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Application of Isogeometric Method in Modeling and Analyzing Crack Growth Problems

Esmaeili, Mir Sardar | 2012

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  1. Type of Document: M.Sc. Thesis
  2. Language: English
  3. Document No: 43489 (53)
  4. University: Sharif University of Technology, International Campus, Kish Island
  5. Department: Science and Engineering
  6. Advisor(s): Khoei, Amir Reza
  7. Abstract:
  8. Isogeometric Analysis method is a newly introduced method for the analysis of problems governed by partial differential equations. The method has some features in common with the finite element method and some in common with the mesh-less methods. This method uses the Non-Uniform Rational B-Splines (NURBS) functions as basis function for analysis. With this basis functions, the refinement procedure is much easier than the classical finite element method by eliminating the need to communicate with the CAD model. Modeling cracks in classical finite element method requires very fine mesh near the crack tip. One can model crack propagation by means of classical finite element, using an updating mesh procedure which is a time consuming task. This is why eXtended Finite Element Method (X-FEM) is introduced. Using this method, no updating for mesh during crack propagation is required. The X-FEM adds some terms into shape function of elements which somehow are involved with crack. This idea is used in this thesis to model the crack and crack propagation. Due to the nature of NURBS function, the precision of calculations is upgraded in comparison with the classical Lagrange shape function and thus, fewer degrees of freedom can be used in order to obtain the desired precision which results in a significant decrease in costs. In the upcoming chapters, the details in modeling, enrichment procedure and crack propagation are presented, and results have a good agreement with those available in the literature
  9. Keywords:
  10. Fracture Mechanics ; Crack Growth ; Extended Finite Element Method ; Non-Uniform Rational B-Splines (NURBS)Functions ; B-Spline ; Isogeometric Method ; T-Splines

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  • Chapter 1 Introduction
    • 1.1 Isogeometric Analysis
    • 1.2 Fracture mechanics and eXtended Finite Element Method
    • 1.3 eXtended Isogeometric Analysis (XIGA)
  • Chapter 2 Introduction to Isogeometric Analysis
    • 2.1 Isogeometric Analysis Method
      • 2.1.1 Computational Geometry
    • 2.2 Basic Terminologies
    • 2.3 B-splines
      • 2.3.1 Knot vector
      • 2.3.2 Basis Functions
      • 2.3.3 Derivatives of B-spline basis functions
      • 2.3.4 B-spline geometries
        • 2.3.4.1 B-spline curves
        • 2.3.4.2 B-spline surfaces
        • 2.3.4.3 B-spline solids
      • 2.3.5 Refinement
        • 2.3.5.1 Knot insertion
        • 2.3.5.2 Order Elevation
    • 2.4 Non-Uniform Rational B-Splines (NURBS)
    • 2.5 NURBS drawbacks
      • 2.5.1 Local refinement
      • 2.5.2 Patch Assembly
    • 2.6 T-splines
      • 2.6.1 PB-Splines
      • 2.6.2 Defining T-splines
        • 2.6.2.1 Anchors and T-mesh
        • 2.6.2.2 Building a T-spline
    • 2.7 Isoparametric concept
    • 2.8 Obtaining weak form of BVPs
    • 2.9 Similarities and differences for Isogeometric analysis and Finite element analysis
  • Chapter 3 Discretization and Linear Elasticity Problems
    • 3.1 Formulating the equation
      • 3.1.1 Strong form of BVP
      • 3.1.2 Weak Form of Equilibrium equation
        • 3.1.2.1 Galerkin’s method
    • 3.2 One Dimensional Poisson’s Equation
    • 3.3 Infinite plate with circular hole under constant in-plane tension
  • Chapter 4 Fracture Mechanics, A review
    • 4.1 Historical perspective
    • 4.2 Westergaard analysis of a sharp crack
    • 4.3 Stress Intensity Factor (SIFs)
    • 4.4 Griffith theory of strength
    • 4.5 Brittle Materials
    • 4.6 Mixed mode crack propagation
    • 4.7 J-integral method for calculation of SIFs
    • 4.8 Some numerical method based on J-integral method
      • 4.8.1 Equivalent domain integral method
      • Li et al [23] proposed the equivalent domain integral method as an alternative approach, which requires only one analysis. According to Fig. 2.17b, the J integral can be defined as ( [23,24]):
      • 4.8.2 Interaction integral method
  • Chapter 5 Extended finite element method
    • 5.1 A review of XFEM development
    • 5.2 Basics of XFEM
      • 5.2.1 Partition of unity
      • 5.2.2 Enrichment
        • 5.2.2.1 Intrinsic enrichment
        • 5.2.2.2 Extrinsic enrichment
    • 5.3 Enrichment in extended finite element method
      • 5.3.1 The Heaviside function
      • 5.3.2 Asymptotic functions
    • 5.4 XFEM discretization
    • 5.5 Element integration
  • Chapter 6 Extended isogeometric analysis for modeling crack growth problems
    • 6.1 Introduction
    • 6.2 Need for XIGA
    • 6.3 Enrichment in XIGA
    • 6.4 Numerical examples
      • 6.4.1 A rectangular plate with an edge crack
      • 6.4.2 An infinite plate with a middle crack under uniform uniaxial in-plane tension
      • 6.4.3 Propagating crack through a T-spline patch of plate with a middle crack under uniform in-plane tension
      • 6.4.4 A plate with two circular holes and two edge crack under displacement control state
  • Chapter 7 Conclusion and future works
    • 7.1 Conclusion
    • 7.2 Future works
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