A method for interpolating on manifolds structural dynamics reduced‐order models

A rigorous method for interpolating a set of parameterized linear structural dynamics reduced-order models (ROMs) is presented. By design, this method does not operate on the underlying set of parameterized full-order models. Hence, it is amenable to an online real-time implementation. It is based on mapping appropriately the ROM data onto a tangent space to the manifold of symmetric positive-definite matrices, interpolating the mapped data in this space and mapping back the result to the aforementioned manifold. Algorithms for computing the forward and backward mappings are offered for the case where the ROMs are derived from a general Galerkin projection method and the case where they are constructed from modal reduction. The proposed interpolation method is illustrated with applications ranging from the fast dynamic characterization of a parameterized structural model to the fast evaluation of its response to a given input. In all cases, good accuracy is demonstrated at real-time processing speeds. Copyright © 2009 John Wiley & Sons, Ltd.

[1]  C. Farhat,et al.  Interpolation Method for Adapting Reduced-Order Models and Application to Aeroelasticity , 2008 .

[2]  Charbel Farhat,et al.  Application of Multiple Methods for Aeroelastic Uncertainty Analysis , 2008 .

[3]  Ira B. Schwartz,et al.  Dynamics of Large Scale Coupled Structural/ Mechanical Systems: A Singular Perturbation/ Proper Orthogonal Decomposition Approach , 1999, SIAM J. Appl. Math..

[4]  Christopher Flanigan.  Development of the IRS component dynamic reduction method for substructure analysis , 1991 .

[5]  Helmuth Späth,et al.  Two-dimensional spline interpolation algorithms , 1993 .

[6]  M. P. Païdoussis,et al.  Reduced-order models for nonlinear vibrations of cylindrical shells via the proper orthogonal decomposition method , 2003 .

[7]  G. Kerschen,et al.  The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview , 2005 .

[8]  S. Volkwein,et al.  Reduced order output feedback control design for PDE systems using proper orthogonal decomposition and nonlinear semidefinite programming , 2006 .

[9]  Taehyoun Kim,et al.  Frequency-Domain Karhunen -Loeve Method and Its Application to Linear Dynamic Systems , 1998 .

[10]  M. Hinze,et al.  Proper Orthogonal Decomposition Surrogate Models for Nonlinear Dynamical Systems: Error Estimates and Suboptimal Control , 2005 .

[11]  Xavier Pennec,et al.  A Riemannian Framework for Tensor Computing , 2005, International Journal of Computer Vision.

[12]  Charbel Farhat,et al.  On-Demand CFD-Based Aeroelastic Predictions Using a Database of Reduced-Order Bases and Models , 2009 .

[13]  Bogdan I. Epureanu,et al.  A parametric analysis of reduced order models of viscous flows in turbomachinery , 2003 .

[14]  E. Carson Yates,et al.  AGARD standard aeroelastic configurations for dynamic response. Candidate configuration I.-wing 445.6 , 1987 .

[15]  B. Feeny,et al.  Enhanced Proper Orthogonal Decomposition for the Modal Analysis of Homogeneous Structures , 2002 .

[16]  R. Guyan.  Reduction of stiffness and mass matrices , 1965 .

[17]  David J. Ewins,et al.  Modal Testing: Theory, Practice, And Application , 2000 .