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TY - JOUR
TI - MIXED FINITE ELEMENT METHOD FOR A DEGENERATE CONVEX VARIATIONAL PROBLEM FROM TOPOLOGY OPTIMIZATION
AU - CARSTENSEN, CARSTEN
AU - GÜNTHER, DAVID
AU - RABUS, HELLA
AB - 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.
C1 - Full publication date: 2012
EP - 543
IS - 2
JO - SIAM Journal on Numerical Analysis
PB - Society for Industrial and Applied Mathematics
PY - 2012
SN - 00361429
SP - 522
UR - http://www.jstor.org/stable/41582748
VL - 50
Y2 - 2021/03/04/
ER -