Isogeometric Analysis Integrating Design and Analysis IGA 2014

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

Outline 1/21 ØMotivations ØModeling strategy by IGABEM ØNumerical examples 2

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

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

Motivation 4/21 Ø BEM is suitable for modeling fracture problems Dimensional reduction Boundary mesh modification High accuracy Ø IGABEM shows more advantages keep the exact geometry Natural fit with BEM Release the mesh burden further Higher order continuity Ø Drawbacks of BEM High complexity in assembling (O(N 2)) and solving (O(N 3)) Full populated matrix Awkward in non-homogeneous, non-linear material 5

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

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

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

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

Evaluation of stress intensity factors 9/21 ØContour integral based methods: • M integral (involving J 1): • J integral (involving J 1 and J 2): Singular in evaluating J 2 10

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

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

Numerical examples: Griffith crack 12/21 13

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 14

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

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

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

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

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

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

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, 209212(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): 12691287, 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): 13351350, 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. 21

Future works 21/21 ØCrack tip enrichment Ø 3 D implementation Acknowledgements: Thanks given to the Framework Programme 7 Initial Training Network Funding under grant number 289361 "Integrating Numerical Simulation and Geometric Design Technology” (FP 7: ITN-INSIST) Thanks for attention 22