American Institute of Mathematical Sciences

Advanced
November  2004, 4(4): 1013-1032. doi: 10.3934/dcdsb.2004.4.1013

Analytical and numerical solutions for a class of optimization problems in elasticity

G. Machado 1, and  L. Trabucho 2,

1. 

Department of Mathematics for Science and Technology, Officina Mathematica, University of Minho, 4800-058 Guimarães, Portugal
2. 

Department of Mathematics, University of Lisbon, 1649-003 Lisboa, Portugal

Received  December 2002 Revised  January 2004 Published  August 2004

The subject of topology optimization methods in structural design has increased rapidly since the publication of [5], where some ideas from homogenization theory were put into practice. Since then, several engineering applications have been developed successfully. However, in the literature, there is a lack of analytical solutions, even for simple cases, which might help in the validation of the numerical results. In this work, we develop analytical solutions for simple minimum compliance problems, in the framework of elasticity theory. We compare these analytical solutions with numerical results obtained via an algorithm proposed in [4].
Keywords: homogenization theory, optimality conditions., Elasticity.
Mathematics Subject Classification: 74Q05, 74P0.
Citation: G. Machado, L. Trabucho. Analytical and numerical solutions for a class of optimization problems in elasticity. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 1013-1032. doi: 10.3934/dcdsb.2004.4.1013
[1]

Antoine Gloria Cermics. A direct approach to numerical homogenization in finite elasticity. Networks and Heterogeneous Media, 2006, 1 (1) : 109-141. doi: 10.3934/nhm.2006.1.109
[2]

B. Bonnard, J.-B. Caillau, E. Trélat. Second order optimality conditions with applications. Conference Publications, 2007, 2007 (Special) : 145-154. doi: 10.3934/proc.2007.2007.145
[3]

Hugo Beirão da Veiga. A challenging open problem: The inviscid limit under slip-type boundary conditions.. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 231-236. doi: 10.3934/dcdss.2010.3.231
[4]

José M. Arrieta, Simone M. Bruschi. Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 327-351. doi: 10.3934/dcdsb.2010.14.327
[5]

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
[6]

Mansoureh Alavi Hejazi, Soghra Nobakhtian. Optimality conditions for multiobjective fractional programming, via convexificators. Journal of Industrial and Management Optimization, 2020, 16 (2) : 623-631. doi: 10.3934/jimo.2018170
[7]

Luong V. Nguyen. A note on optimality conditions for optimal exit time problems. Mathematical Control and Related Fields, 2015, 5 (2) : 291-303. doi: 10.3934/mcrf.2015.5.291
[8]

Piernicola Bettiol, Nathalie Khalil. Necessary optimality conditions for average cost minimization problems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2093-2124. doi: 10.3934/dcdsb.2019086
[9]

Geng-Hua Li, Sheng-Jie Li. Unified optimality conditions for set-valued optimizations. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1101-1116. doi: 10.3934/jimo.2018087
[10]

Ying Gao, Xinmin Yang, Kok Lay Teo. Optimality conditions for approximate solutions of vector optimization problems. Journal of Industrial and Management Optimization, 2011, 7 (2) : 483-496. doi: 10.3934/jimo.2011.7.483
[11]

Adela Capătă. Optimality conditions for vector equilibrium problems and their applications. Journal of Industrial and Management Optimization, 2013, 9 (3) : 659-669. doi: 10.3934/jimo.2013.9.659
[12]

Qiu-Sheng Qiu. Optimality conditions for vector equilibrium problems with constraints. Journal of Industrial and Management Optimization, 2009, 5 (4) : 783-790. doi: 10.3934/jimo.2009.5.783
[13]

Majid E. Abbasov. Generalized exhausters: Existence, construction, optimality conditions. Journal of Industrial and Management Optimization, 2015, 11 (1) : 217-230. doi: 10.3934/jimo.2015.11.217
[14]

Shahlar F. Maharramov. Necessary optimality conditions for switching control problems. Journal of Industrial and Management Optimization, 2010, 6 (1) : 47-55. doi: 10.3934/jimo.2010.6.47
[15]

Gang Li, Yinghong Xu, Zhenhua Qin. Optimality conditions for composite DC infinite programming problems. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022064
[16]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
[17]

Michael Eden, Michael Böhm. Homogenization of a poro-elasticity model coupled with diffusive transport and a first order reaction for concrete. Networks and Heterogeneous Media, 2014, 9 (4) : 599-615. doi: 10.3934/nhm.2014.9.599
[18]

Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1047-1069. doi: 10.3934/dcds.2008.21.1047
[19]

Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control and Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001
[20]

Qilin Wang, Xiao-Bing Li, Guolin Yu. Second-order weak composed epiderivatives and applications to optimality conditions. Journal of Industrial and Management Optimization, 2013, 9 (2) : 455-470. doi: 10.3934/jimo.2013.9.455

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy