A meshfree Hermite point interpolation method for buckling and post-buckling analysis of thin plates

This paper aims to develop a meshfree approach based on the coupling of the Hermite-type point interpolation method (HPIM) and a high order continuation (HOC) Solver for numerical simulation of the geometrically nonlinear problems that require higher order continuity shape functions such as the buckling of a thin plate. The point interpolation method (PIM) and Hermite-type point interpolation method (HPIM) shape functions construction procedure is presented in detail. The first type is used to approximate the in-plane displacements, while the second one is used for the transverse component and its derivatives. The standard Galerkin method is adopted to discretize the governing partial differential equations which were derived from using the Kirchhoff’s plate theory. The resolution of the considered problem is performed thanks to a solver that combines a Taylor series expansion with a continuation procedure. Numerical examples with different geometric shapes, various boundary conditions and loadings are given to verify the efficiency, accuracy, and robustness of the proposed approach.

[1]  Bouazza Braikat,et al.  Effect of regularized functions on the dynamic response of a clutch system using a high-order algorithm , 2017 .

[2]  Bouazza Braikat,et al.  Dynamic analysis of a thin-walled beam with open cross section subjected to dynamic loads using a high-order implicit algorithm , 2016 .

[3]  S. Timoshenko.  Theory of Elastic Stability , 1936 .

[4]  Guirong Liu,et al.  A point interpolation meshless method based on radial basis functions , 2002 .

[5]  G. Dhatt,et al.  Une présentation de la méthode des éléments finis , 1984 .

[6]  Bouazza Braikat,et al.  On the use of Radial Point Interpolation Method (RPIM) in a high order continuation for the resolution of the geometrically nonlinear elasticity problems , 2020 .

[7]  T. Belytschko,et al.  Element-free galerkin methods for static and dynamic fracture , 1995 .

[8]  Hung Nguyen-Xuan,et al.  Isogeometric analysis for functionally graded microplates based on modified couple stress theory , 2016, ArXiv.

[9]  B. Cochelin.  A path-following technique via an asymptotic-numerical method , 1994 .

[10]  Chien H. Thai,et al.  NURBS-based postbuckling analysis of functionally graded carbon nanotube-reinforced composite shells , 2019, Computer Methods in Applied Mechanics and Engineering.

[11]  Guirong Liu,et al.  On the optimal shape parameters of radial basis functions used for 2-D meshless methods , 2002 .

[12]  Weeratunge Malalasekera,et al.  An introduction to computational fluid dynamics - the finite volume method , 2007 .

[13]  J. N. Reddy,et al.  BUCKLING OF SYMMETRICALLY LAMINATED COMPOSITE PLATES USING THE ELEMENT-FREE GALERKIN METHOD , 2002 .

[14]  Bouazza Braikat,et al.  Discussion about parameterization in the asymptotic numerical method: Application to nonlinear elastic shells , 2010 .

[15]  Youssef Hilali,et al.  Nonlinear dynamic response of functionally graded material plates using a high‐order implicit algorithm , 2020, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik.

[16]  Hung Nguyen-Xuan,et al.  Generalized shear deformation theory for functionally graded isotropic and sandwich plates based on isogeometric approach , 2014 .

[17]  Marco Amabili,et al.  Nonlinear Vibrations and Stability of Shells and Plates , 2008 .

[18]  G. Dhatt,et al.  Modélisation des structures par éléments finis , 1990 .

[19]  Ted Belytschko,et al.  A three-dimensional impact-penetration algorithm with erosion , 1987 .

[20]  E. Riks.  An incremental approach to the solution of snapping and buckling problems , 1979 .

[21]  Ping Lin,et al.  Numerical analysis of Biot's consolidation process by radial point interpolation method , 2002 .

[22]  F. Mohri,et al.  Stability analysis of thin-walled beams with open section subject to arbitrary loads , 2016 .

[23]  Oussama Bourihane,et al.  New finite element model for the stability analysis of a functionally graded material thin plate under compressive loadings , 2020 .

[24]  Khadija Mhada,et al.  Buckling and post-buckling analysis of a functionally graded material (FGM) plate by the Asymptotic Numerical Method , 2021 .

[25]  Guirong Liu,et al.  A LOCAL RADIAL POINT INTERPOLATION METHOD (LRPIM) FOR FREE VIBRATION ANALYSES OF 2-D SOLIDS , 2001 .

[26]  O. C. Zienkiewicz,et al.  The Finite Element Method: Its Basis and Fundamentals , 2005 .

[27]  Guirong Liu,et al.  A point interpolation method for two-dimensional solids , 2001 .

[28]  H. Zahrouni,et al.  A 2D mechanical–thermal coupled model to simulate material mixing observed in friction stir welding process , 2017, Engineering with Computers.

[29]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[30]  Michel Potier-Ferry,et al.  Méthode asymptotique numérique , 2008 .

[31]  Hung Nguyen-Xuan,et al.  A novel three-variable shear deformation plate formulation: Theory and Isogeometric implementation , 2017 .

[32]  N. Damil,et al.  Bifurcation points and bifurcated branches in fluids mechanics by high‐order mesh‐free geometric progression algorithms , 2020, International Journal for Numerical Methods in Fluids.

[33]  Guirong Liu,et al.  A meshfree radial point interpolation method (RPIM) for three-dimensional solids , 2005 .

[34]  Youssef Hilali,et al.  A mixed MLS and Hermite-type MLS method for buckling and postbuckling analysis of thin plates , 2021 .

[35]  Gui-Rong Liu,et al.  An Introduction to Meshfree Methods and Their Programming , 2005 .

[36]  Bouazza Braikat,et al.  An implicit algorithm for the dynamic study of nonlinear vibration of spur gear system with backlash , 2018 .

[37]  M. D'Ottavio,et al.  Classical, first order, and advanced theories , 2017 .

[38]  Michel Potier-Ferry,et al.  A New method to compute perturbed bifurcations: Application to the buckling of imperfect elastic structures , 1990 .

[39]  S. Atluri,et al.  A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics , 1998 .