Adaptive finite element method for shape optimization
ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 4, pp. 1122-1149.

We examine shape optimization problems in the context of inexact sequential quadratic programming. Inexactness is a consequence of using adaptive finite element methods (AFEM) to approximate the state and adjoint equations (via the dual weighted residual method), update the boundary, and compute the geometric functional. We present a novel algorithm that equidistributes the errors due to shape optimization and discretization, thereby leading to coarse resolution in the early stages and fine resolution upon convergence, and thus optimizing the computational effort. We discuss the ability of the algorithm to detect whether or not geometric singularities such as corners are genuine to the problem or simply due to lack of resolution - a new paradigm in adaptivity.

DOI : 10.1051/cocv/2011192
Classification : 49M25, 65M60
Mots clés : shape optimization, adaptivity, mesh refinement/coarsening, smoothing
@article{COCV_2012__18_4_1122_0,
     author = {Morin, Pedro and Nochetto, Ricardo H. and Pauletti, Miguel S. and Verani, Marco},
     title = {Adaptive finite element method for shape optimization},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1122--1149},
     publisher = {EDP-Sciences},
     volume = {18},
     number = {4},
     year = {2012},
     doi = {10.1051/cocv/2011192},
     mrnumber = {3019475},
     zbl = {1259.49046},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2011192/}
}
TY  - JOUR
AU  - Morin, Pedro
AU  - Nochetto, Ricardo H.
AU  - Pauletti, Miguel S.
AU  - Verani, Marco
TI  - Adaptive finite element method for shape optimization
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2012
SP  - 1122
EP  - 1149
VL  - 18
IS  - 4
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2011192/
DO  - 10.1051/cocv/2011192
LA  - en
ID  - COCV_2012__18_4_1122_0
ER  - 
%0 Journal Article
%A Morin, Pedro
%A Nochetto, Ricardo H.
%A Pauletti, Miguel S.
%A Verani, Marco
%T Adaptive finite element method for shape optimization
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2012
%P 1122-1149
%V 18
%N 4
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2011192/
%R 10.1051/cocv/2011192
%G en
%F COCV_2012__18_4_1122_0
Morin, Pedro; Nochetto, Ricardo H.; Pauletti, Miguel S.; Verani, Marco. Adaptive finite element method for shape optimization. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 4, pp. 1122-1149. doi : 10.1051/cocv/2011192. http://archive.numdam.org/articles/10.1051/cocv/2011192/

[1] G. Allaire, Conception optimale de structures. Springer-Verlag, Berlin (2007). | MR | Zbl

[2] P. Alotto, P. Girdinio, P. Molfino and M. Nervi, Mesh adaption and optimization techniques in magnet design. IEEE Trans. Magn. 32 (1996) 2954-2957.

[3] W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations, Birkhäuser (2003) | MR | Zbl

[4] N.V. Banichuk, A. Falk and E. Stein, Mesh refinement for shape optimization, Struct. Optim. 9 (1995) 46-51.

[5] E. Bänsch, P. Morin and R.H. Nochetto, Surface diffusion of graphs : variational formulation, error analysis and simulation. SIAM J. Numer. Anal. 42 (2004) 773-799. | MR | Zbl

[6] R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10 (2001) 1-102. | MR | Zbl

[7] J.A. Bello, E. Fernandez-Cara, J. Lemoine and J. Simon, The differentiability of the drag with respect to the variations of a Lipschitz domain in a Navier-Stokes flow. SIAM J. Control Optim. 35 (1997) 626-640. | MR | Zbl

[8] A. Bonito and R.H. Nochetto and M.S. Pauletti, Geometrically consistent mesh modification. SIAM J. Numer. Anal. 48 (2010) 1877-1899. | MR | Zbl

[9] M. Burger, A framework for the construction of level set methods for shape optimization and reconstruction. Interfaces Free Bound. 5 (2003) 301-329. | MR | Zbl

[10] J. Céa, Conception optimale ou identification de formes : calcul rapide de la dérivée directionnelle de la fonction coût. RAIRO Modél. Math. Anal. Numér. 20 (1986) 371-402. | Numdam | MR | Zbl

[11] F. De Gournay, Velocity extension for the level-set method and multiple eigenvalues in shape optimization. SIAM J. Control Optim. 45 (2006) 343-367. | MR | Zbl

[12] M.C. Delfour and J.-P. Zolésio, Shapes and Geometries. SIAM Advances in Design and Control 22 (2011). | MR | Zbl

[13] A. Demlow, Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces. SIAM J. Numer. Anal. 47 (2009) 805-827. | MR | Zbl

[14] A. Demlow and G. Dziuk, An adptive finite element method for the Laplace-Beltrami operator on implicitly defined surfaces. SIAM J. Numer. Anal. 45 (2007) 421-442. | MR | Zbl

[15] G. Dogan, P. Morin, R.H. Nochetto and M. Verani. Discrete gradient flows for shape optimization and applications. Comput. Methods Appl. Mech. Engrg. 196 (2007) 3898-3914. | MR | Zbl

[16] M. Giles and E. Süli, Adjoint methods for PDEs : a posteriori error analysis and postprocessing by duality. Acta Numer. 11 (2002) 145-236. | MR | Zbl

[17] M. Giles, M. Larson, J.M. Levenstam and E. Süli, Adaptive error control for finite element approximation of the lift and drag coefficients in viscous flow. Technical Report 1317 (1997) http://eprints.maths.ox.ac.uk/1317/.

[18] V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations : Theory and Algorithms, Springer Series in Computational Mathematics 5. Springer-Verlag, Berlin (1986) | MR | Zbl

[19] A. Henderson, Paraview Guide, A Parallel Visualization Application. Kitware Inc. (2007).

[20] M. Lei, J.P. Archie and C. Kleinstreuer, Computational design of a bypass graft that minimizes wall shear stress gradients in the region of the distal anastomosis. J. Vasc. Surg. 25 (1997) 637-646.

[21] K. Mekchay, P. Morin, and R.H. Nochetto, AFEM for Laplace Beltrami operator on graphs : design and conditional contraction property. Math. Comp. 80 (2011) 625-648. | MR | Zbl

[22] B. Mohammadi, O. Pironneau, Applied shape optimization for fluids. Oxford University Press, Oxford (2001). | MR | Zbl

[23] M.S. Pauletti, Parametric AFEM for geometric evolution equations and coupled fluid-membrane interaction. Ph.D. thesis, University of Maryland, College Park, ProQuest LLC, Ann Arbor, MI (2008) | MR

[24] M.S. Pauletti, Second order method for surface restoration. Submitted.

[25] O. Pironneau, On optimum profiles in Stokes flow. J. Fluid Mech. 59 (1973) 117-128. | MR | Zbl

[26] O. Pironneau, On optimum design in fluid mechanics. J. Fluid Mech. 64 (1974) 97-110. | MR | Zbl

[27] A. Quarteroni and G. Rozza, Optimal control and shape optimization of aorto-coronaric bypass anastomoses. Math. Models Methods Appl. Sci. 13 (2003) 1801-1823. | MR | Zbl

[28] G. Rozza, Shape design by optimal flow control and reduced basis techniques : applications to bypass configurations in haemodynamics. Ph.D. thesis, École Polytechnique Fédèrale de Lausanne (2005).

[29] J.R. Roche, Adaptive method for shape optimization, 6th World Congresses of Structural and Multidisciplinary Optimization. Rio de Janeiro (2005).

[30] A. Schleupen, K. Maute and E. Ramm, Adaptive FE-procedures in shape optimization. Struct. Multidisc. Optim. 19 (2000) 282-302.

[31] A. Schmidt and K.G. Siebert, Design of Adaptive Finite Element Software, The Finite Element Toolbox ALBERTA, Lecture Notes in Computational Science and Engineering 42. Springer, Berlin (2005). | MR | Zbl

[32] J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization. Springer-Verlag, Berlin (1992). | Zbl

[33] R.S. Taylor, A. Loh, R.J. Mcfarland, M. Cox and J.F. Chester, Improved technique for polytetrafluoroethylene bypass grafting : long-term results using anastomotic vein patches. Br. J. Surg. 79 (1992) 348-354.

Cité par Sources :