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Efficient design optimization strategy for structural dynamic systems using a reduced basis method combined with an equivalent static load

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Abstract

This paper presents an efficient structural design optimization strategy that combines the reduced basis method (RBM) with the equivalent static load (ESL). In dynamic response optimization using ESL, the computation of a static system optimization is repeatedly executed under multiple static loads. In this process, we propose parametrizing the static system and employing the RBM with global proper orthogonal decomposition (POD). In general, the snapshots for the sampling procedure under multiple loads increase proportionally to the number of loads, which results in an inefficient computational procedure. Thus, we propose taking snapshots with a proper orthogonal mode (POM) of the multiple loads rather than for the multiple loads themselves. The number of snapshots then decreases, and the original system can be efficiently reduced. We directly employ the framework of the proposed RBM with the POM of multiple loads to ESL-based design optimization, and the results indicate that the proposed method is more efficient than conventional ESL-based optimization as well as a full order model. Various numerical examples, including comparisons of relative errors and the dynamic response optimizations, support the strength of the proposed strategy.

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Abbreviations

:

Set of the feasible design variables

ℝ:

Real numbers

μ :

Design parameter

K :

Stiffness matrix

M :

Mass matrix

U :

Displacement column matrix

\( \overline{\mathbf{V}} \) :

Displacement column matrix in generalized coordinate system

F :

Force column matrix

X :

Snapshot matrix

Φ :

Proper orthogonal mode

Σ :

Diagonal matrix of singular values

A :

Right singular vectors

W :

Reduced basis space

ζ :

Reduced basis

T :

Transformation matrix

w :

Interpolation function

Γ :

Mode of external forces

C :

Damping matrix

t :

Time variable

\( \mathcal{W} \) :

Objective function

c :

Constraint function

ω :

Natural frequency

M :

Mach number

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Acknowledgments

This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government (MSIP; Ministry of Science, ICT & Future Planning) (No. 2017R1C1B5017595).

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Authors and Affiliations

  1. School of Mechanical Engineering, Kyungnam University, Changwon, 51767, Republic of Korea

    Jaehun Lee

  2. Department of Mechanical and Aerospace Engineering, Seoul National University, Seoul, 08826, Republic of Korea

    Maenghyo Cho

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  1. Jaehun Lee
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Lee, J., Cho, M. Efficient design optimization strategy for structural dynamic systems using a reduced basis method combined with an equivalent static load. Struct Multidisc Optim 58, 1489–1504 (2018). https://doi.org/10.1007/s00158-018-1976-5

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