Luận án tiến sĩ: Mô hình hóa vết nứt lan truyền bằng NURBS-XFEM

ABSTRACT NUMERICAL MODELING OF PROPAGATING CRACKS USING NURBS-BASED EXTENDED FINITE ELEMENT METHOD NGUYEN NGOC MINH The recently proposed concept of Isogeometric Analysis [Cottrell et al. (2009)] and [Hughes et al. (2005)]), which employs the Non-Uniform Rational B-Spline (NURBS) [Piegl and Tiller (1997] as shape functions, is promising in crack modeling because of the exactness of NURBS in approximating the field variables, especially when discontinuities (e. Meanwhile, for solving problems in fracture mechanics, the Extended Finite Element Method (XFEM) is well-known as an accurate, robust and non-remeshing method.

The main idea of XFEM is to embed discontinuities into the solution space by suitable enrichment functions. Cracks are presented as surfaces of discontinuous displacements continuously propagating through the problem domain. In this thesis, the Isogeometric Analysis concept is incorporated into XFEM formulation in an attempt combine their advantages for numerical modeling of crack growth. The maximum circumferential stress criterion is used for predicting the direction of crack propagation.

Performance of the model is shown by selected benchmark examples. i ACKNOWLEDGEMENT This master thesis was written in the year 2011, in the Institute for Structural Mechanics at Ruhr University Bochum. I would like to express my sincere gratitude to all those who offered me various helps during my work on the thesis, especially my supervisors, Professor Dr. Günther Meschke and Dipl.

Dirk Leonhart, who are always very kind and friendly. They are open for questions, discussions and give valuable suggestions which help me to step by step improve this thesis. I also would like to say thanks to Dr. Bui Quoc Tinh, who is a friend, a colleague and a senior, for all the discussions in many fields related and non-related to my study in Germany and my thesis.

Finally, I am really grateful to my parents. Without their unlimited love, support and encouragement, I would have not enough strength to overcome all the difficulties to complete my master thesis. ii Contents Abstract. iii List of Figures.

vi List of Tables. The Extended Finite Element Method (XFEM). The scope of the thesis. Fundamentals of B-Splines and NURBS 2.

Univariate B-Spline basis functions. Multivariate B-Spline functions. Non Uniform Rational B-splines (NURBS). Derivatives of NURBS basis functions.

Incoporating NURBS into Finite Element Analysis. Imposing essential boundary conditions. Integrated NURBS-based XFEM for numerical modeling of crack propagation 3. Extended Finite Element method (XFEM).

The level set method. Strain – Stress relations. Element stiffness matrix. Derivatives of the branch functions.

Crack growth model .1 Maximum circumferential stress criterion .2 Averaged stress criterion. Finite element computation of the interaction integral. Asymptotic fields of pure mode I. Asymptotic fields of pure mode II.

Test of calculation of stress intensity factor. Edge crack under uniform shear. Edge crack with variation of slant angle under uniform tensile loading. Crack growth simulation.

Edge crack with slant angle under uniform tensile loading. Three point bending. Conclusions and Future works 5. 73 v List of Figures 2.

Basis functions of order 1, 2 for the uniform knot vector Ξ = {0,1, 2,3, 4,5,. Basis functions for the knot vector Ξ = {0, 0, 0,1, 2, 2, 2}. Quadractic basis functions for an open, non-uniform knot vector. A quadratic B-Spline curve.

The control net and mesh for the biquadratic B-Spline surface with Ξ1 = {0,0, 0,0. Example of knot insertion. Elevate the order of the curve in Figure 2. An L-shaped panel modeled by one patch and three patches.

Mapping of an element from physical space ( Ωe ) to parametric space ( Ω ˆ e) and to the parent element space ( Ω ɶ e ). Initial mesh and Refined mesh. The index space view of the mesh in Figure 2. Global numbering of control points for Figure 2.5 in parametric space.

Each control point is associated with a shape function. The red and blue lines denote element 1 and element 2, respectively. Location of interpolating points for the linear constraints for NURBS case. Body with internal boundaries subject to loads.

An arbitrary crack placed on a mesh. Signed distance function. Construction of initial level set functions. Level set functions.

Selection of enriched elements and enriched nodes by using level set functions. Examples of identifying split elements not correctly. Wrong identification of split elements and tip elements. Example of sub-elements.

Partial enriched elements around the enriched area. Conventions at crack tip. Domain A is enclosed by Γ, C+, C-, and C0. Unit normal mj = nj on C+, C-, and C0; and mj = -nj on Γ.

Elements selected about the crack tip for calculation of the interaction integral. Weighting function q on the elements. Gaussian weigh function. A rectangular plate with an edge crack under uniform shear loading.

The 21 x 41 mesh for edge crack specimen. The geometry of tensile edge crack problem. Stress fields of the horizontal edge crack specimen under tension (5 x 11 elements, 2nd order NURBS). Stress fields of the horizontal edge crack specimen under tension (31 x 61 elements, 2nd order NURBS).

Variation of stress intensity factors corresponding to crack length (2nd order NURB, 31 x 61 elements). Relative error of the mode I stress intensity factor. Slant edge crack growth for horizontal crack (α = 0°) and inclined crack (α = 15°). Vertical displacement (crack length a = 2.

Vertical displacement (crack length a = 4. Vertical displacement (crack length a = 5. Geometry of the L-shaped panel. Crack path of the L-shaped panel.

Crack path of L-shaped panel in a variation of mesh size. The geometry of three point bending test. Crack path for three point bending test with 21 x 21 elements. Crack path for three point bending test with 33 x 33 elements.

Crack path for three point bending test for eccentric crack. The paths obtained by two criteria. Crack direction becomes unstable. 69 viii List of Tables 2.

Control points for the quadratic B-Spline curve depicted in Figure 2. Control points for the biquadratic B-Spline surface depicted in Figure 2. An example of INC array. An example of IEN array.

Some combinations for shape function, partition of unity and enrichment. Mixed mode SIFs computed with 1st, 2nd, 3rd order NURBS-based XFEM. Mixed mode SIFs computed with various meshes. Pure mode I SIF computed using 2nd order and 3rd order NURBS.

Mixed mode SIFs computed using 2nd order NURBS. INTRODUCTION Chapter 1 INTRODUCTION This chapter provides a brief introduction to the methods developed for the works done in this thesis including Isogeometric Analysis (IGA) and Extended Finite Element Method (XFEM). They will be presented immediately in Section 1. The scope of this thesis and outline are given in the last two sections.

Isogeometric Analysis (IGA) The concept of Isogeometric Analysis was originally proposed by [Hughes et al. In this procedure, Non-Uniform Rational B-Spline (NURBS) functions, widely used in the Computer Aided Design (CAD) are adopted to be used as shape functions for Computer Aided Engineering (CAE), in a fashion similar to the iso-parametric concept of the standard FEM. The use of NURBS in CAE was motivated by its good characteristics such as exactness in reproducing the geometry. Unlike the conventional Lagrangian basis functions used in the standard FEM, the NURBS-based approach essentially provides a better connection between the CAD and CAE and reduces the source of errors caused by the inaccurate approximation of the geometry.

The isogeometric concept also has many great advantages such as it allows the control of the order and the continuity of the basis functions due to the methods of order elevation and knots refinement. In case of second order or higher, the IGA can yield a number of degrees of freedom smaller than that of the standard FEM, provided the same number of elements. NURBS also possesses the capability to approximate the discontinuous characteristics better than the classical FEM, which enables us to model the discontinuities (e. cracks) by means of IGA, e.

see [Verhoosel et al. (2011)] and [de Luycker et al. Recently, the development of T-Spline ([Sederberg et al. (2003)] and [Sederberg et al.

(2004)]) have introduced the ability of local refinement into IA, instead of global refinement as experienced by NURBS. The T-Spline approach seems to be more flexible and gives higher accuracy in comparison to the NURBS on conditions that multiple patches are needed (see [Bazilevs et al. The Extended Finite Element Method (XFEM) The Extended Finite Element Method (see [Moës et al. In order to capture the singular fields at the crack tips, enrichment functions are incorporated into the standard finite element approximation space.

The enrichment functions somehow must be determined appropriately in advances; especially they fully depend upon the materials and the configuration of the problems yet to be solved. Thus, different types of discontinuities such as cracks, voids, inclusions, biomaterials, etc. have different sets of enrichment functions. In order to model the crack geometry, a level set method is employed.

In this method, the crack is modeled as the zero level set of function ψ(x). Another type of level set function, i. φ(x), is used to keep trace of the crack tip. Function ψ(x) is defined as the signed distance from the point x to the crack surface and function φ(x) is defined as the signed distance from the point x to the line contains crack tip and orthogonal to the crack surfae.

The level set functions are updated accordingly to the growth of crack. The scope of the thesis Because of the great advantages of IGA and XFEM in modeling discontinuities and the exactness of the geometry, the main objective of the present thesis is a possible combination of the NURBS basis functions and XFEM (named as NURBS-based XFEM) to model the crack growth problems in 2D solids. To be able to simulate the crack growth, the propagating angle of crack path is determined by employing the maximum circumferential stress criterion. Outline The remainder of this thesis is outlined as follows.

Chapter 2 presents a short review of IGA based on the NURBS. Next, Chapter 3 describes the formulation of the NURBS-based XFEM in the framework of fracture mechanics. Several benchmark problems are numerically investigated by the proposed approach are given in Chapter 4. Finally, the conclusions and future works in the last chapter.

FUNDAMENTALS OF B-SPLINES AND NURBS Chapter 2 FUNDAMENTALS OF B-SPLINES AND NURBS In this chapter, the basic concept of B-Splines and Non-uniform rational B-Splines (NURBS) as well as the usage of NURBS in terms of Isogeometric Analysis is presented. Univariate B-Spline basis functions Univariate B-Spline basis functions are constructed from a given knot vector. A knot vector, defined by Ξ = {ξ1 , ξ 2 , … , ξ n + p +1} , is a non-decreasing sequence of the coordinates within the parametric space. In the knot vector Ξ, ξi ∈ ℝ , represents the i-th knot, , i = 1, 2, …, n+p+1; p stands for the polynomial order, while n is the number of basis functions.

Note that the polynomial order and the polynomial degree have the same meaning in our approach. The knot values can be repeatable, e. The number of appearance of a knot value in the knot vector is called multiplicity of that value. A knot vector is called open knot vector if the multiplicity of both of its first and last knots is p+1.

For example, given polynomial order p = 2, Ξ = {0,0,0,2,3,4,4,5,5,5} is an open knot vector. In this thesis, only the open knot vectors are used.