Shape Optimization for Stokes flows: a finite element convergence analysis

Fumagalli, Ivan; Parolini, Nicola; Verani, Marco

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 921-951.

In this paper we analyze a two-dimensional shape optimization problem, governed by Stokes equations that are defined on a domain with a part of the boundary that is described as the graph of the control function. The state problem formulation is mapped onto a reference domain, which is independent of the control function, and the analysis is mainly led on such domain. The existence of an optimal control function is proved, and optimality conditions are derived. After the analytical inspection of the problem, finite element discretization is considered for both the control function and the state variables, and a priori convergence error estimates are derived. Numerical experiments assess the validity of the theoretical results.

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

P.F. Antonietti, A. Borzì and M. Verani, Multigrid shape optimization governed by elliptic PDEs. SIAM J. Control Optim. 51 (2013) 1417–1440. | DOI | MR | Zbl

F. Ballarin, A. Manzoni, G. Rozza and S. Salsa, Shape Optimization by Free-Form Deformation: Existence Results and Numerical Solution for Stokes Flows. J. Sci. Comput. 60 (2013) 537–563. | DOI | MR | Zbl

D. Begis and R. Glowinski, Application de la méthode des éléments finis à l’approximation d’un problème de domaine optimal. Méthodes de résolution des problèmes approchés. Appl. Math. Optim. 2 (1975) 130–169. | DOI | MR | Zbl

F. Bélahcène and J.A. Desideri, Paramétrisation de Bézier adaptative pour l’optimisation de forme en Aérodynamique. Research Report RR-4943, INRIA (2003).

J.A. Bello, E. Fernández-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. | DOI | MR | Zbl

S.C. Brenner and R. Scott, The mathematical theory of finite element methods. Springer Texts Appl. Math., 3rd edition (2008). | MR | Zbl

F. Brezzi, On the existence uniqueness and approximation of saddle point problems arising from lagrangian multipliers. Rev. Fr. Automat. Infor. 8 (1974) 129–151. | Numdam | MR | Zbl

D. Chenais and E. Zuazua, Controllability of an elliptic equation and its finite difference approximation by the shape of the domain. Numer. Math. 95 (2003) 63–99. | DOI | MR | Zbl

D. Chenais and E. Zuazua, Finite-element approximation of 2D elliptic optimal design. J. Math. Pure Appl. 85 (2006) 225–249. | DOI | MR | Zbl

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. | Numdam | MR | Zbl

M.C. Delfour and J.P. Zolésio, Shapes and geometries. Society for Industrial and Applied Mathematics (SIAM), 2nd edition (2011). | MR | Zbl

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

R.G. Durán, An elementary proof of the continuity from $L_0^2\left(\Omega \right)$ to $H_0^1{\left(\Omega \right)}^n$ of Bogovskii’s right inverse of the divergence. Rev. Un. Mat. Argentina 53 (2013) 59–78. | MR | Zbl

K. Eppler, Second derivatives and sufficient optimality conditions for shape functionals. Control Cybern. 29 (2000) 485–511. | MR | Zbl

K. Eppler, H. Harbrecht and R. Schneider, On Convergence in Elliptic Shape Optimization. SIAM J. Control Optim. 46 (2007) 61–83. | DOI | MR | Zbl

I. Fumagalli, Shape optimization for Stokes flows: a reference-domain approach. M.Sc. thesis, Politecnico di Milano, Italy (2013). Available at http://www.mate.polimi.it/biblioteca/?pp=view&id=537&collezione=tesi&L=i.

V. Girault and P.A. Raviart, Finite element methods for Navier-Stokes equations. Springer-Verlag (1986). | MR | Zbl

M.D. Gunzburger, Perspectives in flow control and optimization. Society for Industrial and Applied Mathematics (SIAM) (2003). | MR | Zbl

M.D. Gunzburger, H. Kim and S. Manservisi, On a shape control problem for the stationary Navier–Stokes equations. ESAIM: M2AN 34 (2000) 1233–1258. | DOI | Numdam | MR | Zbl

B. Guo and C. Schwab, Analytic regularity of Stokes flow on polygonal domains in countably weighted Sobolev spaces. J. Comput. Appl. Math. 190 (2006) 487–519. | DOI | MR | Zbl

J. Haslinger and R.A.E. Mäkinen, Introduction to shape optimization. Society for Industrial and Applied Mathematics (SIAM) (2003). | MR | Zbl

J. Haslinger and P. Neittaanmäki, Finite element approximation for optimal shape design: Theory and applications. Wiley Chichester (1988). | MR | Zbl

K. Ito and K. Kunisch, Lagrange multiplier approach to variational problems and applications. Society for Industrial and Applied Mathematics (SIAM) (2008). | MR | Zbl

A.M. Khludnev and J. Sokolowski, Modelling and control in solid mechanics. Birkhäuser Verlag, Basel (1997). | MR | Zbl

B. Kiniger and B. Vexler, A priori error estimates for finite element discretizations of a shape optimization problem. ESAIM: M2AN 47 (2013) 1733–1763. | DOI | Numdam | MR | Zbl

M. Laumen, Newton’s method for a class of optimal shape design problems. SIAM J. Optim. 10 (2000) 503–533. | DOI | MR | Zbl

J.L. Lions, Optimal control of systems governed by partial differential equations. Springer-Verlag (1971). | MR | Zbl

A. Logg, K.A. Mardal and G.N. Wells, Automated Solution of Differential Equations by the Finite Element Method. Springer (2012). | Zbl

P. Morin, R.H. Nochetto, M.S. Pauletti and M. Verani, Adaptive finite element method for shape optimization. ESAIM: COCV 18 (2012) 1122–1149. | Numdam | MR | Zbl

J. Nocedal and S.J. Wright, Numerical optimization. 2nd edition, Springer (2006). | MR | Zbl

O. Pironneau, On optimum profiles in Stokes flow. J. Fluid Mech. 59 (1973) 117–128. | DOI | MR | Zbl

P. Plotnikov and J. Sokołowski, Compressible Navier-Stokes equations. Theory and shape optimization. Birkhäuser/Springer, Basel (2012). | MR | Zbl

A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Series Comput. Math. Springer (2008). | Zbl

J. Sokolowsky and J.P. Zolésio, Introduction to Shape Optimization. Springer-Verlag (1992). | MR | Zbl