Shape optimization of a noise absorbing wall

Transcription

1 Intro Model Well-posedness Shape design Numerics Conclusion Shape optimization of a noise absorbing wall F. Magoulès, K. Nguyen, P. Omnès, A. Rozanova-Pierrat MICS, CentraleSupélec 18 Mars / 34

2 1 Introduction Motivation 2 Model s validation for a fixed porous medium Damping in volume by a porous material Damping by the boundary 3 Well-posedness of the direct problem Regular case Well-posedness in the class of (ε, δ)-domains 4 Shape design problem in Ê 2 Existence of a local minimum ε-optimal domain in a range of frequencies Shape derivative 5 Numerical shape optimization algorithm and results Algorithm Physical principle Stability and Nonuniqueness ε-optimized wall for a fixed range of frequencies 6 Conclusion

3 1 Introduction Motivation 2 Model s validation for a fixed porous medium Damping in volume by a porous material Damping by the boundary 3 Well-posedness of the direct problem Regular case Well-posedness in the class of (ε, δ)-domains 4 Shape design problem in Ê 2 Existence of a local minimum ε-optimal domain in a range of frequencies Shape derivative 5 Numerical shape optimization algorithm and results Algorithm Physical principle Stability and Nonuniqueness ε-optimized wall for a fixed range of frequencies 6 Conclusion

4 Intro Model Well-posedness Shape design Numerics Conclusion Motivation Traffic noise absorbing wall Fractal wall TM, porous material is the cement-wood (acoustic absorbent) Patent Ecole Polytechnique-Colas, Canadian and US patent 4 / 34

5 Intro Model Well-posedness Shape design Numerics Conclusion Motivation Absorption of the Fractal wall 5 / 34

6 1 Introduction Motivation 2 Model s validation for a fixed porous medium Damping in volume by a porous material Damping by the boundary 3 Well-posedness of the direct problem Regular case Well-posedness in the class of (ε, δ)-domains 4 Shape design problem in Ê 2 Existence of a local minimum ε-optimal domain in a range of frequencies Shape derivative 5 Numerical shape optimization algorithm and results Algorithm Physical principle Stability and Nonuniqueness ε-optimized wall for a fixed range of frequencies 6 Conclusion

7 Intro Model Well-posedness Shape design Numerics Conclusion Damping in volume Damping in volume by a porous material J.-F. Hamet and M. Bérengier, Internoise 1993 Damping by the boundary The wave propagation in the porous material can be models by a damped wave equation involving the physical characteristics of the material: the porosity φ, the tortuosity α h the resistivity to the passage of air σ. For Ω = Ω air Ω wall and the interior boundary Γ 1 c0 2 t 2p p = 0, x Ω air ( ) φγ p 2 c0 2 t p + σφ2 γ p c0 2ρ t p φ 0α p = 0, x Ω h αh wall p t=0 = g(x)½ Ωair, t p t=0 = 0, p [p] Γ = [D(x) p n] Γ = 0, = 0. Ω n 7 / 34

8 Intro Model Well-posedness Shape design Numerics Conclusion Damping in volume Damping by the boundary Damping wave equation: a 0 and supp(a) = Ω wall G. Lebeau 1994, S. Cox, E. Zuazua C.P.D.E. 1994, M. Asch et G. Lebeau Experimental Mathematics 2003,... c 1 (x) t 2u (c 2(x) u) + a(x) t u = 0, u t=0 = g(x)½ Ωair, t u t=0 = 0 u n = 0 Ω x Ω t E(t, u) E(0, u) = 2 a(x) t u 2 dx, 0 Ω wall where the energy of u in the time t E(t, u) is defined by E(t, u) = Ω c 1 u t 2 + c 2 x u 2 dx. 8 / 34

9 Intro Model Well-posedness Shape design Numerics Conclusion Damping in volume Helmholtz boundary-valued problem Damping by the boundary Γ N Γ Γ D Ω Porous Γ N u + ω 2 u = f (x) x Ω, u u = g(x) on Γ D, n = 0 on Γ N, u n + α(x, ω)u = 0 on Γ, Re(α) > 0 and Im(α) < 0 9 / 34

10 Intro Model Well-posedness Shape design Numerics Conclusion Damping in volume Damping by the boundary Time depending model (Re(α) > 0 and Im(α) < 0) C. Bardos, J. Rauch, Asymptotic Analysis, 1994 t 2u u = e iωt f, u u ΓD = 0, n = 0, ΓN R(u) Γ = u n Im(α(x)) tu + Re(α(x))u Γ = 0, u t=0 = u 0, t u t=0 = u 1 X 0 (Ω) = (u, v) 2 X 0 (Ω) = { } u H 1 (Ω) u ΓD = 0, R(u) Γ = 0 L 2 (Ω) Ω t ( (u, t u) 2 X 0 (Ω) ( x u 2 + v 2) dx + ) Γ Re(α(x)) u 2 dσ. = 2 Im(α(x)) t u 2 ds. Γ 10 / 34

11 Intro Model Well-posedness Shape design Numerics Conclusion Damping in volume Damping boundary coefficient α 2d minimization problem Damping by the boundary Air Air Porous 11 / 34

12 Intro Model Well-posedness Shape design Numerics Conclusion Damping in volume Damping boundary coefficient α 2d minimization problem Damping by the boundary in Ω air : (µ 1 u 1 ) + ω 2 ρ 1 u 1 = 0 in Ω wall : (µ 2 u 2 ) + ω 2 ρ 2 (1 + ai ω ) u 2 = 0 on Γ : u 1 = u 2 and µ 1 u 1 n = µ 2 u 2 n on the left boundary: u 1 ( L, y) = g(y) in Ω air : (µ 1 u 3 ) + ω 2 ρ 1 u 3 = 0 on Γ : µ 1 u 3 n + αu 3 = 0 on the left boundary: u 3 ( L, y) = g(y) Find α such that for constant A 0 and B 0 A u 1 u 3 2 L 2 (Ω air ) + B (u 1 u 3 ) 2 L 2 (Ω air ) min. 11 / 34

13 Intro Model Well-posedness Shape design Numerics Conclusion Damping in volume Damping boundary coefficient α 2d minimization problem Damping by the boundary Numerically, for the grid size x, we minimize the finite sum of elementary errors e x (α) := k= nπ L,n, L x n L x e k (α). 6 x 10 4 error ω x / 34

14 Intro Model Well-posedness Shape design Numerics Conclusion Damping in volume Damping boundary coefficient α 2d minimization problem Damping by the boundary Real(α) ω x 10 4 Imag(α) ω x / 34

15 1 Introduction Motivation 2 Model s validation for a fixed porous medium Damping in volume by a porous material Damping by the boundary 3 Well-posedness of the direct problem Regular case Well-posedness in the class of (ε, δ)-domains 4 Shape design problem in Ê 2 Existence of a local minimum ε-optimal domain in a range of frequencies Shape derivative 5 Numerical shape optimization algorithm and results Algorithm Physical principle Stability and Nonuniqueness ε-optimized wall for a fixed range of frequencies 6 Conclusion

16 Intro Model Well-posedness Shape design Numerics Conclusion Regular case Fractals Case of a regular boundary Ω = Γ D Γ N Γ L.C. Evans, G. Allaire, F. Magoulès,... Γ N Γ D Ω Γ Γ N u + ω 2 u = f (x) x Ω, u = g(x) on Γ D, u n = 0 on Γ N, u + α(x)u = 0 on Γ, n (1) Re(α) > 0 and Im(α) < 0 are smooth functions on Γ. 13 / 34

17 Intro Model Well-posedness Shape design Numerics Conclusion Regular case Fractals Case of a regular boundary Ω = Γ D Γ N Γ L.C. Evans, G. Allaire, F. Magoulès,... Theorem f L 2 (Ω), g H 1/2 (Γ D ) and ω > 0! u H 1 (Ω) solution of problem (1), continuously depending on the data: C > 0 independing on f and g such that ( ) u H 1 (Ω) C f L 2 (Ω) + g 1. H 2 (ΓD ) In addition, if, for m Æ, Ω C m+2, f H m (Ω) and g H m+ 3 2 (ΓD ), then the weak solution u belongs to H m+2 (Ω). 13 / 34

18 Intro Model Well-posedness Shape design Numerics Conclusion Regular case d-sets and (ε, δ)-domains P.W. Jones, H. Wallin Fractals Definition (d-set) Let Γ be a closed subset of Ê n and 0 < d n. A positive Borel measure m d with support Γ is called a d-measure of Γ if, for some positive constants c 1, c 2 > 0, c 1 r d m d (Γ U r (x)) c 2 r d, for x Γ, 0 < r 1, where U r (x) Ê n denotes the Euclidean ball centered at x and of radius r. The set Γ is a d-set if there exists a d-measure on Γ. Any two d-measures on Γ are equivalent. 14 / 34

19 Intro Model Well-posedness Shape design Numerics Conclusion Regular case d-sets and (ε, δ)-domains P.W. Jones, H. Wallin Fractals Definition ((ǫ, δ)-domain) An open connected subset Ω of Ê n is an (ε, δ)-domain, ε > 0, 0 < δ, if whenever x, y Ω and x y < δ, there is a rectifiable arc γ Ω with length l(γ) joining x to y and satisfying 1 l(γ) x y ε and 2 d(z, Ω) ε x z y z x y for z γ. A Lipschitz domain Ω is an (ε, δ)-domain and also a n-set (i.e., a d-set with d = n). Self-similar fractals (e.g., von Koch s snowflake domain) are examples of (ε, )-domains with the d-set boundary, d > n / 34

20 Intro Model Well-posedness Shape design Numerics Conclusion Regular case d-sets and (ε, δ)-domains P.W. Jones, H. Wallin Fractals Definition For an open set Ω of Ê n, the trace operator Tr is defined for u L 1 loc (Ω) by 1 Tru(x) = lim u(y)dy, r 0 m(ω U r (x)) Ω U r (x) where m denotes the Lebesgue measure. The trace operator Tr is considered for all x Ω for which the limit exists. 14 / 34

21 Intro Model Well-posedness Shape design Numerics Conclusion Regular case d-sets and (ε, δ)-domains P.W. Jones, H. Wallin Fractals For a bounded (ε, δ)-domain Ω Ê n with a closed d-set boundary with β = 1 n d 2 > 0 1 Tr : H 1 (Ω) B 2,2 β ( Ω) L2 ( Ω) is linear continuous and surjective; 2 the Green formula holds (whatever u H 1 (Ω) such that u L 2 (Ω) and v H 1 (Ω)) Ω v udx = u ν, Trv ((B 2,2 β ( Ω)),B 2,2 β Ω ( Ω)) v udx 3 if d = n 1 then β = 1 2 and B2,2 1 2 ( Ω) = H 1 2 ( Ω). 14 / 34

22 Intro Model Well-posedness Shape design Numerics Conclusion Regular case General well-posedness Fractals Theorem Let Ω be a bounded (ε, δ)-domain with a closed d-set boundary Ω = Γ D Γ N Γ. By m d is denoted the d-measure on Ω. Let in addition Re(α) > 0, Im(α) < 0 be smooth functions on Γ. Then f L 2 (Ω), g B 2,2 β (Γ D) and ω > 0! u H 1 (Ω) solution of the following weak problem: v H 1 (Ω) Ω = u vdx dy + αu vdm d Ω Γ f vdx dy + g vdm d. Γ D u vdx dy ω 2 Ω C > 0 independing on f and g such that ( ) u H 1 (Ω) C f L 2 (Ω) + g B 2,2 β (Γ. D) 15 / 34

23 Intro Model Well-posedness Shape design Numerics Conclusion Regular case Well-posedness of an adjoint problem Fractals Let V (Ω) = {u H 1 (Ω) u = 0 on Γ D }. u + ω 2 u = f (x) x Ω, u u = 0 on Γ D, n = 0 on Γ N, u + α(x)u = h(x) on Γ, n 1 f L 2 (Ω) and h V (Ω)!u V (Ω) such that v V (Ω) u vdx dy ω 2 u vdx dy + αu vdm d Ω Ω Γ = f vdx dy + h vdm d. Ω Γ ) 2 C > 0 such that u V (Ω) C ( f L 2 (Ω) + h V (Ω). 3 For m Æ, if (2) Ω C m+2, f H m (Ω) and h H m+1 (Ω) V (Ω), then the weak solution u H m+2 (Ω) V (Ω). 16 / 34

24 1 Introduction Motivation 2 Model s validation for a fixed porous medium Damping in volume by a porous material Damping by the boundary 3 Well-posedness of the direct problem Regular case Well-posedness in the class of (ε, δ)-domains 4 Shape design problem in Ê 2 Existence of a local minimum ε-optimal domain in a range of frequencies Shape derivative 5 Numerical shape optimization algorithm and results Algorithm Physical principle Stability and Nonuniqueness ε-optimized wall for a fixed range of frequencies 6 Conclusion

25 Intro Model Well-posedness Shape design Numerics Conclusion Existence ε-optimal domain Shape derivative Admissible shapes G. Allaire, F. Murat and J. Simon,... x Γ 0 (x + θ(x)) Γ = (Id+θ)Γ 0 following θ W 1, (Ê 2, Ê 2 ). C(Ω 0 ) = {Ω Ê 2 θ W 1, (Ê 2, Ê 2 ), θ W 1, (Ê 2,Ê 2 ) < 1 such that Ω = (Id + θ)ω 0 }. d(ω, Ω 0 ) is the psedo-distance on C(Ω 0 ): d(ω, Ω 0 ) = inf T T T(Ω 0 )=Ω ( T Id W 1, (Ê 2,Ê 2 )+ T 1 Id W 1, (Ê 2,Ê 2 )) with the following space T of diffeomorphisms on Ê 2 : T = Id + θ with θ W 1, (Ê 2,Ê 2 ) < / 34

26 Intro Model Well-posedness Shape design Numerics Conclusion Existence ε-optimal domain Shape derivative Admissible shapes G. Allaire, F. Murat and J. Simon,... J(Ω, u) = A u 2 dx + B u 2 dx + C u 2 dσ min Ω Ω Γ for the domains Ω U ad for a fixed R > 0 U ad = {Ω C(Ω 0 ) d(ω, Ω 0 ) R, Γ D Γ N Ω and Vol(Ω) = Vol(Ω 0 )} with A 0, B 0, C 0 positive constants for all fixed ω > / 34

27 Intro Model Well-posedness Shape design Numerics Conclusion Existence ε-optimal domain Shape derivative Existence of an optimal shape Since u and J are continuous functions of Ω and U ad is closed, we have Theorem For a fixed frequency ω > 0, the shape optimization problem inf J(Ω) Ω U ad has at least one minimum point (there exists at least one optimal shape of Γ). Approach of F. Murat and J. Simon [Optimal Design, Optimization Techniques Modeling and Optimization in the Service of Man (2005)], see also A. Henrot, M. Pierre, [Springer, 2005], D. Bucur and A. Giacomini, [Ann. Inst. H. Poincaré Anal. Non Linéaire (2015)]. 19 / 34

28 Intro Model Well-posedness Shape design Numerics Conclusion Existence ε-optimal domain Shape derivative ε-optimal domain in a range of frequencies Definition The domain Ω U ad is called the ε-optimal domain for the range of frequencies [ω 0, ω 1 ], if for all ω [ω 0, ω 1 ] it holds min Ω U ad J(Ω)(ω) J(Ω )(ω) < ε, (3) where by J(Ω )(ω) is denoted the value of the functional J calculated for the domain Ω at the frequency ω. ε > 0 δ > 0 such that since ω ω < δ we have min Ω U ad J(Ω)(ω) J(Ω )(ω) min Ω U ad J(Ω)(ω) J(Ω )(ω ) + J(Ω )(ω ) J(Ω )(ω) η+ˆη = ε. 20 / 34

29 Intro Model Well-posedness Shape design Numerics Conclusion Existence ε-optimal domain Shape derivative ε-optimal domain: motivation Let ω, ω [ω 0, ω 1 ] for which Ω and Ω are optimal. 1 u depends continuously on ω. 2 J is equicontinuous on the compact [ω 0, ω 1 ] Ê + : η > 0 δ 1 (η) > 0 : ω ω < δ 1 J(Ω )(ω ) J(Ω )(ω) < η. 3 J is continuous as a function of Ω: ˆη > 0 δ 2 > 0 : ω ω < δ 2 d(ω, Ω ) < δ 3 J(Ω )(ω ) J(Ω )(ω ) < ˆη. ε > 0 δ > 0 such that since ω ω < δ we have min Ω U ad J(Ω)(ω) J(Ω )(ω) min Ω U ad J(Ω)(w) J(Ω )(ω ) + J(Ω )(ω ) J(Ω )(ω) ˆη+η = ε. 21 / 34

30 Intro Model Well-posedness Shape design Numerics Conclusion Existence ε-optimal domain Shape derivative Physical principle: caracteristic optimal shape size is compared to the wavelength Physical principle A wave with a wavelength λ 0 does not fit into the shape of the caracterestic scale much smaller than λ 0. Therefore, we update the definition of the admissible domains. ω 1 < ω 2 <... < ω k <... λ 1 > λ 2 >... > λ k >... ω k +, k λ k = 2π 0 ω k U ad (λ k ) = {Ω C(Ω k 1 ) d(ω, Ω k 1 ) R(λ k ), and Vol(Ω) = Vol(Ω k 1 )} Γ D Γ N Ω 22 / 34

31 Intro Model Well-posedness Shape design Numerics Conclusion Existence ε-optimal domain Shape derivative Physical principle: caracteristic optimal shape size is compared to the wavelength Assumption for the optimization processus Let Ω o be a ε-optimal domain in [ω a, ω b ]. There exists ω big ω b such that for all ω ω big, if Ω is the optimal domain for the fixed frequency ω, it is ε-optimal on [ω a, ω b ]: ω [ω a, ω b ] J(Ω o, ω) J(Ω, ω) < ε. Numerically, we have ω big = 2ω b 22 / 34

32 Intro Model Well-posedness Shape design Numerics Conclusion Existence ε-optimal domain Shape derivative Physical principle: caracteristic optimal shape size is compared to the wavelength Let l(ω ) be the biggest characteristic geometry size of Γ. Biggest scale of ε-optimal shape If Ω is ε-optimal for the frequency ω, i.e. δ > 0 : ω ω < δ J(Ω )(ω) inf Ω U ad J(Ω)(ω ) < ε, then either l(ω ) λ 2 = π ω, or Ω is in a small neighbourhood of a domain Ω λ with l(ω λ ) λ / 34

33 Intro Model Well-posedness Shape design Numerics Conclusion Existence ε-optimal domain Shape derivative Numerical illustration of the physical principle Ω = Ω 0 Ω 1 = [0, 1] [ 2, 2] Energy 1 Γ Γ Γ Time 23 / 34

34 Intro Model Well-posedness Shape design Numerics Conclusion Existence ε-optimal domain Shape derivative Existence of global ε-optimal domains Theorem Let us consider the following minimization problem: for a fixed ε > 0 to find a ε-optimal domain for all frequency ω [ω 1, ω max ] with ω 1 > 0. If ω max < then there exists at least one ε-optimal domain Ω ε U ad(λ min ). If ω max = then there exists at least one ε-optimal domain Ω ε belonging to the class of (ε, δ)-domain. 24 / 34

35 Intro Model Well-posedness Shape design Numerics Conclusion Existence ε-optimal domain Shape derivative Existence of global ε-optimal domains 1 For ω 1 Ω 1 optimal for J(Ω, ω 1 ) in U ad (Ω 0, λ 1 ). J is continuous on ω and Ω ω 2 > ω 1 s. t. Ω 1 is ε-optimal on [ω 1, ω 2 ]. 2 For k 2 until ω k < ω max If ω k ω big (ω k ω k 1 ) then Ω k is optimal for J(Ω, ω k ) in U ad (Ω k 1, λ k ) and it is ε-optimal on [ω 1, ω k+1 ]. Else, we update Ω k 1 until ω k ω big by J is equicontinuous on [ω k 1, ω big ] N Æ s.t. for ω i+1 = ω i + ω big ω k 1 N a solution Ω k of the optimization problem inf Ω U ad (Ω k 1, λ k 1 +λ k +...+λ k+n 1 N 1 is ε-optimal on [ω 1, ω big ]. k+n 1 ) [ i=k 1 J(Ω)(ω i )] 24 / 34

36 Intro Model Well-posedness Shape design Numerics Conclusion Existence ε-optimal domain Shape derivative Shape derivative of the functional for a fixed frequency Definition The shape derivative of a function K(Ω) : C(Ω 0 ) Ê at Ω 0 is defined as the Frechet derivative in W 1, (Ê 2, Ê 2 ) at 0 of the function θ K (Id + θ) (Ω 0 ), i.e., K (Id + θ) (Ω 0 ) = K(Ω 0 ) + K (Ω 0 )(θ) + o(θ) with o(θ) L lim (Ê 2 ) θ 0 θ W 1, (Ê 2,Ê 2 ) = 0, where K (Ω 0 ) is a continuous linear form on W 1, (Ê 2, Ê 2 ). 25 / 34

37 Intro Model Well-posedness Shape design Numerics Conclusion Existence ε-optimal domain Shape derivative Shape derivative of the functional for a fixed frequency J 1 (Ω, u) =A u 2 dx + B u 2 dx + C u 2 dσ Ω Ω Γ + µ(vol(ω) Vol(Ω 0 )) 2 Direct problem x Ω 0 Adjoint problem x Ω 0 u + ω 2 u = f w + ω 2 w = 2 (Aū(Ω 0 ) B ū(ω 0 )) u ΓD = g w ΓD u = 0 w n = 0 ΓN n = 0 ΓN u n + αu w = 0 Γ0 n + αw = 2Bᾱū(Ω 0 ) + 2Cū(Ω 0 ) Γ0 25 / 34

38 Intro Model Well-posedness Shape design Numerics Conclusion Existence ε-optimal domain Shape derivative Shape derivative of the functional for a fixed frequency Theorem ( Ω 0 = Γ 0 Γ D Γ N C 3, Ω C(Ω 0 ) : Γ = (Id + θ)γ 0 ) Let u(ω 0 ) H 3 (Ω 0 ) with g H 5 2 (Γ D ) and f H 1 (Ê 2 ). Then ( J 1 (Ω 0)(θ) = θ n A u(ω 0 ) 2 + B u(ω 0 ) 2 + 2B α 2 u(ω 0 ) 2 Γ 0 4CRe(α) u(ω 0 ) 2 + CH u(ω 0 ) 2) ds ( + θ nre u(ω 0 ) w + ω 2 u(ω 0 )w fw αhu(ω 0 )w Γ 0 ) +2α 2 u(ω 0 )w ds + 2µ θ n(vol(ω) Vol(Ω 0 ))ds Γ 0 with n the exterior normal vector and H the curvature on Γ 0, w V (Ω 0 ) the unique solution of the adjoint problem. 1) Lagrangian approach 2) using of the Eurien and the Lagrangian derivatives [A. Henrot, M. Pierre, [Springer, 2005]; G. Allaire, [Springer, 2007]] 25 / 34

39 1 Introduction Motivation 2 Model s validation for a fixed porous medium Damping in volume by a porous material Damping by the boundary 3 Well-posedness of the direct problem Regular case Well-posedness in the class of (ε, δ)-domains 4 Shape design problem in Ê 2 Existence of a local minimum ε-optimal domain in a range of frequencies Shape derivative 5 Numerical shape optimization algorithm and results Algorithm Physical principle Stability and Nonuniqueness ε-optimized wall for a fixed range of frequencies 6 Conclusion

40 Intro Model Well-posedness Shape design Numerics Conclusion Algorithm Physical principle Properties ε-optimized wall Numerical optimization algorithm Conjugate descend gradient method J 1(Ω 0 )(θ) = (θ n)( V )ds, Γ 0 if V is the velocity of the outward normal direction to the Robin boundary, θ n = V, then J 1 (Ω 0)(θ) = V 2 ds < 0. Γ 0 Consequently, the optimality condition can be given by J 1 (Ω 0)(θ) / 34

41 Intro Model Well-posedness Shape design Numerics Conclusion Algorithm Physical principle Properties ε-optimized wall Numerical optimization algorithm Conjugate descend gradient method 1 Solving of the Helmholtz problem on a fine mesh with a center finite difference scheme in a fixed domain D including all minimizing domains Ω; 2 Calculation of the velocity V for the movement of the Robin boundary Γ and extension of the velocity to all D using a level set method on a coarse mesh; 3 Solving level set equation (of Hamilton-Jacobi type), update the shape of the initial domain using the coarse mesh. 27 / 34

42 Intro Model Well-posedness Shape design Numerics Conclusion Algorithm Physical principle Properties ε-optimized wall Numerical parameters Let l = 1. Global domain: D = [0, 3l] [0, l] Flat shape: Ω 0 = [0, 2l] [0, l] Helmholtz equation for the wave number k = ω c 0 and f = 0, i.e, u + k 2 u = 0, where c 0 is the sound speed propagation in the air g(y) = 1 2π exp ( (y 1/2)2 2 ) for y [0, l] α(ω) from the minimization flat problem for ISOREL as a porous material 28 / 34

43 Intro Model Well-posedness Shape design Numerics Conclusion Algorithm Physical principle Properties ε-optimized wall Numerical parameters Let l = 1. Global domain: D = [0, 3l] [0, l] Flat shape: Ω 0 = [0, 2l] [0, l] Helmholtz equation for the wave number k = ω c 0 and f = 0, i.e, u + k 2 u = 0, where c 0 is the sound speed propagation in the air g(y) = 1 2π exp ( (y 1/2)2 2 ) for y [0, l] α(ω) from the minimization flat problem for ISOREL as a porous material The smallest wavelength excited by g is λ = l 2 Fine mesh: h = l 64 (for the Helmholtz system) Coarse mesh: κ = 2h = l 32 (for the level set) Note that κ λ Γ [ 3l 2, 2l] [0, l] 28 / 34

44 Intro Model Well-posedness Shape design Numerics Conclusion Algorithm Physical principle Properties ε-optimized wall Numerical of the physical principle R R R illustration J(Ω)(ω) = Ω u 2 dx + Ω u 2 dx + Re(α) Γ u 2 ds, ω0 = 3170 Ω1 Ω10 Ω19 Ω26 Ω1 Ω10 Ω19 Ω26 Ω1 Ω10 Ω19 Ω26 u 2 u 2 u 2 + u 2 29 / 34

45 Intro Model Well-posedness Shape design Numerics Conclusion Algorithm Physical principle Properties ε-optimized wall Numerical of the physical principle R R R illustration J(Ω)(ω) = Ω u 2 dx + Ω u 2 dx + Re(α) Γ u 2 ds, ω0 = 3170 Ω 1 Ω 3 Ω 5 Ω 7 Ω 1 Ω 3 Ω 5 Ω 7 Ω 1 Ω 3 Ω 5 Ω 7 u 2 u 2 u 2 + u 2 29 / 34

46 Intro Model Well-posedness Shape design Numerics Conclusion Algorithm Physical principle Properties ε-optimized wall Numerical illustration of the physical principle J(Ω)(ω) = Ω u 2 dx + Ω u 2 dx + Re(α) Γ u 2 ds, ω 0 = X: 3170 Y: X: 3170 Y: ω Blue J(Ω 0 )(ω), Green J(Ω 26 )(ω), Red J(ˆΩ 7 )(ω) J(Ω 0 )(ω 0 )/J(ˆΩ 7 )(ω 0 ) = / 34

47 Intro Model Well-posedness Shape design Numerics Conclusion Algorithm Physical principle Properties ε-optimized wall Stability of the optimization algorithm J(Ω)(ω) = Ω u 2 dx, ω 0 = 3170, J(Ω a )(ω 0) = J(Ω b )(ω 0) = , J(Ω 0)(ω 0)/J(Ω a )(ω 0) = Ω a 1 Ω a Ω b 1 Ω b J 1 J original J 1 J original x Ω b 18 5 x vol vol vol vol Ω b / 34

48 Intro Model Well-posedness Shape design Numerics Conclusion Algorithm Physical principle Properties ε-optimized wall Non unicity of the optimimal shape J(Ω)(ω) = Ω u 2 dx, ω 0 = 3170, J(Ω 0 )(ω 0) = J(Ω c )(ω 0) = Ω 0 1 Ω 0 Ω c 1 Ω c 31 / 34

49 Intro Model Well-posedness Shape design Numerics Conclusion Algorithm Physical principle Properties ε-optimized wall Non unicity of the optimimal shape J(Ω)(ω) = Ω u 2 dx, ω 0 = 3170, J(Ω 0 )(ω 0) = J(Ω c )(ω 0) = X: 3170 Y: objective function X: Y: ω Blue J(Ω 0 1 )(ω), Green J(Ω0 )(ω), Red J(Ω c )(ω) 31 / 34

50 Intro Model Well-posedness Shape design Numerics Conclusion Algorithm Physical principle Properties ε-optimized wall ε-optimized simple wall J(Ω)(ω) = Ω u 2 dx, ω [3000, 6000]. A = 1; B = 0; C =0; 4.5 X: 3170 Y: flat shape shape 2 4 shape 3 shape X: 3410 Y: shape 5 shape 6 shape 7 objective function X: 3380 Y: X: 3540 Y: X: 3625 Y: X: 4025 Y: X: 4120 Y: X: 4555 Y: X: 4240 Y: ω 32 / 34

51 Intro Model Well-posedness Shape design Numerics Conclusion Algorithm Physical principle Properties ε-optimized wall ε-optimized simple wall J(Ω)(ω) = Ω u 2 dx, ω [3000, 6000]. 32 / 34

52 Intro Model Well-posedness Shape design Numerics Conclusion Algorithm Physical principle Properties ε-optimized wall ε-optimized simple wall J(Ω)(ω) = Ω u 2 dx, ω [3000, 6000]. 5 A = 1; B = 0; C =0; flat shape shape 7 shape 8 objective function ω 32 / 34

53 Intro Model Well-posedness Shape design Numerics Conclusion Algorithm Physical principle Properties ε-optimized wall ε-optimized simple wall J(Ω)(ω) = Ω u 2 dx, ω [3000, 6000]. 5 A = 1; B = 0; C =0; flat shape shape 7 shape objective function ω 32 / 34

54 1 Introduction Motivation 2 Model s validation for a fixed porous medium Damping in volume by a porous material Damping by the boundary 3 Well-posedness of the direct problem Regular case Well-posedness in the class of (ε, δ)-domains 4 Shape design problem in Ê 2 Existence of a local minimum ε-optimal domain in a range of frequencies Shape derivative 5 Numerical shape optimization algorithm and results Algorithm Physical principle Stability and Nonuniqueness ε-optimized wall for a fixed range of frequencies 6 Conclusion

55 Intro Model Well-posedness Shape design Numerics Conclusion Conclusion Optimal shape for a fixed λ has λ/2 caracterestic scaling ε-optimality of the shape multi-scale of the boundary Existence of a global ε-optimal shape in the class of d-sets Ill-posedness (nonunicity of the (ε-) optimal shape) 34 / 34