Asymptotics of an optimal compliance-network problem
-
Received:
01 October 2012
Revised:
01 March 2013
-
-
Primary: 49J45; Secondary: 49Q10, 74P05.
-
-
We consider the problem of the optimal location of a Dirichlet
region in a $d$-dimensional domain $\Omega$ subjected to a
given force $f$ in order to minimize the $p$-compliance of the
configuration. We look for the optimal region among the class
of all closed connected sets of assigned length $l.$ Then
we let the length $l$ tend to infinity and we look at
the $\Gamma$-limit of a suitable rescaled functional, from which
we get information of the asymptotic distribution of the
optimal region. We also study the case where the Dirichlet
region is a discrete set of finite cardinality.
Citation: Al-hassem Nayam. Asymptotics of an optimal compliance-network problem[J]. Networks and Heterogeneous Media, 2013, 8(2): 573-589. doi: 10.3934/nhm.2013.8.573
Related Papers:
| [1] |
Giuseppe Buttazzo, Filippo Santambrogio .
Asymptotical compliance optimization for connected networks. Networks and Heterogeneous Media, 2007, 2(4): 761-777.
doi: 10.3934/nhm.2007.2.761
|
| [2] |
Al-hassem Nayam .
Asymptotics of an optimal compliance-network problem. Networks and Heterogeneous Media, 2013, 8(2): 573-589.
doi: 10.3934/nhm.2013.8.573
|
| [3] |
Paolo Tilli .
Compliance estimates for two-dimensional
problems with Dirichlet region of prescribed length. Networks and Heterogeneous Media, 2012, 7(1): 127-136.
doi: 10.3934/nhm.2012.7.127
|
| [4] |
Al-hassem Nayam .
Constant in two-dimensional $p$-compliance-network problem. Networks and Heterogeneous Media, 2014, 9(1): 161-168.
doi: 10.3934/nhm.2014.9.161
|
| [5] |
Jean-Bernard Baillon, Guillaume Carlier .
From discrete to continuous Wardrop equilibria. Networks and Heterogeneous Media, 2012, 7(2): 219-241.
doi: 10.3934/nhm.2012.7.219
|
| [6] |
Alessandra Pluda .
Evolution of spoon-shaped networks. Networks and Heterogeneous Media, 2016, 11(3): 509-526.
doi: 10.3934/nhm.2016007
|
| [7] |
Laura Sigalotti .
Homogenization of pinning conditions on periodic networks. Networks and Heterogeneous Media, 2012, 7(3): 543-582.
doi: 10.3934/nhm.2012.7.543
|
| [8] |
Yicheng Liu, Yipeng Chen, Jun Wu, Xiao Wang .
Periodic consensus in network systems with general distributed processing delays. Networks and Heterogeneous Media, 2021, 16(1): 139-153.
doi: 10.3934/nhm.2021002
|
| [9] |
Alberto Bressan, Sondre Tesdal Galtung .
A 2-dimensional shape optimization problem for tree branches. Networks and Heterogeneous Media, 2021, 16(1): 1-29.
doi: 10.3934/nhm.2020031
|
| [10] |
Michael Herty, Veronika Sachers .
Adjoint calculus for optimization of gas networks. Networks and Heterogeneous Media, 2007, 2(4): 733-750.
doi: 10.3934/nhm.2007.2.733
|
-
Abstract
We consider the problem of the optimal location of a Dirichlet
region in a $d$-dimensional domain $\Omega$ subjected to a
given force $f$ in order to minimize the $p$-compliance of the
configuration. We look for the optimal region among the class
of all closed connected sets of assigned length $l.$ Then
we let the length $l$ tend to infinity and we look at
the $\Gamma$-limit of a suitable rescaled functional, from which
we get information of the asymptotic distribution of the
optimal region. We also study the case where the Dirichlet
region is a discrete set of finite cardinality.
References
|
[1]
|
G. Allaire, "Shape Optimization by the Homogenization Method," Applied Mathematical Sciences, 146, Springer-Verlag, New York 2002.
|
|
[2]
|
G. Bouchité, C.Jimenez and M. Rajesh, Asymptotique d'un problème de positionnement optimal, C. R. Acad. Sci. Paris Sér. I, 335 (2002), 1-6.
|
|
[3]
|
D. Bucur and G. Buttazzo, "Variational Methods in Shape Optimization Problems," Progress in Nonlinear Differential Equation and their Applications, 65, Birkhäuser Boston, Inc., Boston, MA, 2005.
|
|
[4]
|
D. Bucur and P. Trebeschi, Shape optimization governed by nonlinear state equations, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 945-963. doi: 10.1017/S0308210500030006
|
|
[5]
|
G. Buttazzo and F. Santambrogio, Asymptotical compliance optimization for connected networks, Networks and Heterogeneous Media, 2 (2007), 761-777. doi: 10.3934/nhm.2007.2.761
|
|
[6]
|
G. Buttazzo, F. Santambrogio and N. Varchon, Asymptotics of an optimal compliance-location problem, ESAIM Control Optimization and Calculus of Variations, 12 (2006), 752-769. doi: 10.1051/cocv:2006020
|
|
[7]
|
G. Dal Maso, "An Introduction to $\Gamma$-Convergence," Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8
|
|
[8]
|
G. Dal Maso and F. Murat, Asymptotic behavior and corrector for the Dirichlet problem in perforated domains with homogeneous monotone operators, Ann. Sc. Norm. Sup. Pisa Cl. Sci Ser. (4), 24 (1997), 239-290.
|
|
[9]
|
V. G. Maz'ja, "Sobolev Spaces," Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985.
|
|
[10]
|
S. Mosconi and P. Tilli, $\Gamma$-convergence for the irrigation problem, J. Conv. Anal., 12 (2005), 145-158.
|
|
[11]
|
V. Šverak, On optimal shape design, J. Math. Pures Appl. (9), 72 (1993), 537-551.
|
-
-
This article has been cited by:
| 1.
|
Bohdan Bulanyi,
On the importance of the connectedness assumption in the statement of the optimal p-compliance problem,
2021,
499,
0022247X,
125064,
10.1016/j.jmaa.2021.125064
|
|
| 2.
|
Bohdan Bulanyi,
Partial regularity for the optimal p-compliance problem with length penalization,
2022,
61,
0944-2669,
10.1007/s00526-021-02073-8
|
|
| 3.
|
Bohdan Bulanyi, Antoine Lemenant,
Regularity for the planar optimalp-compliance problem,
2021,
27,
1292-8119,
35,
10.1051/cocv/2021035
|
|
-
-
DownLoad: