Differential Equations , 1992, SIAM J. Sci. Comput..

[3]  Graham Horton,et al.  A Space-Time Multigrid Method for Parabolic Partial Differential Equations , 1995, SIAM J. Sci. Comput..

[4]  Charbel Farhat,et al.  6. A Time-Parallel Implicit Methodology for the Near-Real-Time Solution of Systems of Linear Oscillators , 2007 .

[5]  Kevin Burrage,et al.  Parallel methods for initial value problems , 1993 .

[6]  Andrew Lumsdaine,et al.  Krylov Subspace Acceleration of Waveform Relaxation , 2003, SIAM J. Numer. Anal..

[7]  L. Brugnano,et al.  Solving differential problems by multistep initial and boundary value methods , 1998 .

[8]  J. Lions,et al.  Résolution d'EDP par un schéma en temps « pararéel » , 2001 .

[9]  Charbel Farhat,et al.  Time‐decomposed parallel time‐integrators: theory and feasibility studies for fluid, structure, and fluid–structure applications , 2003 .

[10]  Pierluigi Amodio,et al.  Parallel implementation of block boundary value methods for ODEs , 1997 .

[11]  Y. Maday,et al.  A “Parareal” Time Discretization for Non-Linear PDE’s with Application to the Pricing of an American Put , 2002 .

[12]  Graham Horton,et al.  An Algorithm with Polylog Parallel Complexity for Solving Parabolic Partial Differential Equations , 1995, SIAM J. Sci. Comput..

[13]  Joachim Stadel,et al.  A Parallel Integration Method for Solar System Dynamics , 1997 .

[14]  Guillaume Bal,et al.  On the Convergence and the Stability of the Parareal Algorithm to Solve Partial Differential Equations , 2005 .

[15]  Shlomo Ta'asan,et al.  Fourier-Laplace analysis of the multigrid waveform relaxation method for hyperbolic equations , 1996 .

[16]  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 .

[17]  M. Géradin,et al.  ON THE GENERAL SOLUTION BY A DIRECT METHOD OF A LARGE-SCALE SINGULAR SYSTEM OF LINEAR EQUATIONS : APPLICATION TO THE ANALYSIS OF FLOATING STRUCTURES , 1998 .