Advertisement

Reduction process based on proper orthogonal decomposition for dual formulation of dynamic substructures

  • Sunyoung Im
  • Euiyoung Kim
  • Maenghyo ChoEmail author
Original Paper
  • 15 Downloads

Abstract

Novel reduction processes utilizing the proper orthogonal decomposition method have been proposed to facilitate dual formulation of dynamic substructures. Dual formulation is significant owing to its capability of effectively solving ill-conditioned problems related to parallel-processing computers. Dual-formulation techniques employ the Lagrange multiplier method to couple substructures. In the proposed approach, substructures are reduced under constraint conditions via use of the Lagrange multiplier method, and each domain is divided into internal and interface parts—each handled separately during the reduction process. The interface to which the coupling condition is imposed is preserved without further reduction (using the internal reduction method) or reduced to a different ratio from the internal part (via internal and interface reduction technique). Once all substructures are reduced, Lagrange multipliers are applied to preserved or reduced interface parts. Since Boolean matrices that preserve constraint conditions are not reduced but constructed instead, the stability and accuracy of the reduced system are highly enhanced.

Keywords

Dynamic substructures Lagrange multiplier method Reduced-order model (ROM) Proper orthogonal decomposition (POD) Internal and interface reduction method (IIRM) Internal reduction method (IRM) 

Notes

Acknowledgements

This work was supported by the National Research Foundation (NRF) of Korea funded by the Korea government (MSIP) (Grant No. 2012R1A3A2048841).

References

  1. 1.
    Klerk D, Rixen DJ, Voormeeren SN (2008) General framework for dynamic substructuring: history, review and classification of techniques. AIAA J 46(5):1169–1181CrossRefGoogle Scholar
  2. 2.
    Craig Jr, Roy R, Ching-Jone C (1977) Substructure coupling for dynamic analysis and testingGoogle Scholar
  3. 3.
    Magoulès F, Roux F-X (2006) Lagrangian formulation of domain decomposition methods: a unified theory. Appl Math Model 30(7):593–615CrossRefzbMATHGoogle Scholar
  4. 4.
    Smith B et al (2004) Domain decomposition: parallel multilevel methods for elliptic partial differential equations. Cambridge University Press, CambridgeGoogle Scholar
  5. 5.
    Valli A, Quarteroni A (1999) Domain decomposition methods for partial differential equations. Numerical Mathematics and Scientific Computation. The Clarendon Press, Oxford University Press, New YorkzbMATHGoogle Scholar
  6. 6.
    Toselli A, Widlund OB (2005) Domain decomposition methods: algorithms and theory, vol 34. Springer, BerlinCrossRefzbMATHGoogle Scholar
  7. 7.
    Lai CH, Bjørstad PE, Cross M, Widlund O (1998) Eleventh international conference on domain decomposition methods. In: Proceedings of the 11th international conference on domain decomposition methods in Greenwich, England, 20–24 July 1998Google Scholar
  8. 8.
    Dryja M, Widlund OB (1990) Towards a unified theory of domain decomposition algorithms for elliptic problems, In: Proceedings of the third international symposium on domain decomposition methods for partial differential, held in Houston, Texas, 20–22 March 1989, SIAMGoogle Scholar
  9. 9.
    Collino F, Ghanemi S, Joly P (2000) Domain decomposition method for harmonic wave propagation: a general presentation. Comput Methods Appl Mech Eng 184(2):171–211MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Farhat C, Le Tallec P, eds (2000) Vistas in domain decomposition and parallel processing in computational mechanicsGoogle Scholar
  11. 11.
    Lai C-H et al (1999) Accuracy of a domain decomposition method for the recovering of discontinuous heat sources in metal sheet cutting. Comput Vis Sci 2(2):149–152CrossRefzbMATHGoogle Scholar
  12. 12.
    Keyes DE (2002) Domain decomposition in the mainstream of computational science. In: Proceedings of the 14 international conference on domain decomposition methodsGoogle Scholar
  13. 13.
    Hurty WC (1965) Dynamic analysis of structural systems using component modes. AIAA J 3(4):678–685CrossRefGoogle Scholar
  14. 14.
    Bampton MCC, Craig RR (1968) Coupling of substructures for dynamic analyses. AIAA J 6(7):1313–1319CrossRefzbMATHGoogle Scholar
  15. 15.
    MacNeal RH (1971) A hybrid method of component mode synthesis. Comput Struct 1(4):581–601CrossRefGoogle Scholar
  16. 16.
    Benfield WA, Hruda RF (1971) Vibration analysis of structures by component mode substitution. AIAA J 9(7):1255–1261CrossRefzbMATHGoogle Scholar
  17. 17.
    Rubin S (1975) Improved component-mode representation for structural dynamic analysis. AIAA J 13(8):995–1006CrossRefzbMATHGoogle Scholar
  18. 18.
    Robert Morris Hintz (1975) Analytical methods in component modal synthesis. AIAA J 13(8):1007–1016CrossRefzbMATHGoogle Scholar
  19. 19.
    Craig R, Chang CJ (1977) On the use of attachment modes in substructure coupling for dynamic analysis. In: 18th structural dynamics and materials conferenceGoogle Scholar
  20. 20.
    Meirovitch L, Hale AL (1981) On the substructure synthesis method. AIAA J 19(7):940–947CrossRefGoogle Scholar
  21. 21.
    Spanos JT, Tsuha WS (1991) Selection of component modes for flexible multibody simulation. J Guid Control Dyn 14(2):278–286CrossRefGoogle Scholar
  22. 22.
    Kim H, Cho M (2007) Improvement of reduction method combined with sub-domain scheme in large-scale problem. Int J Numer Meth Eng 70(2):206–251CrossRefzbMATHGoogle Scholar
  23. 23.
    Kim H, Cho M (2008) Sub-domain reduction method in non-matched interface problems. J Mech Sci Technol 22(2):203CrossRefGoogle Scholar
  24. 24.
    Choi D, Kim H, Cho M (2008) Improvement of substructuring reduction technique for large eigenproblems using an efficient dynamic condensation method. J Mech Sci Technol 22(2):255CrossRefGoogle Scholar
  25. 25.
    Choi D, Cho M, Kim H (2008) Efficient dynamic response analysis using substructuring reduction method for discrete linear system with proportional and nonproportional damping. Int J Aeronaut Space Sci 9(1):85–99CrossRefGoogle Scholar
  26. 26.
    Lee J, Cho M (2017) An interpolation-based parametric reduced order model combined with component mode synthesis. Comput Methods Appl Mech Eng 319:258–286MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kerschen G, Golinval J-C (2002) Physical interpretation of the proper orthogonal modes using the singular value decomposition. J Sound Vib 249(5):849–865MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Chatterjee A (2000) An introduction to the proper orthogonal decomposition. Curr Sci 78(7):808–817Google Scholar
  29. 29.
    Kerschen G et al (2005) The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview. Nonlinear Dyn 41(1):147–169MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kerfriden P et al (2013) A partitioned model order reduction approach to rationalise computational expenses in nonlinear fracture mechanics. Comput Methods Appl Mech Eng 256:169–188MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Corigliano A, Dossi M, Mariani S (2015) Model Order Reduction and domain decomposition strategies for the solution of the dynamic elastic–plastic structural problem. Comput Methods Appl Mech Eng 290:127–155MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Confalonieri F et al (2013) A domain decomposition technique applied to the solution of the coupled electro-mechanical problem. Int J Numer Meth Eng 93(2):137–159CrossRefzbMATHGoogle Scholar
  33. 33.
    Markovic D, Park KC, Ibrahimbegovic A (2007) Reduction of substructural interface degrees of freedom in flexibility-based component mode synthesis. Int J Numer Methods Eng 70(2):163–180CrossRefzbMATHGoogle Scholar
  34. 34.
    van der Valk PLC (2010) Model reduction and interface modeling in dynamic substructuring. Master’s Thesis, Delft University of Technology, DelftGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Multiscale Mechanical Design Division, Department of Mechanical and Aerospace EngineeringSeoul National UniversitySeoulRepublic of Korea
  2. 2.Korea Institute of Machinery and MaterialsDaejeonRepublic of Korea

Personalised recommendations