Top

Journal of Engineering Mathematics

Published in:

01-08-2013

A Dirichlet–Neumann cost functional approach for the Bernoulli problem

Authors: A. Ben Abda, F. Bouchon, G. H. Peichl, M. Sayeh, R. Touzani

Published in: Journal of Engineering Mathematics | Issue 1/2013

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

The Bernoulli problem is rephrased into a shape optimization problem. In particular, the cost function, which turns out to be a constitutive law gap functional, is borrowed from inverse problem formulations. The shape derivative of the cost functional is explicitly determined. The gradient information is combined with the level set method in a steepest descent algorithm to solve the shape optimization problem. The efficiency of this approach is illustrated by numerical results for both interior and exterior Bernoulli problems.

previous article Effect of a non-constant magnetic field on natural convection in a horizontal porous layer heated from the bottom

next article An inverse method for the design of TIR collimators to achieve a uniform color light beam

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Lacey AA, Shillor M (1987) Electrochemical and electro-discharge machining with a threshold current. IMA J Appl Math 39:131–142MathSciNetCrossRef

Friedman F (1984) Free boundary problem in fluid dynamics, Astérisque. Soc Math France 118:55–67

Acker A (1977) Heat flow inequalities with applications to heat flow optimization problems. SIAM J Math Anal 8:604–618MathSciNetMATHCrossRef

Acker A (1981) An extremal problem involving distributed resistance. SIAM J Math Anal 12:169–172MathSciNetMATHCrossRef

Fasano A (1992) Some free boundary problems with industrial applications. In: Delfour M (ed) Shape optimization and free boundaries, vol 380. Kluwer, Dordrecht

Flucher M, Rumpf M (1997) Bernoulli’s free-boundary problem, qualitative theory and numerical approximation. J Reine Angew Math 486:165–204MathSciNetMATH

Alt A, Caffarelli LA (1981) Existence and regularity for a minimum problem with free boundary. J Reine Angew Math 325:105–144MathSciNetMATH

Beurling A (1957) On free boundary problems for the Laplace equation, Seminars on analytic functions, vol 1. Institute of Advance Studies Seminars, Princeton, pp 248–263

Henrot A, Shahgholian H (1997) Convexity of free boundaries with Bernoulli type boundary condition. Nonlinear Anal Theory Methods Appl 28(5):815–823MathSciNetMATHCrossRef

10.

Acker A, Meyer R (1995) A free boundary problem for the p-Laplacian: uniqueness, convexity and successive approximation of solutions. Electron J Differ Equ 8:1–20

11.

Henrot A, Shahgholian H (2000) Existence of classical solution to a free boundary problem for the p-Laplace operator: (II) the interior convex case. Indiana Univ Math J 49(1):311–323MathSciNetMATHCrossRef

12.

Cardaliaguet P, Tahraoui R (2002) Some uniqueness results for the Bernoulli interior free-boundary problems in convex domains. Electron J Differ Equ 102:1–16MathSciNet

13.

Bouchon F, Clain S, Touzani R (2005) Numerical solution of the free boundary Bernoulli problem using a level set formulation. Comput Method Appl Mech Eng 194:3934–3948MathSciNetADSMATHCrossRef

14.

Haslinger J, Kozubek T, Kunish K, Peichl G (2003) Shape optimization and fictitious domain approach for solving free-boundary value problems of Bernoulli type. Comput Optim Appl 26:231–251MathSciNetMATHCrossRef

15.

Ito K, Kunisch K, Peichl G (2006) Variational approach to shape derivative for a class of Bernoulli problem. J Math Anal Appl 314:126–149MathSciNetMATHCrossRef

16.

Tiihonen T (1997) Shape optimization and trial methods for free-boundary problems, RAIRO Model. Math Anal Numer 31(7):805–825MathSciNetMATH

17.

Sokolowski J, Zolésio J-P (1992) Introduction to shape optimization: shape sensitivity analysis. Springer, BerlinMATHCrossRef

18.

Novruzi A, Roche J-R (2000) Newton’s method in shape optimisation: a three-dimensional case. BIT Numer Math 40(1):102–120MathSciNetMATHCrossRef

19.

Simon J (1989) Second variation for domain optimization problems. In: Kappel F, Kunish K and Schappacher W (eds) Control and estimation of distributed parameter systems. International Series of Numerical Mathematics, no 91 . Birkhäuser pp 361–378

20.

Osher S, Sethian J (1988) Fronts propagating with curvature dependent speed: algorithms based on Hamilton–Jacobi formulations. J Comp Phys 56:12–49MathSciNetADSCrossRef

21.

Pj Ladeveze, Leguillon D (1983) Error estimate procedure in the finite element method and applications. SIAM J Numer Anal 20(3):485–509MathSciNetCrossRef

22.

Kohn RV, McKenney A (1990) Numerical implementation of a variational method for electrical impedance tomography. Inverse Probl 6(3):389MathSciNetADSMATHCrossRef

23.

Chaabane S, Jaoua M (1999) Identification of Robin coefficients by means of boundary measurements. Inverse Probl 15(6):1425MathSciNetADSMATHCrossRef

24.

Ben Abda A, Hassine M, Jaoua M, Masmoudi M (2009) Topological sensitivity analysis for the location of small cavities in stokes flows. SIAM J Control Opt 48(5):2871–2900MathSciNetCrossRef

25.

Eppler K, Harbrecht H (2012) On a Kohn–Vogelius like formulation of free boundary problems. Comput Optim Appl 52:69–85MathSciNetMATHCrossRef

26.

Eppler K (2000) Boundary integral representations of second derivatives in shape optimization. Discuss Math Differ Incl Control Optim 20:63–78MathSciNetMATHCrossRef

27.

Eppler K (2000) Optimal shape design for elliptic equations via BIE-methods. J Appl Math Comput Sci 10:487–516MathSciNetMATH

28.

Delfour MC, Zolesio J-P (2001) Shapes and geometries. SIAM

29.

Murat F, Simon J (1976) Étude de problèmes d’optimal design. Lect Notes Comput Sci 41:54–62CrossRef

30.

Murat F, Simon J (1976) Sur le controle par un domaine géométrique Research report of the Laboratoire d’Analyse Numérique, University of Paris 6

31.

Simon J (1980) Differentiation with respect to the domain in boundary value problems. Numer Funct Anal Optim 2:649–687MathSciNetMATHCrossRef

32.

Henrot A, Pierre M (2000) Variation et optimization de formes, Une analyse géométrique. Springer, Berlin

33.

Burger M (2001) A framework for the construction of level set methods for shape optimization and reconstruction. Inverse Probl 17:1327–1356MathSciNetADSMATHCrossRef

34.

Osher S, Santosa F (2001) Level set methods for optimization problems involving geometry and constraints. J Comput Phys 171:272–288MathSciNetADSMATHCrossRef

35.

Sethian JA (1999) Level set methods and fast marching methods. Cambridge University Press, CambridgeMATH

36.

Osher S, Fedkiw R (2002) Level set methods and dynamic implicit surfaces. Springer, New York

37.

Bouchon F, Clain S, Touzani R (2008) A perturbation method for the numerical solution of the Bernoulli problem. J Comput Math 26:23–36MathSciNet

38.

Adalsteinsson A, Sethian JA (1999) The fast construction of extension velocities in level set methods. J Comput Phys 148:2–22MathSciNetADSMATHCrossRef

Title: A Dirichlet–Neumann cost functional approach for the Bernoulli problem
Authors: A. Ben Abda
F. Bouchon
G. H. Peichl
M. Sayeh
R. Touzani
Publication date: 01-08-2013
Publisher: Springer Netherlands
Published in: Journal of Engineering Mathematics / Issue 1/2013
Print ISSN: 0022-0833
Electronic ISSN: 1573-2703
DOI: https://doi.org/10.1007/s10665-012-9608-3

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Other articles of this Issue 1/2013

A note on waveless subcritical flow past a submerged semi-ellipse

Effect of anisotropic permeability on convective heat transfer through a porous river bed underlying a fluid layer

Water-wave propagation through an infinite array of floating structures

Tangential contact problems for several transversely isotropic elastic layers bonded to an elastic foundation

A new approximate formulation of periodic gravity waves

An inverse method for the design of TIR collimators to achieve a uniform color light beam

Premium Partners