M Institut de Recherche en G nie C ivil et M canique , N. - PowerPoint PPT Presentation

Context and tool box Theoritical development Numerical example Conclusion and upcoming Inequality level set for contact A new coupled X-FEM/ Level-Set strategy to solve contact problems Nicolas Chevaugeon, Matthieu Graveleau, Nicolas Moës from: GeM, Nantes (FRANCE) for ICCCM 2013 07/12/2013 G M Institut de Recherche en G énie C ivil et M écanique , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique 2 ⁄ 28 Context and tool box 1 2 Theoritical development Numerical example 3 Conclusion and upcoming 4 , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique Summary 3 ⁄ 28 1 Context and tool box 2 Theoritical development 3 Numerical example 4 Conclusion and upcoming , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique 4 ⁄ 28 Contact is an active research domain! Several studies and methods : ◮ theoretical * , ◮ practical † . Still studied, why? ◮ multiple contact zones, ◮ represent discontinuities. * . † . , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique 4 ⁄ 28 Contact is an active research To contact or not to contact, that domain! is the question? y Several studies and methods : undeformable ◮ theoretical * , ◮ practical † . x z imposed Still studied, why? efforts ◮ multiple contact zones, deformable ◮ represent discontinuities. * . † . , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique 5 ⁄ 28 y First let’s agree on some notations : x ◮ Ω domain, z ◮ ∂Ω its frontier, f ∂ Ω t ◮ ∂Ω c the part in contact, Ω ◮ Γ c its boundary, ∂ Ω u ◮ ∂Ω t imposed effort t d , Γ c ◮ ∂Ω u imposed displacement u d , ∂ Ω c ◮ f volumic forces , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique y 6 ⁄ 28 Our original problem : x z ∂ Ω c ◮ what is the equilibrium state? ( u, σ ) ◮ what is the contact zone? Hard to solve! , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique y 6 ⁄ 28 Our original problem : x z ∂ Ω c ◮ what is the equilibrium state? ( u, σ ) ◮ what is the contact zone? y Hard to solve! Idea : split the problem x z ∂ Ω c ◮ solve elastic problem guessing ∂Ω c , ( u, σ ) ◮ find ∂Ω c which verifies the contact conditions. y ⇒ iterative process x z ∂ Ω c ( u, σ ) , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique y 6 ⁄ 28 Our original problem : x z ∂ Ω c ◮ what is the equilibrium state? ( u, σ ) ◮ what is the contact zone? y Hard to solve! Idea : split the problem x z ∂ Ω c ◮ solve elastic problem guessing ∂Ω c , ( u, σ ) ◮ find ∂Ω c which verifies the contact conditions. y ⇒ iterative process Question : how to make ∂Ω c evolve? x z ∂ Ω c ( u, σ ) , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique 7 ⁄ 28 Question : what does our method need to do in order to make ∂Ω c evolve? , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique 7 ⁄ 28 Question : what does our method need to do in order to make ∂Ω c evolve? , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique 7 ⁄ 28 Question : what does our method need to do in order to make ∂Ω c evolve? , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique 7 ⁄ 28 Question : what does our method need to do in order to make ∂Ω c evolve? , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique 7 ⁄ 28 Question : what does our method need to do in order to make ∂Ω c evolve? , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique 7 ⁄ 28 Question : what does our method need to do in order to make ∂Ω c evolve? Criteria : ◮ good representation of Γ c and the phenomena on it, ◮ having a criterion for the “good position” of the contact zone, ◮ fast convergence, ◮ adaptability. , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique 7 ⁄ 28 Question : what does our method need to do in order to make ∂Ω c evolve? * Criteria : ◮ good representation of Γ c and the phenomena on it, level-set/XFEM ◮ having a criterion for the “good position” of the contact zone, configurational mechanic ◮ fast convergence, configurational mechanic ◮ adaptability. level-set * . , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique 8 ⁄ 28 Φ ≥ 0 level-set * signed function : Γ c Φ ≤ 0 φ ( x ) = ± min x c ∈ Γ c � x c − x � (1) thus the iso-zero level-set represents Γ c = { x / φ ( x ) = 0 } * . † . ‡ . , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique 8 ⁄ 28 level-set * XFEM † signed function : exploits partition of unity ‡ : φ ( x ) = ± min x c ∈ Γ c � x c − x � (1) → take into account discontinuities on Γ c → differentiate contact or non thus the iso-zero level-set represents Γ c = { x / φ ( x ) = 0 } contact domain * . † . ‡ . , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique Summary 9 ⁄ 28 1 Context and tool box 2 Theoritical development 3 Numerical example 4 Conclusion and upcoming , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique 10 ⁄ 28 Hypothesis ◮ elastic linear material Strong formulation for 2 bodies ◮ small displacement ◮ div σ + f = 0 in Ω ◮ frictionless contact ◮ u KA and σ SA with σ = C ε Ω 1 Karush-Kuhn-Tucker(KKT) conditions : ◮ ( x 2 − x 1 ) · n 1 � 0 on Γ c ◮ F n 1 = − F n 2 � 0 on Γ c n 1 ◮ ( x 2 − x 1 ) · n 1 F n 1 = 0 on Γ c Ω 2 , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique 11 ⁄ 28 Variationnal formulation * Find u KA + CC a ( u , v ) − l ( v ) � 0 ∀ v KA 0 (2) with : � ◮ a ( u , v ) = Ω σ : ε ( v ) d Ω � � Ω f ( v ) d Ω + t d v d Γ ◮ l ( v ) = Γ t → variational inequality * . , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique 11 ⁄ 28 Variationnal formulation * If ∂Ω c is given we can relax the inequality with Lagrange multipliers (2) becames : Find u KA and λ ∈ Λ such as : � a ( u , v )+ b ( λ , v ) − l ( v ) = 0 ∀ v KA 0 b ( q , u ) = l b ( q ) ∀ q ∈ Λ Thus we will have a solution which fulfills the static and kinematic conditions but not the contact conditions. * . , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique Evolution of the interface 12 ⁄ 28 To fulfill the contact conditions − → find the right ∂Ω c . Γ k Γ exact c c , N. Chevaugeon ILS for contact

Context and tool box Theoritical development Numerical example Conclusion and upcoming G M Institut de Recherche en G énie C ivil et M écanique Evolution of the interface 12 ⁄ 28 Let’s define a positioning criteria g such as : g ( x ) = 0 on Γ c if Γ c = Γ ex c g ( x ) � = 0 Γ k Γ k +1 Γ exact c c c , N. Chevaugeon ILS for contact