Development of isogeometric finite element methods

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
1. Assoc. Prof. Dr. Nguyen Xuan Hung

2. 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
1

2

3

Introduction
1.1 Review of Isogeometric Analysis
1.2 Review of plate theories . . . . .
1.3 Goal of the thesis . . . . . . . .
1.4 Outline . . . . . . . . . . . . .

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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 . . . . . . . . .

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3
5

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14
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21

Isogeometric analysis of laminated composite and sandwich Mindlin plates1 22
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1 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
4

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 . . . . . . . . . . .
1

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63
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67
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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 . . . . . . . . . . . . . . . . . . . . . . . . .

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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 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

6

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
1

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.
2 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
7

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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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
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 loading . . . . . . . . . . . . . . . . . . . . . . . . . . 172
1

based on H. Nguyen-Xuan, Chien H. Thai, J. Bleyer, Vinh Phu Nguyen. Upper bound limit analysis
of plates using a rotation-free isogeometric approach. Asia Pacific Journal on Computational Engineering, 1:12, 2014.

vii


CONTENTS

7.4.4.2

7.5
8

Circular plate subjected to non-uniform transverse
loading . . . . . . . . . . . . . . . . . . . . . . . . 172
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

Conclusions, discussions and future works
176
8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.2 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
8.3 Future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

List of publications

181

References

184

viii


List of Figures
2.1
2.2
2.3

1D and 2D quadratic B-spline basis functions. . . . . . . . . . . . . .
1D and 2D cubic B-spline basis functions. . . . . . . . . . . . . . . .
A quadratic (p = 2) B-spline curve with a uniform open knot vector
Ξ = {0, 0, 0, 1, 2, 3, 4, 5, 5, 5}. . . . . . . . . . . . . . . . . . . . . . .
2.4 Knot insertion (h-refinement) on a quadratic B-spline curve. . . . . .
2.5 Order elevation of a quadratic B-spline curve to cubic : B-spline curves
and associated basis functions. . . . . . . . . . . . . . . . . . . . . .
2.6 The use of a NURBS
√ curve to construct a quarter circle (with w1 = w3
= 1 and w2 = 1/ 2). . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Example of a NURBS curve for a circle of unit radius. . . . . . . . .
2.8 Effect of decreasing weights on a NURBS curve: the weights at control
point P4 are 1, 0.6 and 0.3 for the red, blue and cyan curves, respectively.
2.9 NURBS surface and control mesh. . . . . . . . . . . . . . . . . . . .
2.10 Parametric and physical space with quadratic B-splines. . . . . . . . .
2.11 Illustration of imposing Dirichlet BCs. Black points denote corner
control points where the NURBS basis satisfy the Kronecker delta
property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8

Geometry of a typical Mindlin-Reissner plate . . . . . . . . . . . . .
Fully clamped, simply supported square plate models and control points.

Normalized deflection of simply supported and clamped square isotropic
plates subjected to uniformly distributed load . . . . . . . . . . . . .
Normalized strain energy of simply supported and clamped square
isotropic plates subjected to uniformly distributed load . . . . . . . .
Performance of present element with various ratios L/h of clamped
and simple supported isotropic plates . . . . . . . . . . . . . . . . . .
Geometry and control points of a L-shape isotropic plate . . . . . . .
A square plate is subjected to axial in-plane edge loadings . . . . . .
Normalized deflection wc : (a) R=5; (b) R=10; (c) R=15 and (d) Relative errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

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34
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LIST OF FIGURES

3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14

5.1
5.2
5.3

Geometry and control points of a four-layer square laminated plate
under sinusoidal load . . . . . . . . . . . . . . . . . . . . . . . . . .
The distribution of stresses through thickness of the plate with a/h =
4, 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Geometry and control points of a three-layer square laminated plate .
Mode shapes 1-6 of the three-layer clamped laminated square plate . .
Geometry of a circular plate . . . . . . . . . . . . . . . . . . . . . .
Mesh and control net for a disk of radius 0.5 . . . . . . . . . . . . . .
Meshes produced by h-refinement (knot insertion) . . . . . . . . . . .
The first six mode shapes of a 4-layer clamped laminated circular plate
Geometry of laminated composite plates under axial and biaxial compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fundamental buckling modes of the 10-layer square plate with various
mixed boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1D representation of the layerwise kinematics . . . . . . . . . . . .
Geometry of a sandwich plate . . . . . . . . . . . . . . . . . . . . .
Meshes and control net of the square plate . . . . . . . . . . . . . . .
Geometry of a four-layer square laminated plate under sinusoidal load
The distribution of stresses through the thickness of the four-layer laminated composite square plate under a sinusoidally distributed load . .
The distribution of stresses through the thickness of the sandwich square
plate under a sinusoidally distributed load . . . . . . . . . . . . . . .
Geometry and control points of a three-layer laminated square plate .
Modes shape 1-6 of a three-layer clamped laminated square plate with
a/b= 1 and b/h= 10. . . . . . . . . . . . . . . . . . . . . . . . . . .
Geometry of a circular plate . . . . . . . . . . . . . . . . . . . . . .
a) Coarse mesh and control points of a circular plate; b) mesh element
13 × 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Modes shape 1-6 of a four-layer clamped laminated circular plate with
R/h= 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Geometry and element mesh of a clamped ellipse plate . . . . . . . .
Modes shape 1-8 of a three-layer clamped laminated ellipse plate with
a/h= 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Geometry of laminated composite plates under axial and biaxial compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37
40
41
44
45
46
46
48
49
51
57
65
66
69
69
73
74
75
78
79
79
81
83

84

Geometry of a plate . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Shape functions f (z) and their derivatives across the thickness of the
plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Geometry of a laminated square plate under sinusoidally distributed load 97

x


LIST OF FIGURES

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 square
plate 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 compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.1

6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13

6.14

A typical configuration of FGM plate . . . . . . . . . . . . . . . . .
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). . . .
The sandwich plate with homogeneous skins and FGM core. . . . . .
The sandwich plate with FGM skins and homogeneous core . . . . .
Shape functions f (z) and their derivatives across the thickness of the
plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The square plate geometry. . . . . . . . . . . . . . . . . . . . . . . .
Comparison of the present result with the analytical solution of Vel and
Batra for power index n = 1 and 6. . . . . . . . . . . . . . . . . . . .
Cubic element mesh and control net of the circular plate. . . . . . . .
The normalized center displacement for various ratios of radius-tothickness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The stresses through thickness of a Al/Al2O3 FG plate under sinusoidal load with a/h=4, n=1, for other HSDT models. . . . . . . . . .
The stresses through thickness of a Al/Al2O3 FG plate under sinusoidal load with a/h=4, for various power indices n. . . . . . . . . . .
The normalized deflection of a Al/ZrO2 -1 FGM plate for various power

indexes and boundary conditions. . . . . . . . . . . . . . . . . . . . .
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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. .

xi

121
122
123
124
124
129
130
131
132
132
134
136

138
141


LIST OF FIGURES

6.15 Geometry of the circular plate under a uniform radial pressure. . . . . 145
7.1

7.2

7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
7.11
7.12
7.13
7.14
7.15
7.16
7.17
7.18
7.19

Clamped boundary conditions in a rotation-free IGA formulation: simply fixing the deflections of two rows of control points along the clamped
boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A fully clamped square plate: a quarter of plate is modeled. Along the
symmetry lines, the normal rotation is fixed which can be achieved by
enforcing the deflection of two rows of control points that define the
tangent of the plate to have the same value. . . . . . . . . . . . . . .
Full model of clamped and simply supported square plates. . . . . . .
A quarter of plate is modeled: meshes of cubic elements. . . . . . . .
Comparison of numerical results of the clamped square plate using two

Gaussian rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of computational times of the clamped square plate using
quadratic and cubic elements. . . . . . . . . . . . . . . . . . . . . . .
Relative error to the reference upper bound of the clamped square plate.
The plastic dissipation of a square plate: a) SSSS; b) CCCC. . . . . .
Convergence of the limit load factor (qab
¯ 2 /m p ) of a clamped square
plate using k-refinement. . . . . . . . . . . . . . . . . . . . . . . . .
The plastic dissipation of a rectangular plate: a) SSSS; b) CCCC; c)
CCCsF; d) CCClF; e) CCsFF. . . . . . . . . . . . . . . . . . . . . .
Geometry of a rhombic plate. . . . . . . . . . . . . . . . . . . . . . .
Rhombic plates: meshes of B-spline elements. . . . . . . . . . . . . .
Limit load factor of a skew plate with simply supported and clamped
boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . .
The plastic dissipation of a rhombic plate with α = 300 : a) simply
supported plate; b) clamped plate. . . . . . . . . . . . . . . . . . . .
L-shaped plate models. . . . . . . . . . . . . . . . . . . . . . . . . .
Relative error to the reference upper bound of a simply supported Lshape plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Geometry of a clamped circular plate. . . . . . . . . . . . . . . . . .
Relative error of the limit load factor (qR
¯ 2 /m p ) of a circular plate. . .
An illustration of a circular plate subjected to a non-uniform load. . .

xii

156

156
160
162

163
163
164
164
165
166
167
168
169
170
171
171
172
173
174


List of Tables
2.1

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

16

A non-dimensional frequency parameter ϖ = (ω a) (ρ /G)1/2 of a clamped
isotropic L-shaped plate . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Buckling load parameters λ¯ = λcr b/D0 of a supported isotropic square
plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 The normalized displacement w¯ and stresses of the square sandwich plate 36
3.4 The normalized displacement w¯ of a four-layer simply supported laminated square plate under sinusoidal load . . . . . . . . . . . . . . . . 39

3.5 A non-dimensional frequency parameter ϖ = ω a2 /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 = ω b2 /π 2 (ρ h/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 parameter ϖ = ω a2 /h (ρ /E2 )1/2 of a
circular 4-layer [θ / − 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 /E2 ratios . . . . . . . . 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
3.1

4.1
4.2
4.3

The convergence of the normalized displacement and stresses of the
three-layer sandwich square plate laminated plate (a/h = 10) . . . . . 65
The normalized displacement and stresses of the square sandwich plate
under uniform load . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
The normalized displacement and stresses of the four-layer [00 /900 /900 /00 ]
laminated square plate under a sinusoidally distributed load . . . . . . 70

xiii

LIST OF TABLES

4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
5.1
5.2
5.3

5.4
5.5

5.6
5.7
5.8

The normalized displacement and stresses of the sandwich (00 /core/00 )
simply supported square plate under sinusoidally distributed load . . . 72
The convergence of non-dimensional frequencies parameter ϖ of the
three-layer [00 /900 /00 ] clamped laminated plate (b/h = 5 and a/b=1)
76
0

0
0
Normalized frequencies ϖ of a [0 /90 /0 ] clamped laminated square
plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
A non-dimensional frequency parameter ϖ = ω a2 /h (ρ /E2 )1/2 of a
[00 /900 /900 /00 ] SSSS laminated square plate (a/h=5) . . . . . . . . 78
Normalized frequencies ϖ = ω a2 /h (ρ /E2 )1/2 of a circular 4-layer
[θ 0 / − θ 0 / − θ 0 /θ 0 ] clamped laminated plate . . . . . . . . . . . . . 80
Normalized frequencies ϖ of a [00 /900 /00 ] the fully clamped laminated ellipse plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
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
Normalized critical buckling load of a simply supported [00 /900 /900 /00 ]
square plate with various E1 /E2 ratios . . . . . . . . . . . . . . . . . 84
Normalized critical buckling load of a simply supported [00 /900 /900 /00 ]
square plate with various ratios a/h . . . . . . . . . . . . . . . . . . 85
Biaxial critical buckling load of a simply supported cross-ply [00 /900 /00 ]
square plate with various modulus ratios . . . . . . . . . . . . . . . . 85
Several trigonometric shear deformation theories . . . . . . . . . . . 90
The convergence of the normalized displacement and stresses of a fourlayer [00 /900 /900 /00 ] laminated composite square plate (a/h = 4) . . 98
The normalized displacement and stresses of a four-layer [00 /900 /900 /00 ]
simply supported laminated square plate under a sinusoidally distributed
load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
The normalized displacement and stresses of a sandwich (00 /core/00 )
simply supported square plate under a sinusoidally distributed load . . 102
The convergence of non-dimensional frequency parameter ϖ of a fourlayer [00/900 /900 /00 ] simply supported laminated square plate (a/h =
5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
A non-dimensional frequency parameter ϖ = ω a2 /h (ρ /E2 )1/2 of a
[00 /900 /900 /00 ] simply supported laminated square plate (a/h=5) . . 105
A non-dimensional frequency parameter ϖ = ω a2 /h (ρ /E2 )1/2 of a
[00 /900 /900 /00 ] simply supported laminated square plate (E1 /E2 = 40) 105

The first normalized frequency ϖ = ω b2 /h
(ρ /E2 ) f of an antisymmetry (00 /900 /core/00/900 ) sandwich square plate with hc /h f = 10106

xiv


LIST OF TABLES

5.9

Normalized frequencies ϖ = ω b2 /h

(ρ /E2 ) f of an antisymmetry

(00 /900 /core/00/900 ) sandwich simply supported square plate with
hc /h f = 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.10 The first normalized frequency ϖ = ω b2 /h
5.11
5.12
5.13
5.14
5.15
5.16
5.17

(ρ /E2 ) f of an anti-

symmetry (00 /900 /core/00 /900 ) sandwich square plate with a/h = 10 107
Normalized frequencies ϖ = ω a2 /h (ρ /E2 )1/2 of a circular 4-layer
[θ 0 / − θ 0 / − θ 0 /θ 0 ] clamped laminated plate . . . . . . . . . . . . . 110

Normalized frequencies ϖ = ω a2 (ρ h/D0 )1/2 of a [00 /900 /00 ] clamped
laminated elliptical plate . . . . . . . . . . . . . . . . . . . . . . . . 112
Normalized critical buckling load of a simply supported [00 /900 /900 /00 ]
square plate with various E1 /E2 ratios and a/h=10 . . . . . . . . . . 113
Normalized critical buckling load of a simply supported [00 /900 /900 /00 ]
square plate with various ratios a/h and E1 /E2 =40 . . . . . . . . . . 114
Normalized critical buckling load of eleven and twenty-one layer sandwich simply supported square plates . . . . . . . . . . . . . . . . . . 115
Biaxial critical buckling load of a simply supported cross-ply [00 /900 /00 ]
square plate with various modulus ratios . . . . . . . . . . . . . . . . 116
Biaxial critical buckling load of a simply supported cross-ply [00 /900 /00 ]
square plate with various ratios a/h . . . . . . . . . . . . . . . . . . 117

6.1
6.2
6.3

Various forms of shape functions and their derivatives . . . . . . . . .
Material properties . . . . . . . . . . . . . . . . . . . . . . . . . . .
The non-dimensional deflection and axial stress of a simply supported
(SSSS) Al/Al2O3 square plate under sinusoidal load. . . . . . . . . .
6.4 The non-dimension deflection of a Al/ZrO2 − 1 plate under uniform
load with a/h=5 for different boundary conditions. . . . . . . . . . .
6.5 The non-dimensional deflection and transverse shear stress of a SSSS
square sandwich plate with core FGM type B under sinusoidal load . .
6.6 The natural frequency ω¯ =ω h ρm /Em of a SSSS Al/ZrO2 − 1 plate
with a/h=5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7 The natural frequency ω¯ of a SSSS Al/ZrO2 -1 plate with various ratios
a/h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.8 The first five non-dimensionalized frequencies ω¯ of a SSSS sandwich
plate 2-1-2 with a/h=10. . . . . . . . . . . . . . . . . . . . . . . . .

6.9 Comparisons of the natural frequency ω¯ of a SSSS sandwich plate with
other theories (a/h=10). . . . . . . . . . . . . . . . . . . . . . . . .
6.10 Comparison of the buckling load parameter of a clamped thick circular
Al/ZrO2-2 plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

125
132
133
135
137
139
140
141
143
144


LIST OF TABLES

6.11 Uni-axial critical buckling load a SSSS sandwich plate with FGM
skins and isotropic core. . . . . . . . . . . . . . . . . . . . . . . . . 146
6.12 Bi-axial critical buckling load of a SSSS sandwich plate with FGM
skins and isotropic core. . . . . . . . . . . . . . . . . . . . . . . . . 147
7.1
7.2
7.3
7.4
7.5

7.6
7.7
7.8

The convergence of the limit load factor (qa
¯ 2 /m p ) for a clamped square
plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The convergence of the limit load factor (qa
¯ 2 /m p ) for a simply supported square plate . . . . . . . . . . . . . . . . . . . . . . . . . . .
A comparison of the limit load factor (qa
¯ 2 /m p ) for a square plate . . .
The limit load factor (qab/m
¯
p ) for a rectangular plate with a/b = 2 and
various boundary conditions . . . . . . . . . . . . . . . . . . . . . .
Results of the limit load factor (qR
¯ 2 /m p ) for the rhombic plate . . . .
2
The limit load factor (qL
¯ /m p ) for a L-shaped plate . . . . . . . . . .
The limit load factor λcr /m p for a clamped circular plate subjected to
a linear load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The limit load factor λcr /m p for a clamped circular plate subjected to
a parabolic load . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xvi

161
162
165

167
168
172
173
174


Nomenclature
Latin Symbols
J

Jacobian matrix

B

Gradient matrix

D

Matrix of material

u¨

Acceleration

u˙

Velocity

f

Global force vector

K

Global stiffness matrix

M

Mass matrix

Ni,p

B-splines basis functions

P

Control points

R

Rational basis function

u

Displacement field

E

Young’s modulus

t

The thickness

w

Weights

Greek Symbols

λcr

Critical buckling load

ν

Poisson’s ratio

xvii


LIST OF TABLES

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 in ξ 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

xviii


LIST OF TABLES

DQM Differential quadrature method
dTrSDTs Different trigonometric shear deformation theories
EFG 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

xix


LIST OF TABLES

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

xx

Chapter 1
Introduction
The main objective of this thesis is to develop an isogeometric analysis for elasticity
and plasticity plate structures. This chapter gives a literature review on isogeometric
approach as well as plate theories for the analysis of structures.

1.1 Review of Isogeometric Analysis
Before the advent of computers, all engineering drawing was done manually by using
pencil and pen on paper or other substrate (e.g., vellum, mylar). Since the advent of
computer-aided design (CAD), engineering drawing has been done more and more in
the electronic medium. Today most engineering drawing in industries area such as
shipbuilding, automobile, aerospace, industrial and architectural design are done with
CAD [177]. The B-spline basis functions have been used to represent the curves by
the designers since 1972 [45]. NURBS (Non-uniform rational B-splines) are a generalization of Bezier splines (B-spline), and have been used in CAD programs since
1975 [45]. NURBS are a mathematical model commonly used in computer graphics
for generating and representing curves and surfaces. NURBS offer great flexibility
and precision for handling both analytic (surfaces defined by common mathematical
formulae) and modeled shapes [177]. By using NURBS, conic sections like circles,
cylinders and spheres can be represented exactly. The most used basis functions to
represent geometries are NURBS, which were in the beginning only used in proprietary CAD packages of car companies, but are today used in all standard CAD packages [177]. Today, there exist many efficient numerical stable algorithms to generate
NURBS objects [149].
Computer Aided Engineering (CAE) is the broad usage of computer software to aid in
engineering 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-

1

1.1 Review of Isogeometric Analysis

nomial. The Finite Element Method (FEM) started developing in the 1950s [45], with
all analyses made by hand. The method was therefore only applied on small and easy
systems. In the decades that followed, as analysts got experience with the method on
different problems, they started improving the algorithms and developing new basis
function elements(Hermite polynomials). To solve partial differential equations using
the FEM one usually uses variational or weak form formulations. When computers
were first used to perform the analysis, the computational efficiency was a very critical
issue. In practical applications, computational efficiency is still an issue. However, it
turns out that higher order element uses more work per degree of freedom but fewer
degree of freedom converge.
With the advancement in technology comes the desire to create more and more complex constructions, resulting in the need of more efficient and accurate methods for
design and analysis. It is a difficult task to improve efficiency, as there exists a gap between FEA and CAD. The design and the analysis communities evolved independently
of each other, as they had different goals and needs. In order to integrate CAD and
FEA 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 converting the data generated in CAD to a data set suitable for FEA, where B-spline or
NURBS are the most widely used computational geometry technology in engineering
design. Geometry domains having conic sections like circles, cylinders, spheres, ellipsoids, etc... can be represented exactly. Using NURBS lets us easily control continuity,
as C p−1 -continuity is obtained using p-th order NURBS. A monograph of the isogeometric analysis has been published entirely on the subject [45] and applications have
been found in several fields including structural mechanics, solid mechanics, fluid mechanics and contact mechanics.
Isogeometric analysis has been applied to a wide range of mechanics problems. The
smoothness of NURBS basis functions is attractive for analysis of fluids [71, 137] and
for fluid-structure interaction problems [17, 18]. In contact formulations using conventional geometry discretisations, faceted surfaces are often found on contact surfaces
that 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 the
contact surface is now smooth, leading to more physically accurate contact stresses.
Recent work in this area includes [105, 193]. Another area where IGA has shown
advantages over traditional approaches is optimization problems [111, 216]. IGA is
particularly suited to such problems due to the tight coupling with CAD models and

offers an extremely attractive approach for industrial applications. In addition, due
to the ease of constructing high order continuous basis functions, IGA has been used
with 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 damage models [215] and gradient elasticity [66]. NURBS have also shown advantageous
properties for structural vibration problems [47] with the mathematical properties of

2