VIETNAM NATIONAL UNIVERSITY – HO CHI MINH CITY UNIVERSITY OF SCIENCE THAI HOANG CHIEN DEVELOPMENT OF ISOGEOMETRIC FINITE ELEMENT METHODS PHD THESIS IN MATHEMATICS Ho Chi Minh City - 2015 VIETNAM NATIONAL UNIVERSITY – HO CHI MINH CITY UNIVERSITY OF SCIENCE THAI HOANG CHIEN DEVELOPMENT OF ISOGEOMETRIC FINITE ELEMENT METHODS Major: Solid Mechanics Codes: 62 44 21 01 Referee 1: Assoc Prof Dr Nguyen Hoai Son Referee 2: Assoc Prof Dr Truong Tich Thien Referee 3: Dr Nguyen Van Hieu Independent Referee 1: Dr Nguyen Trong Phuoc Independent Referee 2: Dr Vu Duy Thang SCIENTIFIC SUPERVISORS Assoc Prof Dr Nguyen Xuan Hung Professor Dr Timon Rabczuk Ho Chi Minh City - 2015 DEVELOPMENT OF ISOGEOMETRIC FINITE ELEMENT METHODS Ph.D Thesis Presented at Vietnam National University - Ho Chi Minh City University of Science - Ho Chi Minh City Faculty of Mathematics and Computer Science Department of Mechanics by Thai Hoang Chien Supervisor: Assoc Prof Dr Nguyen Xuan Hung Prof Dr Timon Rabczuk Ho Chi Minh City, March 2015 Acknowledgements This dissertation was written from 2010 to 2014 during my time as a researcher at the Division of Computational Mechanics (DCM) at Ton Duc Thang University I would like to sincerely thank Assoc Prof Nguyen Xuan Hung for giving me the opportunity to work in his research group and for his helpful guidance as my principal doctoral supervisor I also want to express my thanks to Prof Timon Rabczuk from the Institute of Structural Mechanics, Bauhaus-University-Weimar, for his devotion as a co-supervisor for my PhD thesis I would like also to acknowledge The National Foundation for Science and Technology Development (NAFOSTED, Vietnam) and Vietnam National University-Ho Chi Minh City for their financial assistance throughout the research project; without their help this thesis would not have been completed on time I am truly grateful to my colleagues at the Division of Computational Mechanics for their help and friendly supports I would also like to thank Assoc Prof Nguyen Thoi Trung, Msc Tran Vinh Loc and Msc Phung Van Phuc for their research insights and collaborations I would like to express my sincere acknowledgement to Dr Nguyen Thanh Nhon from the Institute of Applied Mechanics, Technical University of Braunschweig, Prof Stephane Bordas from the Faculty of Science Technology and Communication, University of Luxembourg, Prof A.J.M Ferreira, from the Department of Mechanical Engineering, University of Porto for their assistance, insightful suggestions, and collaborations in research Finally, my sincere thanks go to my family, especially to my wife Vu Thi Thanh Nga and my daughter Thai Man Ngoc, for their emotional support and encouragement throughout my study Ho Chi Minh City, March 2015 Thai Hoang Chien Originality statement ”I hereby declare that this submission is my own work, done under the supervision of Assoc Prof Dr Nguyen Xuan Hung and Prof Dr Timon Rabczuk, and, to the best of my knowledge, it contains no materials previously published or written by another person” Ho Chi Minh City, March 2015 Thai Hoang Chien Abstract Isogeometric analysis (IGA) is a recent method of computational analysis with the main objective of integrating Computer Aided Design (CAD) and Finite Element Analysis (FEA) into one model It means that the IGA uses Non-Uniform Rational B-Splines (NURBS), which are commonly used in CAD in order to describe both the geometry and the unknown variables for analysis problems Therefore, the process of remeshing in IGA can be omitted In this thesis, the isogeometric approach is applied to the elasticity and plasticity analysis of plate structures A Reissner-Mindlin plate theory (RMPT) based on isogeometric approach has been applied for static, free vibration and bucking analysis of the laminated composite plates In order to alleviate the locking phenomenon, a stabilization technique is introduced to modify the shear terms of the constitutive matrix Next, a novel numerical approach using a NURBS-based isogeometric approach associated with the layerwise deformation theory is formulated for static, free vibration and buckling analysis of laminated composite and sandwich plate structures In addition, a rotation-free isogeometric finite element approach for upper bound limit analysis of thin plate structures is presented for the first time A new higher order shear deformation theory (HSDT) is proposed using NURBS as basis functions for the analysis of laminated composite and functionally graded plates Under this higher-order shear deformation theory, the classical plate theory (CPT) and the Reissner-Mindlin plate theory are included as special cases by setting shape function determining the distribution of the transverse shear strains and stresses across the thickness of plates All CPT, RMPT and HSDT based on the isogeometric approach for the analysis of plate structures are presented in this thesis Numerical examples are provided to illustrate the effectiveness of the present method compared with other methods introduced in the literature Contents Introduction 1.1 Review of Isogeometric Analysis 1.2 Review of plate theories 1.3 Goal of the thesis 1.4 Outline Isogeometric analysis framework 2.1 B-spline 2.1.1 Properties 2.1.2 Derivatives 2.1.3 B-spline curves 2.1.4 h-, p- and k-refinements 2.1.4.1 Knot insertion (h-refinement) 2.1.4.2 p-refinement 2.1.4.3 k-refinement 2.1.5 B-spline surfaces 2.2 NURBS 2.2.1 NURBS basis functions 2.2.2 NURBS curves 2.2.3 NURBS surfaces 2.3 Isoparametric discretisation 2.4 Spatial derivatives of shape functions 2.5 Numerical integration 2.6 Essential boundary conditions 1 8 10 10 10 11 12 12 12 14 14 15 17 17 18 19 21 Isogeometric analysis of laminated composite and sandwich Mindlin plates1 22 3.1 Introduction 22 based on Chien H Thai, H Nguyen-Xuan, N Nguyen-Thanh, T.H Le, T Nguyen-Thoi, T Rabczuk Static, free vibration and buckling analyses of laminated composite Reissner-Mindlin plates using NURBS-based isogeometric approach, International Journal for Numerical Methods in Engineering, 91:571-603, 2012 iv CONTENTS 3.2 3.3 3.4 3.5 An isogeometric formulation for laminated composite Reissner-Mindlin plates 3.2.1 The displacements, strains and stresses of plates 3.2.2 Weak form equation of plates An improved technique on shear terms Numerical results 3.4.1 Isotropic plate 3.4.1.1 Static analysis 3.4.1.2 Free vibration analysis 3.4.1.3 Buckling analysis of rectangular plates subjected to partial in-plane edge loads 3.4.2 Static analysis of laminated composite plates 3.4.2.1 Three-layer square sandwich plate, under uniform load 3.4.2.2 Four-layer [0/90/90/0] square laminated plate under sinusoidal load 3.4.3 Free vibration analysis of laminated composite plates 3.4.3.1 Square laminated plates 3.4.3.2 Circular plates 3.4.4 Buckling analysis of composite plate 3.4.4.1 Square plate under uniaxial compression 3.4.4.2 Square plate under biaxial compression Conclusion Isogeometric analysis of laminated composite and sandwich plates using a layerwise deformation theory1 4.1 Introduction 4.2 An isogeometric formulation for laminated composite and sandwich plates using layerwise theory 4.2.1 The displacements, strains and stresses in plates 4.2.2 Weak form 4.3 Numerical results 4.3.1 Static analysis 4.3.1.1 Three-layer sandwich square plate subjected to a uniform load 4.3.1.2 Four-layer [00 /900 /900 /00 ] square laminated plate under sinusoidally distributed load 4.3.1.3 The sandwich (00 /core/00 ) square plate subjected to sinusoidally distributed load 24 24 25 28 28 28 28 31 31 35 35 38 38 38 41 45 45 52 52 54 54 56 56 59 63 64 64 67 67 based on Chien H Thai, A.J.M Ferreira, E Carrera, H Nguyen-Xuan Isogeometric analysis of laminated composite and sandwich plates using a layerwise deformation theory Composite Structures, 104: 196-214, 2013 v CONTENTS 4.3.2 4.4 Free vibration analysis 4.3.2.1 Square laminated plates 4.3.2.2 Circular plates 4.3.2.3 Ellipse plates 4.3.3 Buckling analysis 4.3.3.1 Square plate under uniaxial compression 4.3.3.2 Square plate under biaxial compression Conclusion 71 71 76 81 81 81 83 85 Isogeometric analysis of laminated composite and sandwich plates using a new higher order shear deformation theory1 87 5.1 Introduction 87 5.2 An isogeometric formulation for composite and sandwich plates using the higher-order shear deformation theory 89 5.2.1 The displacements, strains and stresses in plates 89 5.2.2 Weak form 92 5.3 Numerical examples and discussion 95 5.3.1 Static analysis 96 5.3.1.1 Four-layer [00 /900 /900 /00 ] square laminated plate under sinusoidally distributed load 96 5.3.1.2 Sandwich (00 /core/00 ) square plate subjected under sinusoidally distributed load 101 5.3.2 Free vibration analysis 101 5.3.2.1 Square plates 101 5.3.2.2 Circular plates 108 5.3.2.3 Elliptical plates 108 5.3.3 Buckling analysis 112 5.3.3.1 Square plate under uniaxial compression 112 5.3.3.2 Square plate under biaxial compression 114 5.4 Conclusions 116 Generalized shear deformation theory for functionally graded isotropic 118 and sandwich plates based on isogeometric approach2 6.1 Introduction 118 6.2 The novel higher order shear deformation theory for FGM plates 120 based on Chien H Thai, A.J.M Ferreira, T Rabczuk, S.P.A Bordas, H Nguyen-Xuan Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory European Journal of Mechanics- A/Solids,43:89-108, 2014 based on Chien H Thai, S Kulasegaram, Loc V Tran, H Nguyen-Xuan Generalized shear deformation theory for functionally graded isotropic and sandwich plates based on isogeometric approach Computer and Structures, 141:94-112, 2014 vi CONTENTS 6.2.1 6.3 6.4 Problem formulation 6.2.1.1 Isotropic FGM plates (type A) 6.2.1.2 Sandwich plate with FGM core and isotropic skins (type B) 6.2.1.3 Sandwich plates with isotropic core and FGM skins (type C) 6.2.2 The generalized shear deformation plate theory Numerical examples and discussion 6.3.1 Convergence study 6.3.2 Static analysis 6.3.2.1 Isotropic FGM plates 6.3.2.2 Sandwich plates with FGM core 6.3.3 Free vibration analysis 6.3.3.1 Isotropic FGM plates 6.3.3.2 Sandwich plate with FGM skins and isotropic core 6.3.4 Buckling analysis 6.3.4.1 Isotropic FGM plates 6.3.4.2 Sandwich plate with FGM skins and isotropic core Conclusions 120 121 122 123 123 129 129 130 130 134 136 136 138 142 142 142 145 Upper bound limit analysis of plates using a rotation-free isogeometric 149 approach1 7.1 Introduction 149 7.2 Rotation-free isogeometric formulation for upper bound limit analysis of plates 151 7.2.1 A background of limit analysis theorems of thin plates 151 7.2.2 NURBS-based approximate formulation 154 7.2.3 Essential boundary conditions 155 7.3 Solution procedure of the discrete problem 157 7.3.1 Second-Order Cone Programming (SOCP) 157 7.3.2 Solution procedure using Second-Order Cone Programming 158 7.4 Numerical results 159 7.4.1 Rectangular plates 159 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