1. A 2D isogeometric boundary element method for linear elastic fracture
Isogeometric Analysis:
Integrating Design and Analysis (IGA 2014)
Austin, Texas, USA
A 2D isogeometric boundary element method for linear elastic fracture
Xuan Peng, Elena Atroshchenko, Robert Simpson,
Stephane Bordas, Sivakumar Kulasegaram
Cardiff University, UK
January 2014 1

2. Modeling strategy by IGABEM
Outline
1/21
Motivations
Modeling strategy by IGABEM
Numerical examples

3. Motivation
2/21
Goal
develop a robust, efficient and accurate program system for 3D fracture simulation and fatigue life prediction based on isogeometric boundary element method (IGABEM)

4. Expensive computational cost
Motivation
3/21
Bordas&Moran (2006)
Mesh burden
Expensive computational cost

5. BEM is suitable for modeling fracture problems Dimensional reduction
Motivation
4/21
BEM is suitable for modeling fracture problems
Dimensional reduction
Boundary mesh modification
High accuracy
IGABEM shows more advantages
keep the exact geometry
Natural fit with BEM
Release the mesh burden further
Higher order continuity
Drawbacks of BEM
High complexity in assembling (O(N2)) and solving (O(N3))
Full populated matrix
Awkward in non-homogeneous, non-linear material

6. Displacement BIE: non–crack boundary and one crack surface
Dual BEM for crack modeling
5/21
Displacement BIE: non–crack boundary and one crack surface
Traction BIE: the other crack surface

7. Greville Abscissae: Discretised BIEs
NURBS discretisation and collocation
6/21
Greville Abscissae:
NURBS(B-Spline)
p=2
Discretised BIEs
Discontinous Lagange
p=2

8. Singular integration for source points on non-crack surfaces
7/21
Rigid body motion:

9. Singularity subtraction technique:
Singular integration for source points on crack surfaces
8/21
Singularity subtraction technique:

10. Contour integral based methods:
Evaluation of stress intensity factors
9/21
Contour integral based methods:
M integral (involving J1):
J integral (involving J1 and J2):
Singular in evaluating J2

11. Space constraint , parametric constraint
Algorithm for crack propagation
10/21
Space constraint , parametric constraint
Maximum hoop stress criterion
Localization constraint function
Calculate the moving vector

12. Uniform mesh refinement No special treatment for crack tip
Numerical examples: Griffith crack
11/21
NURBS(B-Spline)
p=2
Mode I:
Mode II:
Discontinous Lagange
p=2
Uniform mesh refinement
No special treatment for crack tip

13. Numerical examples: Griffith crack
12/21

14. Numerical examples: inclined centre crack 13/21
IGABEM(r) :Uniform mesh (refined tip element)
LBEM: discontinuous Lagrange BEM
SGBEM: symmetric Galerkin BEM, Sutrahar&Paulino (2004)
m: number of elements in uniform mesh along the crack surface

15. Investigation with varied angle:
Numerical examples: inclined centre crack
14/21
Investigation with varied angle:

16. Numerical examples: arc crack 15/21
Uniform mesh + refined tip element
Splitting parameter in J integral:

17. Numerical examples: crack growth from rivet holes 16/21
12 elements for each circle
3 elements for initial cracks with tip refinement

18. Contour of Mises stress in elastostatic analysis
Numerical examples: crack growth in a spanner
17/21
Simpson et al (2012)
Contour of Mises stress in elastostatic analysis

19. Numerical examples: crack growth in a spanner
18/21

20. Numerical examples: crack growth in a spanner
19/21
IGABEM
XFEM

21. References
20/21
R N Simpson, S P A Bordas, J Trevelyan, and T Rabczuk. A two-dimensional Isogeometric Boundary Element Method for elastostatic analysis. Computer Methods in Applied Mechanics and Engineering, (0):87100, 2012.
M A Scott, R N Simpson, J A Evans, S Lipton, S P A Bordas, T J R Hughes, and T W Sederberg. Isogeometric boundary element analysis using unstructured T-splines. Computer Methods in Applied Mechanics and Engineering, 254(0):197221, 2013.
A Portela, M H Aliabadi, and D P Rooke. The dual boundary element method: Effective implementation for crack problems. International Journal for Numerical Methods in Engineering, 33(6): , 1992.
A Sutradhar and G H Paulino. Symmetric Galerkin boundary element computation of T-stress and stress intensity factors for mixed-mode cracks by the interaction integral method. Engineering Analysis with Boundary Elements, 8(11): , 2004.
N Moës, J Dolbow, and T Belytschko. A Finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 46(1):131150, 1999.

22. Crack tip enrichment 3D implementation Acknowledgements:
Future works
21/21
Crack tip enrichment
3D implementation
Acknowledgements:
Thanks given to the Framework Programme 7 Initial Training Network Funding under grant number "Integrating Numerical Simulation and Geometric Design Technology” (FP7: ITN-INSIST)
Thanks for attention