Extended Finite Element Method with Global Enrichment K. Agathos 1 E. Chatzi 2 S. P. A. Bordas 3 , 4 D. Talaslidis 1 1 Institute of Structural Analysis and Dynamics of Structures Aristotle University Thessaloniki 2 Institute of Structural Engineering ETH Z¨ urich 3 Research Unit in Engineering Sciences Luxembourg University 4 Institute of Theoretical, Applied and Computational Mechanics Cardiff University 2015 K. Agathos et al. GE-XFEM 2015 1 / 82
Outline Problem statement Governing equations Weak Form Global enrichment XFEM Motivation Related works Crack representation Tip enrichment Jump enrichment Point-wise matching Integral matching Displacement approximation Definition of the Front Elements Numerical examples 2D convergence study 3D convergence study Conclusions References K. Agathos et al. GE-XFEM 2015 2 / 82
Problem statement Governing equations 3D body geomery Γ ¯ 0 t ¯ y t Γ c t t Γ Γ = Γ 0 ∪ Γ u ∪ Γ t ∪ Γ c c Γ 0 c Γ c = Γ t c ∪ Γ 0 c Γ u x z K. Agathos et al. GE-XFEM 2015 3 / 82
Problem statement Governing equations Governing equations Equilibrium equations and boundary conditions: ∇ · σ + b = 0 in Ω u = ¯ on Γ u u σ · n = ¯ t on Γ t Γ 0 σ · n = 0 on c Γ t σ · n = ¯ on t c c Kinematic equations: ǫ = ∇ s u Constitutive equations: σ = D : ǫ K. Agathos et al. GE-XFEM 2015 4 / 82
Problem statement Weak Form Weak form of equilibrium equations Find u ∈ U such that ∀ v ∈ V 0 � � � � t c · v d Γ t ¯ ¯ σ ( u ) : ǫ ( v ) d Ω = b · v d Ω + t · v d Γ + c Γ t Ω Ω Γ t c where : � 3 , u = ¯ � � H 1 (Ω) � U = u | u ∈ u on Γ u and � 3 , v = 0 on Γ u � � H 1 (Ω) � V = v | v ∈ K. Agathos et al. GE-XFEM 2015 5 / 82
Global enrichment XFEM Motivation Motivation ◮ XFEM for industrially relevant (3D) crack problems ◮ Requires robust methods for stress intensity evaluation. ◮ Requires low solution times and ease of use. ◮ but standard XFEM leads to ◮ Ill-conditioning of the stiffness matrix for “large” enrichment domains. ◮ Lack of smoothness and accuracy of the stress intensity factor field along the crack front. ◮ Blending issues close at the boundary of the enriched region. ◮ Problem size for propagating cracks (“old” front-dofs must be kept for stability of time integration schemes). K. Agathos et al. GE-XFEM 2015 6 / 82
Global enrichment XFEM Motivation Global enrichment XFEM There exists different approaches to alleviate the above difficulties: ◮ Preconditioning (e.g. Mo¨ es; Menk and Bordas) ◮ Ghost penalty (Burman) ◮ Stable XFEM/GFEM (Banerjee, Duarte, Babuˇ ska, Paladim, Bordas) - behaviour for realistic 3D crack not clear. ◮ Corrected XFEM/GFEM (Fries, Loehnert) ◮ SIF-oriented (goal-oriented) error estimation methods for SIFs (R´ odenas, Estrada, Ladev` eze, Chamoin, Bordas) ◮ Restrict the variability of the enrichment within the enriched domain: doc-gathering, cut-off XFEM (Laborde, Renard, Chahine, Sal¨ un and the French team ;-) K. Agathos et al. GE-XFEM 2015 7 / 82
Global enrichment XFEM Motivation Global enrichment XFEM An XFEM variant is introduced which: ◮ Extends dof gathering to 3D through global enrichment. ◮ Employs point-wise matching of displacements. ◮ Employs integral matching of displacements. ◮ Enables the application of geometrical enrichment to 3D. K. Agathos et al. GE-XFEM 2015 8 / 82
Global enrichment XFEM Related works Related works Similar concepts to the ones introduced herein can be found: ◮ In the work of Laborde et al. → dof gathering → point-wise matching (Laborde, Pommier, Renard, & Sala¨ un, 2005) ◮ In the work of Chahine et al. → integral matching (Chahine, Laborde, & Renard, 2011) K. Agathos et al. GE-XFEM 2015 9 / 82
Global enrichment XFEM Related works Related works ◮ In the work of Langlois et al. → discretization along the crack front (Langlois, Gravouil, Baieto, & R´ ethor´ e, 2014) ◮ In the s-finite element method → superimposed mesh (Fish, 1992) K. Agathos et al. GE-XFEM 2015 10 / 82
Global enrichment XFEM Crack representation Crack representation Level set functions: ◮ φ ( x ) is the signed distance from the crack surface. ◮ ψ ( x ) is a signed distance function such that: → ∇ φ · ∇ ψ = 0 → φ ( x ) = 0 and ψ ( x ) = 0 defines the crack front Polar coordinates: � φ � � φ 2 + ψ 2 , r = θ = arctan ψ K. Agathos et al. GE-XFEM 2015 11 / 82
Global enrichment XFEM Crack representation Crack representation ( ) = 0 ψ x φ ( ) = 0 ( ) ψ x x x r ( ) φ x θ crack surface crack extension crack front K. Agathos et al. GE-XFEM 2015 12 / 82
Global enrichment XFEM Tip enrichment Tip enrichment Enriched part of the approximation for tip elements: N g � � u te ( x ) = K ( x ) F j ( x ) c Kj j K N g K are the global shape functions to be defined. Tip enrichment functions: � √ 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 K. Agathos et al. GE-XFEM 2015 13 / 82
Global enrichment XFEM Tip enrichment Geometrical enrichment ◮ Enrichment radius r e is defined. ◮ Nodal values r i of variable r are computed. ◮ The condition r i < r e is tested. ◮ If true for all nodes of an element, the element is tip enriched. K. Agathos et al. GE-XFEM 2015 14 / 82
Global enrichment XFEM Jump enrichment Jump enrichment Jump enrichment function definition: � 1 for φ > 0 H ( φ ) = − 1 for φ < 0 Shifted jump enrichment functions are used throughout this work. 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 - 0.5 - 0.5 - 0.5 - 1.0 - 1.0 - 1.0 1.5 1.0 0.5 0.5 1.0 0.0 0.0 0.5 - 0.5 - 0.5 - 1.0 - 1.0 0.0 - 1.0 - 1.0 - 1.0 - 0.5 - 0.5 - 0.5 0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0 H ( φ ) ( ) N ( H ( φ ) H ) N H φ 1 − 1 1 K. Agathos et al. GE-XFEM 2015 15 / 82
Global enrichment XFEM Jump enrichment Enrichment strategy Motivation for an alternative enrichment strategy: ◮ Tip enrichment functions are derived from the first term of the Williams expansion. ◮ Displacements consist of higher order terms as well. ◮ Those terms are represented by: → the FE part → spatial variation of the tip enrichment functions K. Agathos et al. GE-XFEM 2015 16 / 82
Global enrichment XFEM Jump enrichment Enrichment strategy ◮ In the proposed method: → no spatial variation is allowed → higher order terms can only be approximated by the FE part ◮ Higher order displacement jumps can not be represented in tip elements. K. Agathos et al. GE-XFEM 2015 17 / 82
Global enrichment XFEM Jump enrichment Enrichment strategy Proposed enrichment strategy: crack surface crack front Tip enriched node Tip and jump enriched node r Jump enriched node e Tip enriched elements Jump enriched element Both tip and jump enrichment is used for tip elements that contain the crack. K. Agathos et al. GE-XFEM 2015 18 / 82
Global enrichment XFEM Point-wise matching Tip and Regular Elements Tip enriched element Regular element g � N � F ( x ) c a j 2 Kj K K j 2 a 1 u u 2 2 g u u � N � F ( x ) c j 1 Kj 1 1 K K j 1 2 1 2 Displacement approximations of regular and tip elements: � � u r ( x ) = N I ( x ) u I + N J ( x ) a J I J N g � � � u t ( x ) = N I ( x ) u I + K ( x ) F j ( x ) c Kj j I K K. Agathos et al. GE-XFEM 2015 19 / 82
Global enrichment XFEM Point-wise matching Tip and Regular Elements Tip enriched element Regular element g � � ( ) a N F x c j Kj 2 2 K K j a 1 u u 2 2 g � � ( ) u u N F x c 1 j 1 Kj 1 K K j 1 1 2 2 Displacements are matched by imposing the condition: u r ( x I ) = u t ( x I ) K. Agathos et al. GE-XFEM 2015 20 / 82
Global enrichment XFEM Point-wise matching Tip and Regular Elements Tip enriched element Regular element g � � ( ) N F x c a j 2 Kj 2 K K j a 1 u u 2 2 g u u � N � F ( x ) c j 1 Kj 1 1 K K j 1 2 1 2 Parameters a I are obtained: N g � � a I = K ( X I ) F j ( X I ) c Kj K j K. Agathos et al. GE-XFEM 2015 21 / 82
Global enrichment XFEM Point-wise matching Tip and Regular Elements Tip enriched element Regular element g � � ( ) N F x c a j 2 Kj 2 K K j a 1 u u 2 2 g u u � N � F ( x ) c j 1 Kj 1 1 K K j 1 2 1 2 Parameters a I can be expressed as: T t − r � � a I = IKj c Kj K j K. Agathos et al. GE-XFEM 2015 22 / 82
Global enrichment XFEM Point-wise matching Tip and Jump Elements Displacement approximations of tip and jump elements: � � � u j ( x ) = N I ( x ) u I + N J ( x ) a J + N L ( x ) ( H ( x ) − H L ) b L + I J L � N M ( x ) ( H ( x ) − H M ) b t + M , M � � u t ( x ) = N I ( x ) u I + N J ( x ) ( H ( x ) − H J ) b J + I J N g � � + K ( x ) F j ( x ) c Kj K j K. Agathos et al. GE-XFEM 2015 23 / 82
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