Directional C¹ Continuity without Global Knot Propagation in Kirchhoff–Love Shells: A Hermite–NURBS Strategy for Membrane Locking Reduction and Elastoplastic Validation

Document Type : Research Paper

Authors
1 Faculty of Science and Technology, Sidi Mohamed Ben Abdellah University, Fez 30000, Morocco
2 Department of Mechanical Engineering, Transylvania University of Brasov, B-dul Eroilor 3, 500036 Brasov, Romania
3 Institute of Solid Mechanics, Romanian Academy, Str. C. Mille 7, Bucharest 050564, Romania
Abstract
To accurately simulate elongated thin-walled shells, including cylindrical pressure vessels, submarine hulls, and aerospace fuselage sections, a numerical formulation must combine precise geometric representation with directional C¹ continuity. This work proposes a hybrid Hermite × NURBS Kirchhoff–Love shell formulation that enforces C¹ continuity in the axial direction through cubic Hermite interpolation, without global knot propagation, while preserving exact circumferential geometry through NURBS. Quantified results validate the approach: on the Scordelis–Lo roof benchmark the formulation converges monotonically to within 0.02% of the reference free-edge displacement (0.3024) at the finest mesh considered, with a convergence rate of approximately five. The framework provides directional C¹ control with explicit nodal derivative degrees of freedom and full compatibility with standard NURBS CAD workflows, at a degree-of-freedom cost comparable to single-basis isogeometric discretizations.
Keywords
Subjects

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[1] Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement, Computer Methods in Applied Mechanics and Engineering, 194, 2005, 4135-4195.
[2] Kiendl, J., Bletzinger, K.-U., Linhard, J., Wüchner, R., Isogeometric shell analysis with Kirchhoff–Love elements, Computer Methods in Applied Mechanics and Engineering, 198, 2009, 3902-3914.
[3] Kim, M.G., Lee, G.H., Lee, H., Koo, B., Isogeometric analysis for geometrically exact shell elements using Bézier extraction of NURBS with assumed natural strain method, Thin-Walled Structures, 172, 2022, 108846.
[4] Liu, C., Tian, Q., Hu, H., New spatial curved beam and cylindrical shell elements, Nonlinear Dynamics, 70, 2012, 1903-1918.
[5] Matikainen, M.K., Mikkola, A., Schwab, A.L., The quadrilateral fully-parametrized plate elements based on the absolute nodal coordinate formulation, Journal of Structural Mechanics, 42, 2009, 138-148.
[6] Ambati, M., Kiendl, J., De Lorenzis, L., Isogeometric Kirchhoff–Love shell formulation for elasto-plasticity, Computer Methods in Applied Mechanics and Engineering, 340, 2018, 320-339.
[7] Guo, M., Wang, W., Zhao, G., Du, X., Zhang, R., Yang, J., T-splines for isogeometric analysis of the large deformation of elastoplastic Kirchhoff–Love shells, Applied Sciences, 13, 2023, 1709.
[8] Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y., Isogeometric Analysis: Toward Integration of CAD and FEA, Wiley, 2009.
[9] Casquero, H., Liu, L., Zhang, Y., Reali, A., Kiendl, J., Gomez, H., Arbitrary-degree T-splines for isogeometric analysis of fully nonlinear Kirchhoff–Love shells, Computer-Aided Design, 82, 2017, 140-153.
[10] Coradello, L., D'Angelo, P., Vázquez, R., Buffa, A., Adaptive isogeometric analysis on two-dimensional trimmed domains based on a hierarchical approach, Computer Methods in Applied Mechanics and Engineering, 380, 2021, 113782.
[11] Hirschler, T., Bouclier, R., Duval, A., Elguedj, T., Morlier, J., The embedded isogeometric Kirchhoff–Love shell: From design to shape optimization of non-conforming stiffened multipatch structures, Computer Methods in Applied Mechanics and Engineering, 349, 2019, 774-797.
[12] Liu, C., Wang, S., Xie, X., Adaptive isogeometric topology optimization of shell structures based on truncated hierarchical B-splines, Frontiers of Mechanical Engineering, 20(6), 2025, 49.
[13] Rabczuk, T., Areias, P.M.A., A meshfree thin shell for arbitrary evolving cracks based on an extrinsic basis, CMES-Computer Modeling in Engineering & Sciences, 8, 2006, 115-130.
[14] Rabczuk, T., Areias, P.M.A., Belytschko, T., A meshfree thin shell method for nonlinear dynamic fracture, International Journal for Numerical Methods in Engineering, 72, 2007, 524-548.
[15] Nguyen-Thanh, N., Nguyen-Xuan, H., Bordas, S., Rabczuk, T., Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids, Computer Methods in Applied Mechanics and Engineering, 200, 2011, 1892-1908.
[16] Samaniego, E., Anitescu, C., Goswami, S., Nguyen-Thanh, V.M., Guo, H., Hamdia, K., Zhuang, X., Rabczuk, T., An energy approach to the solution of partial differential equations in computational mechanics via machine learning, Computer Methods in Applied Mechanics and Engineering, 362, 2020, 112790.
[17] Nguyen-Thanh, V.M., Anitescu, C., Alajlan, N., Rabczuk, T., Zhuang, X., Parametric deep energy approach for elasticity accounting for strain gradient effects, European Journal of Mechanics - A/Solids, 87, 2021, 104225.
[18] Petera, J., Pittman, J.F.T., Isoparametric Hermite elements, International Journal for Numerical Methods in Engineering, 37, 1994, 3489-3519.
[19] Attia, S., Mohareb, M., Adeeb, S., Nonlinear finite element formulation for thin-walled conical shells using Hermitian interpolation and Kirchhoff–Love kinematics, Thin-Walled Structures, 205, 2024, 112617.
[20] Oumellal, F., Lamnii, A., Curve and surface construction using Hermite trigonometric interpolant, Mathematical and Computational Applications, 26, 2021, 11.
[21] Riffnaller-Schiefer, A., Augsdörfer, U.H., Fellner, D.W., Isogeometric shell analysis with NURBS compatible subdivision surfaces, Applied Mathematics and Computation, 272, 2016, 139-147.
[22] Brank, B., Perić, D., On large deformations of thin elastoplastic shells, International Journal for Numerical Methods in Engineering, 40, 1997, 689-726.
[23] Areias, P.M.A., Ritto-Corrêa, M.C., Martins, J.A.C., Finite strain plasticity, the stress condition, and a complete shell model, Computational Mechanics, 45, 2010, 189-209.
[24] Piegl, L., Tiller, W., The NURBS Book, 2nd ed., Springer, 1997.
[25] Krysl, P., Chen, J.-S., Benchmarking computational shell models, Archives of Computational Methods in Engineering, 30, 2023, 301-315.
[26] Belytschko, T., Stolarski, H., Liu, W.K., Carpenter, N., Ong, J.S.J., Stress projection for membrane and shear locking in shell finite elements, Computer Methods in Applied Mechanics and Engineering, 51, 1985, 221-258.
[27] Riks, E., An incremental approach to the solution of snapping and buckling problems, International Journal of Solids and Structures, 7, 1979, 529-551.
[28] Cortivo, N.D., Felippa, C.A., Bavestrello, H., Silva, W.T.M., Plastic buckling and collapse of thin shell structures, Computer Methods in Applied Mechanics and Engineering, 198, 2009, 785-798.
[29] MacNeal, R.H., Harder, R.L., A proposed standard set of problems to test finite element accuracy, Finite Elements in Analysis and Design, 1, 1985, 3-20.
[30] Veldin, T., Brank, B., Brojan, M., Discrete Kirchhoff–Love four-node shell finite element based on the Hermite interpolation and membrane locking treatment, Thin-Walled Structures, 168, 2021, 108268.
[31] Tornabene, F., Viscoti, M., Dimitri, R., Higher-order weak formulation for vibration analysis of anisotropic doubly-curved shells, Engineering Analysis with Boundary Elements, 180, 2025, 106480.
[32] Hermite, C., Sur quelques formules relatives à l'interpolation, Journal de Mathématiques Pures et Appliquées, 6, 1869, 1-10.

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