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Surrogate model reduction for linear dynamic systems based on a frequency domain modal analysis

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Abstract

A novel model reduction methodology for linear dynamic systems with parameter variations is presented based on a frequency domain formulation and use of the proper orthogonal decomposition. For an efficient treatment of parameter variations, the system matrices are divided into a nominal and an incremental part. It is shown that the perturbed part is modally equivalent to a new system where the incremental matrices are isolated into the forcing term. To account for the continuous changes in the parameters, the single-composite-input is invoked with a finite number of predetermined incremental matrices. The frequency-domain Karhunen–Loeve procedure is used to calculate a rich set of basis modes accounting for the variations. For demonstration, the new procedure is applied to a finite element model of the Goland wing undergoing oscillations and shown to produce extremely accurate reduced-order surrogate model for a wide range of parameter variations.

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Abbreviations

\({\varvec{A}}, {\varvec{B}}, {\varvec{C}}\) :

Linear system matrices

\({\varvec{A}}_0, {\varvec{B}}_0, {\varvec{C}}_0 \) :

Nominal system matrices

\({\varvec{A}}_R, {\varvec{B}}_R, {\varvec{C}}_R \) :

Reduced-order system matrices

\({\varvec{\Delta }}{\varvec{A}}, {\varvec{\Delta }}{\varvec{B}}, {\varvec{\Delta }}{\varvec{C}}\) :

Perturbed incremental system matrices

\({\varvec{p}}\) :

\(\left( {P\times 1} \right) \) generalized coord. for the second ROSM

\({\varvec{q}}_{0}\) :

\(\left( {R_0 \times 1} \right) \) generalized coord. for the nominal ROM

\({\varvec{q}}\) :

\(\left( {R\times 1} \right) \) generalized coord. for the first ROSM

\({\varvec{r}}_0\) :

\(\left( {R_0 \times 1} \right) \) statistically uncorrelated signals

\({\varvec{{\mathcal {R}}}}_0\) :

Fourier transform of \({\varvec{r}}_0\)

\({\varvec{s}}\) :

\(\left( {I\times 1} \right) \) statistically uncorrelated signals

\({\varvec{{\mathcal {S}}}}\) :

Fourier transform of \({\varvec{s}}\)

\({\varvec{u}}\) :

\(\left( {I\times 1} \right) \) system inputs

u, v, w :

nodal displacements

\({\varvec{{\mathcal {U}}}}\) :

Fourier transform of \({\varvec{u}}\)

\({\varvec{V}}, {\varvec{W}}\) :

Right and left eigenvectors of (15) and (19)

\({\mathcal {W}}\) :

Fourier transform of the vertical displacement w

\({\varvec{x}}\) :

\(\left( {N\times 1} \right) \) system states

\({\varvec{x}}_0\) :

Nominal solution of \({\varvec{x}}\)

\({\varvec{\Delta }}{\varvec{x}}\) :

Perturbed solution of \({\varvec{x}}\)

\({\varvec{\Delta }}{\varvec{{\mathcal {X}}}}\) :

Fourier transform of \({\varvec{\Delta }}{\varvec{x}}\)

\({\varvec{y}}\) :

\(\left( {L\times 1} \right) \) system outputs

\({\varvec{\alpha }}\) :

Eigenvector of the covariance matrix in (6)

\({\lambda }\) :

Eigenvalue of the covariance matrix in (6) or (15)

\({\varvec{\varLambda }}\) :

Diagonal matrix containing eigenvalues of (15)

\({\varvec{\varLambda }}^{\prime }\) :

Diagonal matrix containing eigenvalues of (19)

\(\lambda ^{\prime }\) :

Eigenvalue of (19)

\({\varvec{\mu }}\) :

\(\left( {H\times 1} \right) \) system parameters

\({\varvec{\phi }}\) :

\(\left( {N\times 1} \right) \) mode vector for the first ROSM

\({\varvec{\varPhi }}_{0}\) :

\(\left( {N\times R_0 } \right) \) modes set for the nominal system

\({\varvec{\varPhi }}_{1}\) :

\(\left( {N\times R_1 } \right) \) modes set for the perturbed system

\({\varvec{\varPhi }}\) :

\(\left( {N\times R} \right) \) modes set for the first ROSM

\({\varvec{\psi }}\) :

\(\left( {R\times 1} \right) \) mode vector for the second ROSM

\({\varvec{\varPsi }}\) :

\(\left( {R\times P} \right) \) modes set for the second ROSM

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Acknowledgments

The author is grateful to Yu Qijing, a graduate research assistant, who kindly prepared the Goland wing finite element model using NASTRAN.

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Authors and Affiliations

  1. Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore, 117576, Singapore

    T. Kim

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Appendix: MEPS for structural dynamic system

Appendix: MEPS for structural dynamic system

Given a structural dynamic equation of motion with mass, damping, and stiffness matrices subject to changes in parameters:

$$\begin{aligned} {\varvec{M}}\left( {\varvec{\mu }} \right) \ddot{{\varvec{x}}}+ {\varvec{B}}\left( {\varvec{\mu }} \right) \dot{{\varvec{x}}}+ {\varvec{K}}\left( {\varvec{\mu }} \right) {\varvec{x}}={\varvec{f}} \end{aligned}$$
(49)

Expanding the solution and system matrices in nominal and perturbed parts

$$\begin{aligned}&{\varvec{\mu }} ={\varvec{\mu }}_{\mathbf{0}} +{\varvec{\Delta }}{\varvec{\mu }}\nonumber \\&{\varvec{x}}\left( {{\varvec{\mu }}, t} \right) ={\varvec{x}}_{\mathbf{0}} \left( {{\varvec{\mu }}_0, t} \right) +{\varvec{\Delta }}{\varvec{x}}\left( {{\varvec{\mu }}, t} \right) \nonumber \\&{\varvec{M}}\left( {\varvec{\mu }} \right) ={\varvec{M}}_{\mathbf{0}} \left( {{\varvec{\mu }}_0 } \right) + {\varvec{\Delta }}{\varvec{M}}\left( {\varvec{\mu }} \right) \nonumber \\&{\varvec{B}}\left( {\varvec{\mu }} \right) ={\varvec{B}}_{\mathbf{0}} \left( {{\varvec{\mu }}_0} \right) +{\varvec{\Delta }}{\varvec{B}}\left( {\varvec{\mu }} \right) \nonumber \\&{\varvec{K}}\left( {\varvec{\mu }} \right) ={\varvec{K}}_{\mathbf{0}} \left( {{\varvec{\mu }}_0 } \right) + {\varvec{\Delta }}{\varvec{K}}\left( {\varvec{\mu }} \right) \end{aligned}$$
(50)

it can be shown that inserting (50) into (49) and following the previous modal analysis results in the following nominal, perturbed, and MEPS equations analogous to the equations (10), (12), and (25):

Nominal system

$$\begin{aligned} \ddot{{\varvec{x}}}_0 +\bar{{\varvec{B}}}_0 \dot{{\varvec{x}}}_0 +\bar{{\varvec{K}}}_0 {\varvec{x}}_0 =\bar{{\varvec{f}}}_0 \end{aligned}$$
(51)

Perturbed system

$$\begin{aligned}&{\varvec{\Delta }}\ddot{{\varvec{x}}}+\bar{{\varvec{B}}}_0 {\varvec{\Delta }}\dot{{\varvec{x}}}+\bar{{\varvec{K}}}_0 {\varvec{\Delta }}{\varvec{x}} +{\varvec{\Delta }}\bar{{\varvec{B}}} \left( {\dot{{\varvec{x}}}_{0} +{\varvec{\Delta }}{\varvec{x}}} \right) + {\varvec{\Delta }}\bar{{\varvec{K}}} \left( {{\varvec{x}}_0 +{\varvec{\Delta }}{\varvec{x}}} \right) \nonumber \\&\quad ={\varvec{\Delta }}\bar{{\varvec{f}}} \end{aligned}$$
(52)

MEPS

$$\begin{aligned} {\varvec{\Delta }}\ddot{{\varvec{x}}}+\bar{{\varvec{B}}}_0 {\varvec{\Delta }}\dot{{\varvec{x}}}+ \bar{{\varvec{K}}}_0 {\varvec{\Delta }}{\varvec{x}}+{\varvec{\Delta }}\bar{{\varvec{B}}} \dot{{\varvec{x}}}_0 +{\varvec{\Delta }}\bar{{\varvec{K}}} {\varvec{x}}_0 ={\varvec{\Delta }}\bar{{\varvec{f}}} \end{aligned}$$
(53)

where

$$\begin{aligned} \bar{{\varvec{B}}}_{0}\equiv & {} {\varvec{M}}_{0}^{-1} {\varvec{B}}_0, \qquad {\varvec{\Delta }}\bar{{\varvec{B}}} \equiv {\varvec{M}}^{-1}{\varvec{B}}-\bar{{\varvec{B}}}_0\nonumber \\ \bar{{\varvec{K}}}_0\equiv & {} {\varvec{M}}_0^{-1} {\varvec{K}}_0, \qquad {\varvec{\Delta }}\bar{{\varvec{K}}} \equiv {\varvec{M}}^{-1}{\varvec{K}}-\bar{{\varvec{K}}}_0\nonumber \\ \bar{{\varvec{f}}}_0\equiv & {} {\varvec{M}}_0^{-1} {\varvec{f}}, \qquad {\varvec{\Delta }}\bar{{\varvec{f}}} \equiv {\varvec{M}}^{-1}{\varvec{f}}-\bar{{\varvec{f}}}_0 \end{aligned}$$
(54)

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Kim, T. Surrogate model reduction for linear dynamic systems based on a frequency domain modal analysis. Comput Mech 56, 709–723 (2015). https://doi.org/10.1007/s00466-015-1196-4

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