Three-Dimensional Crack Propagation with Global Enrichment XFEM and Vector Level Sets K. Agathos 1 G. Ventura 2 E. Chatzi 3 S. P. A. Bordas 4,5 1 Institute of Structural Analysis and Dynamics of Structures Aristotle University Thessaloniki 2 Department of Structural and Geotechnical Engineering Politecnico di Torino 3 Institute of Structural Engineering ETH Z¨ urich 4 Research Unit in Engineering Science Luxembourg University 5 Institute of Theoretical, Applied and Computational Mechanics Cardiff University September 9, 2015 K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 1 / 29
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Three-Dimensional Crack Propagation with GlobalEnrichment XFEM and Vector Level Sets
K. Agathos1 G. Ventura2 E. Chatzi3 S. P. A. Bordas4,5
1Institute of Structural Analysis and Dynamics of StructuresAristotle University Thessaloniki
2Department of Structural and Geotechnical EngineeringPolitecnico di Torino
3Institute of Structural EngineeringETH Zurich
4Research Unit in Engineering ScienceLuxembourg University
5Institute of Theoretical, Applied and Computational MechanicsCardiff University
September 9, 2015K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 1 / 29
Outline
1 Global enrichment XFEMDefinition of the Front ElementsTip enrichmentWeight function blendingDisplacement approximation
2 Vector Level SetsCrack representationLevel set functionsPoint projectionEvaluation of the level set functions
3 Numerical ExamplesEdge crack in a beamSemi circular crack in a beam
4 Conclusions5 References
K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 2 / 29
Far from the crackh = 0.02 unitsIn the vicinity of thecrack h = 0.005 units
K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 26 / 29
Semi circular crack in a beam
K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 27 / 29
Semi circular crack in a beam
K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 27 / 29
Semi circular crack in a beam
K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 27 / 29
Semi circular crack in a beam
K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 27 / 29
Semi circular crack in a beam
K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 27 / 29
Semi circular crack in a beam
K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 27 / 29
Semi circular crack in a beam
K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 27 / 29
Semi circular crack in a beam
K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 27 / 29
Semi circular crack in a beam
K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 27 / 29
Semi circular crack in a beam
K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 27 / 29
Semi circular crack in a beam
K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 27 / 29
Semi circular crack in a beam
K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 27 / 29
Semi circular crack in a beam
K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 27 / 29
Semi circular crack in a beam
K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 27 / 29
Semi circular crack in a beam
K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 27 / 29
Conclusions
A method for 3D fracture mechanics was presented which:
Enables the use of geometrical enrichment in 3D.Eliminates blending errors.
A method for the representation of 3D cracks was presented which:
Avoids the solution of evolution equations.Utilizes only simple geometrical operations.
The methods were combined to solve 3D crack propagation problems.
K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 28 / 29
Bibliography
Agathos, K., Chatzi, E., Bordas, S., & Talaslidis, D. (2015). Awell-conditioned and optimally convergent xfem for 3d linear elasticfracture. International Journal for Numerical Methods inEngineering.
Fries, T. (2008). A corrected XFEM approximation without problems inblending elements. International Journal for Numerical Methods inEngineering.
Fries, T., & Baydoun, M. (2012). Crack propagation with the extendedfinite element method and a hybrid explicit-implicit crack description.International Journal for Numerical Methods in Engineering.
Ventura, G., Budyn, E., & Belytschko, T. (2003). Vector level sets fordescription of propagating cracks in finite elements. InternationalJournal for Numerical Methods in Engineering.
Ventura, G., Gracie, R., & Belytschko, T. (2009). Fast integration andweight function blending in the extended finite element method.International journal for numerical methods in engineering.K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 29 / 29