Strong Formulation Isogeometric Analysis (SFIGA) for laminated composite arbitrarily shaped plates

Abstract Engineering applications in conventional numerical modeling are investigated by domain decomposition techniques because, generally, structural components do not have a regular shape. The most common and versatile numerical approach, in this regard, is the Finite Element (FE) method, which uses low order polynomials and the variational formulation for the solution of the differential problem. In contrast with the classic approach, the present paper shows a strong form collocation method for solving laminated composite plates wherein discontinuities arise. Both approaches need mapping technique for converting arbitrarily shaped elements into the computational space. Since the present strong form technique can deal with variable order approximating polynomials within each element, advanced blending function mapping must be employed for treating any discontinuity and distortion effect using the smallest number of elements. The Differential Quadrature (DQ) method is utilized for solving the mathematical problem and isogeometric mapping is implemented for the nonlinear mapping of complex shapes. The present methodology belongs to the framework of Isogeometric Analysis (IGA), therefore this approach is termed Strong Formulation Isogeometric Analysis (SFIGA). Numerical applications show the accuracy, stability and reliability of the present methodology.

[1]  R. Lal,et al.  Axisymmetric vibrations and buckling analysis of functionally graded circular plates via differential transform method , 2015 .

[2]  K. M. Liew,et al.  Analysis of laminated CNT reinforced functionally graded plates using the element-free kp-Ritz method , 2016 .

[3]  P. Wriggers,et al.  Isogeometric large deformation frictionless contact using T-splines , 2014 .

[4]  Charles W. Bert,et al.  The differential quadrature method for irregular domains and application to plate vibration , 1996 .

[5]  E. Viola,et al.  Static analysis of completely doubly-curved laminated shells and panels using general higher-order shear deformation theories , 2013 .

[6]  Erasmo Viola,et al.  Analytical and numerical results for vibration analysis of multi-stepped and multi-damaged circular arches , 2007 .

[7]  F. Tornabene,et al.  The local GDQ method for the natural frequencies of doubly-curved shells with variable thickness: A general formulation , 2016 .

[8]  Chien H. Thai,et al.  A simple four-unknown shear and normal deformations theory for functionally graded isotropic and sandwich plates based on isogeometric analysis , 2016 .

[9]  Trung Nguyen-Thoi,et al.  A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) using layerwise theory based on the C0-HSDT for analyses of composite plates , 2014 .

[10]  Nicholas Fantuzzi,et al.  Accurate inter-laminar recovery for plates and doubly-curved shells with variable radii of curvature using layer-wise theories , 2015 .

[11]  Xinwei Wang,et al.  Accurate buckling analysis of thin rectangular plates under locally distributed compressive edge stresses , 2016 .

[12]  R. Luciano,et al.  Exact solutions of isotropic viscoelastic functionally graded Kirchhoff plates , 2014 .

[13]  Hung Nguyen-Xuan,et al.  Static, free vibration, and buckling analysis of laminated composite Reissner–Mindlin plates using NURBS‐based isogeometric approach , 2012 .

[14]  Salvatore Brischetto,et al.  3D exact and 2D generalized differential quadrature models for free vibration analysis of functionally graded plates and cylinders , 2016 .

[15]  A. Ceruti,et al.  Mixed Static and Dynamic Optimization of Four-Parameter Functionally Graded Completely Doubly Curved and Degenerate Shells and Panels Using GDQ Method , 2013 .

[16]  E. Viola,et al.  Radial basis functions based on differential quadrature method for the free vibration analysis of laminated composite arbitrarily shaped plates , 2015 .

[17]  Stéphane Bordas,et al.  Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory , 2014 .

[18]  L. Dozio.  Refined 2-D theories for free vibration analysis of annular plates , 2015 .

[19]  Nicholas Fantuzzi,et al.  Four-parameter functionally graded cracked plates of arbitrary shape: A GDQFEM solution for free vibrations , 2016 .

[20]  K. K. Choong,et al.  Vibration of Rectangular Orthotropic Quarter Elliptic Plates with Simply-Supported Curved Boundary and Other Complicated Effects , 2006 .

[21]  Jae-Hoon Kang,et al.  Exact solutions for stresses, strains, and displacements of a rectangular plate with an arbitrarily located circular hole subjected to in-plane bending moment , 2014 .

[22]  Elena Ferretti,et al.  GDQFEM Numerical Simulations of Continuous Mediawith Cracks and Discontinuities , 2013 .

[23]  T. Irie,et al.  Free vibration of cross-shaped, I-shaped and L-shaped plates clamped at all edges , 1978 .

[24]  Nicholas Fantuzzi,et al.  Generalized Differential Quadrature Finite Element Method for vibration analysis of arbitrarily shaped membranes , 2014 .

[25]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[26]  Erasmo Viola,et al.  2-D solution for free vibrations of parabolic shells using generalized differential quadrature method , 2008 .

[27]  Alfredo Liverani,et al.  Laminated composite rectangular and annular plates: A GDQ solution for static analysis with a posteriori shear and normal stress recovery , 2012 .

[28]  F. Fazzolari.  Natural frequencies and critical temperatures of functionally graded sandwich plates subjected to uniform and non-uniform temperature distributions , 2015 .

[29]  K. M. Liew,et al.  Buckling and vibration analysis of isotropic and laminated plates by radial basis functions , 2011 .

[30]  A. Kalamkarov,et al.  Analysis of Smart Piezo-Magneto-Thermo-Elastic Composite and Reinforced Plates: Part I – Model Development , 2014 .

[31]  Nicholas Fantuzzi,et al.  The local GDQ method applied to general higher-order theories of doubly-curved laminated composite shells and panels: The free vibration analysis , 2014 .

[32]  Erasmo Viola,et al.  Vibration Analysis of Damaged Circular Arches with Varying Cross-section , 2005 .

[33]  Moshe Eisenberger,et al.  Vibration analysis of variable thickness plates and shells by the Generalized Differential Quadrature method , 2016 .

[34]  Liz G. Nallim,et al.  Natural frequencies of symmetrically laminated elliptical and circular plates , 2008 .

[35]  Nicholas Fantuzzi,et al.  A new approach for treating concentrated loads in doubly-curved composite deep shells with variable radii of curvature , 2015 .

[36]  Salvatore Brischetto,et al.  Numerical and exact models for free vibration analysis of cylindrical and spherical shell panels , 2015 .

[37]  Alessandro Reali,et al.  Isogeometric Analysis of Structural Vibrations , 2006 .

[38]  Francesco Tornabene,et al.  Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations , 2011 .

[39]  Nicholas Fantuzzi,et al.  Free vibrations of free-form doubly-curved shells made of functionally graded materials using higher-order equivalent single layer theories , 2014 .

[40]  F. Tornabene,et al.  Higher-order structural theories for the static analysis of doubly-curved laminated composite panels reinforced by curvilinear fibers , 2016 .

[41]  E. Viola,et al.  General higher-order shear deformation theories for the free vibration analysis of completely doubly-curved laminated shells and panels , 2013 .

[42]  Loc V. Tran,et al.  Isogeometric analysis of functionally graded plates using higher-order shear deformation theory , 2013 .

[43]  Nicholas Fantuzzi,et al.  Radial basis function method applied to doubly-curved laminated composite shells and panels with a General Higher-order Equivalent Single Layer formulation , 2013 .

[44]  Ying-Te Lee,et al.  A meshless method for free vibration analysis of circular and rectangular clamped plates using radial basis function , 2004 .

[45]  Nicholas Fantuzzi,et al.  Dynamic analysis of thick and thin elliptic shell structures made of laminated composite materials , 2015 .

[46]  Erasmo Viola,et al.  Static analysis of functionally graded doubly-curved shells and panels of revolution , 2013 .

[47]  Les A. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communication.

[48]  Tinh Quoc Bui,et al.  On the thermal buckling analysis of functionally graded plates with internal defects using extended isogeometric analysis , 2016 .

[49]  Francesco Tornabene,et al.  2-D GDQ solution for free vibrations of anisotropic doubly-curved shells and panels of revolution , 2011 .

[50]  Bo Liu,et al.  High‐accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain , 2009 .

[51]  Salvatore Brischetto,et al.  Exact elasticity solution for natural frequencies of functionally graded simply-supported structures , 2013 .

[52]  Nicholas Fantuzzi,et al.  A SFEM-based evaluation of mode-I Stress Intensity Factor in composite structures , 2016 .

[53]  Y. M. Ghugal,et al.  On the free vibration analysis of laminated composite and sandwich plates: A review of recent literature with some numerical results , 2015 .

[54]  O. G. McGee,et al.  Comprehensive exact solutions for free vibrations of thick annular sectorial plates with simply supported radial edges , 1995 .

[55]  S. M. Hasheminejad,et al.  Free vibration analysis of an elliptical plate with eccentric elliptical cut-outs , 2014 .

[56]  R. Dimitri,et al.  Free vibration analysis of conical shells reinforced with agglomerated Carbon Nanotubes , 2016 .

[57]  Nicholas Fantuzzi,et al.  Higher-order theories for the free vibrations of doubly-curved laminated panels with curvilinear reinforcing fibers by means of a local version of the GDQ method , 2015 .

[58]  A. Neves,et al.  Free vibrations and buckling analysis of laminated plates by oscillatory radial basis functions , 2015 .

[59]  Alfredo Liverani,et al.  FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations , 2011 .

[60]  A. Alibeigloo.  Thermo elasticity solution of sandwich circular plate with functionally graded core using generalized differential quadrature method , 2016 .

[61]  Prediction of natural frequencies of laminated curved panels using refined 2-D theories in the spectral collocation method , 2014 .

[62]  Nicholas Fantuzzi,et al.  Generalized differential quadrature finite element method for cracked composite structures of arbitrary shape , 2013 .

[63]  Francesco Tornabene,et al.  Free vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ method , 2011 .

[64]  G. Zavarise,et al.  T‐splines discretizations for large deformation contact problems , 2015 .

[65]  Nicholas Fantuzzi,et al.  Stress and strain recovery for functionally graded free-form and doubly-curved sandwich shells using higher-order equivalent single layer theory , 2015 .

[66]  Guang Liu,et al.  A Differential Quadrature Finite Element Method , 2010, A Differential Quadrature Hierarchical Finite Element Method.

[67]  H. Nguyen-Xuan,et al.  Analysis of cross-ply laminated plates using isogeometric analysis and unified formulation , 2014 .

[68]  Erasmo Viola,et al.  Free vibration analysis of functionally graded panels and shells of revolution , 2009 .

[69]  L. Dozio,et al.  Exact refined buckling solutions for laminated plates under uniaxial and biaxial loads , 2015 .

[70]  Erasmo Carrera,et al.  Static analysis of doubly-curved anisotropic shells and panels using CUF approach, differential geometry and differential quadrature method , 2014 .

[71]  M. Shariyat,et al.  Three-dimensional magneto-elastic analysis of asymmetric variable thickness porous FGM circular plates with non-uniform tractions and Kerr elastic foundations , 2015 .

[72]  Salvatore Brischetto,et al.  Exact 3D solutions and finite element 2D models for free vibration analysis of plates and cylinders , 2014 .

[73]  J. N. Reddy,et al.  Winkler–Pasternak foundation effect on the static and dynamic analyses of laminated doubly-curved and degenerate shells and panels , 2014, Composites Part B: Engineering.

[74]  Salvatore Brischetto,et al.  Three-dimensional exact free vibration analysis of spherical, cylindrical, and flat one-layered panels , 2014 .

[75]  Nicholas Fantuzzi,et al.  Effect of agglomeration on the natural frequencies of functionally graded carbon nanotube-reinforced laminated composite doubly-curved shells , 2016 .

[76]  M. Shariyat,et al.  Biaxial thermo-mechanical buckling of orthotropic auxetic FGM plates with temperature and moisture dependent material properties on elastic foundations , 2015 .

[77]  Daniel J. Inman,et al.  2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures , 2009 .

[78]  R. Luciano,et al.  Some analytical solutions of functionally graded Kirchhoff plates , 2015 .

[79]  Nicholas Fantuzzi,et al.  New insights into the strong formulation finite element method for solving elastostatic and elastodynamic problems , 2014 .

[80]  Kenneth James Versprille.  Computer-aided design applications of the rational b-spline approximation form. , 1975 .

[81]  Elena Ferretti,et al.  Soft Core Plane State Structures Under Static Loads UsingGDQFEM and Cell Method , 2013 .

[82]  C. Shu,et al.  Free Vibration Analysis of Curvilinear Quadrilateral Plates by the Differential Quadrature Method , 2000 .

[83]  W. Becker,et al.  Analysis of sandwich plates with compressible core using layerwise refined plate theory and interface stress continuity , 2016 .

[84]  K. Prabakar,et al.  Free vibration of anti-symmetric angle-ply plates with variable thickness , 2016 .

[85]  Francesco Ubertini,et al.  Strong Formulation Finite Element Method Based on Differential Quadrature: A Survey , 2015 .

[86]  C. Bert,et al.  VIBRATION ANALYSIS OF PLATES WITH CURVILINEAR QUADRILATERAL PLANFORMS BY DQM USING BLENDING FUNCTIONS , 2000 .

[87]  K. Liew,et al.  Free vibration of symmetrically laminated thick-perforated plates , 2000 .

[88]  Alfredo Liverani,et al.  Static analysis of laminated composite curved shells and panels of revolution with a posteriori shear and normal stress recovery using generalized differential quadrature method , 2012 .

[89]  Ashraf M. Zenkour,et al.  Analysis of Sandwich Plates by Generalized Differential Quadrature Method , 2013 .

[90]  Les A. Piegl,et al.  On NURBS: A Survey , 2004 .

[91]  A. Kalamkarov,et al.  Analysis of Smart Piezo-Magneto-Thermo-Elastic Composite and Reinforced Plates: Part II – Applications , 2014 .

[92]  Erasmo Carrera,et al.  Analysis of thick isotropic and cross-ply laminated plates by generalized differential quadrature method and a Unified Formulation , 2014 .

[93]  A. Razek,et al.  A two dimensions modeling of non-collocated piezoelectric patches bonded on thin structure , 2014 .

[94]  Tiangui Ye,et al.  Structural vibration : a uniform accurate solution for laminated beams, plates and shells with general boundary conditions , 2015 .

[95]  J. R. Banerjee,et al.  An exact spectral-dynamic stiffness method for free flexural vibration analysis of orthotropic composite plate assemblies – Part I: Theory , 2015 .

[96]  Alfredo Liverani,et al.  General anisotropic doubly-curved shell theory: A differential quadrature solution for free vibrations of shells and panels of revolution with a free-form meridian , 2012 .

[97]  Francesco Tornabene,et al.  General higher-order layer-wise theory for free vibrations of doubly-curved laminated composite shells and panels , 2016 .

[98]  E. Viola,et al.  General higher-order equivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panels , 2013 .

[99]  Nicholas Fantuzzi,et al.  A strong formulation finite element method (SFEM) based on RBF and GDQ techniques for the static and dynamic analyses of laminated plates of arbitrary shape , 2014 .

[100]  Ö. Civalek.  Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns , 2004 .

[101]  Joe Schallan.  VLSI and New Markets Drive CG Developments , 1983, IEEE Computer Graphics and Applications.

[102]  R. Dimitri.  Isogeometric treatment of large deformation contact and debonding problems with T-splines: a review , 2015 .

[103]  P. Wriggers,et al.  NURBS- and T-spline-based isogeometric cohesive zone modeling of interface debonding , 2014 .

[104]  Erasmo Viola,et al.  Free vibrations of three parameter functionally graded parabolic panels of revolution , 2009 .

[105]  Francesco Tornabene,et al.  Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution , 2009 .

[106]  M. Aslami,et al.  Wavelet-based finite element method for multilevel local plate analysis , 2016 .

[107]  L. Dozio,et al.  A spectral collocation solution for in-plane eigenvalue analysis of skew plates , 2015 .

[108]  Ö. Civalek.  Geometrically nonlinear dynamic analysis of doubly curved isotropic shells resting on elastic foundation by a combination of harmonic differential quadrature-finite difference methods , 2005 .

[109]  Salvatore Brischetto,et al.  AN EXACT 3D SOLUTION FOR FREE VIBRATIONS OF MULTILAYERED CROSS-PLY COMPOSITE AND SANDWICH PLATES AND SHELLS , 2014 .

[110]  Zhi Zong,et al.  Advanced Differential Quadrature Methods , 2009 .

[111]  Erasmo Viola,et al.  FREE VIBRATIONS OF FOUR-PARAMETER FUNCTIONALLY GRADED PARABOLIC PANELS AND SHELLS OF REVOLUTION , 2009 .

[112]  M. Petyt,et al.  Natural frequencies for free vibration of nonhomogeneous elliptic and circular plates using two-dimensional orthogonal polynomials , 1997 .

[113]  A. Ceruti,et al.  Free-Form Laminated Doubly-Curved Shells and Panels of Revolution Resting on Winkler-Pasternak Elastic Foundations: A 2-D GDQ Solution for Static and Free Vibration Analysis , 2013 .

[114]  J. N. Reddy,et al.  FGM and Laminated Doubly-Curved and Degenerate Shells Resting on Nonlinear Elastic Foundations: A GDQ Solution for Static Analysis with a Posteriori Stress and Strain Recovery , 2013 .

[115]  Hung Nguyen-Xuan,et al.  Isogeometric analysis of laminated composite and sandwich plates using a layerwise deformation theory , 2013 .

[116]  Erasmo Viola,et al.  Inter-laminar stress recovery procedure for doubly-curved, singly-curved, revolution shells with variable radii of curvature and plates using generalized higher-order theories and the local GDQ method , 2016 .

[117]  R. Dimitri,et al.  Free vibrations of composite oval and elliptic cylinders by the generalized differential quadrature method , 2015 .

[118]  Salvatore Brischetto,et al.  Refined 2D and Exact 3D Shell Models for the Free Vibration Analysis of Single- and Double-Walled Carbon Nanotubes , 2015 .

[119]  E. Carrera,et al.  Coupled thermoelastic effect in free vibration analysis of anisotropic multilayered plates and FGM plates by using a variable-kinematics Ritz formulation , 2014 .