Influence of singularities in rounded corners (Les coins ronds) - PowerPoint PPT Presentation

e IRMAR ` ` e R´ egularit´ e et singularit´ es en optimisation de forme et fronti` eres libres GDR “Applications nouvelles de l’optimisation de forme" 21-23 octobre 2004, Ker Lann Influence of singularities in rounded corners (Les coins ronds) Monique D AUGE . Adaptation libre d’un article avec Gabriel C ALOZ , Martin C OSTABEL et Gr´ egory V IAL Institut de Recherche MAth´ ematique de Rennes http://perso.univ-rennes1.fr/Monique.Dauge

e IRMAR ` ` eLaplace operator in polygonal domains 1 Let Ω be a polygonal domain in R 2 . Let u be solution of the Dirichlet problem ♣ Polygons ♥ Rounded corn. ∆ u = f in Ω u = 0 on ∂ Ω . and ♥ Convergence? At each of the corners c of Ω , the solution u has singular parts: ♥ Starting • If f is smooth and flat enough, and if the sides of Ω are straight ♥ Profiles K sin kπ � a k r kπ/ω ω θ c + O ( r Kπ/ω u = ) , r c → 0 . ♥ Substitution c c k =1 ♥ Multiscale Here: ( r c , θ c ) polar coord. centered in c and ω = ω c opening of Ω at c . ♥ Estimates The coefficients a k depend on f . • If the sides of Ω are curved around c and exponents kπ ♥ Zig-zag ω are not integers: ♥ Cracked K L k � � r ℓ + kπ/ω ϕ k,ℓ ( θ c ) + O ( r Kπ/ω u = a k ) , r c → 0 . c c ♥ Generalizations k =1 ℓ =0 ♥ Coefficients Here the angular functions ϕ k,ℓ ( θ c ) depend on the curved sides. ♥ Conclusion kπ • If some exponents are integers logarithmic terms may appear, except ω in the situation of a curved crack, despite the fact that kπ ω = k 2 can be integer! The above splittings can be realized between Sobolev spaces...

e IRMAR ` ` eRounded corners 2 ♥ Polygons The same Dirichlet problem has no singularities if Ω is smooth... Thus if the corners of the polygon are “rounded" by small arcs of circles. ♣ Rounded corn. Let us define the radius of these circles as a small parameter ε , and index the ♥ Convergence? domain accordingly: Ω ε . The limiting polygon is denoted by Ω 0 . ♥ Starting Example of “the" corner of a circular sector of opening 270 ◦ , rounded with two ♥ Profiles values of ε . ♥ Substitution ♥ Multiscale 1 1 ♥ Estimates Ω ε , ε = Ω ε , ε = 10 20 ♥ Zig-zag ♥ Cracked ♥ Generalizations ♥ Coefficients Let u ε be the solution of the Dirichlet pb in Ω ε and u 0 the solution in Ω 0 . ♥ Conclusion In principle, we expect that u ε → u 0 in energy as ε → 0 . How are the singularities of u 0 hiding inside u ε ?

e IRMAR ` ` eRounded corners... from inside or outside? 3 Example of “the" corner of a circular sector of opening 270 ◦ , ♥ Polygons rounded by two different procedures. ♥ Rounded corn. From inside ( C 1 Bezier curve). ♣ Convergence? From outside (exterior arc of circle). Ω 0 ⊂ Ω ε . Ω ε ⊂ Ω 0 . ♥ Starting In red Ω ε \ Ω 0 . In blue Ω 0 \ Ω ε . ♥ Profiles ♥ Substitution ♥ Multiscale Ω ε Ω ε ♥ Estimates ♥ Zig-zag ♥ Cracked ♥ Generalizations ♥ Coefficients For convex angles (the exterior of the sector) the situation is reversed. ♥ Conclusion What is the meaning of a convergence of u ε towards u 0 ? We try to give answers with the help of multi-scale expansions.

e IRMAR ` ` eStarting the expansion 4 ♥ Polygons Fix the rhs f . Decompose ( “deconstruct" ) u 0 : ♥ Rounded corn. � � u 0 ,N = O ( r N ) . π 2 π ω + a 2 π ω + · · · + u 0 ,N , u 0 = χ ( r ) a π ω S ω S ♥ Convergence? Here χ ≡ 1 near the corner c and χ ≡ 0 outside a region where Ω 0 ♣ Starting kπ coincides with a plane sector of opening ω . The function S is the k -th ω singularity. ♥ Profiles By a dilatation of ratio ε − 1 and Near c , the domains Ω ε are self-similar. ♥ Substitution making ε → 0 ( “blow-up" ) we obtain an infinite domain Q which coincides ♥ Multiscale with a infinite sector of opening ω when R → ∞ and reproduces the pattern of Ω ε at finite distance. ♥ Estimates ♥ Zig-zag ♥ Cracked R → ∞ ♥ Generalizations 1 Ω ε , ε = Q 10 ♥ Coefficients ♥ Conclusion

e IRMAR ` ` eThe fundamental result on profiles 5 ♥ Polygons Let R be the distance to the origin in Q . Relations between slow ( r ) and rapid ( R ) variables through the singularities ♥ Rounded corn. R = r S λ ( r ) = ε λ S λ � r � = ε λ S λ ( R ) ♥ Convergence? and ε ε ♥ Starting The singularities S λ ( r ) can be viewed as profiles at infinity S λ ( R ) , with the singularity exponents λ ∈ { π ω , 2 π ω , 3 π ♣ Profiles ω , · · · } ♥ Substitution Theorem 1. ω , · · · } , there exists a solution K λ ∈ H 1 For λ ∈ { π ω , 2 π ω , 3 π loc ( Q ) to: ♥ Multiscale ∆ K λ = 0 ♥ Estimates  in Q,    K λ = 0 ♥ Zig-zag ∂Q, on  K λ − S λ = O ( R λ )  R → ∞ . ♥ Cracked  ♥ Generalizations Moreover, there exist homogeneous functions K λ, − µ of degree − µ , µ ∈ { π ω , 2 π ω , 3 π ω , · · · } such that for all M > 0 ♥ Coefficients K λ − S λ = K λ, − µ + O ( R − M ) , � ♥ Conclusion R → ∞ . µ = π ω , 2 π ω , ··· , j π ω ≤ M Proof: (i) cut-off by ψ , (ii) variational formulation on Q , (iii) Mellin transform.

e IRMAR ` ` eThe substitution trick 6 ♥ Polygons Recall: χ ≡ 1 near the corner, with support where Ω 0 coincides with a sector ♥ Rounded corn. a λ S λ + u 0 ,N , u 0 ,N = O ( r N ) . � u 0 = χ ( r ) λ ∈ Λ N ♥ Convergence? � π ω , 2 π ω , · · · , j π � Here Λ N = ω ≤ N . ♥ Starting Cut-off ψ : ψ ≡ 1 near infinity, with support where Q coincides with a sector. � x ♥ Profiles � is defined on Ω ε and is ≡ 1 outside a ball B ( c, κε ) . By convention, ψ ε ♣ Substitution Ansatz for u ε a λ ε λ χ ( r ) K λ � r � r � � u 0 ,N + ρ 1 ♥ Multiscale � u ε = + ψ ε , ε ε λ ∈ Λ N ♥ Estimates Theorem 2. ♥ Zig-zag The remainder ρ 1 ε solves ♥ Cracked  � � ∆ ρ 1 ε λ + µ f λ,µ ( x ) + O ( ε N ) , ε = Ω ε , in ♥ Generalizations   λ ∈ Λ N µ ∈ Λ N ♥ Coefficients ρ 1  ε = 0 , ∂ Ω ε . on  ♥ Conclusion 2 π ρ 1 ω ) . ε = O ( ε Consequence: K λ � x − S λ � x using Theorem 1 and ∆ S λ = 0 . � � � ��� Proof: Develop ∆ χ ε ε

e IRMAR ` ` eA multiscale expansion for u ε 7 ♥ Polygons We fix N > 0 . The solution u ε can be split into pieces according to � r � r ♥ Rounded corn. � u 0 ,N + � � ε λ ψ u λ,N − λ u ε = ψ ε ε ♥ Convergence? λ ∈ Λ N χ ( r ) K λ � r ε λ � a λ,µ ε µ � � � � + O ( ε N ) , + a λ + ♥ Starting ε λ ∈ Λ N µ ∈ Λ N ♥ Profiles ε → 0 . as ♥ Substitution The a λ,µ are coefficients. The functions u λ,N − λ are O ( r N − λ ) as r → 0 . ♣ Multiscale ♥ Estimates � x � Support of ψ Support of χ ( r ) ε ♥ Zig-zag ♥ Cracked ♥ Generalizations ♥ Coefficients ♥ Conclusion

eEnergy ( H 1 ) estimates and H 2 estimates e IRMAR ` ` 8 The three pieces (bleu-blanc-rouge) of the H 1 norm of “ u ε − u 0 " ♥ Polygons The worst term (provided a π ω � = 0 ). ♥ Rounded corn. The strength in ε -power. ♥ Convergence? � u 0 � 1 , Ω 0 \ Ω ε � u ε − u 0 � 1 , Ω ε ∩ Ω 0 � u ε � 1 , Ω ε \ Ω 0 ♥ Starting � r � r � r � � �� � π π π π π π ω ( x ) ω χ ( r ) ω K − S S ε ε K ♥ Profiles ω ω ω ε ε ε π π π ♥ Substitution ω ) ω ) ω ) O ( ε O ( ε O ( ε ♥ Multiscale We have a C 1 dependence in ε if and only if ♣ Estimates the “rounded" corners are convex. ♥ Zig-zag The limit u 0 belongs to H 2 (Ω 0 ) iff π ω ≥ 1 . ω � x ♥ Cracked The H 2 -norm of u ε is equivalent (as ε → 0 ) to � ε π π � ω χ ( r ) K � 2 , Ω ε : ε ♥ Generalizations  π 1 ω ≥ 1 if ω � x ω } =  ♥ Coefficients π π π ω − 1 ε min { 0 , 1 − π ω χ ( r ) K � � ε � 2 , Ω ε ≃ ε ε π π ω − 1 ω < 1 . ε if  ♥ Conclusion What happens if the self-similar perturbation at a corner contains itself one or more corners?

e IRMAR ` ` eZig-zag corners 9 ♥ Polygons The perturbed domain for two values of ε , the limit domain, the profile domain. ♥ Rounded corn. ♥ Convergence? 1 1 Ω ε , ε = Ω ε , ε = 10 20 ♥ Starting ♥ Profiles ♥ Substitution ♥ Multiscale ♥ Estimates ♣ Zig-zag R → ∞ ♥ Cracked Ω 0 Q ♥ Generalizations ♥ Coefficients ♥ Conclusion

e IRMAR ` ` eCracked corners 10 ♥ Polygons The perturbed domain for two values of ε , the limit domain, the profile domain. ♥ Rounded corn. ♥ Convergence? 1 1 Ω ε , ε = Ω ε , ε = 10 20 • ♥ Starting • d ε d ε ♥ Profiles ♥ Substitution ♥ Multiscale ♥ Estimates ♥ Zig-zag R → ∞ • ♣ Cracked d Ω 0 Q ♥ Generalizations ♥ Coefficients ♥ Conclusion