Optimal design and numerics Enrique Zuazua Ikerbasque & BCAM - - PowerPoint PPT Presentation

1

Optimal design and numerics

Enrique Zuazua

Ikerbasque & BCAM & CIMI - Toulouse Bilbao, Basque Country, Spain zuazua@bcamath.org http://www.bcamath.org/zuazua/

Talk # 2, CIMI Lecture Series, April 8, 2014

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

Table of Contents

1

Motivation

2

Finite elements in Optimal Design

3

Numerics for Homogenization Motivation The 1 − d case The continuous Bloch wave decomposition The Discrete Bloch wave decomposition Conclusion

4

Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Finite element approximation

5

Concluding remarks

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

Motivation Optimal Design problems arise in most relevant engineering applications.

Design of flexible structures. Aeronautics. Optical fibers, wave guides. Medicine, Biology,...

A challenge: to develop efficient numerical methods. Optimal design problems often lead to minimizing sequences developing oscillating patterns. Risk of failure of numerical methods because of the resonance phenomena. Generally speaking, there is a big gap between the existing theory for continuum analytical methods for optimal design and the numerical practice.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

The topic is closely related to that of Inverse problems, Control and Optimization. Henrot, Antoine; Pierre, Michel Variation et optimisation de

• formes. Une analyse g´

eom´

• etrique. Math´

ematiques & Applications (Berlin), 48. Springer, Berlin, 2005. Bucur, Dorin; Buttazzo, Giuseppe, Variational methods in shape optimization problems. Progress in Nonlinear Differential Equations and their Applications, 65. Birkh¨ auser Boston, Inc., Boston, MA, 2005. Choulli, Mourad, Une introduction aux probl` emes inverses elliptiques et paraboliques, Math´ ematiques & Applications, 65, Springer, 2009. Mohammadi, Bijan; Pironneau, Olivier, Applied shape

• ptimization for fluids. Second edition. Numerical

Mathematics and Scientific Computation. Oxford University Press, Oxford, 2010.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

A number of analytical tools have been developed to derive

• ptimality conditions. In particular the so-called Hadamard shape

derivative2 and the topological derivatives3 These tools serve also to build gradient descent strategies. But they often fail to be efficient because of the pollution generated by the high frequency spurious numerical oscillations4

• 2J. Simon, Differentiation with Respect to the Domain in Boundary Value

Problems, Numerical Functional Analysis and Optimization Volume 2, Issue 7-8, 1980

3St´

ephane Garreau, Philippe Guillaume, and Mohamed Masmoudi The Topological Asymptotic for PDE Systems: The Elasticity Case SIAM J. Control Optim., 39(6), 1756 D1778.

• 4G. Dogan, P. Morin, R.H. Nochetto, M. Verani. Discrete gradient flows for

shape optimization and applications, Comput. Methods Appl. Mech. Engrg. 196 (2007) 3898 D3914.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Zubi zuri bridge, Bilbao

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

Table of Contents

1

Motivation

2

Finite elements in Optimal Design

3

Numerics for Homogenization Motivation The 1 − d case The continuous Bloch wave decomposition The Discrete Bloch wave decomposition Conclusion

4

Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Finite element approximation

5

Concluding remarks

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

OPTIMAL DESIGN ∼ OPTIMIZATION PROCESS INVOLVING GEOMETRIES AND SHAPES.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

Elliptic optimal design

Control = Shape of the domain5 State equation = Diric Dimension n = 2, V. ˘ Sver` ak6: There exists an optimal domain in the class of all open subsets of a given bounded open set, whose complements have a uniformly bounded number of connected components. Key point: compactness of this class of domains with respect to the complementary-Hausdorff topology and the continuous dependence of the solutions of the Dirichlet laplacian in H1 with respect to it.

5W

• r k

i n c o l l a b o r a t i o n w i t h D . C H E N A I S . F i n i t e E l e m e n t A p p r o x i m a t i o n

• f

2 D E l l i p t i c Op t i m a l D e s i g n , J M P A , 8 5 ( 2 0 0 6 ) , 2 2 5 - 2 4 9 .

6V . ˘S v e r `a k , On

• p t i m

a l s h a p e d e s i g n , J . M a t h . P u r e s . A p p . , 7 2 , 1 9 9 3 , p p . 5 3 7 - 5 5 1 .

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

L’Avion III de Clément Ader, 1897 (Muret 1841 - Toulouse 1925) CNAM Museum, Paris

jueves, 17 de julio de 14

On peut conclure que, ce 14 octobre 1897, le Français Clément Ader aurait peut être effectué un décollage motorisé – mais non contrôlé – d'un plus lourd que l'air. Le ministère de la Guerre cesse de financer Ader, qui est contraint d'arrêter la construction de ses prototypes.

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

An example: To choose the optimal (with respect to some cost function associated to the Dirichlet laplacian) domain Ω within the class of domains, embedded in the pav´ e D and containing the subdomain ω.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

It is well known that, when the number of holes is unlimited, homogenization phenomena arise and the minimum is not

• achieved. Cioranescu-Murat: −∆ → −∆ + µ.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

COMMON COMPUTATIONAL/NUMERICAL PRACTICE: Continous optimal design → discrete finite-element version. Compute the discrete optimal shape (discrete optimization or shape and topological derivatives, level set methos,...) The choice of one method or another depends very much on the expertise and computational capacities. THE PROBLEM: Do these methods converge? IN THIS PARTICULAR CASE, YES!

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

OPTIMAL SHAPE DESIGN+NUMERICS = NUMERICS+OPTIMAL SHAPE DESIGN

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

This is a proof of the efficiency that most methods employed to solve optimal design problem computationally exhibit. One may use different tools at the discrete level:

• Shape derivatives;
• Topological derivatives;
• Discrete Optimization;
• Level set methods.

This is so despite the geometric complexity of optimal shapes.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

We consider a finite-element discrete version of this problem and prove that the discrete optimal domains converge in that topology towards the continuous one. Key point : finite-element approximations of the solution of the Dirichlet laplacian converge in H1 whenever the polygonal domains converge in the sense of Hc-topology. This provides a rigorous justification to the most common engineering to optimal design.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

The triangulation of the pav´ e and the fixed subdomain (constraint) from which all admissible discrete domains have to be built. The class of admissible domains for the discrete problem. This time the admissible domains need to be unions of triangles from the discrete mesh.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

Let D be a bounded open lipschitz connected subset of R2. Denote by ]cΩ the number of connected components of D \ Ω. For a fixed N ∈ N, we consider the family of admissible domains ON = {Ω ⊂ D; Ω open, ]cΩ ≤ N}, and the Hc-topology defined by the metric dHc(Ω, Ω0) = max{ max

x2D\Ω

d(x, Ω0), max

x02D\Ω0 d(x0, Ω)}.

ON is Hc-compact.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

For f 2 H−1(D) and Ω 2 ON, let yΩ be the solution of the Dirichlet problem in Ω: ∆y = f in Ω; y = 0 on ∂Ω;

• r, in variational form,

yΩ 2 H1

0(Ω);

Z

Ω

ryΩ·rz = < f , z >H−1(Ω),H1

0(Ω),

8z 2 H1

0(Ω).

Optimal design problem: min

Ω∈ON j(Ω),

with j(Ω) =< f , yΩ >H−1(Ω),H1

0(Ω)=

Z

Ω

|ry|2dx. ˘ Sver` ak proved that a minimizer Ω? does exist in ON.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

For each h > 0 we introduce a regular triangular mesh (Th)h of the domain D of size h and the family ON

h of polygonal open subsets

• f D union of triangles in (Th)h belonging to the class ON.

The finite-element spaceVh(Ωh) ⇢ H1

0(Ωh) constituted by

continuous and piecewise P1 polynomial (over triangles). The Galerkin finite-element approximation: yh 2 Vh(Ωh); Z

Ωh

ryh · rzh =< f , zh >H−1,H1

0(Ωh),

8zh 2 Vh(Ωh). The discrete optimal design problem: min

Ω∈ON

h

j(Ω), with jh(Ω) = Z

Ωh

|ryh|2dx.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

The ingredients of the proof: Γ-convergence FACT 1: Any Ω ∈ ON, can be approximated by Ωh ∈ ON

h as

follows: F = D \ Ω, Fh = [

T2Th,T\F6=;

T, Ωh = D \ Fh. It is easy to prove that dHc(Ωh, Ω) h!0 − → 0, and ]cΩh ≤ N for any h. We then prove that e yh

h!0

− → f yΩ strong − H1(D) jh(Ωh) :=< f , yh >H−1(Ω ),H1

0(Ω h)

h!0

− → j(Ω). This result guarantees the convergence of the Galerkin finite-element approximations with respect to the Hc- convergence

• f domains.

FACT 2: The same occurs if Ωh is an arbitrary sequence of admissible domains Hc-converging to Ω. FACT 3: The existence of a minimizer Ω?

h for jh in the class ON h

for each h > 0 is obvious since ON

h has a finite number of elements.

FACT 4: The results above and a standard Γ-convergence

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

This theoretical result ignores the computational complexity of the problem of computing optimal shapes. Note that if one proceeds simply by a discrete algorithm the number of possible subdomains to be considered in a rectangular grid of 1/h2 elements is of the order of 21/h2, which is simply out

• f our computing capacities.... The curse of dimensionality....

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

CONCLUSION We have presented a frame in which most optimal design problems can be reformulated at the discrete level and their numerical convergence analyzed. We have proved a convergence result that uses in a critical manner the fact that the boundary conditions are of Dirichlet

• type. The problem is much more sensible in the Neumann

case. This proof of convergence does not provide an efficient computational method. Rather this would require the implementation of gradient like algorithms with the filtering of possible high frequency numerical spurious oscillations.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

• G. Dogan, P

. Morin, R.H. Nochetto, M.

• Verani. Discrete

gradient flows for shape optimization and applications,

• Comput. Methods Appl. Mech. Engrg. 196 (2007) 3898–3914.

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

Further comments Note that the problem of optimal design, through the notion

• f Hadamard shape derivative, is closely linked to that of

boundary control. The methods we have presented here apply also to evolution equations, provided the shapes under consideration are independent of time. Considering shapes that may vary in time would require significant further work.7

7Delfour, M. C.; Zolsio, J.-P. Shapes and geometries. Metrics, analysis,

differential calculus, and optimization. Second edition. Advances in Design and Control, 22. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. xxiv+622 pp.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

Table of Contents

1

Motivation

2

Finite elements in Optimal Design

3

Numerics for Homogenization Motivation The 1 − d case The continuous Bloch wave decomposition The Discrete Bloch wave decomposition Conclusion

4

Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Finite element approximation

5

Concluding remark

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

Motivation Numerical approximation methods for PDEs with rapidly

• scillating coefficients.

There is an extensive literature in which ideas and methods of classical Numerical Analysis (finite differences and elements) and Homogenization Theory are combined: Bensoussan-Lions-Papanicolaou, Sanchez-Palencia, Allaire, Cioranescu-Donato,....

• B. Engquist [1997,1998], Y. Efendiev, Th. Hou, X.Wu

[1998,1999, 2002,2004], M. Matache, Babuˇ ska, Ch. Schwab [2000,2002], G. Allaire, C. Conca[1996], C. Conca, S. Natesan, M. Vanninathan [2001,2005], P. Gerard, P.A. Markowich, N. J. Mauser, F. Poupaud [1997], Kozlov [1986], Piatnitski, Remi [2001], ...

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

Some common facts: Multiscale analysis: Two scales are involved: ε for the size of the microstructure and h for that of the numerical mesh; As usual, three different regimes: h << ε, h ∼ ε, ε << h; Slow convergence of standard approximations (finite elements, finite differences): h << ε. Resonances may occur when ε ∼ h Convergence may be accelerated when the Galerkin method is built on bases adapted to the “topography” of the oscillating medium.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

Two different issues: Compute an efficient numerical approximation of the solution in the highly heterogeneous medium; Homogenization theory is a tool that helps doing that. Analyze the limit behavior as the characteristic size of the medium and the mesh-size tend to zero. BUT A COMPLETE UNDERSTANDING OF THIS COMPLEX ISSUE NEEDS BOTH QUESTIONS TO BE ADDRESSED.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

Convergence of the standard numerical methods improves when the numerical mesh samples the oscillating medium in an “ergodic way”:

• B. Engquist, Th. Hou [1989,1993], M.Avellaneda, Th. Hou, G.

Papanicolaou [1991], Babuˇ ska, Osborn [2000]. In other words: According to classical homogenization theory: uε converges to the homogenized solution u∗ as ε → 0; This is not necessarily the case for the numerical solution uε

h

as both h, ε → 0. Under some ergodicity condition (ε/h = irrational) uε

h → u∗.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

In 8 we explain what is going on when ε/h = rational and how, using diophantine approximation, one can recover convergence for irrational ratios.

• 8R. Orive and E. Z. Finite difference approximations of homogenization

problems for elliptic equations. Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, 4 (1) (2005) pp. 36-87.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

Problem formulation: We consider the periodic elliptic equation associated to the following rapidly oscillating coefficients: A" = − ∂ ∂xk ✓ a"

k`(x) ∂

∂x` ◆ , with a"

k`(x) = ak` (x/ε), and ak` satisfying

8 > > > < > > > : akl ∈ L∞

# (Y ) are Y -periodic, where Y =]0, 1[N,

∃α > 0 s.t.

N

P

k,`=1

akl(y)ηk ¯ ηl ≥ α|η|2, ∀η ∈ CN, akl = alk ∀l, k = 1, ..., N. Homogenization: u∗ limit of the solutions of A"u" = f , satisfies A∗u∗ = − ∂ ∂xk ✓ a∗

k`

∂u∗ ∂x` ◆ = f .

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

Discretization: Let h = (h1, . . . , hd) with hi = 1 ni with ni 2 N. The following is a natural numerical approximation scheme by finite-differences:

d

X

i,j=1

r−h

i

h aε

ij(x(i, j))r+h j

uε

h(x)

i = f (x), x 2 Γh, where Γh is the numerical mesh and x(i, j) = x + 1 2hiei + (1 δij)1 2hjej.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

Classical Numerical Analysis ensures ||uε

h − u⇤|| ≤ c h

ε + c0ε. Note that, in particular, no convergence is guaranteed for h ∼ ε.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

Convergence under ergodicity: In Avellaneda, Hou, Papanicolaou [1991] for the 1 − d problem with Dirichlet conditions the following was proved: Theorem If f is continuous and bounded in (0, 1), then lim

✏,h→0 ||u✏ h − u∗||∞ → 0,

for sequences h, ✏ such that h/✏ = r with r irrational. Our goal: Analyze the behavior when ✏/h=rational; Reprove the same result as in the Theorem above using diophantine approximation. Do it using explicit Bloch wave representations of solutions.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

More precisely: what is the behavior of u✏

h when

h ✏ = q p , with q, p 2 N, H.C.F.(q, p) = 1, and h ! 0????????????. In this case the numerical mesh, despite of the fact that h ! 0,

• nly samples a finite number of values in each periodicity cell of

the coefficient a(x). Thus, it is impossible that the numerical schemes recovers the continuous homogenized limit u∗. One rather expects a discrete homogenized limit u∗

q/p such that

u∗

q/p 6= u∗;

u∗

q/p ! u∗ as q/p ! r, with r irrational.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

Main 1 − d result Theorem Assume that a = a(x) is Lipschitz, 1-periodic and ↵ ≤ a(x) ≤ . Let {u✏

h(xi)}n i=0 the approximation of u✏ with h/✏ = q/p. Then,

||u✏

h − u∗ q/p||∞ ≤ c hp

Moreover, u∗

q/p is a discrete Fourier approximation with mesh-size

h of the solution of 8 < : −a∗

p

@2v @x2 (x) = f (x), 0 < x < 1, v(0) = v(1) = 0, with a∗

p = 1 p p

P

j=1 1 a((j+1/2)/p)

!−1 .

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

Recall that the continuous homogenized solution u⇤ is a solution of the same Dirichlet problem but with a continuous effective coefficient a⇤ defined as a⇤ = ✓Z 1 (1/a(x))dx ◆1 . Furthermore, ||u⇤

q/p − u⇤||1 ≤ c0 1

p. In conclusion, ||u✏

h − u⇤||1 ≤ c hp + c0/p

where c and c0 depend on ↵, , ||a0||1 and ||f ||1. Note that, this estimate, together with diophantine approximation results, allows to recover convergence for h/✏ irrational. A similar result holds in the multi-dimensional case.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

Continuous Bloch wave decomposition Following the presentation by C. Conca & M. Vanninathan9: Spectral problem family with parameter η ∈ Y 0 = [−1/2, 1/2[d: Aψ(·; η) = λ(η)ψ(·; η) in Rd, ψ(·; η) is (η, Y )-periodic, i.e., ψ(y + 2πm; η) = e2πim·ηψ(y; η). ψ(y; η) = eiy·ηφ(y; η), φ being Y -periodic in the variable y.

• 9C. Conca and M. Vanninathan, Homogenization of periodic structures

via Bloch decomposition, SIAM J. Appl. Math., 57 (1997), pp. 1639–1659.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

A discrete sequence of eigenvalues with the following properties exists: ⇢ 0 ≤ λ1(η) ≤ · · · ≤ λn(η) ≤ · · · → ∞, λm(η) is a Lipschitz function of η ∈ Y 0, ∀m ≥ 1. λ2(η) ≥ λ(N)

2

> 0, ∀η ∈ Y 0, where λ(N)

2

> 0 is the second eigenvalue of A in the cell Y with Neumann boundary conditions. The eigenfunctions ψm(·; η) and φm(·; η), form orthonormal bases in the subspaces of L2

loc(Rd) of (η, Y )-periodic and Y -periodic

functions, respectively.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

"

m(⇠) = "−2m("⇠),

"

m(x; ⇠) = m(x

" ; "⇠). Given f , the mth Bloch coefficient of f at the ✏ scale: b f ✏

m(k) =

Z

Y

f (x)e−ik·x✏

m(x; k)dx

∀m ≥ 1, k ∈ Λ✏, Λ✏ = {k = (k1, . . . , kd) ∈ Zd : [−1/2✏] + 1 ≤ ki ≤ [1/2✏]}. f (x) = X

k∈Λ✏

X

m≥1

b f ✏

m(k)eik·x✏ m(x; k).

Z

Y

|f (x)|2dx = X

k∈Λ✏

X

m≥1

|b f ✏

m(k)|2.

✏

m(k)b

u✏

m(k) = b

f ✏

m(k),

∀ m ≥ 1, k ∈ Λ✏.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

u"(x) = X

k∈Λ✏ ∞

X

m=1

b f ✏

m(k)

m(✏k)/✏−2 eik·x"

m(x; k).

u"(x) ∼ ✏2 X

k∈Λ✏

b f ✏

1 (k)

1(✏k)eik·x"

1(x; k).

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

c1|⌘|2 ≤ 1(⌘) ≤ c2|⌘|2, ∀⌘ ∈ Y 0, 1(0) = @k1(0) = 0, k = 1, . . . , N, @2

k`1(0) = 2a⇤ k`, k, ` = 1, . . . , N,

where a⇤

k` are the homogenized coefficients.

⌘ ∈ B → 1(y; ⌘) ∈ L1 ∩ L2

#(Y ) is analytic,

1(y; 0) = (2⇡) d

2 .

b f "

1 (k) ∼ b

fk b u"

1(k) ∼ b

u⇤

k

as ✏ → 0, u"(x) ∼ X

k2Λ✏

b f ✏

1 (k)

1(✏k)/✏2 eik·x"

1(x; k) ∼

X

k2Zd

b f k a⇤

ijkikj

eik·x which is the solution of the homogenized problem in its Fourier representation.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

Discrete Bloch waves In 1 − d one can use the explicit representation formula for discrete solutions. But, of course, this is impossible for multi-dimensional problems. In 1 − d the homogenized coefficient a∗ can be computed explicitly as above. But in several space dimensions, the homogenized coefficients depend on test functions χk that are defined by solving elliptic problems on the unit cell. In several space dimensions Bloch wave expansions can be used to derive explicit representation formulas and to prove

• homogenization. This is the method we shall employ to derive
• ur results.

Enrique Zuazua Optimal design and numerics

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Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

Explicit 1 − d computations. ⇢ −a✏

i u✏ i+1 + (a✏ i + a✏ i−1)u✏ i − a✏ i−1u✏ i−1 = h2fi,

1 ≤ i ≤ n − 1, u✏

0 = b,

u✏

n = c.

Therefore, u✏

i = b + U✏,h i

X

j=1

h a✏

j

−

i

X

j=1

h a✏

j j

X

k=1

hfk 1 ≤ i ≤ n − 1, with U✏,h = a✏,∗

h (c − b) + a✏,∗ h n−1

X

j=1

1 a✏

j j

X

k=1

h2fk ! , and a✏,∗

h

= @

n−1

X

j=0

h a✏

j

1 A

−1

. Using that a✏

p+i = a✏ i , a✏,∗ h

→ a∗

p ( with explicit estimates).

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

DISCRETE BLOCH WAVE METHOD: 1 d Since h/✏ = q/p, a✏(x + ph) = a✏(x), x 2 Γh Γp

h = {x = zh :

0  z < p, z 2 Z} f (x, k) = hp

1 2

X

z∈Γhp

f (x + z)e−i2⇡k·(x+ z), k 2 Λq✏, Λq✏ = ⇢ k 2 Zd, such that  1 2q✏

• + 1  k 

 1 2q✏

• .

The discrete Bloch waves are defined by the family of eigenvalue problems: r−h ⇥ a✏ (x) r+ h(ei2⇡x·⇠✏

h(x, ⇠))

⇤ = (⇠)eix·⇠✏

h(x, ⇠),

x 2 Γp

h,

✏

h(x, ⇠) is ph-periodic in x, i.e.,

✏

h(x + ph, ⇠) = ✏ h(x, ⇠).

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

There exist a sequence 1(⇠), ..., p(⇠) 0 and their eigenfunctions {✏

h,m(x, ⇠)}p m=1.

m(⇠) c ✏2q2 > 0, m 2 ⇠ 2 B 7! (1(⇠), 1(·, ⇠)) 2 R ⇥ Cp is analytic. 1(y, 0) = p−1/2 1(0) = @1(0) = 0, @21(0) = 1 p

p

X

i=1

1 a ((i + 0.5)/p)) !−1 .

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

This method allows obtaining sharp estimates on both ||u✏

h − u∗ q/p||

and ||u∗ − u∗

q/p||.

Indeed, All solutions involved can be represented in a similar form by means of Bloch wave expansions; The contribution of Bloch components m ≥ 2 is uniformly negligible; The dependence of the first Bloch component, both in what concerns the eigenvalue and eigenfunction, can be estimated very precisely in terms of the various parameters.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems Motivation 1 − d Bloch-c Bloch-d Conclusion

Conclusion Discrete Bloch waves allow getting a complete representation formula for the numerical approximations when h/✏ is rational. This allows deriving the discrete homogenized solution with convergence rates. The discrete homogenized problem has the same structure as the continuous one but with different effective coefficients. The distance between the discrete and continuous effective coefficients can be estimated as well. This allows recovering, with convergence rates, results on numerical homogenization under ergodicity conditions.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

Table of Contents

1

Motivation

2

Finite elements in Optimal Design

3

Numerics for Homogenization Motivation The 1 − d case The continuous Bloch wave decomposition The Discrete Bloch wave decomposition Conclusion

4

Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Finite element approximation

5

Concluding remarks

Enrique Zuazua Optimal design and numerics

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Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

An example The finite element approximation to 2 − d optimal design problems for the Dirichlet Laplacian. 10 11

• 10V. ˘

Sver` ak, On optimal shape design, JMPA, 72, 1993, pp. 537-551.

• 11D. Chenais and E. Z. Finite Element Approximation of 2D Elliptic Optimal

Design, JMPA, 85 (2006), 225-249.

Enrique Zuazua Optimal design and numerics

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Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

A model for mixtures For a given bounded open set Ω of RN, N 1, consider the

• ptimization problem 12

(P) 8 < : Find ω0 2 U such that J (ω0) = min

ω∈U J (ω),

with U = {ω ⇢ Ω : ω measurable , |ω|  κ} , J (ω) = Z

ω

F1(x, uω, ruω) dx + Z

Ω\ω

F2(x, uω, ruω) dx, where F1, F2 : Ω ⇥ R ⇥ RN ! R, the source term f : Ω ! R and the material constants α and β are given, and uω 2 H1

0(Ω);

div ⇣ (αχω + β(1 χω)) ruω ⌘ = f in Ω. (1)

• 12J. Casado-D´

ıaz, C. Castro, M. Luna-Laynez and E. Z., Numerical approximation of a one-dimensional elliptic optimal design problem, SIAM J. Multiscale Analysis, to appear.

Enrique Zuazua Optimal design and numerics

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Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

(P) has not a solution in general ! Relaxation. 13 14 Replace the characteristic function χω by a measurable function θ with values in [0, 1]. Replace the function (αχω + β(1 χω)) in the elliptic PDE by a matrix function A in the set K(θ) of matrices constructed by homogenization mixing the materials α and β with respective proportions θ and 1 θ. The corresponding state is denoted by uθ,A. Replace the cost functional J by another one of the form ˆ J (θ, A) = Z

Ω

H(x, uθ,A, ruθ,A, Aruθ,A, θ) dx, where H is a function known explicitly only in a few cases (N = 1, Fi(x, s, ξ) = |ξ|2, . . .).

• 13F. Murat. Un contre-example pour le probl`

eme du contrˆ

• le dans les
• coefficients. C.R.A.S Sci. Paris A 273 (1971), 708-711.
• 14F. Murat, L. Tartar. H-convergence. In Topics in the Mathematical

Modelling of Composite Materials, ed. by L. Cherkaev, R.V. Kohn. Progress in Nonlinear Diff. Equ. and their Appl., 31, Birka¨ user, Boston, 1998, 21-43.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

In practice it would be very natural to: Discretize (by FEM, for instance) the problem using finite element approximations of the PDE and the functional under consideration. Search for the discrete optimal shape design (finite-dimensional problem). “Hope” that, as the mesh-size tends to zero, the discrete

• ptimal shape will converge to the continuous one.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

CONTINUOUS SOLUTION OF THE OPTIMAL DESIGN PROBLEM

+

CONVERGENT ALGORITHM FOR SOLVING THE PDE = CONVERGENT ALGORITHM FOR OPTIMAL SHAPES?????

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

NOT NECESSARILY !!!!

• E. Z., SIAM Review, 47 (2) (2005), 197-243.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

Two (main) strategies for discretization: Discrete approach: Discretize directly the original problem. Continuous approach: Discretize the relaxed formulation. We analyze the 1 − d problem N = 1 showing that the continuous approach provides a better approximation and a faster convergence rate with a lower computational cost.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

Both approaches, discrete and continuous, were successfully developed in 15 but no convergence rates were obtained.

• 15J. Casado-D´

ıaz, J. Couce-Calvo, M. Luna-Laynez, J.D. Mart´ ın-G´

• mez.

Optimal design problems for a non-linear cost in the gradient: numerical results. Applicable Anal. 87 (2008), 1461-1487.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

Theorem A relaxation of problem (P) is given by (ˆ P) Find θ0 ∈ ˆ U such that ˆ J (θ0) = min

θ∈ ˆ U

ˆ J (θ), ˆ U = ⇢ θ ∈ L∞(0, 1; [0, 1]) : Z 1 θdx ≤ κ

• ,

and ˆ J : ˆ U − → R is defined by ˆ J (θ) = Z 1 ✓ θF1 ✓ x, uθ, Mθ α duθ dx ◆ + (1 − θ)F2 ✓ x, uθ, Mθ β duθ dx ◆◆ dx, for every θ ∈ ˆ U, with Mθ = ⇣

θ α + 1−θ β

⌘−1 and uθ ∈ H1

0(0, 1) the

solution of − d dx ✓ Mθ duθ dx ◆ = f in (0, 1).

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

For r > 0, we consider a partition Qr = {yk}mr

k=0 of [0, 1], with

r = max

1≤k≤mr(yk − yk−1) .

We now approximate both optimal design problems, the original and the relaxed one, but we do it in a stratified manner, in two levels, increasing complexity and making the method better adapted to simulation practices. Approximation level #1: Continuous PDE but discrete control sets or coefficients. Approximation level #2: Discrete approximation of PDE and also discrete control sets or coefficients.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

Discrete Approach, Level #1: (Pr) Find ωr

0 ∈ Ur such that J (ωr 0) = min ω∈Ur J (ω)

where Ur =

• ω ∈ U : ∃J ⊂ {1 . . . , mr} such that ω = ∪k∈J(yk−1, yk)

. Continuous Approach, Level #1: (ˆ Pr) Find θr

0 ∈ ˆ

Ur such that ˆ J (θr

0) = min θ∈ ˆ Ur

ˆ J (θ) where ˆ Ur =

• θ ∈ ˆ

U : θ constant in every (yk−1, yk) .

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

Theorem [Discrete Approach, Level #1] Problem (Pr) has a solution for every r > 0, and we have 0 ≤ min

!∈Ur J (!) − inf !∈U J (!) ≤ Cr

1 2 .

Moreover, if for some integer ` ≥ 1, we have that f belongs to W `,1(0, 1) and Fi(x, s, ⇠) is independent of s and belong to C `,1

loc([0, 1] × R), then we have

0 ≤ min

!∈Ur J (!) − inf !∈U J (!) ≤ Cr

`+1 `+2 . Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

Theorem [Continuous Approach, Level #1] Problem (ˆ Pr) has a solution for every r > 0, and we have 0 ≤ min

θ∈ ˆ Ur

ˆ J (θ) − inf

ω∈U J (ω) = o(r).

Moreover, if problem (ˆ P) has a solution θ0 in BV (0, 1), then 0 ≤ min

θ∈ ˆ Ur

ˆ J (θ) − inf

ω∈U J (ω) ≤ Cr2.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

Remark The convergence rate for (ˆ Pr) is better than the one for (Pr). These convergence rates are sharp. Problem (ˆ Pr) is simpler to solve because the set of controls ˆ Ur is convex. This is true even in those cases where problem (P) has a classical solution, and therefore a relaxation is not necessary.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

Now we consider a full discretization of problem (P), where not

• nly we discretize the set of controls but we also approximate the

state equation and the cost functional. For h > 0, we set r = √ h and take two partitions Qh = {xi}nh

i=0,

Qr = {yk}mr

k=0 of [0, 1], with Qr ⊂ Qh and

h = max

1≤i≤nh(xi − xi−1),

r = max

1≤k≤mr(yk − yk−1),

and we define W h = {v ∈ C 0

0 ([0, 1]) : v is affine on every (xi−1, xi)}.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

For θ ∈ ˆ U, constant in every (xi−1, xi), we define ˜ uθ ∈ W h by Z 1 Mθ d˜ uθ dx dv dx dx = Z 1 fvdx, ∀v ∈ W h and ˆ J h(θ) = Z 1 ✓ θF1 ✓ x, ˜ uθ, Mθ α d˜ uθ dx ◆ + (1 − θ)F2 ✓ x, ˜ uθ, Mθ β d˜ uθ dx ◆◆ dx For ω ∈ U, with ω = ∪i∈J(xi−1, xi), J ⊂ {1 . . . , nh}, we denote ˜ uω = ˜ uχω and define J h(ω) = Z

ω

F1 ✓ x, ˜ uω, d˜ uω dx ◆ dx + Z

(0,1)\ω

F2 ✓ x, ˜ uω, d˜ uω dx ◆ dx

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

Discrete Approach – Full Discretization: (Ph

c )

Find ωh

0 ∈ Uh such that J h(ωh 0) = min ω∈Uh J h(ω)

where Uh =

• ω ∈ U : ∃J ⊂ {1 . . . , nh} such that ω = ∪i∈J(xi−1, xi)

Continuous Approach – Full Discretization: (ˆ Ph

c )

Find θ0 ∈ ˆ U

√ h such that ˆ

J h(θ0) = min

θ∈ ˆ Ur

ˆ J h(θ) where ˆ Ur =

• θ ∈ ˆ

U : θ constant in every (yk−1, yk)

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

Theorem [Discrete approach] Problem (Ph

c ) has a solution for every h > 0. Moreover, every

solution !0 satisfies 0 ≤ J (!0) − inf

!∈U J (!) ≤ Ch

1 2 .

Moreover, if for some nonnegative integer `, we have that f belongs to W `,1(0, 1) and F(x, s, ⇠) is independent of s and belong to C `,1

loc([0, 1] × R), then we have

0 ≤ J (!0) − inf

!∈U J (!) ≤ Ch

`+1 `+2 . Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

Theorem [Continuous approach] Problem (ˆ Ph

c ) has a solution for every h > 0. Moreover, if we

assume that exists an optimal control of bounded variation for (ˆ P), then every θ0 solution of (ˆ Ph

c ) satisfies

0 ≤ J (θ0) − inf

ω∈U J (ω) ≤ Ch.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

Remark In the discrete approach, based on the unrelaxed formulation, the PDE and the control are discretized in the same fine grid

• f size h.

The continuous full discretization, implemented on the relaxed version, constitutes a bigrid strategy: The PDE is discretized in the fine grid of size h while the control is discretized in the coarse one of size √

• h. And it gives a faster convergence with

a lower computational cost !

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

The methods of proof combine: Fine properties on the dependence of the 1 − d elliptic problem on its coefficients. Convergence properties of the FEM. Oscillatory constructions of bang-bang oscillating functions towards relaxation.

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Fi

Bang-bang postprocessing of relaxed numerical shapes Relaxed numerical coefficient profiles can be approximated by classical piecewise constant bang-bang functions, repredsenting pure mixtures. Assume f ∈ L∞(0, 1) and θ =

mr

X

k=1

tkχ(yk−1,yk) ∈ ˆ Ur, with tk ∈ [0, 1] for every k ∈ {1, . . . , mr}. Set jk = yk − yk−1 r2

• + 1, sk = yk − yk−1

jk , ∀ k ∈ {1, . . . , mr}, ω =

mr

[

k=1 jk

[

i=1

(yk−1 + (i − 1)sk, yk−1 + (i − 1 + tk)sk). (2) Then, we have

• ˆ

J (θ) − J (ω)

• =
• ˆ

J (θ) − ˆ J (χω)

• ≤ Cr2.

(3)

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

Table of Contents

1

Motivation

2

Finite elements in Optimal Design

3

Numerics for Homogenization Motivation The 1 − d case The continuous Bloch wave decomposition The Discrete Bloch wave decomposition Conclusion

4

Numerics for some (toy) optimal design problems An example A 1 − d model for mixtures 1 − d relaxation Finite element approximation

5

Concluding remarks

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14

Motivation Finite elements in Optimal Design Numerics for Homogenization Numerics for some (toy) optimal design problems

Concluding remarks There is a big gap between the existing theory for continuum analytical methods for optimal design and the numerical practice. We have shown that numerics and oscillating coefficients easily produce resonances. Optimal design problems are often better behaved since numerics is capable of finding the microstructure that minimization sequences develop. There are several ways of discretizing the optimal design problems and the convergence rate may differ from one to another. The use of relaxed formulations may help in improving the convergence rates.. A LOT is still to be done....

Enrique Zuazua Optimal design and numerics

jueves, 17 de julio de 14