Applied Mathematical Sciences, Vol. 6, 2012, no. 12, 553 - 573 Geometrical Behavior in Elasticity Problems by Topological Optimization Coumba DIALLO ∗,+ ∗ Universit´ e Cheikh Anta DIOP de Dakar + Laboratoire de Math´ ematiques de la D´ ecision et d’Analyse Num´ erique Ecole Doctorale de Math´ ematique et Informatique coumbdiallo@yahoo.fr Ibrahima FAYE + Universit´ e de Bambey, Bp 30, Bambey Ecole Doctorale de Math´ ematique et Informatique UMMISCO-UMI,209, IRD, France grandmbodj@hotmail.com Diaraf SECK ∗,+ Ecole Doctorale de Math´ ematique et Informatique UMMISCO-UMI,209, IRD, France diaraf.seck@ucad.edu.sn Abstract In this paper we study thermoelasticity and Michell trusses problems by using topolog- ical optimization tools. After the modelling of the thermoelasticity problem, we give the topological derivative in both Dirichlet and Neumann conditions.We will finish this work by giving a linking between the trusses Michell and topological optimization. And the last part of this work is devoted to numerical simulations. Mathematics Subject Classification: 74P05, 74P10, 74P15, 65F30, 49, 49Q10 Keywords: Topological optimization, Topological gradient,Thermoelasticity, Elasticity, Michell trusses, Numerical simulations 1 Introduction In this work we consider the system as the following form: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ -μΔu Ω - (λ + μ) --→ graddivu Ω - 3kα --→ grad θ Ω = -→ f in Ω -Δθ Ω = g in Ω ∂θ Ω ∂n = h on ∂ Ω Bu Ω = v on ∂ Ω. (1)