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 Motivation 1 Finite elements in Optimal Design 2 Numerics for Homogenization 3 Motivation The 1 − d case The continuous Bloch wave decomposition The Discrete Bloch wave decomposition Conclusion Numerics for some (toy) optimal design problems 4 An example A 1 − d model for mixtures 1 − d relaxation Finite element approximation Concluding remarks 5 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 e ffi cient 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 Di ff erential 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 optimization 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 optimality conditions. In particular the so-called Hadamard shape derivative 2 and the topological derivatives 3 These tools serve also to build gradient descent strategies. But they often fail to be e ffi cient because of the pollution generated by the high frequency spurious numerical oscillations 4 2 J. Simon, Di ff erentiation with Respect to the Domain in Boundary Value Problems, Numerical Functional Analysis and Optimization Volume 2, Issue 7-8, 1980 3 St´ ephane Garreau, Philippe Guillaume, and Mohamed Masmoudi The Topological Asymptotic for PDE Systems: The Elasticity Case SIAM J. Control Optim., 39(6), 1756 D 1778. 4 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 D 3914. 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 Motivation 1 Finite elements in Optimal Design 2 Numerics for Homogenization 3 Motivation The 1 − d case The continuous Bloch wave decomposition The Discrete Bloch wave decomposition Conclusion Numerics for some (toy) optimal design problems 4 An example A 1 − d model for mixtures 1 − d relaxation Finite element approximation Concluding remarks 5 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 domain 5 State equation = Diric Dimension n = 2, V. ˘ ak 6 : There exists an optimal domain in Sver` 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-Hausdor ff topology and the continuous dependence of the solutions of the Dirichlet laplacian in H 1 with respect to it. 5 W o 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 o 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 . 6 V . ˘S v e r `a k , On o 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 e ffi ciency that most methods employed to solve optimal design problem computationally exhibit. One may use di ff erent 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 H 1 whenever the polygonal domains converge in the sense of H c -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
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