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
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 Point projection Evaluation of the level set functions Numerical Examples 3 Edge crack in a beam Semi circular crack in a beam Conclusions 4 References 5 K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 2 / 29
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 9/9/2015 3 / 29
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. K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 4 / 29
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 9/9/2015 5 / 29
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 9/9/2015 6 / 29
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 9/9/2015 7 / 29
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 9/9/2015 8 / 29
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 9/9/2015 9 / 29
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 9/9/2015 10 / 29
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 9/9/2015 11 / 29
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 9/9/2015 12 / 29
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 9/9/2015 13 / 29
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 9/9/2015 14 / 29
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 9/9/2015 15 / 29
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 9/9/2015 16 / 29
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 9/9/2015 17 / 29
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 9/9/2015 18 / 29
Point projection t +1 x i +1 x i v t s i +1 u t s i t +1 x i t +1 x i Element parametric equations φ ( u , v ), u , v ∈ [ − 1 , 1]: i +1 + g 3 ( u , v ) x t +1 i +1 + g 4 ( u , v ) x t +1 = g 1 ( u , v ) x t i + g 2 ( u , v ) x t φ x i i +1 + g 3 ( u , v ) y t +1 i +1 + g 4 ( u , v ) y t +1 = g 1 ( u , v ) y t i + g 2 ( u , v ) y t φ y i i +1 + g 3 ( u , v ) z t +1 i +1 + g 4 ( u , v ) z t +1 = g 1 ( u , v ) z t i + g 2 ( u , v ) z t φ z i where g i ( u , v ), u , v ∈ [ − 1 , 1] are linear shape functions. K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 19 / 29
Point projection Equation of the tangent plane Π 0 at ( u 0 , v 0 ): x − φ x ( u 0 , v 0 ) y − φ y ( u 0 , v 0 ) z − φ z ( u 0 , v 0 ) det φ x , u ( u 0 , v 0 ) φ y , u ( u 0 , v 0 ) φ z , u ( u 0 , v 0 ) = 0 φ x , v ( u 0 , v 0 ) φ y , v ( u 0 , v 0 ) φ z , v ( u 0 , v 0 ) Normal vector to the parametric surface at ( u 0 , v 0 ): n ( u 0 , v 0 ) = ( A , B , C ) where A , B , C are the minors of the previous matrix at ( u 0 , v 0 ). K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 20 / 29
Point projection Point P can be expressed as: P = P ′ ( u , v ) + λ n ( u , v ) where: P ′ the projection of the point to the surface. λ unknown parameter. The above is solved for u , v and λ to obtain the projection. K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 21 / 29
Evaluation of the level set functions At each step t : For each point all crack advance segments are tested. If for a certain element u , v ∈ [ − 1 , 1] then the point is projected on that element. If u / ∈ [ − 1 , 1] for all elements then the projection lies on the advance vector. If v / ∈ [ − 1 , 1] for all elements then the projection lies either: → at a previous crack advance segment → at the crack front at time t − 1 or t K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 22 / 29
Evaluation of the level set functions Level set function f : f = P − P ′ where P ′ is either: Projection to an element of the crack surface Closest point projection to a kink wedge Level set function g : g = P − P ′ where P ′ is a closest point projection to the crack front K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 23 / 29
Edge crack in a beam H a α d L Geometry: Mesh: L = 1 unit Far from the crack h = 0 . 02 units H = 0 . 2 units In the vicinity of the d = 0 . 1 units crack h = 0 . 005 units a = 0 . 05 units α = 45 ◦ K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 24 / 29
Edge crack in a beam K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 25 / 29
Edge crack in a beam K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 25 / 29
Edge crack in a beam K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 25 / 29
Edge crack in a beam K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 25 / 29
Edge crack in a beam K. Agathos et al. GE-XFEM and Vector Level Sets 9/9/2015 25 / 29
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