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

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

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].
Citation: G. Machado, L. Trabucho. Analytical and numerical solutions for a class of optimization problems in elasticity. Discrete & 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 & 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 & 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 & 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 & Heterogeneous Media, 2012, 7 (3) : 543-582. doi: 10.3934/nhm.2012.7.543

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

Jiongmin Yong. Optimality conditions for controls of semilinear evolution systems with mixed constraints. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 371-388. doi: 10.3934/dcds.1995.1.371

[15]

Miniak-Górecka Alicja, Nowakowski Andrzej. Sufficient optimality conditions for a class of epidemic problems with control on the boundary. Mathematical Biosciences & Engineering, 2017, 14 (1) : 263-275. doi: 10.3934/mbe.2017017

[16]

Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417

[17]

Lihua Li, Yan Gao, Hongjie Wang. Second order sufficient optimality conditions for hybrid control problems with state jump. Journal of Industrial & Management Optimization, 2015, 11 (1) : 329-343. doi: 10.3934/jimo.2015.11.329

[18]

Adela Capătă. Optimality conditions for strong vector equilibrium problems under a weak constraint qualification. Journal of Industrial & Management Optimization, 2015, 11 (2) : 563-574. doi: 10.3934/jimo.2015.11.563

[19]

Vladimir Srochko, Vladimir Antonik, Elena Aksenyushkina. Sufficient optimality conditions for extremal controls based on functional increment formulas. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 191-199. doi: 10.3934/naco.2017013

[20]

Jing Quan, Zhiyou Wu, Guoquan Li. Global optimality conditions for some classes of polynomial integer programming problems. Journal of Industrial & Management Optimization, 2011, 7 (1) : 67-78. doi: 10.3934/jimo.2011.7.67

2016 Impact Factor: 0.994

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]