On coupling of complementarity with friction in contact shape - PowerPoint PPT Presentation

On coupling of complementarity with friction in contact shape optimization Jiˇ rí V. Outrata Academy of Sciences of the Czech Republic Based on two joint papers with the coauthors J. Haslinger(Charles University Prague), M. Koˇ cvara (University of Birmingham, UK), R. Kuˇ cera (Technical University Ostrava) P . Beremlijski (Technical University Ostrava) R. Pathó (Charles University Prague). Jiˇ rí V. Outrata (UTIA) 1 / 23

Goal: To find, for and elastic body, an admissible shape of a part of its boundary such that, after applying the body forces and given surface tractions, the variables corresponding to the shape, the displacement and the multiplier associated with the Signorini condition will create a local minimizer of a given objective. We will be dealing with 2 friction models, namely A Coulomb friction with a fixed friction coefficient (3D); B Coulomb friction with a solution-dependent friction ceofficient (2D). Jiˇ rí V. Outrata (UTIA) 2 / 23

Outline: Outline: (i) Backgroung from variational analysis; (ii) Algebraic setting of the problems; (iii) Implicit programming approach (ImP); (iv) Computation of limiting coderivatives of the set-valued parts of the respective GEs; (v) Sensitivity analysis; (vi) Numerical results. Jiˇ rí V. Outrata (UTIA) 3 / 23

Ad (i): Background from variational analysis Consider a closed set A ⊂ R n and ¯ x ∈ A . A − ¯ T A (¯ x is the contingent (Bouligand) cone to A at ¯ x ) := Lim sup x . τ τ ↓ 0 x )) 0 is the regular (Fréchet) normal cone to A at ¯ � N A (¯ x ) := ( T A (¯ x . The limiting (Mordukhovich) normal cone to A at ¯ x is defined by A N ( x ) = { x ∗ ∈ R n |∃ k → x ∗ such that x ∗ � x , x ∗ k ∈ � N A (¯ x k → ¯ x ) := Lim sup N A ( x k ) ∀ k } . A x → ¯ x Jiˇ rí V. Outrata (UTIA) 4 / 23

Background from variational analysis Now consider a closed-graph multifunction Φ[ R n ⇒ R m ] and a point (¯ x , ¯ y ) ∈ gph Φ . The multifunction � y )[ R m ⇒ R n ] defined by D ∗ Φ(¯ x , ¯ y )( y ∗ ) := { x ∗ ∈ R n | ( x ∗ , − y ∗ ) ∈ � � D ∗ Φ(¯ x , ¯ N gph Φ (¯ x , ¯ y ) } is the regular (Fréchet) coderivative of Φ at (¯ x , ¯ y ) . y )[ R m ⇒ R n ] defined by The multifunction D ∗ Φ(¯ x , ¯ y )( y ∗ ) := { x ∗ ∈ R n | ( x ∗ , − y ∗ ) ∈ N gph Φ (¯ D ∗ Φ(¯ x , ¯ x , ¯ y ) } is the limiting (Mordukhovich) coderivative of Φ at (¯ x , ¯ y ) . Jiˇ rí V. Outrata (UTIA) 5 / 23

Geometrical setting Γ P γ Γ u Γ Ω(α) P C 0 Γ (α) c a b Elastic body and its contact boundary: Ω( α ) := { ( x 1 , x 2 ) | a < x 1 < b , α ( x 1 ) < x 2 < γ } , Γ c ( α ) := Gr α, where � � � � 0 ≤ α ≤ C 0 , � α ′ � L ∞ ≤ C 1 � α ∈ C 0 , 1 ([ a , b ]) α ∈ U ad := . � � C 2 ≤ meas Ω( α ) ≤ C 3 Jiˇ rí V. Outrata (UTIA) 6 / 23

Ad (ii): In both considered models the discretized state problems attain the form minimize J ( α, y ) subject to (1) 0 ∈ F ( α, y ) + Q ( y ) α ∈ ω ⊂ R l , where l denotes the number of nodes on the contact boundary, the state variable y amounts to ( u t , u ν , λ ) , where u t , u ν stand for the tangential and normal displacements, respectively, λ is the multiplier associated with the Signorini condition and ω is the discretized set of admissible shapes. Concretely, the GE from (1) takes the form 0 ∈ A tt ( α ) u t + A t ν ( α ) u ν − L t ( α ) + � Q ( u t , u ν , λ ) 0 = A ν t ( α ) u t + A νν ( α ) u ν − L ν ( α ) − λ (2) 0 ∈ u ν + α + N R l + ( λ ) , Jiˇ rí V. Outrata (UTIA) 7 / 23

where the blocks A tt , A t ν , A ν t and A νν correspond to the stiffness matrix and vectors L t , L ν correspond to the body forces and surface tractions. All of them depend on α in a continuously differentiable way. Further, ω = { α ∈ R l | 0 ≤ α i ≤ C 0 , i = 1 , 2 , . . . , l , | α i + 1 − α i | ≤ C 1 h , i = 1 , 2 , . . . , l − 1 , C 2 ≤ meas Ω( α ) ≤ C 3 } , Q ( u t , u ν , λ )) i = F λ i ∂ � u i ( � t � 2 , i = 1 , 2 , . . . l ( in model A ) and Q ( u t , u ν , λ )) i = F ( | u i ( � t | ) λ i ∂ | u i t | , i = 1 , 2 , . . . l ( in model B ) . It is well-known that under suitable assumptions concerning F (in A) or F ( · ) (in B) the solution map S ( α ) := { y | 0 ∈ F ( α, y ) + Q ( y ) is single-valued and Lipschitz. Moreover, for l → ∞ the solutions of (1) (which exist due to the boundedness of ω ) converge to a solution of the original continuous problem in the appropriate function spaces. Jiˇ rí V. Outrata (UTIA) 8 / 23

Ad (iii): Define Θ( α ) := J ( α, S ( α )) . Then (1) amounts to the optimization problem mimimize Θ( α ) subject to (3) α ∈ ω. Assume that J is continuously differentiable. Then Θ is locally Lipschitz and (3) can be numerically solved, e.g., by a bundle method of nonsmooth optimization. To this aim we must be able to compute for each α ∈ ω the value Θ( α ) and a vector ξ ∈ ¯ ∂ Θ( u ) . The latter will be done by using the relationship ¯ ∂ Θ( u ) = conv ∂ Θ( u ) ⊃ ∂ Θ( u ) = { ξ | ξ ∈ ∇ u J ( α, y ) + D ∗ S ( u )( ∇ y J ( α, y )) } , where y = S ( α ) . Furthermore, for a given vector a , one has D ∗ S ( u )( a ) ⊂ { ( ∇ α F ( α, y )) T b | 0 ∈ a + ( ∇ y F ( α, y )) T b + D ∗ Q ( y , − F ( α, y ))( b ) } . The above inclusion becomes equality provided either (i) ∇ α F ( α, y ) is surjective, or (ii) gph Q is (normally) regular at ( y , − F ( α, y )) . Jiˇ rí V. Outrata (UTIA) 9 / 23

Ad(iv): In the computation of ξ the most difficult part consists in the computation of the limiting coderivative of Q . To facilitate this step we regroup GE (2) in such a way that l ¯ Q ( y i ) , X Q ( y ) = i = 1 with the multifunctions     F λ i ∂ � u i F ( | u i t | ) λ i ∂ | u i t � 2 t |     Q ( y i ) = ¯ Q ( y i ) = ¯ 0 0 and (4)     N R + ( λ i ) N R + ( λ i ) in the cases A and B, respectively. It follows that for u ∈ Q ( y ) one has d ∈ D ∗ Q ( y , u )( c ) ⇔ d i ∈ D ∗ ¯ Q ( y i , u i )( c i ) ∀ i . So, everything boils down to analysis of multifunctions ¯ Q which are associated to single nodes lying on the contact part of the boundary. Jiˇ rí V. Outrata (UTIA) 10 / 23

Theorem 1. Consider the multifunction Ψ[ R n × R m × R o ⇒ R p × R s ] defined by � � G ( x , y ) F ( x , y , z ) = , H ( y , z ) where G [ R n × R m ⇒ R p ] and H [ R m × R o ⇒ R s ] are closed-graph multifunctions. Assume that (¯ x , ¯ y , ¯ z , ¯ u , ¯ v ) ∈ gph F and the qualification condition � �  0   ∈ D ∗ G (¯ x , ¯ y , ¯ u )( 0 ) ,    w 2 ⇒ w 2 = 0 (5) � �  − w 2   ∈ D ∗ H (¯  y , ¯ z , ¯ v )( 0 )  0 2 ∈ R p × R s one has holds true. Then for any d ∗ 1 , d ∗ D ∗ F (¯ z )( d ∗ 1 , d ∗ ∈ D ∗ G (¯ u )( d ∗ x , ¯ y , ¯ 2 ) ⊂ { ( w 1 , w 2 + w 3 , w 4 ) | ( w 1 , w 2 ) x , ¯ y , ¯ 1 ) , (6) ∈ D ∗ H (¯ y , ¯ z , ¯ v )( d ∗ ( w 3 , w 4 ) 2 ) } . Remark Qualification condition (5) can be weakened on the basis of the calmness of respective perturbation maps. Jiˇ rí V. Outrata (UTIA) 11 / 23

Theorem 2. Inclusion (6) becomes equality provided (i) G is single-valued and continuously differentiable near (¯ x , ¯ y ) . In this case condition (5) is automatically fulfilled; (i) In addition to the assumptions of Theorem 1, for each sequence y ( i ) → ¯ y and 1 ) ∃ sequences x ( i ) → ¯ x , u ( i ) → ¯ u , d ∗ ( i ) each η ∈ D ∗ G (¯ y )( d ∗ → d ∗ x , ¯ 1 such that 1 ( x ( i ) , y ( i ) , u ( i ) ) ∈ gph G D ∗ G ( x ( i ) , y ( i ) , u ( i ) )( d ∗ ( i ) � η ∈ Lim sup ) . 1 i →∞ In verification of the assumptions in (ii) one may use the following statement. Lemma. Assume that G ( x , y ) = f ( x ) g ( y ) , where f [ R n → R ] and g [ R m → R p ] are Lipschitz near y ) and any d ∗ one has ¯ x and ¯ y , respectively. Then for any ( x , y ) close to (¯ x , ¯ � � � D ∗ f ( x )( � g ( y ) , d ∗ � ) D ∗ G ( x , y )( d ∗ ) = � . � D ∗ g ( y )( f ( x ) d ∗ ) The above assertion enables us to prove that such mapping G fulfills the assumptions in (ii) whenever g is continuously differetiable near ¯ y . Jiˇ rí V. Outrata (UTIA) 12 / 23

Analysis of the friction terms Denote by Φ the friction terms in the definitions of ¯ Q in (4), i.e., Φ( y i ) = F λ i ∂ � u i t � 2 ( in the case A ) Φ( y i ) = F ( | u i t | ) λ i ∂ | u i t | ( in the case B ) . Let ¯ z ∈ gph Φ and ∃ neighborhood O of ¯ z such that gph Φ ∩ O = Γ ∪ Ξ ∪ Λ , where Γ and Ξ are open in the relative topology of gph Φ and ¯ z ∈ Λ ⊂ bd Γ ∩ bd Ξ . Then, by the definition, � � � N gph Φ (¯ z ) = Lim sup N Γ ( z ) ∪ Lim sup N Ξ ( z ) ∪ Lim sup N gph Φ ( z ) = Γ Ξ Λ z → ¯ z z → ¯ z z → ¯ z ( T Γ ( z ) ∪ T Ξ ( z )) ◦ = � � N Γ ( z ) ∪ Lim sup N Ξ ( z ) ∪ Lim sup Lim sup Γ Ξ Λ z → ¯ z → ¯ z → ¯ z z z � � ( � N Γ ( z ) ∩ � N Γ ( z ) ∪ Lim sup N Ξ ( z ) ∪ Lim sup N Ξ ( z )) . Lim sup Γ Ξ Λ z → ¯ z → ¯ z → ¯ z z z Jiˇ rí V. Outrata (UTIA) 13 / 23