MINISTRY OF EDUCATION AND TRAINING HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY AND EDUCATION DO VAN HIEN ISOGEOMETRIC FINITE ELEMENT METHOD FOR LIMIT AND SHAKEDOWN ANALYSIS OF STRUCTURES PhD DISSERTATION MAJOR: ENGINEERING MECHANICS Ho Chi Minh City, October 13, 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 the 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. 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. 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 designing engineering structures but also assessing their safety. The limit load multipliers can be determited by using the 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 necessity for current research in the field of limit and shakedown analysis. The numerical tools involve two steps: a 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 the optimization problem.
Mathematically, the shakedown problem is considered as a nonlinear programming problem. Starting from the 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 the 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 C 0 Bernstein polynomials. Lagrange extraction is similar to Bézier extraction but it sets up a direct connection between NURBS and Lagrange polynomial basis functions instead of using C 0 Bernstein polynomials as a new shape function in the Bézier extraction.
iii Abstract iv Numerical results of structure problems are compared with analytical or other available solutions to prove the reliability and efficiency of these approaches. A 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 1.2 Motivation of the thesis .3 Objectives and Scope of study .4 Outline of the thesis .5 Original contributions of the thesis .6 List of Publications .1 Elastic perfectly plastic and rigid perfectly plastic material models 9 2.2 Drucker’s stability postulate .1 Plastic dissipation function .2 Fundamental of shakedown analysis. 27 v Table of Contents vi 2.5 Primal-dual interior point methods. 28 3 ISOGEOMETRIC FINITE ELEMENT METHOD 30 3.1 B-Splines basis functions .3 NURBS-based isogeometric analysis .4 A brief of NURBS based on Bézier extraction .2 Bézier extraction of NURBS .5 A brief review on Lagrange extraction of smooth splines .2 The Lagrange extraction operator .3 Rational Lagrange basis functions and control points .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 4.2 Isogeometric FEM discretizations .1 Discretization formulation of lower bound .2 Discretization formulation of upper bound and upper bound algorithm .3 Dual relationship between lower bound and upper bound and dual algorithm 76 5 NUMERICAL APPLICATIONS 85 5.2 Limit and shakedown analysis of two dimensional structures .1 Square plate with a central circular hole .2 Grooved rectangular plate subjected to varying tension.
95 Table of Contents vii 5.3 Limit and shakedown analysis of 3D structures .1 Thin square slabs with two different cutout subjected to tension 100 5.2 2D and 3D symmetric continuous beam .3 Thin-walled pipe subjected to internal pressure and axial force .4 Limit and shakedown analysis of pressure vessel components .1 Pressure vessel support skirt .2 Reinforced Axisymmetric Nozzle .5 Limit analysis of crack structures. 124 6 CONCLUSIONS AND FURTHER STUDIES 129 6.2 Limitations and Further studies. 130 References 132 List of Figures 2.2 Material models: (a) Elastic perfectly plastic; (b) Rigid perfectly plastic 10 2.3 Elastic perfectly plastic material model .6 von Mises and Tresca yield conditions in biaxial stress states .8 Load domain with two variable loads .9 Critical cycles of load for shakedown analysis [72; 84; 89] .1 Estimation of the relative time costs .2 The workchart of a design-through-analysis process .3 The concept of mesh in IGA .4 The concept of IGA: .5 Different types of B-Spline basis functions on the same distinct knot vector 35 3.6 The cubic B-Spline functions Ni3 (ξ) and its first and second derivatives 36 3. Control points are denoted by red circular •.
Control points are denoted by red circular •. The knots, which define a mesh by partitioning the curve into elements, are denoted by green square .9 Comparison of refinement strategies: p-refinement and k-refinement .10 A circle as a NURBS curve .11 Bent pipe modeled with a single NURBS patch. (b) NURBS mesh with control points. (c) Geometry with 32 NURBS elements 44 3.12 Flowchart of a classical finite element code .13 Flowchart of a multi-patch isogeometric analysis code.
The basis functions extend over a series of elements 48 h i 3.15 Bézier decomposition of Ξ = 0, 0, 0, 0.16 The Bernstein polynomials for polynomial degree p = 1, 2, 3 and 4. 52 viii List of Figures ix 3.17 Smooth C 2 -continuous curve represented by a B-spline basis .18 Smooth C 2 -continuous curve represented by a nodal Lagrange basis .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 3.20 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.1 Flow chart for the upper bound algorithm for shakedown analysis .2 Flow chart for the primal-dual algorithm for shakedown analysis .1 Square plate with a central hole: Full (a) and symmetric geometry (b).2 Square plate with central circular hole: Quadratic NURBS mesh with 32 elements and control net.3 The load factors of the IGA compared with those of different methods for limit analysis (with P2 = 0) of the square plate with a central circular hole.4 The convergence rate of the IGA with different orders for limit analysis (with P2 = 0) of the square plate with a central circular hole.5 The relative errors of the IGA with the exact solution for limit analysis (with P2 = 0) of the square plate with a central circular hole.6 The limit load domain of the square plate with a central circular hole using the IGA compared with those of other numerical methods.7 Limit and shakedown load factors for square plate with a central hole .8 The influential parameter of ε, c and τ .9 Full geometry and applied load of grooved rectangular plate.10 A symmetry of the grooved rectangular plate: a) A symmetric todel including applied loads and boundary conditions; b) 2D control point net and 40 NURBS quadratic elements.11 Limit load factors of the plate with tension of a strip with semi-circular notches.12 Limit and shakedown load factors for the grooved rectangular plate subjected to both tension and bending loads.13 The influential parameter of ε, c and τ.
99 List of Figures x 5.14 The 2D view geometry of thin square slabs with two different cutouts subjected to biaxial loading.15 The 3D geometry of thin square slabs with two different cutouts subjected to biaxial loading.16 The 3D quadrant NURBS meshes of thin square slabs with two different cutouts: (a)-Circular cutout and (b)-Square cutout .17 Finite element discretization using quartic NURBS elements for thin square slabs with two different cutouts.18 Convergence of limit load factors using the IGA solution in comparison with those of other methods for thin square slabs with two different cutouts: a) circular; b) square.19 The influential parameter of ε, c and τ for 3D circular cutout.20 Geometry and loading of the continuous beam .21 Continuous beam: (a) 2D NURBS mesh and (b) 3D NURBS mesh.22 2D Continuous beam: Convergence of limit and shakedown load factors in comparison with those of two other methods.23 The influency parameter of ε, c and τ .24 A thin-walled pipe subjected to internal pressure and axial force: a) Full model subjected to internal pressure and axial uniform loads; b) Cubic mesh and control net; c) a quarter of the model with symmetric conditions imposed on the oxz, oyz and oxy surface.