Development of isogeometric finite element methods

Next, anovel numerical approach using a NURBS-based isogeometric approachassociated with the layerwise deformation theory is formulated for static,free vibration and buckling analysis of

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

Independent Referee 1: Dr Nguyen Trong Phuoc Independent Referee 2: Dr Vu Duy Thang

SCIENTIFIC SUPERVISORS

1 Assoc Prof Dr Nguyen Xuan Hung

2 Professor Dr Timon Rabczuk

Ho Chi Minh City - 2015

Ph.D Thesis

Presented atVietnam National University - Ho Chi Minh CityUniversity of Science - Ho Chi Minh CityFaculty 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

This dissertation was written from 2010 to 2014 during my time as a searcher at the Division of Computational Mechanics (DCM) at Ton DucThang University I would like to sincerely thank Assoc Prof NguyenXuan Hung for giving me the opportunity to work in his research groupand for his helpful guidance as my principal doctoral supervisor I alsowant to express my thanks to Prof Timon Rabczuk from the Institute ofStructural Mechanics, Bauhaus-University-Weimar, for his devotion as aco-supervisor for my PhD thesis.

re-I would like also to acknowledge The National Foundation for Science andTechnology Development (NAFOSTED, Vietnam) and Vietnam NationalUniversity-Ho Chi Minh City for their financial assistance throughout theresearch project; without their help this thesis would not have been com-pleted on time

I am truly grateful to my colleagues at the Division of Computational chanics for their help and friendly supports I would also like to thankAssoc Prof Nguyen Thoi Trung, Msc Tran Vinh Loc and Msc PhungVan Phuc for their research insights and collaborations

Me-I would like to express my sincere acknowledgement to Dr Nguyen ThanhNhon from the Institute of Applied Mechanics, Technical University ofBraunschweig, Prof Stephane Bordas from the Faculty of Science Tech-nology and Communication, University of Luxembourg, Prof A.J.M.Ferreira, from the Department of Mechanical Engineering, University ofPorto for their assistance, insightful suggestions, and collaborations in re-search

Finally, my sincere thanks go to my family, especially to my wife Vu ThiThanh Nga and my daughter Thai Man Ngoc, for their emotional supportand encouragement throughout my study

Ho Chi Minh City, March 2015

Thai Hoang Chien

”I hereby declare that this submission is my own work, done under thesupervision of Assoc Prof Dr Nguyen Xuan Hung and Prof Dr TimonRabczuk, and, to the best of my knowledge, it contains no materials pre-viously published or written by another person”.

Ho Chi Minh City, March 2015

Thai Hoang Chien

Isogeometric analysis (IGA) is a recent method of computational analysiswith the main objective of integrating Computer Aided Design (CAD) andFinite Element Analysis (FEA) into one model It means that the IGA usesNon-Uniform Rational B-Splines (NURBS), which are commonly used inCAD in order to describe both the geometry and the unknown variablesfor analysis problems Therefore, the process of remeshing in IGA can beomitted.

In this thesis, the isogeometric approach is applied to the elasticity andplasticity analysis of plate structures A Reissner-Mindlin plate theory(RMPT) based on isogeometric approach has been applied for static, freevibration and bucking analysis of the laminated composite plates In or-der to alleviate the locking phenomenon, a stabilization technique is in-troduced to modify the shear terms of the constitutive matrix Next, anovel numerical approach using a NURBS-based isogeometric approachassociated with the layerwise deformation theory is formulated for static,free vibration and buckling analysis of laminated composite and sandwichplate structures In addition, a rotation-free isogeometric finite element ap-proach for upper bound limit analysis of thin plate structures is presentedfor the first time

A new higher order shear deformation theory (HSDT) is proposed usingNURBS as basis functions for the analysis of laminated composite andfunctionally graded plates Under this higher-order shear deformation the-ory, the classical plate theory (CPT) and the Reissner-Mindlin plate theoryare included as special cases by setting shape function determining the dis-tribution of the transverse shear strains and stresses across the thickness ofplates All CPT, RMPT and HSDT based on the isogeometric approachfor the analysis of plate structures are presented in this thesis Numericalexamples are provided to illustrate the effectiveness of the present methodcompared with other methods introduced in the literature

1 Introduction 1

1.1 Review of Isogeometric Analysis 1

1.2 Review of plate theories 3

1.3 Goal of the thesis 5

1.4 Outline 6

2 Isogeometric analysis framework 8 2.1 B-spline 8

2.1.1 Properties 9

2.1.2 Derivatives 10

2.1.3 B-spline curves 10

2.1.4 h -, p- and k-refinements 10

2.1.4.1 Knot insertion (h-refinement) 11

2.1.4.2 p-refinement 12

2.1.4.3 k-refinement 12

2.1.5 B-spline surfaces 12

2.2 NURBS 14

2.2.1 NURBS basis functions 14

2.2.2 NURBS curves 15

2.2.3 NURBS surfaces 17

2.3 Isoparametric discretisation 17

2.4 Spatial derivatives of shape functions 18

2.5 Numerical integration 19

2.6 Essential boundary conditions 21

3 Isogeometric analysis of laminated composite and sandwich Mindlin plates1 22 3.1 Introduction 22

Rabczuk Static, free vibration and buckling analyses of laminated composite Reissner-Mindlin plates using NURBS-based isogeometric approach, International Journal for Numerical Methods in Engineer-ing, 91:571-603, 2012.

3.2 An isogeometric formulation for laminated composite Reissner-Mindlin

plates 24

3.2.1 The displacements, strains and stresses of plates 24

3.2.2 Weak form equation of plates 25

3.3 An improved technique on shear terms 28

3.4 Numerical results 28

3.4.1 Isotropic plate 28

3.4.1.1 Static analysis 28

3.4.1.2 Free vibration analysis 31

3.4.1.3 Buckling analysis of rectangular plates subjected to partial in-plane edge loads 31

3.4.2 Static analysis of laminated composite plates 35

3.4.2.1 Three-layer square sandwich plate, under uniform load 35 3.4.2.2 Four-layer [0/90/90/0] square laminated plate under sinusoidal load 38

3.4.3 Free vibration analysis of laminated composite plates 38

3.4.3.1 Square laminated plates 38

3.4.3.2 Circular plates 41

3.4.4 Buckling analysis of composite plate 45

3.4.4.1 Square plate under uniaxial compression 45

3.4.4.2 Square plate under biaxial compression 52

3.5 Conclusion 52

4 Isogeometric analysis of laminated composite and sandwich plates using a layerwise deformation theory1 54 4.1 Introduction 54

4.2 An isogeometric formulation for laminated composite and sandwich plates using layerwise theory 56

4.2.1 The displacements, strains and stresses in plates 56

4.2.2 Weak form 59

4.3 Numerical results 63

4.3.1 Static analysis 64

4.3.1.1 Three-layer sandwich square plate subjected to a uni-form load 64

4.3.1.2 Four-layer [00/900/900/00] square laminated plate under sinusoidally distributed load 67

4.3.1.3 The sandwich (00/core/00) square plate subjected to sinusoidally distributed load 67

laminated composite and sandwich plates using a layerwise deformation theory Composite Structures, 104: 196-214, 2013.

4.3.2 Free vibration analysis 71

4.3.2.1 Square laminated plates 71

4.3.2.2 Circular plates 76

4.3.2.3 Ellipse plates 81

4.3.3 Buckling analysis 81

4.3.3.1 Square plate under uniaxial compression 81

4.3.3.2 Square plate under biaxial compression 83

4.4 Conclusion 85

5 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 un-der 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

6 Generalized shear deformation theory for functionally graded isotropic and sandwich plates based on isogeometric approach2 118 6.1 Introduction 118

6.2 The novel higher order shear deformation theory for FGM plates 120

Isogeo-metric analysis of laminated composite and sandwich plates using a new inverse trigonoIsogeo-metric shear deformation theory European Journal of Mechanics- A/Solids,43:89-108, 2014.

defor-mation theory for functionally graded isotropic and sandwich plates based on isogeometric approach Computer and Structures, 141:94-112, 2014

6.2.1 Problem formulation 120

6.2.1.1 Isotropic FGM plates (type A) 121

6.2.1.2 Sandwich plate with FGM core and isotropic skins (type B) 122

6.2.1.3 Sandwich plates with isotropic core and FGM skins (type C) 123

6.2.2 The generalized shear deformation plate theory 123

6.3 Numerical examples and discussion 129

6.3.1 Convergence study 129

6.3.2 Static analysis 130

6.3.2.1 Isotropic FGM plates 130

6.3.2.2 Sandwich plates with FGM core 134

6.3.3 Free vibration analysis 136

6.3.3.1 Isotropic FGM plates 136

6.3.3.2 Sandwich plate with FGM skins and isotropic core 138 6.3.4 Buckling analysis 142

6.3.4.1 Isotropic FGM plates 142

6.3.4.2 Sandwich plate with FGM skins and isotropic core 142 6.4 Conclusions 145

7 Upper bound limit analysis of plates using a rotation-free isogeometric approach1 149 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

7.4.2 Rhombic plate 161

7.4.3 L-shaped plate 170

7.4.4 Circular plate 172

7.4.4.1 Circular plate subjected to uniform transverse load-ing 172

of plates using a rotation-free isogeometric approach Asia Pacific Journal on Computational Engineer-ing, 1:12, 2014.

7.4.4.2 Circular plate subjected to non-uniform transverse

2.1 1D and 2D quadratic B-spline basis functions 9

2.2 1D and 2D cubic B-spline basis functions 9

2.3 A quadratic (p = 2) B-spline curve with a uniform open knot vector

Ξ = {0,0,0,1,2,3,4,5,5,5} 11

2.4 Knot insertion (h-refinement) on a quadratic B-spline curve. 12

2.5 Order elevation of a quadratic B-spline curve to cubic : B-spline curvesand associated basis functions 13

2.6 The use of a NURBS curve to construct a quarter circle (with w1= w3

= 1 and w2= 1/√

2) 15

2.7 Example of a NURBS curve for a circle of unit radius 16

2.8 Effect of decreasing weights on a NURBS curve: the weights at control

point P4are 1, 0.6 and 0.3 for the red, blue and cyan curves, respectively 16

2.9 NURBS surface and control mesh 17

2.10 Parametric and physical space with quadratic B-splines 20

2.11 Illustration of imposing Dirichlet BCs Black points denote cornercontrol points where the NURBS basis satisfy the Kronecker deltaproperty 21

3.1 Geometry of a typical Mindlin-Reissner plate 24

3.2 Fully clamped, simply supported square plate models and control points 29

3.3 Normalized deflection of simply supported and clamped square isotropicplates subjected to uniformly distributed load 30

3.4 Normalized strain energy of simply supported and clamped squareisotropic plates subjected to uniformly distributed load 31

3.5 Performance of present element with various ratios L/h of clamped

and simple supported isotropic plates 32

3.6 Geometry and control points of a L-shape isotropic plate 34

3.7 A square plate is subjected to axial in-plane edge loadings 34

3.8 Normalized deflection w c : (a) R=5; (b) R=10; (c) R=15 and (d)

Rela-tive errors 37

3.9 Geometry and control points of a four-layer square laminated plate

under sinusoidal load 37

3.10 The distribution of stresses through thickness of the plate with a/h = 4, 10 40

3.11 Geometry and control points of a three-layer square laminated plate 41 3.12 Mode shapes 1-6 of the three-layer clamped laminated square plate 44

3.13 Geometry of a circular plate 45

3.14 Mesh and control net for a disk of radius 0.5 46

3.15 Meshes produced by h-refinement (knot insertion) 46

3.16 The first six mode shapes of a 4-layer clamped laminated circular plate 48 3.17 Geometry of laminated composite plates under axial and biaxial com-pression 49

3.18 Fundamental buckling modes of the 10-layer square plate with various mixed boundaries 51

4.1 1D representation of the layerwise kinematics 57

4.2 Geometry of a sandwich plate 65

4.3 Meshes and control net of the square plate 66

4.4 Geometry of a four-layer square laminated plate under sinusoidal load 69 4.5 The distribution of stresses through the thickness of the four-layer lam-inated composite square plate under a sinusoidally distributed load 69

4.6 The distribution of stresses through the thickness of the sandwich square plate under a sinusoidally distributed load 73

4.7 Geometry and control points of a three-layer laminated square plate 74 4.8 Modes shape 1-6 of a three-layer clamped laminated square plate with a /b= 1 and b/h= 10 . 75

4.9 Geometry of a circular plate 78

4.10 a) Coarse mesh and control points of a circular plate; b) mesh element 13× 13 79

4.11 Modes shape 1-6 of a four-layer clamped laminated circular plate with R /h= 5 . 79

4.12 Geometry and element mesh of a clamped ellipse plate 81

4.13 Modes shape 1-8 of a three-layer clamped laminated ellipse plate with a /h= 10 . 83

4.14 Geometry of laminated composite plates under axial and biaxial com-pression 84

5.1 Geometry of a plate 89

5.2 Shape functions f (z) and their derivatives across the thickness of the plate 90

5.3 Geometry of a laminated square plate under sinusoidally distributed load 97

5.4 Meshes and control net of a square plate using cubic elements: a) 9×9;b) 13× 13 and c) 17 × 17 97

5.5 The distribution of stresses through the thickness of a four-layer squareplate under a sinusoidally distributed load 100

5.6 The distribution of stresses through the thickness of a sandwich(00/core/00)plate under a sinusoidally distributed load 103

5.7 First six mode shapes of an antisymmetry sandwich(00/900/core/00/900)simply supported square plate 108

5.8 Geometry and element mesh of a circular plate 109

5.9 Six mode shapes of a four-layer clamped laminated circular plate with

R /h= 5 . 109

5.10 Geometry and element mesh of a clamped elliptical plate 111

5.11 Six mode shapes of a three-layer clamped laminated ellipse plate with

a /h= 10 . 111

5.12 Geometry of laminated composite plates under axial and biaxial pression 113

com-6.1 A typical configuration of FGM plate 121

6.2 The effective modulus of a Al/ZrO2 FGM plate computed by the rule

of mixture (in solid line) and the Mori-Tanaka (in dash dot line) 122

6.3 The sandwich plate with homogeneous skins and FGM core 123

6.4 The sandwich plate with FGM skins and homogeneous core 124

6.5 Shape functions f (z) and their derivatives across the thickness of the

plate 124

6.6 The square plate geometry 129

6.7 Comparison of the present result with the analytical solution of Vel and

Batra for power index n = 1 and 6 . 130

6.8 Cubic element mesh and control net of the circular plate 131

6.9 The normalized center displacement for various ratios of thickness 132

radius-to-6.10 The stresses through thickness of a Al/Al2O3 FG plate under

sinu-soidal load with a/h=4, n=1, for other HSDT models . 132

6.11 The stresses through thickness of a Al/Al2O3 FG plate under

sinu-soidal load with a/h=4, for various power indices n . 134

6.12 The normalized deflection of a Al /ZrO2-1 FGM plate for various powerindexes and boundary conditions 136

6.13 The shear stress through thickness of a SSSS sandwich plate of type

B under sinusoidal load with a/h=4,100 and for various power index values of n . 138

6.14 The shear stress through thickness of a SSSS sandwich plate of types

B under sinusoidal load with n=1, 10 and for different plate models . 141

6.15 Geometry of the circular plate under a uniform radial pressure 145

7.1 Clamped boundary conditions in a rotation-free IGA formulation: ply fixing the deflections of two rows of control points along the clampedboundary 156

sim-7.2 A fully clamped square plate: a quarter of plate is modeled Along thesymmetry lines, the normal rotation is fixed which can be achieved byenforcing the deflection of two rows of control points that define thetangent of the plate to have the same value 156

7.3 Full model of clamped and simply supported square plates 160

7.4 A quarter of plate is modeled: meshes of cubic elements 162

7.5 Comparison of numerical results of the clamped square plate using twoGaussian rules 163

7.6 Comparison of computational times of the clamped square plate usingquadratic and cubic elements 163

7.7 Relative error to the reference upper bound of the clamped square plate 164

7.8 The plastic dissipation of a square plate: a) SSSS; b) CCCC 164

7.9 Convergence of the limit load factor ( ¯qab2/m p) of a clamped square

plate using k-refinement . 165

7.10 The plastic dissipation of a rectangular plate: a) SSSS; b) CCCC; c)CCCsF; d) CCClF; e) CCsFF 166

7.11 Geometry of a rhombic plate 167

7.12 Rhombic plates: meshes of B-spline elements 168

7.13 Limit load factor of a skew plate with simply supported and clampedboundary conditions 169

7.14 The plastic dissipation of a rhombic plate with α = 300: a) simplysupported plate; b) clamped plate 170

7.15 L-shaped plate models 171

7.16 Relative error to the reference upper bound of a simply supported shape plate 171

L-7.17 Geometry of a clamped circular plate 172

7.18 Relative error of the limit load factor ( ¯qR2/m p) of a circular plate 173

7.19 An illustration of a circular plate subjected to a non-uniform load 174

2.1 Control points and weights for a circular plate with radius R = 0.5 16

3.1 A non-dimensional frequency parameterϖ= (ωa) (ρ/G)1/2of a clampedisotropic L-shaped plate 33

3.2 Buckling load parameters ¯λ =λcr b /D0of a supported isotropic squareplate 33

3.3 The normalized displacement ¯wand stresses of the square sandwich plate 36

3.4 The normalized displacement ¯wof a four-layer simply supported inated square plate under sinusoidal load 39

lam-3.5 A non-dimensional frequency parameterϖ= ωa2/h
(ρ/E2)1/2 of a

[0/90/90/0] SSSS laminated plate (a/b=1) 42

3.6 A non-dimensional frequency parameter ϖ = ωb2/π2

(ρh /D0)1/2

of a [0/90/0] clamped laminated square plate 43

3.7 Control points and weights for a disk of radius 0.5 45

3.8 A non-dimensional frequency parameterϖ= ωa2/h
(ρ/E2)1/2 of acircular 4-layer[θ/ −θ/ −θ/ ] clamped laminated plate 47

3.9 A normalized critical buckling load of the simply supported cross-ply[00/900/900/00] square plate with various E1/E2ratios 50

3.10 A normalized critical buckling load of the simply supported cross-ply

square plate with various ratios a/h 50

3.11 A normalized critical buckling load of cross-ply[00/900] and [00/900]5

square plates with various mixed boundaries (E1/E2= 40; a/h = 10) 52

3.12 Biaxial buckling load of the simply supported cross-ply [00/900/00]square plate with various modulus ratios 53

4.1 The convergence of the normalized displacement and stresses of the

three-layer sandwich square plate laminated plate (a/h = 10) 65

4.2 The normalized displacement and stresses of the square sandwich plateunder uniform load 68

4.3 The normalized displacement and stresses of the four-layer[00/900/900/00]laminated square plate under a sinusoidally distributed load 70

4.4 The normalized displacement and stresses of the sandwich(00/core/00)simply supported square plate under sinusoidally distributed load 72

4.5 The convergence of non-dimensional frequencies parameter ϖ of thethree-layer[00/900/00] clamped laminated plate (b/h = 5 and a/b=1) 76

4.6 Normalized frequenciesϖ of a[00/900/00] clamped laminated squareplate 77

4.7 A non-dimensional frequency parameterϖ= ωa2/h
(ρ/E2)1/2 of a[00/900/900/00] SSSS laminated square plate (a/h=5) 78

4.8 Normalized frequencies ϖ = ωa2/h
(ρ/E2)1/2 of a circular 4-layer[θ0

/ −θ0

/ −θ0/ 0] clamped laminated plate 80

4.9 Normalized frequencies ϖ of a [00/900/00] the fully clamped nated ellipse plate 82

lami-4.10 The convergence of normalized critical buckling load of a[00/900/900/00]

simply supported cross-ply square plate with a/h = 10 and E1/E2= 40 84

4.11 Normalized critical buckling load of a simply supported[00/900/900/00]

square plate with various E1/E2ratios 84

4.12 Normalized critical buckling load of a simply supported[00/900/900/00]

square plate with various ratios a/h 85

4.13 Biaxial critical buckling load of a simply supported cross-ply[00/900/00]square plate with various modulus ratios 85

5.1 Several trigonometric shear deformation theories 90

5.2 The convergence of the normalized displacement and stresses of a layer[00/900/900/00] laminated composite square plate (a/h = 4) 98

four-5.3 The normalized displacement and stresses of a four-layer[00/900/900/00]simply supported laminated square plate under a sinusoidally distributedload 99

5.4 The normalized displacement and stresses of a sandwich(00/core/00)simply supported square plate under a sinusoidally distributed load 102

5.5 The convergence of non-dimensional frequency parameterϖof a layer[00/900/900/00] simply supported laminated square plate (a/h =

anti-5.9 Normalized frequenciesϖ= ωb2/h
q(ρ/E2)f of an antisymmetry(00/900/core/00/900) sandwich simply supported square plate with

5.13 Normalized critical buckling load of a simply supported[00/900/900/00]

square plate with various E1/E2ratios and a/h=10 113

5.14 Normalized critical buckling load of a simply supported[00/900/900/00]

square plate with various ratios a/h and E1/E2=40 114

5.15 Normalized critical buckling load of eleven and twenty-one layer wich simply supported square plates 115

sand-5.16 Biaxial critical buckling load of a simply supported cross-ply[00/900/00]square plate with various modulus ratios 116

5.17 Biaxial critical buckling load of a simply supported cross-ply[00/900/00]

square plate with various ratios a/h 117

6.1 Various forms of shape functions and their derivatives 125

6.2 Material properties 132

6.3 The non-dimensional deflection and axial stress of a simply supported

(SSSS) Al /Al2O3square plate under sinusoidal load 133

6.4 The non-dimension deflection of a Al /ZrO2− 1 plate under uniform

load with a/h=5 for different boundary conditions . 135

6.5 The non-dimensional deflection and transverse shear stress of a SSSSsquare sandwich plate with core FGM type B under sinusoidal load 137

6.6 The natural frequency ¯ω=ωhpρm /E m of a SSSS Al /ZrO2− 1 plate

with a/h=5 . 139

6.7 The natural frequency ¯ωof a SSSS Al /ZrO2-1 plate with various ratios

a /h . 140

6.8 The first five non-dimensionalized frequencies ¯ω of a SSSS sandwich

plate 2-1-2 with a/h=10 . 141

6.9 Comparisons of the natural frequency ¯ωof a SSSS sandwich plate with

other theories (a/h=10) . 143

6.10 Comparison of the buckling load parameter of a clamped thick circularAl/ZrO2-2 plate 144

6.11 Uni-axial critical buckling load a SSSS sandwich plate with FGMskins and isotropic core 146

6.12 Bi-axial critical buckling load of a SSSS sandwich plate with FGMskins and isotropic core 147

7.1 The convergence of the limit load factor ( ¯qa2/m p) for a clamped squareplate 161

7.2 The convergence of the limit load factor ( ¯qa2/m p) for a simply ported square plate 162

sup-7.3 A comparison of the limit load factor ( ¯qa2/m p) for a square plate 165

7.4 The limit load factor ( ¯qab /m p ) for a rectangular plate with a/b = 2 and

various boundary conditions 167

7.5 Results of the limit load factor ( ¯qR2/m p) for the rhombic plate 168

7.6 The limit load factor ( ¯qL2/m p) for a L-shaped plate 172

7.7 The limit load factor λcr /m pfor a clamped circular plate subjected to

a linear load 173

7.8 The limit load factor λcr /m pfor a clamped circular plate subjected to

a parabolic load 174

f Global force vector

K Global stiffness matrix

ω Natural frequency

ρ Mass density

σ Stress field

σxx Normal stress in x direction

σxy Shear stress in xy direction

σxz Shear stress in xz direction

σyy Normal stress in y direction

σyz Shear stress in yz direction

ε Strain field

εxx Normal stress in x direction

εxy Shear stress in xy direction

εxz Shear stress in xz direction

εyy Normal stress in y direction

εyz Shear stress in yz direction

Ξ Knot vector inξ direction

ξ;η Parametric coordinates

Abbreviations

2D Two dimensional

3D Three dimensional

CAD Computer Aided Design

CAE Computer Aided Engineering

CFS Closed form solution

CLPT Classical laminate plate theory

CPT Classical plate theory

CUF Carrera’s unified formulation

DQM Differential quadrature method

dTrSDTs Different trigonometric shear deformation theoriesEFG Element-free Galerkin

ESDT Exponential shear deformation theory

ESL Equivalent single layer

FEA Finite Element Analysis

FEM Finite Element Method

FGM Functionally graded material

FiSDT Fifth-order shear deformation theory

FSDT First-order shear deformation theory

FSM Finite strip method

GLHOT Global-local higher-order theory

GSDT Generalized shear deformation theory

HCT HsiehCloughTocher element

HOZT Higher-order zigzag theory

HSDT Higher-order shear deformation theory

IGA Isogeometric Analysis

ITSDT Inverse tangent shear deformation theory

LHOT Local higher-order theory

LWT Layer-wise theory

MR Mindlin/Reissner

MRBF Multiquadric Radial Basis Function

NEM Natural element method

NSFEM Node-based Finite Element Method

NURBS Non-Uniform Rational B-splines

PS Pseudospectral

RBF Radial Basis Function

RPIM Radial point pnterpolation method

RPT Refined plate theory

RPT Refined plate theory

SCF Shear correction factors

SCFs Shear correction factors

SSDT Sinusoidal shear deformation theory

TrSDT Trigonometric shear deformation theory

TSDT Third-order shear deformation theory

UTSDT Unconstrained third-order shear deformation theory

The main objective of this thesis is to develop an isogeometric analysis for elasticityand plasticity plate structures This chapter gives a literature review on isogeometricapproach as well as plate theories for the analysis of structures

Before the advent of computers, all engineering drawing was done manually by usingpencil and pen on paper or other substrate (e.g., vellum, mylar) Since the advent ofcomputer-aided design (CAD), engineering drawing has been done more and more inthe electronic medium Today most engineering drawing in industries area such asshipbuilding, automobile, aerospace, industrial and architectural design are done withCAD [177] The B-spline basis functions have been used to represent the curves bythe designers since 1972 [45] NURBS (Non-uniform rational B-splines) are a gen-eralization of Bezier splines (B-spline), and have been used in CAD programs since

1975 [45] NURBS are a mathematical model commonly used in computer graphicsfor generating and representing curves and surfaces NURBS offer great flexibilityand precision for handling both analytic (surfaces defined by common mathematicalformulae) and modeled shapes [177] By using NURBS, conic sections like circles,cylinders and spheres can be represented exactly The most used basis functions torepresent geometries are NURBS, which were in the beginning only used in propri-etary CAD packages of car companies, but are today used in all standard CAD pack-ages [177] Today, there exist many efficient numerical stable algorithms to generateNURBS objects [149]

Computer Aided Engineering (CAE) is the broad usage of computer software to aid inengineering analysis tasks In CAE, the finite element method (FEM) is often used as

an analysis tool to solve partial differential equations by Lagrange interpolating

poly-nomial The Finite Element Method (FEM) started developing in the 1950s [45], withall analyses made by hand The method was therefore only applied on small and easysystems In the decades that followed, as analysts got experience with the method ondifferent problems, they started improving the algorithms and developing new basisfunction elements(Hermite polynomials) To solve partial differential equations usingthe FEM one usually uses variational or weak form formulations When computerswere first used to perform the analysis, the computational efficiency was a very criticalissue In practical applications, computational efficiency is still an issue However, itturns out that higher order element uses more work per degree of freedom but fewerdegree of freedom converge.

With the advancement in technology comes the desire to create more and more plex constructions, resulting in the need of more efficient and accurate methods fordesign and analysis It is a difficult task to improve efficiency, as there exists a gap be-tween FEA and CAD The design and the analysis communities evolved independently

com-of each other, as they had different goals and needs In order to integrate CAD andFEA into one model, isogeometric analysis was proposed by Hughes and co-workers

in 2005 [78] Data generated from CAD are used directly for analysis without verting the data generated in CAD to a data set suitable for FEA, where B-spline orNURBS are the most widely used computational geometry technology in engineeringdesign Geometry domains having conic sections like circles, cylinders, spheres, ellip-soids, etc can be represented exactly Using NURBS lets us easily control continuity,

con-as C p−1-continuity is obtained using p-th order NURBS A monograph of the

isogeo-metric analysis has been published entirely on the subject [45] and applications havebeen found in several fields including structural mechanics, solid mechanics, fluid me-chanics and contact mechanics

Isogeometric analysis has been applied to a wide range of mechanics problems Thesmoothness of NURBS basis functions is attractive for analysis of fluids [71,137] andfor fluid-structure interaction problems [17,18] In contact formulations using conven-tional geometry discretisations, faceted surfaces are often found on contact surfacesthat can lead to jumps and oscillations in tractions unless very fine meshes are used.The benefits of using NURBS over such an approach are therefore evident, since thecontact surface is now smooth, leading to more physically accurate contact stresses.Recent work in this area includes [105, 193] Another area where IGA has shownadvantages over traditional approaches is optimization problems [111, 216] IGA isparticularly suited to such problems due to the tight coupling with CAD models andoffers an extremely attractive approach for industrial applications In addition, due

to the ease of constructing high order continuous basis functions, IGA has been usedwith great success in solving PDEs that incorporate fourth order (or higher) derivatives

of the field variable such as the Hill-Cahnard equation [74], explicit gradient age models [215] and gradient elasticity [66] NURBS have also shown advantageousproperties for structural vibration problems [47] with the mathematical properties of

dam-IGA studied in detail by Evans et al.[53] In addition, IGA has also been applied tofracture mechanics [107,125].

Therefore, for the analysis of structures, the smoothness of the NURBS basis tions allows for a straightforward construction of beam, plate and shell elements Iso-geometric analysis has been applied to a wide range of practical structures such as:the Euler-Bernoulli beam [47,219], the Timoshenko beams [213], the Kirchhoff plates[187,195], the Mindlin-Reissner plates [84,200,211], the plates based on the higher-order shear deformation theory (HSDT) [133, 134,194,198, 199, 210] and the platesbased on the layerwise theory [73,197], the thin shells based on the Kirchhoff theory[89,90], the shells based on FSDT [20] and the rotation-free shells [20]

In the past few decades, developments in science and technology have motivated searchers to find new structural materials such as composite and functionally gradedmaterials These materials have been used in various engineering disciplines includingaerospace engineering, automotive engineering, civil engineering, nuclear plants andsemiconductor technologies Plates are an important part of many structures Com-posite plates are often made of several orthotropic layers bonded together to achievesuperior properties such as high stiffness and strength-to-weight ratios, long fatiguelife, wear resistance, light weight and a number of other attractive properties To usethem effectively, a true understanding of their structures behavior is required, includ-ing the displacements, stress distribution, dynamic and buckling responses and so on.There are several plate theories provided to analyze the accuracy of laminated compos-ite plates, and developed and reported extensively in literature For example, a review

re-of theories and computational models for laminated composite plate was presented byReddy and Robbins [169] An overall view of laminate theories based on displacementhypothesis was given by Liu and Li [103] A review of theories for laminated and sand-wich plates was given by Altenbach [3] A historical review encompassing early andrecent developments of advanced theories for laminated beams, plates and shells wasgiven in [32] A review of shear deformation plate and shell theories was presented byReddy and Arciniega [166] A review of the finite element models developed based onvarious laminated plate theories was presented by Zhang and Yang [230] A literaturereview on computational models for laminated composite and sandwich panels wasgiven by Kreja [92] Generally, the laminated plate theories can be usually dividedinto two kinds: the continuum-based 3D elasticity theory and the equivalent singlelayer (ESL) theories The ESL theories include

• The classical plate theory (CPT)

• The first-order shear deformation theory (FSDT) (also referred to as Mindlinplate theory)

• The higher-order shear deformation theory (HSDT)

• The layer-wise theory (LWT)

Among the ESL plate theories, the classical laminate plate theory (CLPT) [106], based

on the Love-Kirchoff assumptions, was first proposed As it does not take into count the transverse shear deformation, the CLPT can provide acceptable results forthe thin plates only However, it may not produce accurate results for moderatelythick plates In order to improve the CPT, the first order shear deformation theory, theReissner-Mindlin theory [117,172], which takes shear effect into account, was there-fore developed Nevertheless, with the linear in-plane displacement assumption acrossthe plate thickness, the shear strain/stress distribution obtained from the FSDT is inac-curate and does not satisfy the traction-free boundary conditions at the plate surfaces.The shear correction factors (SCF) are hence required to rectify the unrealistic shearstrain energy component The values of the SCF are quite dispersed through each prob-lem and may be difficult to determine [164] To ensure smooth distribution of shearstress, various types of higher-order shear deformable theories (HSDT), which includehigher-order terms in the approximation of the displacement field, were then devised.These higher-order shear deformation theories include the third-order shear deforma-tion theory (TSDT) [160], the fifth-order shear deformation theory (FiSDT)[133], thetrigonometric shear deformation theory [194], the exponential shear deformation the-ory (ESDT) [85], the refined plate theory (RPT) based on two unknown functions oftransverse deflection [183] and so on The classical, first-order and higher-order the-ories use the equivalent single-layer models (ESL), which consider the same degrees

ac-of freedom for all laminate layers However, in the practical application ac-of sandwichplates, the difference of strength between core and face sheets is very large, so thesetheories will encounter difficulties in accurately predicting the bending behavior anddynamic response In fact, most of them do not correctly represent the transverseshear stresses and the higher frequencies In this case, the use of the layerwise theory

is recommended, which considers independent degrees of freedom for each layer Anumber of layerwise theories [121,170,190] have been proposed for analysis of lami-nated composite and sandwich plates The layerwise theory of Reddy [170] is perhapsthe most popular In this work, a layerwise displacement model [57] is presented toanalyze laminated composite and sandwich plates The proposed model assumes afirst-order shear deformation theory in each layer and the imposition of displacementcontinuity at the layer interfaces In addition, several other layerwise models for lam-inated plates have been presented by Mau [114], Chou and Carleone [42], Di Sciuva

[181], Toledano and Murakami [206] and Ren [176] The zigzag or discrete-layer ory has also been presented by Di Sciuva [182] and Carrera [29,30].

the-It should be emphasized that the CPT, FSDT, HSDT and layerwise formulations scribed above are in fact 2D models The multilayered plates can also use the 3Dmodel based on elastic three-dimensional continuum However, practical application

de-of 3D model is very limited due to the increased number de-of unknowns de-of the problem.Analytical 3D solution is first given by Pagano [143] for selected examples of simplysupported laminated composite plates under the distributed transverse loads Manyresearchers have extended Pagano 3D model for the analysis of laminated plates such

as a 3D displacement-assumed variational analysis [22], the problem of delamination[223] and modeled layered composites [50]

Structural plates are widely used in various engineering disciplines The rapid increase

of the industrial use of these structures has necessitated the development of new lytical and numerical tools Structural plates are often described by partial differentialequations, and in most cases, their closed form solutions are difficult to establish As aresult, approximate numerical methods have been widely used to solve partial differen-tial equations that arise in almost all engineering disciplines Finite elements, finite dif-ference, differential quadrature, boundary elements and meshless methods are widelyused for numerical solutions of engineering problems The finite element method iswidely use for the analysis of structural plates and is now fully established In addi-

ana-tion, it is not easy to conveniently construct conformable plate elements (C1continuity)

of high order as required for thin or higher-order plates In order to apply for thin orhigher-order plates, a new method called the meshless method has been developed

In meshless method, the problem domain is discretized by a set of scattered nodes,and element connectivity among the nodes is not required But the shape functionsused in meshless method are usually not satisfied the delta function property, thereforeessential boundary conditions cannot be directly imposed as conveniently as the con-ventional FEM method In recent years, a new numerical method called isogeometricanalysis is proposed that has the yield higher-order continuity and satisfied the deltafunction property These are an advantage over the finite element and meshless meth-

ods Isogeometric analysis is of at least C1continuity and thus attractive for plates.The aim of this thesis is focussing on the development of isogeometric finite elementmethods for the elasticity and plasticity analysis of plate structures First, a new iso-geometric formulation for laminated composite Reissner-Mindlin plates is presented

In this formulation, a stabilization technique is used to modify the shear terms of theconstitutive matrix to alleviate the locking phenomenon in Reissner-Mindlin plates

Second, the layerwise theory for the analysis of laminated composite and sandwich

is also discussed The new higher order shear deformation theory is proposed for theplates based on isogeometric analysis Next, a generalized shear deformation theoryfor functionally graded sandwich plates based on isogeometric approach is studied Fi-nally, a rotation-free isogeometric approach is extended for upper bound limit analysis

of plates

Chapter 2: The basic of isogeometric analysis is presented First, a brief review ofNURBS functions and surfaces are given Secondly, the differences between the tradi-tional finite element method and the isogeometric analysis are discussed

Chapter3: The formulation of the isogeometric Reissner-Mindlin plates is presented.This formulation is tested on a set of benchmark problems for laminated compos-ite plates Several numerical examples are presented to show the performance of themethod, and the results obtained are compared with other available ones

Chapter4: In this chapter, the formulation of the isogeometric Reissner-Mindlin plates

is extended to the layerwise deformation theory This theory assumes a first-ordershear deformation theory in each layer and the imposition of displacement continuity

at the layer interfaces This permits removing shear correction factors and improvesthe accuracy of transverse shear stresses Numerically intensive studies have been con-ducted to show the highly efficient performance of the proposed formulation

Chapter5: A new inverse trigonometric shear deformation theory combined with geometric analysis is proposed to analyze the laminated and sandwich plates The

iso-proposed formulation requires C1-continuity generalized displacements NURBS

ba-sis functions used in IGA are C p−1 continuous and therefore can easily satisfy the

C1-continuity condition Numerical examples are provided to show high efficiency ofthe present method compared with other published solutions

Chapter6: A generalized shear deformation theory combined with isogeometric ysis is given to analyze the functionally graded sandwich plates There are two newdistribution functions proposed in the present formulation IGA based on two proposeddistribution functions and other distribution functions reported in the literature is pre-sented in this chapter Numerical examples are provided to illustrate the effectiveness

anal-of the proposed method compared with other methods introduced in the literature.Chapter7: A rotation-free isogeometric formulation is presented for upper bound limitanalysis of plates In this formulation, only one deflection variable (without rotationaldegrees of freedom) is used for each control point The optimization formulation oflimit analysis is transformed into the form of a second-order cone programming prob-lem so that it can be solved using highly efficient interior-point solvers Several nu-

merical examples are provided to demonstrate the performance of the present method

in comparison with other published methods

Chapter 8: The concluding remarks and suggested further work are provided in thischapter

Isogeometric analysis framework

A brief discussion of B-splines, NURBS and implemented isogeometric analysis ispresented in this chapter We begin with some basic concepts and define commonlyused notations and introduce B-spline curves and surfaces Most of the algorithmsused to implement B-spline and NURBS are illustrated in the book [149] And thealgorithms for isogeometric analysis can be found in [45]

Let Ξ =ξ1,ξ2, ,ξn +p+1



be a nondecreasing sequence of parameter values, ξi ≤

ξi+1, i = 1, , n + p Theξi values are called knots, andΞ is the set of coordinates inthe parametric space If all knots are equally spaced, the knot vector is called uniform

If the first and the last knots are repeated p+ 1 times, the knot vector is described

as open A B-spline basis function is C∞ continuous inside a knot span and C p−1

continuous at a single knot A knot value can appear more than once and is then called

a multiple knot At a knot of multiplicity k the continuity is C p −k

The B-spline basis functions N i ,p(ξ) of order p = 0 are defined as follows

For p = 0 and 1, the basis functions of isogeometric analysis are identical to those of

the standard piecewise constant and linear finite elements, respectively In IGA, the

basis functions with p≥ 2 are considered [78] Fig.2.1 and Fig 2.2 illustrate a set

of one-dimensional and two-dimensional quadratic and cubic B-spline basis functionsfor open uniform knot vectorsΞ = {0,0,0,12, 1, 1, 1} and H = {0,0,0,0,12, 1, 1, 1, 1},respectively

2 Each basis function is nonnegative over the entire domain N i ,p(ξ) ≥ 0,∀ξ

3 B-spline basis functions are linearly independent i.e.,

n

∑

i=1

αi N i ,p(ξ) = 0 ⇔αk = 0, k =

1, 2, n.

4 The support of a B-spline function of order p is p + 1 knot spans i.e., N i ,p is

non-zero overξi,ξi +p+1

5 Basis functions of order p have p - m icontinuous derivatives across knotξi where

m iis the multiplicity of knotξi

6 Scaling or translating the knot vector does not alter the basis functions

7 B-spline basis are generally only approximants and not interpolants That is, they

do not satisfy the Kronecker delta property N i ,p(ξj) 6=δi j Only in the case m i = p, then N i ,p(ξj) = 1

where Pi is the control points, n denotes the number of control points and N i ,p(ξ) is

the p th-degree B-spline basis function defined on the open knot vector

Fig 2.3gives an example of B-spline curves B-spline curves inherit all of the nuity properties of their underlying bases The use of open knots ensures that the firstand last points are interpolated

The B-spline basis can be enriched by three types of refinements which are termed h-,

p - and k- refinements In computer-aided geometric design notation, these are referred

0 1 2 3 4 5 0

0.5 1 1.5 2 2.5 3 3.5 4

Figure 2.3: A quadratic (p = 2) B-spline curve with a uniform open knot vector Ξ ={0,0,0,1,2,3,4,5,5,5}

to as knot insertion, degree elevation and degree and continuity elevation respectively

h - and p- refinements have a direct analogue of standard FEM while k- refinement has

no equivalence Details of three algorithms can be found in [45]

2.1.4.1 Knot insertion (h-refinement)

Knot insertion is found to preserve the geometry of B-spline curves and surfaces buthas a direct influence on the continuity of the approximation where knots are re-peated Let us consider a knot vector defined by Ξ =ξ1,ξ2, ,ξn +p+1 with the

corresponding set of control points denoted by B i A new extended knot vector given

by ¯Ξ = ¯ξ

1=ξ1, ¯ξ2, ¯ξn +m+p+1=ξn +p+1 is formed where m knots are added The

n +m new control points ¯Piare formed from the original control points by

¯

Pi=αiPi+ (1 −αi) Pi−1 (2.6)where

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

Figure 2.4: Knot insertion (h-refinement) on a quadratic B-spline curve.

2.1.4.2 p-refinement

p-refinement (order elevation) raises the polynomial order of the basis functions ing this process, the multiplicity of each knot is increased by one but no new knots are

Dur-added As with h-refinement, the geometry and the parametrization are not changed.

We refer to [149] for the mathematical details but as an example of p-refinement, let

us consider the quadratic B-spline curve shown in Fig.2.5(left) We raise the order ofthe basis from quadratic to cubic with the new basis functions shown in Fig.2.5(right)

0 0.2 0.4 0.6 0.8 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(a) Original curve

0 0.2 0.4 0.6 0.8 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(d) New basis functions

Figure 2.5: Order elevation of a quadratic B-spline curve to cubic : B-spline curvesand associated basis functions

where N i ,p(ξ) and M j ,q(η) are the B-spline basis functions defined on the knot vectors

NURBS basis functions are defined as

case in which w i = c ∀i, the NURBS basis is reduced to the B-spline basis Note that

for simple geometries, the weights are defined by the analytic method (see e.g., [149]).For complex geometries, the weights are obtained from CAD packages such as Rhino[Rhi] and can be defined by the user

Derivatives of NURBS basis functions: the first derivative of a NURBS basis function

R i ,p(ξ) is computed using the quotient rule That is,

d

dξR i ,p(ξ) = w i

N′i ,p(ξ)W (ξ) − N i ,p(ξ)W′(ξ)

W2(ξ) (2.11)

where Pidenotes the set of control points.

As a first example to illustrate NURBS curves, let us construct a quarter of a circlehaving a unit radius, as shown in Fig 2.6, using only one NURBS curve For thisproblem, a NURBS quadratic basis function is enough to model a circle exactly KnotvectorsΞ of one element are defined as Ξ = {0,0,0,1,1,2,2,3,3,4,4,4} The controlpoints for a complete circle are given in Fig.2.7and Tab.2.1

The effect of a decreasing weight on a NURBS curve is demonstrated in Fig.2.8 As

00.10.20.30.40.50.60.70.80.91

Figure 2.6: The use of a NURBS curve to construct a quarter circle (with w1= w3= 1

and w2= 1/√

2)

can be seen, as the weight w idecreases, the curve is pulled towards point Pi Likewise,

as a weight is increased, the curved is pulled towards the associated control point

Figure 2.7: Example of a NURBS curve for a circle of unit radius.

0 0.2 0.4 0.6 0.8 1

Figure 2.8: Effect of decreasing weights on a NURBS curve: the weights at control

point P4are 1, 0.6 and 0.3 for the red, blue and cyan curves, respectively

Table 2.1: Control points and weights for a circular plate with radius R = 0.5

An example of a quadratic NURBS surface with 4× 4 elements is demonstrated inFig.2.9.

Figure 2.9: NURBS surface and control mesh

The isoparametric concept refers to the use of the same basis functions for both the ometry and unknown field discretisation The isoparametric concept is utilized for bothFEM and IGA However, the difference between FEM and IGA lies in the purposes ofusing basis functions That is,

ge-• Finite elements: the basis which is chosen to approximate the unknown field isalso used to approximate the known geometry This most commonly takes theform of polynomial functions and the geometry is in most cases only approxi-mated