Efficient model reduction in non-linear dynamics using the Karhunen-Loève expansion and dual-weighted-residual methods

Abstract We look at the task of computing the time-evolution of a non-linear system for a long time, in our case under random external influences. Our specific example is the fatigue evaluation of a wind turbine. To facilitate such a computation, we look at a reduction of the computational effort by projecting everything on a low-dimensional basis. In this case we take the Karhunen-Loève basis generated from running the model a little while under the random loading. It is important that the error which is caused by this reduction process can be controlled. We estimate the error by dual or adjoint methods. This in turn allows the process of model reduction to be performed adaptively.

[1]  D. Estep.  A posteriori error bounds and global error control for approximation of ordinary differential equations , 1995 .

[2]  Antony Jameson,et al.  Aerodynamic design via control theory , 1988, J. Sci. Comput..

[3]  W. Rheinboldt,et al.  On the Discretization Error of Parametrized Nonlinear Equations , 1983 .

[4]  P. Holmes,et al.  The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows , 1993 .

[5]  Ahmed K. Noor,et al.  Reduced Basis Technique for Nonlinear Analysis of Structures , 1980 .

[6]  Edwin Kreuzer,et al.  Analysis of long torsional strings by proper orthogonal decomposition , 1996 .

[7]  E. Wilson,et al.  Dynamic analysis by direct superposition of Ritz vectors , 1982 .

[8]  Ray W. Clough,et al.  Dynamic analysis of structures using lanczos co‐ordinates , 1984 .

[9]  Rolf Rannacher,et al.  On Error Control in CFD , 1994 .

[10]  Alberto Cardona,et al.  A reduction method for nonlinear structural dynamic analysis , 1985 .

[11]  Lawrence Sirovich,et al.  Management and Analysis of Large Scientific Datasets , 1992 .

[12]  Kenneth A. Kline,et al.  Dynamic analysis using a reduced basis of exact modes and Ritz vectors , 1986 .

[13]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[14]  J. Remke,et al.  Eine modale Reduktionsmethode zur geometrisch nichtlinearen statischen und dynamischen Finite-Element-Berechnung , 1993 .

[15]  P. Stern,et al.  Automatic choice of global shape functions in structural analysis , 1978 .

[16]  P. Holmes,et al.  Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 1996 .

[17]  Anders Logg,et al.  A Multi-Adaptive ODE-Solver , 2000 .

[18]  Senol Utku,et al.  Errors in reduction methods , 1985 .

[19]  D. Sorensen.  Numerical methods for large eigenvalue problems , 2002, Acta Numerica.

[20]  Rolf Rannacher,et al.  An optimal control approach to a posteriori error estimation in finite element methods , 2001, Acta Numerica.

[21]  Rolf Rannacher,et al.  A Feed-Back Approach to Error Control in Finite Element Methods: Basic Analysis and Examples , 1996 .

[22]  Rolf Rannacher,et al.  Finite element approximation of the acoustic wave equation: error control and mesh adaptation , 1999 .

[23]  Peter Wriggers,et al.  Nichtlineare Finite-Element-Methoden , 2001 .

[24]  R. E. Nickell,et al.  Nonlinear dynamics by mode superposition , 1976 .

[25]  Rolf Rannacher,et al.  The Dual-Weighted-Residual Method for Error Control and Mesh Adaptation in Finite Element Methods , 2000 .

[26]  M. Géradin,et al.  Flexible Multibody Dynamics: A Finite Element Approach , 2001 .

[27]  Ahmed K. Noor,et al.  Recent Advances and Applications of Reduction Methods , 1994 .

[28]  J. Marsden,et al.  Dimensional model reduction in non‐linear finite element dynamics of solids and structures , 2001 .

[29]  L. Sirovich.  Turbulence and the dynamics of coherent structures. I. Coherent structures , 1987 .

[30]  Rolf Rannacher,et al.  An Optimal Control Approach to A-Posteriori Error Estimation , 2001 .

[31]  Claes Johnson,et al.  Computational Differential Equations , 1996 .

[32]  Claes Johnson,et al.  Introduction to Adaptive Methods for Differential Equations , 1995, Acta Numerica.

[33]  Pierre Léger,et al.  Ritz vectors and generation criteria for mode superposition analysis , 1989 .

[34]  Werner C. Rheinboldt,et al.  On the Error Behavior of the Reduced Basis Technique for Nonlinear Finite Element Approximations , 1983 .

[35]  O. Baysal,et al.  Aerodynamic Sensitivity Analysis Methods for the Compressible Euler Equations , 1991 .

[36]  Guriĭ Ivanovich Marchuk,et al.  Adjoint Equations and Analysis of Complex Systems , 1995 .

[37]  K. Karhunen.  Zur Spektraltheorie stochastischer prozesse , 1946 .

[38]  H. Harman.  Modern factor analysis , 1961 .

[39]  Pierre Léger,et al.  Non‐linear seismic response analysis using vector superposition methods , 1992 .

[40]  Anders Logg,et al.  Multi-Adaptive Error Control for ODEs , 1998 .

[41]  Claes Johnson.  Error Estimates and Adaptive Time-Step Control for a Class of One-Step Methods for Stiff Ordinary Differential Equations , 1988 .

[42]  H. Hotelling.  Analysis of a complex of statistical variables into principal components. , 1933 .

[43]  Michel Loève,et al.  Probability Theory I , 1977 .

[44]  Christian Cabos.  Error Bounds for Dynamic Responses in Forced Vibration Problems , 1994, SIAM J. Sci. Comput..