Robust optimal shape design for an elliptic PDE with uncertainty in its input data
ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 4, pp. 901-923.

We consider a shape optimization problem for an elliptic partial differential equation with uncertainty in its input data. The design variable enters the lower-order term of the state equation and is modeled through the characteristic function of a measurable subset of the spatial domain. As usual, a measure constraint is imposed on the design variable. In order to compute a robust optimal shape, the objective function involves a weighted sum of both the mean and the variance of the compliance. Since the optimization problem is not convex, a full relaxation of it is first obtained. The relaxed problem is then solved numerically by using a gradient-based optimization algorithm. To this end, the adjoint method is used to compute the continuous gradient of the cost function. Since the variance enters the cost function, the underlying adjoint equation is non-local in the probabilistic space. Both the direct and adjoint equations are solved numerically by using a sparse grid stochastic collocation method. Three numerical experiments in 2D illustrate the theoretical results and show the computational issues which arise when uncertainty is quantified through random fields.

Reçu le :
DOI : 10.1051/cocv/2014049
Classification : 35J20, 49J20, 49M20, 65K10
Mots clés : Robust shape optimization, average approach, stochastic elliptic partial differential equation, relaxation method, Gaussian random fields, elastic membrane
Martínez-Frutos, Jesús 1 ; Kessler, Mathieu 2 ; Periago, Francisco 2

1 Departamento de Estructuras y Construcción, Universidad Politécnica de Cartagena (UPCT), Campus Muralla del Mar, 30202 Cartagena (Murcia), Spain
2 Departamento de Matemática Aplicada y Estadística. UPCT  
@article{COCV_2015__21_4_901_0,
     author = {Mart{\'\i}nez-Frutos, Jes\'us and Kessler, Mathieu and Periago, Francisco},
     title = {Robust optimal shape design for an elliptic {PDE} with uncertainty in its input data},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {901--923},
     publisher = {EDP-Sciences},
     volume = {21},
     number = {4},
     year = {2015},
     doi = {10.1051/cocv/2014049},
     zbl = {1323.49029},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2014049/}
}
TY  - JOUR
AU  - Martínez-Frutos, Jesús
AU  - Kessler, Mathieu
AU  - Periago, Francisco
TI  - Robust optimal shape design for an elliptic PDE with uncertainty in its input data
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2015
SP  - 901
EP  - 923
VL  - 21
IS  - 4
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2014049/
DO  - 10.1051/cocv/2014049
LA  - en
ID  - COCV_2015__21_4_901_0
ER  - 
%0 Journal Article
%A Martínez-Frutos, Jesús
%A Kessler, Mathieu
%A Periago, Francisco
%T Robust optimal shape design for an elliptic PDE with uncertainty in its input data
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2015
%P 901-923
%V 21
%N 4
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2014049/
%R 10.1051/cocv/2014049
%G en
%F COCV_2015__21_4_901_0
Martínez-Frutos, Jesús; Kessler, Mathieu; Periago, Francisco. Robust optimal shape design for an elliptic PDE with uncertainty in its input data. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 4, pp. 901-923. doi : 10.1051/cocv/2014049. http://archive.numdam.org/articles/10.1051/cocv/2014049/

G. Allaire, Shape Optimization by the Homogenization Method. Vol. 146 of Appl. Math. Sci. Springer-Verlag, New York (2002). | Zbl

M. Allen and K. Maute, Reliability-based shape optimization of structures undergoing fluid-structure interaction phenomena. Comput. Methods Appl. Mech. Engrg. 194 (2005) 3472–3495. | DOI | Zbl

F. Alvarez and M. Carrasco, Minimization of the expected compliance as an alternative approach to multiload truss optimization. Struct. Multidisc. Optim. 29 (2005) 470–476. | DOI | Zbl

A. Asadpoure, M. Tootkaboni and J.K. Guest, Robust topology optimization of structures with uncertainties in stiffness: application to truss structures. Comput. Struct. 89 (2011) 1131–1141. | DOI

I. Babuška, F. Novile and R. Tempone, A Stochastic collocation method for elliptic partial differential equations with random input data. SIAM Rev. 52 (2010) 317–355. | DOI | Zbl

M.P. Bensøe and O. Sigmund, Topology Optimization: Theory, Methods and Applications. Springer-Verlag, Berlin (2003). | Zbl

H.-G. Beyer and B. Sendhoff, Robust optimization â a comprehensive survey. Comput. Methods Appl. Mech. Engrg. 196 (2007) 3190–3218. | DOI | Zbl

S.I. Birbil, S.C. Fang, J.B.G. Frenk and S. Zhang, Recursive approximation of the high dimensional max function. Oper. Res. Lett. 33 (2005) 450–458. | DOI | Zbl

D. Buccur and G. Buttazzo, Variational Methods in Shape Optimization Problems. Progr. Non Lin. Differ. Eq. Appl. Birkhäuser Boston, Inc., Boston, MA (2005). | Zbl

G. Buttazzo, N. Varchon and H. Zoubairi, Optimal measures for elliptic problems. Ann. Mat. Pura Appl. 185 (2006) 207–221. | DOI | Zbl

G. Buttazzo and F. Maestre, Optimal shape for elliptic problems with random perturbations. Discrete Contin. Dyn. Syst. 31 (2011) 1115–1128. | DOI | Zbl

G.C. Calafiore and F. Dabbene, Optimization under uncertainty with applications to design of truss structures. Struct. Multidisc. Optim. 35 (2008) 189–200. | DOI | Zbl

J. Casado-Díaz, C. Castro, M. Luna-Laynez and E. Zuazua, Numerical approximation of a one-dimensional elliptic optimal design problem. Multiscale Model. Simul. 9 (2011) 1181–1216. | DOI | Zbl

S. Chen, W. Chen and S. Lee, Level set based robust shape and topology optimization under random field uncertainties. Struct. Multidisc. Optim. 41 (2010) 507–524. | DOI | Zbl

A. Cherkaev, Variational Methods for Structural Optimization. Vol. 140 of Appl. Math. Sci. Springer-Verlag, New York (2000). | Zbl

A. Cherkaev and E. Cherkaeva, Principal compliance and robust optimal design. J. Elasticity 72 (2003) 71–98. | DOI | Zbl

J. Cohon, Multiobjective Programming and Planning. Dover Publications (2004). | Zbl

S. Conti, H. Held, M. Pach, M. Rumpf and R. Schultz, Shape optimization under uncertainty – A stochastic programming approach. SIAM J. Optim. 19 (2009) 1610–1632. | DOI | Zbl

S. Conti, H. Held, M. Pach, M. Rumpf and R. Schultz, Risk Averse Shape Optimization. SIAM J. Control Optim. 49 (2011) 927–947. | DOI | Zbl

M. Davis, Production of Conditional Simulations via the LU Triangular Decomposition of the Covariance Matrix. J. Math. Geol. 19 (1987) 91–98.

F. De Gournay, G. Allaire and F. Jouve, Shape and topology optimization of the robust compliance via the level set method. ESAIM:COCV 14 (2008) 43–70. | Zbl

X. Guo, W. Zhang and L. Zhang, Robust structural topology optimization considering boundary uncertainties. Comput. Methods Appl. Mech. Engrg. 253 (2013) 356–368. | DOI | Zbl

E. de Rocquigny, N. Devictor, S. Tarantola. Uncertainty in Industrial Practice: A Guide to Quantitative Uncertainty Management. John Wiley (2008). | Zbl

I. Doltsinis and Z. Kang, Robust design of structures using optimization methods. Comput. Methods Appl. Mech. Engrg. 193 (2004) 2221–2237. | DOI | Zbl

P.D. Dunning and H.A. Kim, Robust Topology Optimization: Minimization of Expected and Variance of Compliance. AIAA J. 51 (2013) 2656–2664. | DOI

I. Enevoldsen and J.D. Sørensen, Reliability-based optimization in structural engineering. Struct. Safety 15 (1994) 169–196. | DOI

B. Geihe, M. Lenz, M. Rumpf and R. Schultz, Risk averse elastic shape optimization with parametrized fine scale geometry. Math. Program. 141 (2013) 1–2. | DOI | Zbl

W. Hager, Runge–Kutta methods in optimal control and the transformed adjoint system. Numer. Math. 87 (2000) 247–282. | DOI | Zbl

A. Henrot and H. Maillot, Optimization of the shape and the location of the actuators in an internal control problem. Bolletino U.M.I. 8 (2001) 737–757. | Zbl

A. Henrot and M. Pierre, Variation et optimization de formes. Math. Appl. Springer (2005).

D.W. Kim and B.M. Kwak, Reliability-based shape optimization of two-dimensional elastic problems using BEM. Comput. Struct. 60 (1996) 743–750. | DOI | Zbl

M. Loève, Probability Theory. I, 4th edition. Vol. 45 of Grad. Texts Math. Springer-Verlag, New York (1977). | Zbl

M. Loève, Probability Theory. II, 4th edition. Vol. 46 of Grad. Texts Math. Springer-Verlag, New York (1978). | Zbl

R.T. Marler and J.S. Arora, Survey of multi-objective optimization methods for engineering. Struct. Multidisc. Optim. 26 (2004) 369–395. | DOI | Zbl

G.I. Schuëller and H.A. Jensen, Computational methods in optimization considering uncertainties – An overview. Comput. Methods Appl. Mech. Engrg. 198 (2008) 2–13. | DOI | Zbl

A. Shapiro, D. Dentcheva and A. Ruszczynski, Lectures on Stochastic Programming. Modelling and Theory. MPS-SIAM Series on Optimization (2009). | Zbl

C. Shikui and C. Wei, A new level-set based approach to shape and topology optimization under geometric uncertainty. Struct. Multidisc. Optim. 44 (2011) 1–18. | DOI | Zbl

S.A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions. Doklady Akademii Nauk SSSR 4 (1963) 240–243. | Zbl

J. Sokolowski and J.P. Zolésio, Introduction to Shape Optimization: Shape Sensitivity Analysis. Vol. 16 of Springer Series Comput. Math. Springer-Verlag, Berlin (1992). | Zbl

K. Svanberg, The method of moving asymptotes – a new method for structural optimization. Int. J. Numer. Methods Engrg. 24 (1987) 359–373. | DOI | Zbl

K. Svanberg, A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J. Optim. 12 (2002) 555–573. | DOI | Zbl

N. Varchon, Optimal measures for nonlinear cost functionals. Appl. Math. Optim. 54 (2006) 205–221. | DOI | Zbl

N. Wiener, The homogeneous chaos. Amer. J. Math. 60 (1938) 897–936. | DOI | JFM

E. Zuazua, Propagation, Observation, Control and Numerical Approximation of Waves approximated by finite difference method. SIAM Rev. 47 (2005) 197–243. | DOI | Zbl

Cité par Sources :