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A family of Craig-Bampton methods considering residual mode compensation. (English) Zbl 1433.74105
Summary: In this paper, we investigate the formulation of a family of Craig-Bampton (CB) methods considering residual modes by O’Callahan’s approximation and adding generalized coordinate vectors containing unknown eigenvalues. In addition, we propose an $$n$$th-order higher-order CB+ (HCBn+) method for compensating the $$(n + 1)$$th residual flexibility in the $$n$$th-order HCB (HCBn) method by O’Callahan’s approach. Therefore, various CB methods with improved performance, such as the enhanced Craig-Bampton (ECB) method, which uses O’Callahan’s approximation, and the higher-order Craig-Bampton (HCB) method, which adds generalized coordinate vectors, and HCB+ are employed for the comparison of performance with the aid of multiprecision computing. Through three benchmark examples, it is revealed that the HCB1+ method, a modified version of the first-order HCB method (HCB1) with the aid of O’Callahan’s approximation proposed, shows better performance than HCB1 with the same number of retained modes. However, HCB2+ and HCB3+, modified versions of the second- and third-order HCB method, respectively, cannot be improved further. From the results, we concluded that this was due to the limitation of O’Callahan’s approach, which many researchers have fundamentally questioned.
##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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##### References:
 [1] Craig, R. R.; Bampton, M. C.C., Coupling of substructures for dynamic analyses, AIAA J., 6, 1313-1319 (1968) · Zbl 0159.56202 [2] Kim, J. G.; Lee, P. S., An enhanced Craig-Bampton method, Int. J. Numer. Methods Eng., 103, 79-93 (2015) · Zbl 1352.74378 [3] Kim, J.-G.; Boo, S.-H.; Lee, P.-S., Performance of the enhanced Craig-Bampton method, (Proceedings of the World Congress on Advances in Structural Engineering and Mechanics (ASEM15). Proceedings of the World Congress on Advances in Structural Engineering and Mechanics (ASEM15), Incheon, Korea (2015)) [4] Boo, S.-H.; Kim, J.-H.; Lee, P.-S., Towards improving the enhanced Craig-Bampton method, Comput. Struct., 196, 63-75 (2018) [5] Kim, J.; Boo, S.-H.; Lee, P.-S., Considering the higher-order effect of residual modes in the Craig-Bampton method, AIAA J., 56, 403-412 (2018) [6] Gruber, F. M.; Rixen, D. J., Dual Craig-Bampton component mode synthesis method for model order reduction of nonclassically damped linear systems, Mech. Syst. Sig. Process., 111, 678-698 (2018) [7] Rixen, D. J., A dual Craig-Bampton method for dynamic substructuring, J. Comput. Appl. Math., 168, 383-391 (2004) · Zbl 1107.70303 [8] Gruber, F. M.; Rixen, D. J., A dual Craig-Bampton state‐space approach for model reduction of damped systems, Proc. Appl. Math. Mech., 17, 303-304 (2017) [9] Kim, J.-H.; Kim, J.; Lee, P.-S., Improving the accuracy of the dual Craig-Bampton method, Comput. Struct., 191, 22-32 (2017) [10] Rixen, D. J., Interface reduction in the dual Craig-Bampton method based on dual interface modes, Linking Models and Experiments, 2, 311-328 (2011), Springer [11] Boo, S.-H.; Lee, P.-S., A dynamic condensation method using algebraic substructuring, Int. J. Numer. Methods Eng., 109, 1701-1720 (2017) [12] Boo, S.-H.; Lee, P.-S., An iterative algebraic dynamic condensation method and its performance, Comput. Struct., 182, 419-429 (2017) [13] Kim, J.-H.; Boo, S.-H.; Lee, P.-S., A dynamic condensation method with free interface substructuring, Mech. Syst. Sig. Process., 129, 218-234 (2019) [14] Kim, S. M.; Kim, J.-G.; Park, K. C.; Chae, S.-W., A component mode selection method based on a consistent perturbation expansion of interface displacement, Comput. Methods Appl. Mech. Eng., 330, 578-597 (2018) [15] Besselink, B.; Tabak, U.; Lutowska, A.; Van de Wouw, N.; Nijmeijer, H.; Rixen, D. J.; Hochstenbach, M.; Schilders, W., A comparison of model reduction techniques from structural dynamics, numerical mathematics and systems and control, J. Sound Vib., 332, 4403-4422 (2013) [16] Rixen, D., High order static correction modes for component mode synthesis, (Proceedings of the Fifth World Congress on Computational Mechanics. Proceedings of the Fifth World Congress on Computational Mechanics, Vienna, Austria (2002)) [17] Kim, J.-G.; Lee, K.-H.; Lee, P.-S., Estimating relative eigenvalue errors in the Craig-Bampton method, Comput. Struct., 139, 54-64 (2014) [18] Kim, J.-G.; Park, Y.-J.; Lee, G. H.; Kim, D.-N., A general model reduction with primal assembly in structural dynamics, Comput. Methods Appl. Mech. Eng., 324, 1-28 (2017) [19] Junge, M.; Brunner, D.; Becker, J.; Gaul, L., Interface‐reduction for the Craig-Bampton and Rubin method applied to FE-BE coupling with a large fluid-structure interface, Int. J. Numer. Methods Eng., 77, 1731-1752 (2009) · Zbl 1158.74490 [20] Kim, J.-G.; Lee, P.-S.; Park, K.-C., A mode selection algorithm for the flexibility-based component mode synthesis, (Proceedings of the Fifth International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2015) (2015)), 3162-3171 [21] Kim, J.-G.; Seo, J.; Lim, J. H., Novel modal methods for transient analysis with a reduced order model based on enhanced Craig-Bampton formulation, Appl. Math. Comput., 344-345, 30-45 (2019) · Zbl 1428.74207 [22] Lim, J. H.; Hwang, D.-S.; Kim, K.-W.; Lee, G. H.; Kim, J.-G., A coupled dynamic loads analysis of satellites with an enhanced Craig-Bampton approach, Aerosp. Sci. Technol., 69, 114-122 (2017) [23] Kim, J.-G.; Lee, P.-S., Posteriori error estimation method for flexibility-based component mode synthesis, AIAA J., 53, 2828-2837 (2015) [24] Boo, S.-H.; Kim, J.-G.; Lee, P.-S., A simplified error estimator for the CB method and its application to error control, Comput. Struct., 164, 53-62 (2016) [25] Chung, I. S.; Kim, J.-G.; Chae, S.-W.; Park, K., Formulation of flexibility-based component mode synthesis for transient analysis, AIAA J., 57, 858-869 (2018) [26] Yang, C.; Gao, W.; Bai, Z.; Li, X. S.; Lee, L.-Q.; Husbands, P.; Ng, E., An algebraic substructuring method for large-scale eigenvalue calculation, SIAM J. Sci. Comput., 27, 873-892 (2005) · Zbl 1092.65032 [27] Rixen, D., Generalized mode acceleration methods and modal truncation augmentation, (Proceedings of the Nineteenth AIAA Applied Aerodynamics Conference (2001)), 1300 [28] Hurty, W., A Criterion For Selecting Realistic Natural Modes of a Structure (1967), Jet Propulsion Lab.: Jet Propulsion Lab. Pasadena, CA, TM33-364 [29] O’Callahan, J., A procedure for an improved reduced system (IRS) model, (Proceedings of the Seventh International Modal Analysis Conference, Union College. Proceedings of the Seventh International Modal Analysis Conference, Union College, Las Vegas, NV (1989)), 17-21 [30] Allemang, R. J., The modal assurance criterion-twenty years of use and abuse, Sound Vib., 37, 14-23 (2003) [31] Pastor, M.; Binda, M.; Harčarik, T., Modal assurance criterion, Proc. Eng., 48, 543-548 (2012) [32] Multiprecision Computing Toolbox for MATLAB, in, ADVANPIX LLC, Yokohama, Japan. [33] Abaqus Analysis User’s Manual (6.14) (2014), Dassault Systèmes [34] El Ghaoui, L., Inversion error, condition number, and approximate inverses of uncertain matrices, Linear Algebra Appl., 343-344, 171-193 (2002) · Zbl 0997.65049
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