A deterministic approximation method in shape optimization under random uncertainties
The SMAI journal of computational mathematics, Volume 1 (2015), pp. 83-143.

This paper is concerned with the treatment of uncertainties in shape optimization. We consider uncertainties in the loadings, the material properties, the geometry and the vibration frequency, both in the parametric and geometric optimization setting. We minimize objective functions which are mean values, variances or failure probabilities of standard cost functions under random uncertainties. By assuming that the uncertainties are small and generated by a finite number N of random variables, and using first- or second-order Taylor expansions, we propose a deterministic approach to optimize approximate objective functions. The computational cost is similar to that of a multiple load problems where the number of loads is N. We demonstrate the effectiveness of our approach on various parametric and geometric optimization problems in two space dimensions.

DOI: 10.5802/smai-jcm.5
Classification: 65C20,  65K10,  93C95
Keywords: Shape optimization, random uncertainties, Level Set method
Grégoire Allaire 1; Charles Dapogny 2

1 Centre de Mathématiques Appliquées (UMR 7641), École Polytechnique 91128 Palaiseau, France
2 Laboratoire Jean Kuntzmann, CNRS Université Joseph Fourier, Grenoble INP Université Pierre Mendès France BP 53, 38041 Grenoble Cedex 9, France
     author = {Gr\'egoire Allaire and Charles Dapogny},
     title = {A deterministic approximation method in shape optimization under random uncertainties},
     journal = {The SMAI journal of computational mathematics},
     pages = {83--143},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {1},
     year = {2015},
     doi = {10.5802/smai-jcm.5},
     language = {en},
     url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.5/}
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Grégoire Allaire; Charles Dapogny. A deterministic approximation method in shape optimization under random uncertainties. The SMAI journal of computational mathematics, Volume 1 (2015), pp. 83-143. doi : 10.5802/smai-jcm.5. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.5/

[1] G. Allaire Conception optimale de structures, Mathématiques & Applications (Berlin) [Mathematics & Applications], 58, Springer-Verlag, Berlin, 2007, pp. xii+278 With the collaboration of Marc Schoenauer (INRIA) in the writing of Chapter 8 | MR | Zbl

[2] G. Allaire; C. Dapogny A deterministic approximation method in shape optimization under random uncertainties: supplementary material (Allaire-Dapogny-supp.pdf)

[3] G. Allaire; C. Dapogny A linearized approach to worst-case design in parametric and geometric shape optimization, Math. Models Methods Appl. Sci., Volume 24 (2014) no. 11, pp. 2199-2257 | DOI | MR | Zbl

[4] G. Allaire; F. Jouve A level-set method for vibration and multiple loads structural optimization, Comput. Methods Appl. Mech. Engrg., Volume 194 (2005) no. 30-33, pp. 3269-3290 | DOI | MR | Zbl

[5] G. Allaire; F. Jouve Minimum stress optimal design with the level set method, Engineering Analysis with Boundary Elements, Volume 32 (2008), pp. 909-918 | DOI | Zbl

[6] G. Allaire; F. Jouve; A.-M. Toader Structural optimization using shape sensitivity analysis and a level-set method, J. Comput. Phys., Volume 194 (2004), pp. 363-393 | DOI | MR | Zbl

[7] S. Amstutz; M. Ciligot-Travain A notion of compliance robustness in topology optimization (2014) (accepted for publication in ESAIM: Control, Optimization and Calculus of Variations)

[8] I. Babuška; F. Nobile; R. Tempone A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., Volume 45 (2007) no. 3, pp. 1005-1034 | DOI | MR | Zbl

[9] M. Bendsøe; O. Sigmund Topology optimization. Theory, methods and applications, Springer-Verlag, Berlin, 2003, pp. xiv+370 | Zbl

[10] W. Betz; I. Papaioannou; D. Straub Numerical methods for the discretization of random fields by means of the Karhunen-Loève expansion, Comput. Methods Appl. Mech. Engrg., Volume 271 (2014), pp. 109-129 | DOI | MR | Zbl

[11] C. Bui; C. Dapogny; P. Frey An accurate anisotropic adaptation method for solving the level set advection equation, Internat. J. Numer. Methods Fluids, Volume 70 (2012) no. 7, pp. 899-922 | DOI | MR

[12] J. Céa Conception optimale ou identification de formes: calcul rapide de la dérivée directionnelle de la fonction coût, RAIRO Modél. Math. Anal. Numér., Volume 20 (1986) no. 3, pp. 371-402 | Numdam | MR | Zbl

[13] S. Chen; W. Chen A new level-set based approach to shape and topology optimization under geometric uncertainty, Struct. Multidiscip. Optim., Volume 44 (2011) no. 1, pp. 1-18 | DOI | MR | Zbl

[14] S. Chen; W. Chen; S. Lee Level set based robust shape and topology optimization under random field uncertainties, Struct. Multidiscip. Optim., Volume 41 (2010) no. 4, pp. 507-524 | DOI | MR | Zbl

[15] A. Cherkaev; E. Cherkaev Principal compliance and robust optimal design, J. Elasticity, Volume 72 (2003) no. 1-3, pp. 71-98 | DOI | MR | Zbl

[16] A. Chkifa; A. Cohen; C. Schwab Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs, J. Math. Pures Appl. (9), Volume 103 (2015) no. 2, pp. 400-428 | DOI | MR

[17] S.-K. Choi; R Grandhi; R.A. Canfield Reliability-based Structural Design, Springer, 2007 | Zbl

[18] P.G. Ciarlet Mathematical elasticity. Vol. I: Three-Dimensional Elasticity, North-Holland Publishing Co., Amsterdam, 1988, pp. xlii+451 | MR | Zbl

[19] S. Conti; H. Held; M. Pach; M. Rumpf; R. Schultz Shape optimization under uncertainty—a stochastic programming perspective, SIAM J. Optim., Volume 19 (2008) no. 4, pp. 1610-1632 | DOI | MR | Zbl

[20] M. Dambrine; C. Dapogny; H. Harbrecht Shape optimization for quadratic functionals and states with random right-hand sides, SIAM J. Control Optim., Volume 53 (2015) no. 5, pp. 3081-3103 | DOI | MR

[21] M. Dambrine; H. Harbrecht; B. Puig Computing quantities of interest for random domains with second order shape sensitivity analysis, ESAIM: Math. Model. Numer. Anal., Volume 49 (2015), pp. 1285-1302 | DOI

[22] C. Dapogny Shape optimization, level set methods on unstructured meshes and mesh evolution, University Pierre et Marie Curie (2013) (Ph. D. Thesis)

[23] C. Dapogny; P. Frey Computation of the signed distance function to a discrete contour on adapted triangulation, Calcolo, Volume 49 (2012) no. 3, pp. 193-219 | DOI | MR | Zbl

[24] F. de Gournay; G. Allaire; F. Jouve Shape and topology optimization of the robust compliance via the level set method, ESAIM Control Optim. Calc. Var., Volume 14 (2008) no. 1, pp. 43-70 | DOI | Numdam | MR | Zbl

[25] M.C. Delfour; J.-P. Zolésio Shapes and geometries: Metrics, analysis, differential calculus, and optimization, Advances in Design and Control, 22, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011, pp. xxiv+622 | DOI | Zbl

[26] P.D. Dunning; H.A. Kim Robust Topology Optimization: Minimization of Expected and Variance of Compliance, AIAA Journal, Volume 51 (2013), pp. 2656-2664 | DOI

[27] X. Guo; W. Bai; W. Zhang Confidence extremal structural response analysis of truss structures under static load uncertainty via SDP relaxation, Computers and Structures, Volume 87 (2009), pp. 246-253 | DOI

[28] F. Hecht New development in FreeFem++, J. Numer. Math., Volume 20 (2012), pp. 251-265 | DOI | MR | Zbl

[29] F. Hecht; A. Le Hyaric; O. Pironneau FreeFem++ version 2.15-1 (http://www.freefem.org/ff++/)

[30] A. Henrot; M. Pierre Variation et optimisation de formes, une analyse géométrique, Mathématiques & Applications (Berlin), 48, Springer, Berlin, 2005, pp. xii+334 | DOI | MR

[31] B.S. Lazarov; M. Schevenels; O. Sigmund Topology optimization with geometric uncertainties by perturbation techniques, Int. J. Numer. Meth. Engng., Volume 90 (2012), pp. 1321-1336 | DOI | Zbl

[32] M. Loève Probability theory. II, Graduate Texts in Mathematics, 46, Springer-Verlag, 1977, pp. xvi+413 | MR | Zbl

[33] J. Martínez-Frutos; M. Kessler; F. Periago Robust optimal shape design for an elliptic PDE with uncertainty in its input data (2015) (submitted)

[34] K. Maute Topology optimization under uncertainty, Topology optimization in structural and continuum mechanics (CISM Courses and Lectures), Volume 549, Springer, Vienna, 2014, pp. 457-471 | DOI | MR

[35] F. Murat; J. Simon Sur le contrôle par un domaine géométrique, Technical Report RR-76015, Laboratoire d’Analyse Numérique (1976)

[36] J. Nocedal; S.J. Wright Numerical optimization, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006, pp. xxii+664 | MR | Zbl

[37] S.J. Osher; J.A. Sethian Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., Volume 79 (1988) no. 1, pp. 12-49 | DOI | MR | Zbl

[38] Y. Privat; E. Trélat; E. Zuazua Optimal shape and location of sensors for parabolic equations with random initial data, Arch. Ration. Mech. Anal., Volume 216 (2015) no. 3, pp. 921-981 | DOI | MR

[39] G. Rozvany Structural design via optimality criteria, Kluwer Academic Publishers Group, Dordrecht, 1989, pp. xxvi+463 | DOI | MR | Zbl

[40] C. Schillings; S. Schmidt; V. Schulz Efficient shape optimization for certain and uncertain aerodynamic design, Comput. & Fluids, Volume 46 (2011), pp. 78-87 | DOI | MR

[41] V. Schulz; C. Schillings Optimal Aerodynamic Design under Uncertainty, Management and Minimisation of Uncertainties and Errors in Numerical Aerodynamics, ed. B. Eisfeld et al (2013), pp. 297-338 | DOI

[42] O. Sigmund On the design of compliant mechanisms using topology optimization, Mech. Struct. Mach., Volume 25 (1997), pp. 493-524 | DOI

[43] J. Simon Second variations for domain optimization problems, Control and estimation of distributed parameter systems (Vorau, 1988) (Internat. Ser. Numer. Math.), Volume 91, Birkhäuser, Basel, 1989, pp. 361-378 | MR | Zbl

[44] J. Sokołowski; J.-P. Zolésio Introduction to shape optimization: Shape sensitivity analysis, Springer Series in Computational Mathematics, 10, Springer-Verlag, Berlin, 1992, pp. ii+250 | DOI | Zbl

[45] M.Y. Wang; X. Wang; D. Guo A level set method for structural topology optimization, Comput. Methods Appl. Mech. Engrg., Volume 192 (2003) no. 1-2, pp. 227-246 | DOI | MR | Zbl

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