December  2011, 31(4): 1115-1128. doi: 10.3934/dcds.2011.31.1115

Optimal shape for elliptic problems with random perturbations

1. 

Dipartimento di Matematica, Largo B. Pontecorvo, 5, 56127 Pisa, Italy

2. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Tarfia s/n, E-41012 Sevilla, Spain

Received  October 2009 Revised  March 2010 Published  September 2011

In this paper we analyze the relaxed form of a shape optimization problem with state equation $$ \begin{equation} \left\{\begin{array}{ll} -div\big(a(x)Du\big)=f\qquad\hbox{in }D\\ \hbox{boundary conditions on }\partial D. \end{array} \right. \end{equation} $$ The new fact is that the term $f$ is only known up to a random perturbation $\xi(x,\omega)$. The goal is to find an optimal coefficient $a(x)$, fulfilling the usual constraints $\alpha\le a\le\beta$ and $\displaystyle\int_D a(x)\,dx\le m$, which minimizes a cost function of the form $$\int_\Omega\int_Dj\big(x,\omega,u_a(x,\omega)\big)\,dx\,dP(\omega).$$ Some numerical examples are shown in the last section, which evidence the previous analytical results.
Citation: Giuseppe Buttazzo, Faustino Maestre. Optimal shape for elliptic problems with random perturbations. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1115-1128. doi: 10.3934/dcds.2011.31.1115
References:
[1]

G. Allaire, "Shape Optimization by the Homogenization Method,", Applied Mathematical Sciences, 146 (2002).   Google Scholar

[2]

F. Alvarez and M. Carrasco, Minimization of the expected compliance as an alternative approach to multiload truss optimization,, Struct. Multidiscip. Optim., 29 (2005), 470.  doi: 10.1007/s00158-004-0488-7.  Google Scholar

[3]

M. P. Bendsøe and O. Sigmund, "Topology Optimization: Theory, Methods and Aplications,", Springer-Verlag, (2003).   Google Scholar

[4]

D. Bucur and G. Buttazzo, "Variational Methods in Shape Optimization Problems,", Progress in Nonlinear Differential Equations and their Applications, 65 (2005).   Google Scholar

[5]

J. Casado-Díaz, J. Couce-Calvo, M. Luna-Laynez and J. D. Martín-Gómez, Optimal design problems for a non-linear cost in the gradient: numerical results,, Applicable Analysis, 87 (2008), 1461.   Google Scholar

[6]

A. Cherkaev, "Variational Methods for Structural Optimization,", Applied Mathematical Sciences, 140 (2000).   Google Scholar

[7]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization and ergodic theory,, J. Reine Angew. Math., 368 (1986), 28.   Google Scholar

[8]

A. Donoso and P. Pedregal, Optimal design of 2-$D$ conducting graded materials by minimizing quadratic funtionals in the field,, Struc. Multidisc Optim., 30 (2005), 360.  doi: 10.1007/s00158-005-0521-5.  Google Scholar

[9]

A. Henrot and M. Pierre, "Variation et Optimisation de Formes. Une Analyse Géométrique,", Mathématiques et Applications, 48 (2005).   Google Scholar

[10]

K. A. Lurie and A. V. Cherkaev, Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion,, Proc. R. Soc. Edinb. A, 99 (1984), 71.   Google Scholar

[11]

F. Maestre, A. Münch and P. Pedregal, A spatio-temporal design problem for a damped wave equation,, SIAM J. Appl. Math., 68 (2007), 109.  doi: 10.1137/07067965X.  Google Scholar

[12]

G. W. Milton, "The Theory of Composites,", Cambridge Monographs on Applied and Computational Mathematics, 6 (2002).   Google Scholar

[13]

F. Murat, Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients,, Ann. Mat. Pura Appl. (4), 112 (1977), 49.   Google Scholar

[14]

F. Murat and L. Tartar, Optimality conditions and homogenization,, in, 127 (1985), 1.   Google Scholar

[15]

F. Murat and L. Tartar, $H$-convergence,, in, 31 (1997), 21.   Google Scholar

[16]

F. Murat and L. Tartar, Calculus of variations and homogenization,, in, 31 (1997), 139.   Google Scholar

[17]

P. Pedregal, Vector variational problems and applications to optimal design,, ESAIM Control Optim. Calc. Var., 11 (2005), 357.  doi: 10.1051/cocv:2005010.  Google Scholar

[18]

P. Pedregal, "Parametrized Measures and Variational Principles,", Progress Nonlinear Differential Equations and Their Applications, 30 (1997).   Google Scholar

[19]

J. Sokołowski and J.-P. Zolésio, "Introduction to Shape Optimization: Shape Sensitivity Analysis,", Springer Series in Computational Mathematics, 16 (1992).   Google Scholar

[20]

L. Tartar, An introduction to the homogenization method in optimal design,, in, 1740 (2000), 47.   Google Scholar

show all references

References:
[1]

G. Allaire, "Shape Optimization by the Homogenization Method,", Applied Mathematical Sciences, 146 (2002).   Google Scholar

[2]

F. Alvarez and M. Carrasco, Minimization of the expected compliance as an alternative approach to multiload truss optimization,, Struct. Multidiscip. Optim., 29 (2005), 470.  doi: 10.1007/s00158-004-0488-7.  Google Scholar

[3]

M. P. Bendsøe and O. Sigmund, "Topology Optimization: Theory, Methods and Aplications,", Springer-Verlag, (2003).   Google Scholar

[4]

D. Bucur and G. Buttazzo, "Variational Methods in Shape Optimization Problems,", Progress in Nonlinear Differential Equations and their Applications, 65 (2005).   Google Scholar

[5]

J. Casado-Díaz, J. Couce-Calvo, M. Luna-Laynez and J. D. Martín-Gómez, Optimal design problems for a non-linear cost in the gradient: numerical results,, Applicable Analysis, 87 (2008), 1461.   Google Scholar

[6]

A. Cherkaev, "Variational Methods for Structural Optimization,", Applied Mathematical Sciences, 140 (2000).   Google Scholar

[7]

G. Dal Maso and L. Modica, Nonlinear stochastic homogenization and ergodic theory,, J. Reine Angew. Math., 368 (1986), 28.   Google Scholar

[8]

A. Donoso and P. Pedregal, Optimal design of 2-$D$ conducting graded materials by minimizing quadratic funtionals in the field,, Struc. Multidisc Optim., 30 (2005), 360.  doi: 10.1007/s00158-005-0521-5.  Google Scholar

[9]

A. Henrot and M. Pierre, "Variation et Optimisation de Formes. Une Analyse Géométrique,", Mathématiques et Applications, 48 (2005).   Google Scholar

[10]

K. A. Lurie and A. V. Cherkaev, Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion,, Proc. R. Soc. Edinb. A, 99 (1984), 71.   Google Scholar

[11]

F. Maestre, A. Münch and P. Pedregal, A spatio-temporal design problem for a damped wave equation,, SIAM J. Appl. Math., 68 (2007), 109.  doi: 10.1137/07067965X.  Google Scholar

[12]

G. W. Milton, "The Theory of Composites,", Cambridge Monographs on Applied and Computational Mathematics, 6 (2002).   Google Scholar

[13]

F. Murat, Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients,, Ann. Mat. Pura Appl. (4), 112 (1977), 49.   Google Scholar

[14]

F. Murat and L. Tartar, Optimality conditions and homogenization,, in, 127 (1985), 1.   Google Scholar

[15]

F. Murat and L. Tartar, $H$-convergence,, in, 31 (1997), 21.   Google Scholar

[16]

F. Murat and L. Tartar, Calculus of variations and homogenization,, in, 31 (1997), 139.   Google Scholar

[17]

P. Pedregal, Vector variational problems and applications to optimal design,, ESAIM Control Optim. Calc. Var., 11 (2005), 357.  doi: 10.1051/cocv:2005010.  Google Scholar

[18]

P. Pedregal, "Parametrized Measures and Variational Principles,", Progress Nonlinear Differential Equations and Their Applications, 30 (1997).   Google Scholar

[19]

J. Sokołowski and J.-P. Zolésio, "Introduction to Shape Optimization: Shape Sensitivity Analysis,", Springer Series in Computational Mathematics, 16 (1992).   Google Scholar

[20]

L. Tartar, An introduction to the homogenization method in optimal design,, in, 1740 (2000), 47.   Google Scholar

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