Phương pháp phần tử hữu hạn đẳng hình học cho phân tích giới hạn và thích nghi của kết cấu

MINISTRY OF EDUCATION AND TRAINING
UNIVERSITY OF TECHNOLOGY AND EDUCATION
HO CHI MINH CITY
DO VAN HIEN
ISOGEOMETRIC FINITE ELEMENT METHOD
FOR LIMIT AND SHAKEDOWN ANALYSIS OF
STRUCTURES
DOCTORAL THESIS
MAJOR: ENGINEERING MECHANICS
Ho Chi Minh City, June 16, 2020
Declaration
I, Do Van Hien, declare that this thesis entitled, "Isogeometric finite element method
for limit and shakedown analysis of structures" is a presentation of my original research
work. I confirm that:
• Wherever contributions of others are involved, every effort is made to indicate this
clearly, with due reference to the literature,and acknowledgement of collaborative
research and discussions.
• The work was done under the guidance of Prof. Nguyen Xuan Hung at the Ho
Chi Minh City University of Technology and Education.
i
Acknowledgements
This thesis summarizes my research carried out during the past five years at the Doctoral
Program "Engineering Mechanics" at Ho Chi Minh City University of Technology and
Education in Ho Chi Minh City. This thesis would not have been possible without help
of many, and I would like to acknowledge their kind efforts and assistance.
First of all I would like to express my deep gratitude to my supervisor Prof. Nguyen
Xuan Hung, for his guidance, support and encouragement during the past five years. I
appreciate that he left a lot of freedom for me to pursue my own ideas, set the right
direction when it was necessary and contributed valuable advice.
I am also very grateful to Assoc.Prof. Van Huu Thinh, who has been my second
advisor at HCMUTE for many years.
I am indebted to Prof. Timon Rabczuk for giving me the chance to spend a
one-year research visit at the Bauhaus-Universität Weimar, and I also want to thank
Prof. Tom Lahmer and Prof. Xiaoying Zhuang for the fruitful discussions and their
support.
I also would like to thank the research group members at GACES (at HCMUTE),
CIRTECH (at HUTECH) and ISM (at Bauhaus-Universität Weimar, Germany) for
their helpful supports.
I would like to thank from the bottom of my heart to Assoc.Prof. Nguyen Hoai
Son, Assoc.Prof Nguyen Trung Kien, Assoc.Prof Chau Dinh Thanh and other colleagues
at HCMUTE for their kind supports and advice.
I am immensely indebted to my father Do Tang, my mother Pham Thi Nghe and
my parents in-law who have been the source of love and discipline for their inspiration
and encouragement throughout the course of my education including this Doctoral
Program.
Last but not least, I am extremely grateful to my wife Mrs. Nguyen Thi Nhu Lan
who has been the source of love, companionship and encouragement, to my sons, Do
Quang Khai and Do Minh Nhat, who has been the source of joy and love.
ii
Abstract
The structural safety such as nuclear power plants, chemical industry, pressure vessel
industry and so on can commonly be evaluated with the help of limit and shakedown
analysis. Nowadays, the limit and shakedown analysis plays a well-known role in not
only assessing the safety of engineering structures but also designing of the engineering
structures. The limit load multipliers can be determinated by using lower or upper
bound method. In order to ultilize the limit and shakedown analysis in many practical
engineering areas, the development of numerical tools which are sufficiently efficient and
robust is a neccessary of current research in the field of limit and shakedown analysis.
The numerical tools involve the two steps: finite element discretisation strategy and
constrained optimization.
In this research, the isogeometric finite element method is used to discretise the
displacement domain of strutures in the first step. The primal-dual algorithm based
upon the von Mises yield criterion and a Newton-like iteration is used in the second step
to solve optimization problem. Mathematically, the shakedown problem is considered
as a nonlinear programming problem. Starting from upper bound theorem, shakedown
bound is the minimum of the plastic dissipation function, which is based on von Mises
yield criterion, subjected to compatibility, incompressibility and normalized constraints.
This constraint nonlinear optimization problem is solved by combined penalty function
and Lagrange multiplier methods.
The isogeometric analysis (IGA) uses NURBS basis functions for both the repre-
sentation of the geometry and the approximation of solutions. The main aim of the
IGA was to integrate Finite Element Analysis (FEA) into NURBS based Computer Aid
Design (CAD) design tools. The Bézier and Lagrange extraction of NURBS was used
in the analysis due to The computational aspects of the NURBS function increase the
question of how to implement efficiently the NURBS function in the existing FEM codes
due to a significant differences between the NURBS basis function and the Lagrange
function. The Bézier extraction is founded on the NURBS basis functions in terms of C0
Bernstein polynomials. Lagrange extraction is similar to Bézier extraction but it sets up
a direct connection between NURBS and Lagrange polynomial basis functions instead
iii
Abstract iv
of using C0 Bernstein polynomials as a new shape function in the Bézier extraction.
Numerical results of structure problems are compared with analytical or other available
solutions to prove the reliability and efficiency of these approaches.
Pressure vessel which is designed to hold liquids or gases contains various parts
such as thin walled vessels, thick walled cylinders, nozzle, head, nozzle head, skirt
support and so on. Two types of defects, axial and circumferential cracks, are commonly
found in pressure vessel and piping. The application of shakedown analysis in pressure
vessel engineering is illustrated in this study.
Table of Contents
Contents Page
Acknowledgments iii
Abstract v
List of Figures viii
List of Tables xii
Notations xii
1 INTRODUCTION 1
1.1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Objectives and Scope of study . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Original contributions of the thesis . . . . . . . . . . . . . . . . . . . . 6
1.6 List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 FUNDAMENTALS 9
2.1 Material model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Elastic perfectly plastic and rigid perfectly plastic material models 9
2.1.2 Drucker’s stability postulate . . . . . . . . . . . . . . . . . . . . 12
2.1.3 Normal rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Yield condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Plastic dissipation function . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Variational principles . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Shakedown analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.2 Fundamental of shakedown analysis . . . . . . . . . . . . . . . . 19
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
v
Table of Contents vi
2.5 Primal-dual interior point methods . . . . . . . . . . . . . . . . . . . . 28
3 ISOGEOMETRIC FINITE ELEMENT METHOD 30
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 NURBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.1 B-Splines basis functions . . . . . . . . . . . . . . . . . . . . . . 34
3.2.2 B-Spline Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.3 B-Spline Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.4 B-Spline Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.5 Refinement techniques . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.6 NURBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 NURBS-based isogeometric analysis . . . . . . . . . . . . . . . . . . . . 44
3.3.1 Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3.2 Mesh refinement . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.3 Stiffness matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 A brief of NURBS based on Bézier extraction . . . . . . . . . . . . . . 49
3.4.1 Bézier decomposition . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.2 Bézier extraction of NURBS . . . . . . . . . . . . . . . . . . . . 50
3.5 A brief review on Lagrange extraction of smooth splines . . . . . . . . 54
3.5.1 Lagrange decomposition . . . . . . . . . . . . . . . . . . . . . . 54
3.5.2 The Lagrange extraction operator . . . . . . . . . . . . . . . . . 56
3.5.3 Rational Lagrange basis functions and control points . . . . . . 57
3.5.4 Using Lagrange extraction operators in a finite element code . . 60
4 THE ISOGEOMETRIC FINITE ELEMENT METHOD AP-
PROACH TO LIMIT AND SHAKEDOWN ANALYSIS 61
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Isogeometric FEM discretizations . . . . . . . . . . . . . . . . . . . . . 62
4.2.1 Discretization formulation of lower bound . . . . . . . . . . . . 62
4.2.2 Discretization formulation of upper bound and upper bound
algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3 Dual relationship between lower bound and upper bound and dual algorithm 76
5 NUMERICAL APPLICATIONS 85
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 Limit and shakedown analysis of two dimensional structures . . . . . . 85
5.2.1 Square plate with a central circular hole . . . . . . . . . . . . . 85
5.2.2 Grooved rectangular plate subjected to varying tension . . . . . 94
Table of Contents vii
5.3 Limit and shakedown analysis of 3D structures . . . . . . . . . . . . . . 99
5.3.1 Thin square slabs with two different cutout subjected to tension 99
5.3.2 2D and 3D symmetric continuous beam . . . . . . . . . . . . . . 104
5.3.3 Thin-walled pipe subjected to internal pressure and axial force . 109
5.4 Limit and shakedown analysis of pressure vessel components . . . . . . 113
5.4.1 Pressure vessel support skirt . . . . . . . . . . . . . . . . . . . . 113
5.4.2 Reinforced Axisymmetric Nozzle . . . . . . . . . . . . . . . . . . 119
5.5 Limit analysis of crack structures . . . . . . . . . . . . . . . . . . . . . 123
6 CONCLUSIONS AND FURTHER STUDIES 128
6.1 Consclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.2 Limitations and Further studies . . . . . . . . . . . . . . . . . . . . . . 129
References 131
List of Figures
2.1 Structure model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Material models: (a) Elastic perfectly plastic; (b) Rigid perfectly plastic 10
2.3 Elastic perfectly plastic material model . . . . . . . . . . . . . . . . . . 11
2.4 Stable (a) and unstable (b, c) materials . . . . . . . . . . . . . . . . . . 12
2.5 Normality rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.6 von Mises and Tresca yield conditions in biaxial stress states . . . . . . 15
2.7 Interaction diagram (Bree diagram) . . . . . . . . . . . . . . . . . . . . 18
2.8 Load domain with two variable loads . . . . . . . . . . . . . . . . . . . 20
2.9 Critical cycles of load for shakedown analysis [72; 84; 89] . . . . . . . . 24
3.1 Estimation of the relative time costs . . . . . . . . . . . . . . . . . . . 31
3.2 The workchart of a design-through-analysis process . . . . . . . . . . . 32
3.3 The concept of mesh in IGA . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 The concept of IGA: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5 Different types of B-Spline basis functions on the same distinct knot vector 35
3.6 The cubic B-Spline functions N3i (ξ) and its first and second derivatives 36
3.7 Knot insertion. Control points are denoted by red circular • . . . . . . 39
3.8 Knot insertion. Control points are denoted by red circular •. The knots,
which define a mesh by partitioning the curve into elements, are denoted
by green square  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.9 Comparison of refinement strategies: p-refinement and k-refinement . . 41
3.10 A circle as a NURBS curve . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.11 Bent pipe modeled with a single NURBS patch. (a) Geometry. (b)
NURBS mesh with control points. (c) Geometry with 32 NURBS elements 44
3.12 Flowchart of a classical finite element code . . . . . . . . . . . . . . . . 45
3.13 Flowchart of a multi-patch isogeometric analysis code . . . . . . . . . . 46
3.14 Isogeometric elements. The basis functions extend over a series of elements 48
3.15 Bézier decomposition of Ξ =
[
0, 0, 0, 0.25, 0.5, 0.75, 1, 1, 1
]
. . . . 50
3.16 The Bernstein polynomials for polynomial degree p = 1, 2, 3 and 4. . . 52
viii
List of Figures ix
3.17 Smooth C2-continuous curve represented by a B-spline basis . . . . . . 54
3.18 Smooth C2-continuous curve represented by a nodal Lagrange basis . . 55
3.19 Demonstration of the Lagrange extraction operators in 1D case and their
inverse for the transformation of B-spline, Lagrange on an element level.
The second B-Splines element of the example curve is shown in Fig 3.17 57
3.20 Demonstration of the Lagrange extraction operators in 2D case and their
inverse for the transformation of NURBS and Lagrange on an element
level. The first NURBS element of 2D case example is shown in Fig.
3.20(a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1 Flow chart for the upper bound algorithm for shakedown analysis . . . 75
4.2 Flow chart for the primal-dual algorithm for shakedown analysis . . . . 84
5.1 Square plate with a central hole: Full (a) and symmetric geometry (b). 86
5.2 Square plate with central circular hole: Quadratic NURBS mesh with 32
elements and control net. . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3 The convergence of the IGA compared with those of different methods
for limit analysis (with P2 = 0) of the square plate with a central circular
hole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.4 The limit load domain of the square plate with a central circular hole
using the IGA compared with those of other numerical methods. . . . . 88
5.5 Limit and shakedown load factors for square plate with a central hole . 89
5.6 Influency parameter of ε, c and τ . . . . . . . . . . . . . . . . . . . . . 92
5.7 Full geometry and applied load of grooved rectangular plate. . . . . . . 93
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