Skip to main content
SpringerLink
Log in

A Review of the Accuracy of Primal Assembly Model Order Reduction Techniques

  • Review
  • Published:
Multiscale Science and Engineering Aims and scope Submit manuscript

Abstract

Model order reduction (MOR) techniques have been used in structural dynamics for over sixty years as a quick and effective way to examine the dynamic behavior of complex finite element (FE) models. Various macroscopic phenomena arise from the physics and mechanics of the underlying microstructure, and MOR approaches can be employed in conjunction with the multiscale computational homogenization approach, which is deriving local macroscopic constitutive responses from underlying microstructure; to predict the dynamic behavior of heterogeneous macrostructures. The accuracy and efficiency of the most representative approaches from dynamic condensation, component mode synthesis (CMS), and general model reduction (GMR) methods are compared in this paper. The techniques considered are Guyan and the improved reduced system (IRS) from classical dynamic condensation, Craig-Bampton (CB) and enhanced Craig-Bampton (ECB) from CMS, IRS-based substructuring, GMR, and GMRPlus. All the substructuring methods employ primal assembly at the boundary interface of the substructures. The accuracy and computing efficiency of the techniques are compared in-depth using real-world engineering problems, and the effect of the number of substructures on the accuracy and computational cost is also examined.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

References

  1. R.J. Guyan, Reduction of stiffness and mass matrices. AIAA J. 3(2), 380 (1965). https://doi.org/10.2514/3.2874

    Article  Google Scholar 

  2. B. Irons, Structural eigenvalue problems: elimination of unwanted variables. AIAA J. 3(5), 961–962 (1965). https://doi.org/10.2514/3.3027

    Article  Google Scholar 

  3. J. C. O’Callahan, A procedure for an improved reduced system (IRS) Model. (1989).

  4. D.C. Kammer, Test-analysis-model development using an exact modal reduction. Int. J. Anal. Exp. Modal Anal. 2(4), 174–179 (1987)

    Google Scholar 

  5. J. C. O’Callahan, P. Avitabile, R. Riemer, System Equivalent Reduction Expansion Process (SEREP), Seventh International Modal Analysis Conference, Las Vegas, Nevadarocess, (1988)

  6. H. Sung, S. Chang, M. Cho, Efficient model updating method for system identification using a convolutional neural network. AIAA J. 59(9), 3480–3489 (2021). https://doi.org/10.2514/1.J059964

    Article  Google Scholar 

  7. S. Chang, S. Baek, K.-O. Kim, M. Cho, Structural system identification using degree of freedom-based reduction and hierarchical clustering algorithm. J. Sound Vib. 346(1), 139–152 (2015). https://doi.org/10.1016/j.jsv.2015.02.031

    Article  Google Scholar 

  8. H. Sung, S. Chang, M. Cho, Component model synthesis using model updating with neural networks. Mech. Adv. Mater. Struct. (2021). https://doi.org/10.1080/15376494.2021.2015495

    Article  Google Scholar 

  9. J. H. Gordis, “An analysis of the Improved Reduced System (IRS) model reduction procedure,” in Proceedings of the 10th International Modal Analysis Conference, 471–479 (1992).

  10. L.E. Suarez, M.P. Singh, Dynamic condensation method for structural eigenvalue analysis. AIAA J. 30(4), 1046–1054 (1992). https://doi.org/10.2514/3.11026

    Article  Google Scholar 

  11. M.I. Friswell, S.D. Garvey, J.E.T. Penny, Model reduction using dynamic and iterated IRS techniques. J. Sound Vib. 186(2), 311–323 (1995). https://doi.org/10.1006/jsvi.1995.0451

    Article  MATH  Google Scholar 

  12. M.I. Friswell, S.D. Garvey, J.E.T. Penny, The convergence of the iterated IRS method. J. Sound Vib. 211(1), 123–132 (1998). https://doi.org/10.1006/jsvi.1997.1368

    Article  MathSciNet  MATH  Google Scholar 

  13. Z.-Q. Qu, Z. Fu, New structural dynamic condensation method for finite element models. AIAA J. 36(7), 1320–1324 (1998). https://doi.org/10.2514/2.517

    Article  Google Scholar 

  14. Z.-Q. Qu, R.P. Selvam, Efficient method for dynamic condensation of nonclassically damped vibration systems. AIAA J. 40(2), 368–375 (2002). https://doi.org/10.2514/2.1655

    Article  Google Scholar 

  15. Z.-Q. Qu, Y. Jung, R.P. Selvam, Model Condensation for non-classically damped systems—part I: static condensation. Mech. Syst. Signal Process. 17(5), 1003–1016 (2003). https://doi.org/10.1006/mssp.2002.1526

    Article  Google Scholar 

  16. Z.-Q. Qu, R.P. Selvam, Y. Jung, Model condensation for non-classically damped systems—part II: iterative schemes for dynamic condensation. Mech. Syst. Signal Process. 17(5), 1017–1032 (2003). https://doi.org/10.1006/mssp.2002.1527

    Article  Google Scholar 

  17. Y. Xia, R. Lin, Improvement on the iterated IRS method for structural eigensolutions. J. Sound Vib. 270(4–5), 713–727 (2004). https://doi.org/10.1016/S0022-460X(03)00188-3

    Article  Google Scholar 

  18. K.H. Lee, S. Chang, J.-G. Kim, Iterative improved reduced system method of fluid-structure interaction with free surface. J. Sound Vib. (2021). https://doi.org/10.1016/j.jsv.2021.116445

    Article  Google Scholar 

  19. K. Ahn, K.-H. Lee, J.-S. Lee, S. Chang, 3D-based equivalent model of SMART control rod drive mechanism using dynamic condensation method. Nucl. Eng. Technol. 54(3), 1109–1114 (2022). https://doi.org/10.1016/j.net.2021.08.037

    Article  Google Scholar 

  20. H. Kim, M. Cho, Two-level scheme for selection of primary degrees of freedom and semi-analytic sensitivity based on the reduced system. Comput. Methods Appl. Mech. Eng. 195(33–36), 4244–4268 (2006). https://doi.org/10.1016/j.cma.2005.08.004

    Article  MATH  Google Scholar 

  21. M. Cho, H. Kim, Element-based node selection method. AIAA J. 42(8), 1677–1684 (2004). https://doi.org/10.2514/1.5407

    Article  Google Scholar 

  22. R.R. Craig Jr., A.J. Kurdila, Fundamentals of structural dynamics, 2nd edn. (Wiley, New Jersey, 2006)

    MATH  Google Scholar 

  23. D. De Klerk, D.J. Rixen, S.N. Voormeeren, General framework for dynamic substructuring: history, review, and classification of techniques. AIAA J. 46(5), 1169–1181 (2008). https://doi.org/10.2514/1.33274

    Article  Google Scholar 

  24. M.S. Allen, D. Rixen, M. van der Seijs, P. Tiso, T. Abrahamsson, R.L. Mayes, Substructuring in engineering dynamics, vol. 594 (Springer, 2020)

    Google Scholar 

  25. R.W. Hagos, G. Choi, H. Sung, S. Chang, Substructuring-based dynamic reduction method for vibration analysis of periodic composite structures. Compos. Mater. Eng. An. Intl. J. 4(1), 43–62 (2022). https://doi.org/10.12989/cme.2022.4.1.043

    Article  Google Scholar 

  26. M. A. Aminpour, J. B. Ransom, and S. L. McCleary, A coupled analysis method for structures with independently modeled finite element subdomains. (1992). https://doi.org/10.2514/6.1992-2235

  27. N. Bouhaddi, R. Fillod, Substructuring using a linearized dynamic condensation method. Comput. Struct. 45(4), 679–683 (1992). https://doi.org/10.1016/0045-7949(92)90486-J

    Article  MATH  Google Scholar 

  28. N. Bouhaddi, R. Fillod, Substructuring by a two level dynamic condensation method. Comput. Struct. 60(3), 403–409 (1996). https://doi.org/10.1016/0045-7949(95)00400-9

    Article  MathSciNet  MATH  Google Scholar 

  29. H. Kim, M. Cho, Improvement of reduction method combined with sub-domain scheme in large-scale problem. Int. J. Numer. Methods Eng. 70, 206–251 (2007). https://doi.org/10.1002/nme

    Article  MATH  Google Scholar 

  30. D. Choi, H. Kim, M. Cho, Iterated improved reduced system (IIRS) method combined with sub-structuring scheme (I)—undamped structural systems. Trans. Korean Soc. Mech. Eng. 31(2), 211–220 (2007). https://doi.org/10.3795/KSME-A.2007.31.2.211

    Article  Google Scholar 

  31. D. Choi, H. Kim, M. Cho, Iterated improved reduced system (IIRS) method combined with sub-structuring scheme (II)—nonclassically damped structural systems. Trans. Korean Soc. Mech. Eng. 31(2), 221–230 (2007). https://doi.org/10.3795/KSME-A.2007.31.2.221

    Article  Google Scholar 

  32. D. Choi, H. Kim, M. Cho, Improvement of substructuring reduction technique for large eigenproblems using an efficient dynamic condensation method. J. Mech. Sci. Technol. 22(2), 255–268 (2008). https://doi.org/10.1007/s12206-007-1040-7

    Article  Google Scholar 

  33. J.H. Kim, S.H. Boo, P.S. Lee, A dynamic condensation method with free interface substructuring. Mech. Syst. Signal Process. 129, 218–234 (2019). https://doi.org/10.1016/j.ymssp.2019.04.021

    Article  Google Scholar 

  34. W.C. Hurty, Vibrations of structural systems by component mode synthesis. Trans. Am. Soc. Civ. Eng. 126(1), 157–175 (1961). https://doi.org/10.1061/TACEAT.0008073

    Article  Google Scholar 

  35. W.C. Hurty, Dynamic analysis of structural systems using component modes. AIAA J. 3(4), 678–685 (1965). https://doi.org/10.2514/3.2947

    Article  Google Scholar 

  36. J.I. Kim, S. Na, K. Eom, Large protein dynamics described by hierarchical-component mode synthesis. J. Chem. Theory Comput. 5(7), 1931–1939 (2009). https://doi.org/10.1021/ct900027h

    Article  Google Scholar 

  37. D. Ming, Y. Kong, Y. Wu, J. Ma, Substructure synthesis method for simulating large molecular complexes. Proc. Natl. Acad. Sci. USA. 100, 104–109 (2002). https://doi.org/10.1073/pnas.232588999

    Article  Google Scholar 

  38. J.A. McCammon, S.C. Harvey, Dynamics of proteins and nucleic acids (Cambridge University Press, 1987)

    Book  Google Scholar 

  39. K. Eom, S.-C. Baek, J.-H. Ahn, S. Na, Coarse-graining of protein structures for the normal mode studies. J. Comput. Chem. 28(8), 1400–1410 (2007). https://doi.org/10.1002/jcc.20672

    Article  Google Scholar 

  40. M. Lu, D. Ming, J. Ma, FSUB: normal mode analysis with flexible substructures. J. Phys. Chem. B 116(29), 8636–8645 (2012). https://doi.org/10.1021/jp300312u

    Article  Google Scholar 

  41. D. C. Kammer and M. J. Triller, “Selection of component modes for Craig-Bampton substructure representations,” in AIAA/ASME Structures, Structural Dynamics and Materials Conference. Publishing by AIAA. 2, 1218–1228 (1995). https://doi.org/10.2514/6.1995-1299

  42. D. Givoli, P.E. Barbone, I. Patlashenko, Which are the important modes of a subsystem? Int. J. Numer. Methods Eng. 59(12), 1657–1678 (2004). https://doi.org/10.1002/nme.935

    Article  MathSciNet  MATH  Google Scholar 

  43. B.-S. Liao, Z. Bai, W. Gao, The important modes of subsystems: a moment-matching approach. Int. J. Numer. Methods Eng. 70(13), 1581–1597 (2007). https://doi.org/10.1002/nme.1940

    Article  MathSciNet  MATH  Google Scholar 

  44. K.H. Lee, R.W. Hagos, S. Chang, J.-G. Kim, Multiphysics mode synthesis of fluid–structure interaction with free surface. Eng. Comput. (2022). https://doi.org/10.1007/s00366-022-01676-9

    Article  Google Scholar 

  45. R.R. Craig Jr., M.C.C. Bampton, Coupling of substructures for dynamic analyses. AIAA J. 6(7), 1313–1319 (1968). https://doi.org/10.2514/3.4741

    Article  MATH  Google Scholar 

  46. J.-B. Qiu, Z.-G. Ying, F.W. Williams, Exact modal synthesis techniques using residual constraint mode. Int. J. Numer. Methods Eng. 40(13), 2475–2492 (1997). https://doi.org/10.1002/(SICI)1097-0207(19970715)40:13%3c2475::AID-NME176%3e3.0.CO;2-L

    Article  MATH  Google Scholar 

  47. J.-G. Kim, P. Lee, An enhanced Craig—Bampton method. Int. J. Numer. Methods Eng. 103, 79–93 (2015). https://doi.org/10.1002/nme

    Article  MathSciNet  MATH  Google Scholar 

  48. J.-G. Kim, S.-H. Boo, P.-S. Lee, An enhanced AMLS method and its performance. Comput. Methods Appl. Mech. Eng. 287, 90–111 (2015). https://doi.org/10.1016/j.cma.2015.01.004

    Article  MathSciNet  MATH  Google Scholar 

  49. J. Kim, Development of a component mode synthesis method with higher-order residual flexibility (Korean Advanced Institute of Science and Technology (KAIST), 2016). http://hdl.handle.net/10203/222039

  50. S. Baek, Study on the multi-level substructuring scheme and system condensation for the large-scaled structural dynamic analysis (Seoul National University, 2012). https://s-space.snu.ac.kr/handle/10371/156269

  51. S.H. Boo, J.H. Kim, P.S. Lee, Towards improving the enhanced Craig-Bampton method. Comput. Struct. 196, 63–75 (2018). https://doi.org/10.1016/j.compstruc.2017.10.017

    Article  Google Scholar 

  52. R.H. MacNeal, A hybrid method of component mode synthesis. Comput. Struct. 1(4), 581–601 (1971). https://doi.org/10.1016/0045-7949(71)90031-9

    Article  Google Scholar 

  53. S. Rubin, Improved component-mode representation for structural dynamic analysis. AIAA J. 13(8), 995–1006 (1975). https://doi.org/10.2514/3.60497

    Article  MATH  Google Scholar 

  54. R. R. Craig Jr. and C.-J. Chang, “On the use of attachement modes in substructure coupling for dynamic analysis,” in AIAA/ASME Structures, Structural Dynamics and Materials Conference. Publ by AIAA. 89–99 (1977). https://doi.org/10.2514/6.1977-405

  55. M. Géradin, D.J. Rixen, A ‘nodeless’ dual superelement formulation for structural and multibody dynamics application to reduction of contact problems. Int. J. Numer. Methods Eng. 106(10), 773–798 (2016). https://doi.org/10.1002/nme.5136

    Article  MathSciNet  MATH  Google Scholar 

  56. B. Blachowski, W. Gutkowski, Effect of damaged circular flange-bolted connections on behaviour of tall towers, modelled by multilevel substructuring. Eng. Struct. 111, 93–103 (2016). https://doi.org/10.1016/j.engstruct.2015.12.018

    Article  Google Scholar 

  57. C. Brecher, M. Fey, C. Tenbrock, M. Daniels, Multipoint Constraints for Modeling of Machine Tool Dynamics. J. Manuf. Sci. Eng. Trans. ASME 138(5), 1–8 (2016). https://doi.org/10.1115/1.4031771

    Article  Google Scholar 

  58. D.J. Rixen, A dual Craig-Bampton method for dynamic substructuring. J. Comput. Appl. Math. 168(1–2), 383–391 (2004). https://doi.org/10.1016/j.cam.2003.12.014

    Article  MathSciNet  MATH  Google Scholar 

  59. A.D. PerdahcIoǧlu, M.H.M. Ellenbroek, P.J.M. Van Der Hoogt, A. De Boer, An optimization method for dynamics of structures with repetitive component patterns. Struct. Multidiscip. Optim. 39(6), 557–567 (2009). https://doi.org/10.1007/s00158-009-0399-8

    Article  Google Scholar 

  60. J.-H. Kim, J. Kim, P.-S. Lee, Improving the accuracy of the dual Craig-Bampton method. Comput. Struct. 191, 22–32 (2017). https://doi.org/10.1016/j.compstruc.2017.05.010

    Article  Google Scholar 

  61. S.-H. Boo, P.-S. Lee, A dynamic condensation method using algebraic substructuring. Int. J. Numer. Methods Eng. 109(12), 1701–1720 (2017). https://doi.org/10.1002/nme.5349

    Article  MathSciNet  Google Scholar 

  62. S.-H. Boo, P.-S. Lee, An iterative algebraic dynamic condensation method and its performance. Comput. Struct. 182, 419–429 (2017). https://doi.org/10.1016/j.compstruc.2016.12.011

    Article  Google Scholar 

  63. A. George, Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal. 10(2), 345–363 (1973). https://doi.org/10.1137/0710032

    Article  MathSciNet  MATH  Google Scholar 

  64. B. Hendrickson, E. Rothberg, Effective sparse matrix ordering: Just around the BEND (United States, 1997). https://www.osti.gov/servlets/purl/448063

  65. G. Karypis, V. Kumar, A software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices (Minneapolis, Minnesota, 1998)

    Google Scholar 

  66. C. Yang et al., An algebraic substructuring method for large-scale eigenvalue calculation. SIAM J. Sci. Comput. 27(3), 873–892 (2005). https://doi.org/10.1137/040613767

    Article  MathSciNet  MATH  Google Scholar 

  67. M.F. Kaplan, Implementation of automated multilevel substructuring for frequency response analysis of structures (University of Texas at Austin, Cham, 2001)

    Google Scholar 

  68. J.K. Bennighof, R.B. Lehoucq, An automated multilevel substructuring method for eigenspace computation in linear elastodynamics. SIAM J. Sci. Comput. 25(6), 2084–2106 (2004). https://doi.org/10.1137/s1064827502400650

    Article  MathSciNet  MATH  Google Scholar 

  69. S.-H. Boo, J.-G. Kim, P.-S. Lee, Error estimation for the automated multi-level substructuring method. Int. J. Numer. Methods Eng. 106, 927–950 (2016). https://doi.org/10.1002/nme.5161

    Article  MathSciNet  MATH  Google Scholar 

  70. J.-G. Kim, Y.J. Park, G.H. Lee, D.N. Kim, A general model reduction with primal assembly in structural dynamics. Comput. Methods Appl. Mech. Eng. 324, 1–28 (2017). https://doi.org/10.1016/j.cma.2017.06.007

    Article  MathSciNet  MATH  Google Scholar 

  71. R.L. Kidder, Reduction of structural frequency equations. AIAA J. 11(6), 892 (1973). https://doi.org/10.2514/3.6852

    Article  Google Scholar 

  72. D. Habault, Chapter 5—analytic expansions and approximation methods, in Acoustics. ed. by P. Filippi, D. Habault, J.-P. Lefebvre, A. Bergassoli (Academic Press, London, 1999), pp.159–188

    Chapter  MATH  Google Scholar 

  73. C.W. Groetsch, Functional analysis, in Encyclopedia of physical science and technology, 3rd edn., ed. by R.A. Meyers (Academic Press, New York, 2003), pp.337–353

    Chapter  Google Scholar 

  74. I. Chavel, B. Randol, J. Dodziuk, Pure and applied mathematics, in Eigenvalues in Riemannian geometry, vol. 115, ed. by I. Chavel, B. Randol, J. Dodziuk (Elsevier, 1984), pp.363–364

    Google Scholar 

  75. S. Mazumder, Chapter 2—the finite difference method, in Numerical methods for partial differential equations. ed. by S. Mazumder (Academic Press, 2016), pp.51–101

    Chapter  Google Scholar 

  76. W.C. Hurty, A criterion for selecting realistic natural modes of a structure (Technical Memorandum, Jet Propulsion, Laboratory California Institute of Technology, Pasadena, 1967)

    Google Scholar 

  77. M. Pastor, M. Binda, T. Harčarik, Modal assurance criterion. Procedia Eng. 48, 543–548 (2012). https://doi.org/10.1016/j.proeng.2012.09.551

    Article  Google Scholar 

  78. R.J. Allemang, The modal assurance criterion—twenty years of use and abuse. Sound Vib. 37(8), 14–21 (2003)

    Google Scholar 

  79. R.D. Henshell, J.H. Ong, Automatic masters for eigenvalue economization. Earthq. Eng. Struct. Dyn. 3(4), 375–383 (1975). https://doi.org/10.1002/eqe.4290030408

    Article  Google Scholar 

  80. J.H. Ong, Improved automatic masters for eigenvalue economization. Finite Elem. Anal. Des. 3(2), 149–160 (1987). https://doi.org/10.1016/0168-874x(87)90006-0

    Article  MATH  Google Scholar 

Download references

Acknowledgements

This research was supported by Kumoh National Institute of Technology (202001210001).

Author information

Authors and Affiliations

  1. Department of Mechanical Design Engineering, Kumoh National Institute of Technology, Daehak-Ro 61, Gumi, Gyeongbuk, 39177, Republic of Korea

    Robel Weldebrhan Hagos & Seongmin Chang

Authors
  1. Robel Weldebrhan Hagos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Seongmin Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Seongmin Chang.

Ethics declarations

Conflicts of Interest

Conflicts of Interest The authors declare no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hagos, R.W., Chang, S. A Review of the Accuracy of Primal Assembly Model Order Reduction Techniques. Multiscale Sci. Eng. 4, 179–201 (2022). https://doi.org/10.1007/s42493-022-00088-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s42493-022-00088-7

Keywords

Search

Navigation