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Isogeometric analysis of multi-patch solid-shells in large deformation

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Abstract

In the context of isogeometric analysis (IGA) of shell structures, the popularity of the solid-shell elements benefit from formulation simplicity and full 3D stress state. However some basic questions remain unresolved when using solid-shell element, especially for large deformation cases with patch coupling, which is a common scene in real-life simulations. In this research, after introduction of the solid-shell nonlinear formulation and a fundamental 3D model construction method, we present a non-symmetric variant of the standard Nitsche’s formulation for multi-patch coupling in association with an empirical formula for its stabilization parameter. An selective and reduced integration scheme is also presented to address the locking syndrome. In addition, the quasi-Newton iteration format is derived as solver, together with a step length control method. The second order derivatives are totally neglected by the adoption of the non-symmetric Nitsche’s formulation and the quasi-Newton solver. The solid-shell elements are numerically studied by a linear elastic plate example, then we demonstrate the performance of the proposed formulation in large deformation, in terms of result verification, iteration history and continuity of displacement across the coupling interface.

Graphic Abstract

In the context of isogeometric analysis (IGA), after introduction of the solid-shell nonlinear formulation and a fundamental 3D model construction method, we present a non-symmetric variant of the standard Nitsche’s formulation for multi-patch coupling in association with an empirical formula for its stabilization parameter. An selective and reduced integration scheme is also introduced to address the locking syndrome. In addition, the quasi-Newton iteration format is derived as solver, together with a step length control method. The second order derivatives are totally neglected by the adoption of the non-symmetric Nitsche’s formulation and the quasi-Newton solver. The performance of the proposed formulation in large deformation is demonstrated by several examples.

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Notes

  1. http://abaqus.software.polimi.it/v2016/books/usi/pt03ch17s06s01.html

  2. https://doi.org/10.6084/m9.figshare.11858136.v1

References

  1. Morganti, S., Auricchio, F., Benson, D., et al.: Patient-specific isogeometric structural analysis of aortic valve closure. Comput. Meth. Appl. Mech. Eng. 284, 508–520 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Yang, G., Hu, D., Long, S.: A reconstructed edge-based smoothed DSG element based on global coordinates for analysis of Reissner-Mindlin plates. Acta Mech. Sin. 33, 83–105 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cui, T., Sun, Z., Liu, C., et al.: Topology optimization of plate structures using plate element-based moving morphable component (MMC) approach. Acta Mech. Sin. 36, 412–421 (2020)

    Article  MathSciNet  Google Scholar 

  4. Quy, N.D., Matzenmiller, A.: A solid-shell element with enhanced assumed strains for higher order shear deformations in laminates. Technische Mechanik - Scientific Journal for Fundamentals and Applications of Engineering Mechanics 28, 334–355 (2008)

    Google Scholar 

  5. Hughes, T.J., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: Cad, finite elements, NURBS, exact geometry and mesh refinement. Comput. Meth. Appl. Mech. Eng. 194, 4135–4195 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lipton, S., Evans, J.A., Bazilevs, Y., et al.: Robustness of isogeometric structural discretizations under severe mesh distortion. Comput. Meth. Appl. Mech. Eng. 199, 357–373 (2010)

    Article  MATH  Google Scholar 

  7. Hu, Q., Xia, Y., Natarajan, S., et al.: Isogeometric analysis of thin Reissner-Mindlin shells: locking phenomena and b-bar method. Comput. Mech. 65, 1323–1341 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  8. Du, X., Zhao, G., Wang, W., et al.: Nitsches method for non-conforming multipatch coupling in hyperelastic isogeometric analysis. Comput. Mech. 65, 685–710 (2020)

    Article  MathSciNet  Google Scholar 

  9. Dornisch, W., Stoeckler, J., Mller, R.: Dual and approximate dual basis functions for B-splines and NURBS-comparison and application for an efficient coupling of patches with the isogeometric mortar method. Comput. Meth. Appl. Mech. Eng. 316, 449–496 (2017)

    Article  MathSciNet  Google Scholar 

  10. Schub, S., Dittmann, M., Wohlmuth, B., et al.: Multi-patch isogeometric analysis for Kirchhoff-Love shell elements. Comput. Meth. Appl. Mech. Eng. 349, 91–116 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hu, Q., Chouly, F., Hu, P., et al.: Skew-symmetric Nitsches formulation in isogeometric analysis: Dirichlet and symmetry conditions, patch coupling and frictionless contact. Comput. Meth. Appl. Mech. Eng. 341, 188–220 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  12. Elfverson, D., Larson, M.G., Larsson, K.: CutIGA with basis function removal. Advanced Modeling and Simulation in Engineering Sciences 5, 1–19 (2018)

    Article  Google Scholar 

  13. Nguyen, V.P., Kerfriden, P., Brino, M., et al.: Nitsches method for two and three dimensional NURBS patch coupling. Comput. Mech. 53, 1163–1182 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Adam, C., Bouabdallah, S., Zarroug, M., et al.: Improved numerical integration for locking treatment in isogeometric structural elements. part II: Plates and shells. Comput. Meth. Appl. Mech. Eng. 284, 106–137 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z.: The Finite Element Method: Its Basis and Fundamentals. Elsevier Butterworth-Heinemann, Oxford (2005)

    MATH  Google Scholar 

  16. Bischoff, M., Wall, W., Bletzinger, K., et al.: Chapter 3: Models and finite elements for thin-walled structures. Encyclopedia of Computational Mechanics 2, 59–137 (2004)

    Google Scholar 

  17. Hauptmann, R., Schweizerhof, K.: A systematic development of solid-shellelement formulations for linear and non-linear analyses employing only displacement degrees of freedom. Int. J. Numer. Methods Eng. 42, 49–69 (1998)

    Article  MATH  Google Scholar 

  18. Parisch, H.: A continuum-based shell theory for non-linear applications. Int. J. Numer. Methods Eng. 38, 1855–1883 (1995)

    Article  MATH  Google Scholar 

  19. Remmers, J.J., Wells, G.N., Borst, R.D.: A solid-like shell element allowing for arbitrary delaminations. Int. J. Numer. Methods Eng. 58, 2013–2040 (2003)

    Article  MATH  Google Scholar 

  20. Hosseini, S., Remmers, J.J., Verhoosel, C.V., et al.: An isogeometric solid-like shell element for nonlinear analysis. Int. J. Numer. Methods Eng. 95, 238–256 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Caseiro, J., Valente, R.F., Reali, A., et al.: On the assumed natural strain method to alleviate locking in solid-shell NURBS-based finite elements. Comput. Mech. 53, 1341–1353 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bouclier, R., Elguedj, T., Combescure, A.: An isogeometric locking-free NURBS-based solid-shell element for geometrically nonlinear analysis. Int. J. Numer. Methods Eng. 101, 774–808 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Antolin, P., Kiendl, J., Pingaro, M., et al.: A simple and effective method based on strain projections to alleviate locking in isogeometric solid shells. Comput. Mech. 65, 1621–1631 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mlika, R., Renard, Y., Chouly, F.: An unbiased Nitsches formulation of large deformation frictional contact and self-contact. Comput. Meth. Appl. Mech. Eng. 325, 265–288 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method for Solid and Structural Mechanics. Elsevier Butterworth-Heinemann, Oxford (2005)

    MATH  Google Scholar 

  26. Kim, N.H.: Introduction to Nonlinear Finite Element Analysis. Springer Science & Business Media, New York (2014)

    Google Scholar 

  27. Cottrell, J.A., Hughes, T.J., Bazilevs, Y.: Isogeometric analysis: toward integration of CAD and FEA. John Wiley & Sons, West Sussex (2009)

    Book  MATH  Google Scholar 

  28. De Boor, C.: A Practical Guide to Splines, revised edn. Springer, New York (2001)

    MATH  Google Scholar 

  29. Reali, A., Hughes, T.J.: An Introduction to Isogeometric Collocation Methods. In: Isogeometric Methods for Numerical Simulation, Springer, Vienna (2015)

  30. Nitsche, J.: ber ein variationsprinzip zur l?sung von Dirichlet-problemen bei verwendung von teilr?umen, die keinen randbedingungen unterworfen sind. In: Abhandlungen aus dem mathematischen Seminar der Universit?t Hamburg, Springer-Verlag, (1971)

  31. Apostolatos, A., Schmidt, R., Wchner, R., et al.: A Nitsche-type formulation and comparison of the most common domain decomposition methods in isogeometric analysis. Int. J. Numer. Methods Eng. 97, 473–504 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Guo, Y., Ruess, M.: Nitsches method for a coupling of isogeometric thin shells and blended shell structures. Comput. Meth. Appl. Mech. Eng. 284, 881–905 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Guo, Y., Ruess, M., Schillinger, D.: A parameter-free variational coupling approach for trimmed isogeometric thin shells. Comput. Mech. 59, 693–715 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  34. Guo, Y., Heller, J., Hughes, T.J., et al.: Variationally consistent isogeometric analysis of trimmed thin shells at finite deformations, based on the step exchange format. Comput. Meth. Appl. Mech. Eng. 336, 39–79 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  35. Du, X., Zhao, G., Wang, W., et al.: Nitsches method for non-conforming multipatch coupling in hyperelastic isogeometric analysis. Comput. Mech. 65, 687–710 (2020)

    Article  MathSciNet  Google Scholar 

  36. Shahbazi, K.: An explicit expression for the penalty parameter of the interior penalty method. J. Comput. Phys. 205, 401–407 (2005)

    Article  MATH  Google Scholar 

  37. Langer, U., Moore, S.E.: Discontinuous Galerkin Isogeometric Analysis Of Elliptic PDEs on Surfaces. In: Domain decomposition methods in science and engineering XXII, Springer, (2016)

  38. Juettler, B., Langer, U., Mantzaflaris, A., et al.: Geometry + simulation modules: Implementing isogeometric analysis. Proc. Appl. Math. Mech. 14, 961–962 (2014)

    Article  Google Scholar 

  39. Hosseini, S.F., Hashemian, A., Moetakef-Imani, B., et al.: Isogeometric analysis of free-form Timoshenko curved beams including the nonlinear effects of large deformations. Acta Mech. Sin. 34, 728–743 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  40. Baroli, D., Quarteroni, A., Ruiz-Baier, R.: Convergence of a stabilized discontinuous Galerkin method for incompressible nonlinear elasticity. Adv. Comput. Math. 39, 425–443 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  41. Noels, L., Radovitzky, R.: A general discontinuous Galerkin method for finite hyperelasticity. formulation and numerical applications. Int. J. Numer. Methods Eng. 68, 64–97 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  42. Matthies, H., Strang, G.: The solution of nonlinear finite element equations. Int. J. Numer. Methods Eng. 14, 1613–1626 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  43. Gabriel, D., Plesek, J., Ulbin, M.: Symmetry preserving algorithm for large displacement frictionless contact by the pre-discretization penalty method. Int. J. Numer. Methods Eng. 61, 2615–2638 (2004)

    Article  MATH  Google Scholar 

  44. Nocedal, J.: Updating quasi-Newton matrices with limited storage. Math. Comput. 35, 773–782 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  45. Bouclier, R., Elguedj, T., Combescure, A.: Efficient isogeometric NURBS-based solid-shell elements: Mixed formulation and B-bar method. Comput. Meth. Appl. Mech. Eng. 267, 86–110 (2013)

    Article  MATH  Google Scholar 

  46. Hu, Q., Xia, Y., Zou, R., et al.: A global formulation for complex rod structures in isogeometric analysis. Int. J. Mech. Sci. 115, 736–745 (2016)

    Article  Google Scholar 

  47. Sze, K., Liu, X., Lo, S.: Popular benchmark problems for geometric nonlinear analysis of shells. Finite Elem. Anal. Des. 40, 1551–1569 (2004)

    Article  Google Scholar 

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Acknowledgements

This work was supported by the Fundamental Research Funds for the Central Universities (Grant JUSRP12038) and the Natural Science Foundation of Jiangsu Province (Grant BK20200611).

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Authors and Affiliations

  1. School of Science, Jiangnan University, Wuxi, 214122, China

    Qingyuan Hu & Shuzhen Rao

  2. AICES - Aachen Institute for Advanced Study in Computational Engineering Science, 52062, Aachen, Germany

    Davide Baroli

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  1. Qingyuan Hu
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Correspondence to Qingyuan Hu.

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Executive Editor: Xu Guo.

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Hu, Q., Baroli, D. & Rao, S. Isogeometric analysis of multi-patch solid-shells in large deformation. Acta Mech. Sin. 37, 844–860 (2021). https://doi.org/10.1007/s10409-020-01046-y

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