Well Conditioned and Optimally Convergent Extended Finite Elements - PowerPoint PPT Presentation

Well Conditioned and Optimally Convergent Extended Finite Elements and Vector Level Sets for Three-Dimensional Crack Propagation K. Agathos 1 G. Ventura 2 E. Chatzi 3 S. P. A. Bordas 1 , 4 , 5 1 Research Unit in Engineering Science Luxembourg University 2 Department of Structural and Geotechnical Engineering Politecnico di Torino 3 Institute of Structural Engineering ETH Z¨ urich 4 Institute of Theoretical, Applied and Computational Mechanics Cardiff University Adjunct Professor, Intelligent Systems for Medicine Laboratory, The University of Western Australia 5 K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 1 / 23 June 8, 2016

Outline Global enrichment XFEM 1 Definition of the Front Elements Tip enrichment Weight function blending Displacement approximation Vector Level Sets 2 Crack representation Level set functions Numerical Examples 3 Edge crack in a beam Conclusions 4 References 5 K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 2 / 23

Global enrichment XFEM An XFEM variant (Agathos, Chatzi, Bordas, & Talaslidis, 2015) is introduced which: Enables the application of geometrical enrichment to 3D. Extends dof gathering to 3D through global enrichment. Employs weight function blending. Employs enrichment function shifting. K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 3 / 23

Front elements A superimposed mesh is used to provide a p.u. basis. Desired properties: Satisfaction of the partition of unity property. Spatial variation only along the direction of the crack front. No variation on the plane normal to the crack front. K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 4 / 23

Front elements tip enriched elements crack front FE mesh A set of nodes along the crack front is defined. Each element is defined by two nodes. front element A good starting point for front element thickness is h . front element boundaries front element node K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 5 / 23

Front elements Volume corresponding to two consecutive front elements. front element boundary crack surface crack front Different element colors correspond to different front elements. K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 6 / 23

Front element shape functions Linear 1D shape functions are used: � 1 − ξ 1 + ξ � N g ( ξ ) = 2 2 where ξ is the local coordinate of the superimposed element. Those functions: form a partition of unity. are used to weight tip enrichment functions. K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 7 / 23

Front element shape functions Definition of the front element parameter used for shape function evaluation. front element boundary ξ = 1 ξ = 0 . 5 ξ = 0 ξ x 1 ξ = 0 . 5 x − x m x 0 = 1 ξ 2 − n i e i n i +1 front element node K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 8 / 23

Tip enrichment functions Tip enrichment functions used: � √ r sin θ 2 , √ r cos θ 2 , √ r sin θ 2 sin θ, √ r cos θ � F j ( x ) ≡ F j ( r , θ ) = 2 sin θ Tip enriched part of the displacements: N g � � u t ( x ) = K ( x ) F j ( x ) c Kj K ∈N s j where N g K are the global shape functions N s is the set of superimposed nodes K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 9 / 23

Weight functions Weight functions for a) topological (Fries, 2008) and b) geometrical enrichment (Ventura, Gracie, & Belytschko, 2009). r i r e ( ) ( ) ϕ x ϕ x a) b) K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 10 / 23

Displacement approximation � � u ( x ) = N I ( x ) u I + ¯ ϕ ( x ) N J ( x ) ( H ( x ) − H J ) b J + I ∈N J ∈N j  N g  � � F j ( x ) − + ϕ ( x ) K ( x ) K ∈N s j  N g � � �  c Kj − N T ( x ) K ( x T ) F j ( x T ) T ∈N t K ∈N s j where: N is the set of all nodes in the FE mesh. N j is the set of jump enriched nodes. N t is the set of tip enriched nodes. N s is the set of nodes in the superimposed mesh. K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 11 / 23

Weight functions Enrichment strategies used for tip and jump enrichment. Topological enrichment Geometrical enrichment crack surface crack surface crack front crack front r i r e a) b) Tip enriched element Blending element Jump enriched element Tip enriched node Tip and jump enriched node Jump enriched node K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 12 / 23

Vector Level Sets A method for the representation of 3D cracks is introduced which: Produces level set functions using geometric operations. Does not require integration of evolution equations. Similar methods: 2D Vector level sets (Ventura, Budyn, & Belytschko, 2003). Hybrid implicit-explicit crack representation (Fries & Baydoun, 2012). K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 13 / 23

Crack front x i+1 i +1 Crack front at time t : t i x i i Ordered series of line segments t i Set of points x i x 2 x 2 1 1 K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 14 / 23

Crack front advance t +1 x t s i +1 i +1 x i +1 Crack front at time t + 1: i+1 t i t s t +1 x Crack advance vectors s t i i i at points x i x i i New set of points x t +1 = x t i + s t i i x 2 x 2 1 1 K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 15 / 23

Crack surface advance t +1 x i +1 t s Crack surface advance: i +1 x i +1 Sequence of four sided bilinear t +1 t segments. i t i i +1 , x t +1 i +1 , x t +1 Vertexes: x t i , x t i t +1 x i t s x i i K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 16 / 23

Kink wedges Discontinuities ( kink wedges ) are present: Along the crack front (a). Along the advance vectors (b). kink wedge kink wedge crack front crack front advance vector crack front crack front advance vector a) b) K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 17 / 23

Level set functions crack front crack surface Definition of the level set functions at a point P : t +1 s i +1 g P f distance from the crack surface. f t +1 s i t s i +1 g distance from the crack front. t s i K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 18 / 23

Edge crack in a beam H a α d L Geometry: Mesh: L = 2 unit h = 0 . 02 units H = 0 . 4 units d = 0 . 2 units a = 0 . 1 units α = 45 ◦ K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 19 / 23

Edge crack in a beam K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 20 / 23

Edge crack in a beam K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 20 / 23

Edge crack in a beam K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 20 / 23

Edge crack in a beam K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 20 / 23

Edge crack in a beam K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 20 / 23

Edge crack in a beam K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 20 / 23

Edge crack in a beam K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 20 / 23

Edge crack in a beam K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 20 / 23

Edge crack in a beam K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 20 / 23

Edge crack in a beam K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 20 / 23

Edge crack in a beam K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 20 / 23

Edge crack in a beam K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 20 / 23

Edge crack in a beam K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 20 / 23

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 8/6/2016 21 / 23

Future work Possibilities for future work: Strain smoothing-error estimation. Alternative enrichment functions. Dynamic crack propagation. K. Agathos et al. GE-XFEM and Vector Level Sets 8/6/2016 22 / 23