Erschienen:
Society for Industrial and Applied Mathematics, 2012
Erschienen in:
SIAM Journal on Numerical Analysis, 50 (2012) 2, Seite 522-543
Sprache:
Englisch
DOI:
10.1137/100806837
ISSN:
0036-1429
Entstehung:
Anmerkungen:
Beschreibung:
The optimal design task of this paper seeks the distribution of two materials of prescribed amounts for maximal torsion stiffness of an infinite bar of a given cross section. This example of relaxation in topology optimization leads to a degenerate convex minimization problem $E(v): = \int_\Omega {{\varphi _0}} (|\nabla v|)dx - \int_\Omega {fvdx} $ for $v \in V: = H_0^1\left( \Omega \right)$ with possibly multiple primal solutions u, but with unique stress $\sigma : = \varphi _0^1\left( {|\nabla u|} \right)$ sign ∇u. The mixed finite element method is motivated by the smoothness of the stress variable $\sigma \in H_{loc}^1\left( {\Omega ;{\mathbb{R}^2}} \right)$ , while the primal variables are uncontrollable and possibly nonunique. The corresponding nonlinear mixed finite element method is introduced, analyzed, and implemented. The striking result of this paper is a sharp a posteriori error estimation in the dual formulation, while the a posteriori error analysis in the primal problem suffers from the reliability-efficiency gap. An empirical comparison of that primal formulation with the new mixed discretization schemes is intended for uniform and adaptive mesh refinements.