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Reanalysis-based fast solution algorithm for flexible multi-body system dynamic analysis with floating frame of reference formulation


In order to improve the computational efficiency of flexible multi-body system dynamic analysis with floating frame of reference formulation (FFRF), a reanalysis-based fast solution algorithm is developed here. The data of FFRF analysis process can be divided into two parts: unchanged mass and stiffness matrices part kept by deformation, and changed mass and stiffness matrices part caused by rigid motion and joint constraints. In the proposed method, the factorization of the unchanged part is reused in the entire solution process via employing the reanalysis concept; and the changed part is treated as structural modification. Meanwhile, the joint constraints are handled with an exact reanalysis method—the Sherman–Morrison–Woodbury (SMW) formula, which is also beneficial for saving the computational cost. Numerical examples demonstrate that the computational efficiency of the proposed method is higher than that of full analysis, especially in large scale problems. Moreover, since the proposed fast FFRF solution algorithm is-based on exact reanalysis methods, there is no theoretical error between the results obtained by the fast solution algorithm and full analysis method.

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  1. 1.

    Huston, R.L.: Multibody dynamics—modeling and analysis methods. Appl. Mech. Rev. 44(3), 149–173 (1991)

    Article  Google Scholar 

  2. 2.

    Likins, P.W.: Finite element appendage equations for hybrid coordinate dynamic analysis. Int. J. Solids Struct. 8(5), 709–731 (1972)

    MATH  Article  Google Scholar 

  3. 3.

    Wu, L., Tiso, P.: Nonlinear model order reduction for flexible multibody dynamics: a modal derivatives approach. Multibody Syst. Dyn. 36(4), 405–425 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Shabana, A.A.: Definition of the slopes and the finite element absolute nodal coordinate formulation. Multibody Syst. Dyn. 1(3), 339–348 (1997)

    MATH  Article  Google Scholar 

  5. 5.

    García-Vallejo, D., Mikkola, A.M., Escalona, J.L.: A new locking-free shear deformable finite element based on absolute nodal coordinates. Nonlinear Dyn. 50(1–2), 249–264 (2007)

    MATH  Article  Google Scholar 

  6. 6.

    Kubler, L., Eberhard, P., Geisler, J.: Flexible multibody systems with large deformations using absolute nodal coordinates for isoparameteric solid brick elements. In: Proceedings of DETC’03 ASME 2003 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Chicago, IL, USA (2003)

    Google Scholar 

  7. 7.

    Shabana, A.A., Christensen, A.P.: The Sherman–Morrison–Woodbury absolute nodal co-ordinate formulation: plate problem. Int. J. Numer. Methods Eng. 40(15), 2775–2790 (1997)

    MATH  Article  Google Scholar 

  8. 8.

    Wu, J., Luo, Z., Zhang, N., Zhang, Y.: Dynamic computation of flexible multibody system with uncertain material properties. Nonlinear Dyn. 85(2), 1–24 (2016)

    MATH  Google Scholar 

  9. 9.

    Patel, M., Orzechowski, G., Tian, Q., Shabana, A.A.: A new multibody system approach for tire modeling using ANCF finite elements. Proc. Inst. Mech. Eng., Proc., Part K, J. Multi-Body Dyn. 230(1), 69–84 (2016)

    Google Scholar 

  10. 10.

    Sugiyama, H., Yamashita, H.: Spatial joint constraints for the absolute nodal coordinate formulation using the non-generalized intermediate coordinates. Multibody Syst. Dyn. 26(1), 15–36 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Hussein, B.A., Weed, D., Shabana, A.A.: Clamped end conditions and cross section deformation in the finite element absolute nodal coordinate formulation. Multibody Syst. Dyn. 21(4), 375–393 (2009)

    MATH  Article  Google Scholar 

  12. 12.

    Lee, S.-H., Park, T.-W., Seo, J.-H., Yoon, J.-W., Jun, K.-J.: The development of a sliding joint for very flexible multibody dynamics using absolute nodal coordinate formulation. Multibody Syst. Dyn. 20(3), 223–237 (2008)

    MATH  Article  Google Scholar 

  13. 13.

    Garcia-Vallejo, D., Escalona, J., Mayo, J., Dominguez, J.: Describing rigid-flexible multibody systems using absolute coordinates. Nonlinear Dyn. 34(1), 75–94 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Shabana, A., Hussien, H., Escanola, J.: Application of the absolute nodal coordinate formulation to large rotation and large deformation problems. Trans. Am. Soc. Mech. Eng. J. Mech. Des. 120, 188–195 (1998)

    Google Scholar 

  15. 15.

    Suarez, L.E., Singh, M.P.: Dynamic condensation method for structural eigenvalue analysis. AIAA J. 30(4), 1046–1054 (1992)

    Article  Google Scholar 

  16. 16.

    Pesheck, E., Pierre, C., Shaw, S.W.: Modal reduction of a nonlinear rotating beam through nonlinear normal modes. J. Vib. Acoust. 124(2), 229–236 (2002)

    Article  Google Scholar 

  17. 17.

    Khulief, Y., Mohiuddin, M.: On the dynamic analysis of rotors using modal reduction. Finite Elem. Anal. Des. 26(1), 41–55 (1997)

    MATH  Article  Google Scholar 

  18. 18.

    Benner, P., Breiten, T.: Krylov-subspace based model reduction of nonlinear circuit models using bilinear and quadratic-linear approximations. Prog. Ind. Math. ECMI 2010, 153–159 (2012)

    MATH  Google Scholar 

  19. 19.

    Bai, Z.: Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Appl. Numer. Math. 43(1–2), 9–44 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Freund, R.W.: Krylov-subspace methods for reduced-order modeling in circuit simulation. J. Comput. Appl. Math. 123(1), 395–421 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Wu, L., Tiso, P., Tatsis, K., Chatzi, E., van Keulen, F.: A modal derivatives enhanced Rubin substructuring method for geometrically nonlinear multibody systems. Multibody Syst. Dyn. 45(1), 57–85 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Lehner, M., Eberhard, P.: A two-step approach for model reduction in flexible multibody dynamics. Multibody Syst. Dyn. 17(2), 157–176 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Fischer, M., Eberhard, P.: Linear model reduction of large scale industrial models in elastic multibody dynamics. Multibody Syst. Dyn. 31(1), 27–46 (2014)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Fehr, J., Eberhard, P.: Simulation process of flexible multibody systems with non-modal model order reduction techniques. Multibody Syst. Dyn. 25(3), 313–334 (2011)

    MATH  Article  Google Scholar 

  25. 25.

    Sulitka, M., Šindler, J., Sušeň, J., Smolík, J.: Application of Krylov reduction technique for a machine tool multibody modelling. Adv. Mech. Eng. 2014(1), 65–70 (2014)

    Google Scholar 

  26. 26.

    Huang, G., Wang, H., Li, G.: A novel multi-grid assisted reanalysis for re-meshed finite element models. Comput. Methods Appl. Mech. Eng. 313(1), 817–833 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Wu, B., Li, Z.: Static reanalysis of structures with added degrees of freedom. Commun. Numer. Methods Eng. 22(4), 269–281 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Kirsch, U.: Efficient reanalysis for topological optimization. Struct. Optim. 6(3), 143–150 (1993)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Jian-jun, H., Xiang-Zi, C., Bin, X.: Structural modal reanalysis for large, simultaneous and multiple type modifications. Mech. Syst. Signal Process. 62, 207–217 (2015)

    Article  Google Scholar 

  30. 30.

    He, J.J., Jiang, J.S., Xu, B.: Modal reanalysis methods for structural large topological modifications with added degrees of freedom and non-classical damping. Finite Elem. Anal. Des. 44(1–2), 75–85 (2007)

    Article  Google Scholar 

  31. 31.

    Chen, S.H., Rong, F.: A new method of structural modal reanalysis for topological modifications. Finite Elem. Anal. Des. 38(11), 1015–1028 (2002)

    MATH  Article  Google Scholar 

  32. 32.

    Huang, C., Chen, S.H., Liu, Z.: Structural modal reanalysis for topological modifications of finite element systems. Eng. Struct. 22(4), 304–310 (2000)

    Article  Google Scholar 

  33. 33.

    Yang, Z.J., Chen, S.H., Wu, X.M.: A method for modal reanalysis of topological modifications of structures. Int. J. Numer. Methods Eng. 65(13), 2203–2220 (2006)

    MATH  Article  Google Scholar 

  34. 34.

    Gao, G., Wang, H., Li, G.: An adaptive time-based global method for dynamic reanalysis. Struct. Multidiscip. Optim. 48(2), 355–365 (2013)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Chen, S.H., Ma, L., Meng, G.: Dynamic response reanalysis for modified structures under arbitrary excitation using epsilon-algorithm. Comput. Struct. 86(23–24), 2095–2101 (2008)

    Article  Google Scholar 

  36. 36.

    Materna, D., Kalpakides, V.K.: Nonlinear reanalysis for structural modifications based on residual increment approximations. Comput. Mech. 57(1), 1–18 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Kirsch, U., Bogomolni, M.: Nonlinear and dynamic structural analysis using combined approximations. Comput. Struct. 85(10), 566–578 (2007)

    Article  Google Scholar 

  38. 38.

    Kirsch, U.: A unified reanalysis approach for structural analysis, design, and optimization. Struct. Multidiscip. Optim. 25(2), 67–85 (2003)

    Article  Google Scholar 

  39. 39.

    Yang, Z., Chen, X., Kelly, R.: An adaptive static reanalysis method for structural modifications using epsilon algorithm. In: CSO ’09 Proceedings of the 2009 International Joint Conference on Computational Sciences and Optimization, vol. 2, pp. 897–899 (2009)

    Chapter  Google Scholar 

  40. 40.

    Kołakowski, P., Wikło, M., Holnicki-Szulc, J.: The virtual distortion method—a versatile reanalysis tool for structures and systems. Struct. Multidiscip. Optim. 36(3), 217–234 (2007)

    Article  Google Scholar 

  41. 41.

    Akgün, M.A., Garcelon, J.H., Haftka, R.T.: Fast exact linear and non-linear structural reanalysis and the Sherman–Morrison–Woodbury formulas. Int. J. Numer. Methods Eng. 50(7), 1587–1606 (2001)

    MATH  Article  Google Scholar 

  42. 42.

    Sherman, J., Morrison, W.J.: Adjustment of an inverse matrix corresponding to a change in one element of a given matrix. Ann. Math. Stat. 21(1), 124–127 (1950)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Hager, W.W.: Updating the inverse of a matrix. SIAM Rev. 31(2), 221–239 (1989)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Liu, H., Wu, B., Li, Z.: Method of updating the Cholesky factorization for structural reanalysis with added degrees of freedom. J. Eng. Mech. 140(2), 384–392 (2013)

    Article  Google Scholar 

  45. 45.

    Davis, T.A., Hager, W.W.: Multiple-rank modifications of a sparse Cholesky factorization. SIAM J. Matrix Anal. Appl. 22(4), 997–1013 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Davis, T.A., Hager, W.W.: Modifying a sparse Cholesky factorization. SIAM J. Matrix Anal. Appl. 20(3), 606–627 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    Huang, G., Wang, H., Li, G.: An exact reanalysis method for structures with local modifications. Struct. Multidiscip. Optim. 54(3), 499–509 (2016)

    MathSciNet  Article  Google Scholar 

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This work was supported by the National Natural Science Foundation of China [Grant Nos. 11702065, 91648108, 51875108, 51675106 and 11772100], Guangdong Natural Science Foundation [Grant Nos. 2015A030312008 and 2016A030308016], Guangdong Science and Technology Plan [Grant Nos. 2015B010104006, 2015B010133005, 2015B010104008 and 2015B090921007], National key Research and Develop Program of China [Grant No. 2017YFF0105902], and China Postdoctoral Science Foundation [Grant No. 2017M622623].

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Correspondence to Zhijun Yang.

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Huang, G., Zhu, W., Yang, Z. et al. Reanalysis-based fast solution algorithm for flexible multi-body system dynamic analysis with floating frame of reference formulation. Multibody Syst Dyn 49, 271–289 (2020).

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  • Flexible multi-body system dynamics
  • Floating frame of reference formulation
  • Reanalysis
  • Joint constraints