Mixed Finite Element Method for a Degenerate Convex Variational Problem from Topology Optimization

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 $E\left( v \right):= \int_\Omega \varphi_0\left( \left\lvert\nabla v\right\rvert \right)\operatorname{dx} - \int_\Omega fv\operatorname{dx}\text{for } v \in V:=H^1_0( \Omega )$ with possibly multiple primal solutions $u$, but with unique stress $\sigma:=\varphi_0'\left( \left\lvert\nabla u\right\rvert \right)\operatorname{sign}\nabla u.$ The mixed finite element method is motivated by the smoothness of the stress variable $\sigma \in H^1_{\operatorname{loc}}( \Omega ; \mathbb R^2)$, 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.