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 given cross section.
This example of relaxation in topology optimisation
leads to a degenerate convex minimisation problem
E(v)
:=
∫Ω φ0(|∇v|) dx
-
∫ fv dx
for v∈V:=H01(Ω)
with possibly multiple primal solutions u,
but with unique stress
σ
:=
φ0(|∇u|) sign ∇u
The mixed finite element method is motivated
by the smoothness of the stress variable
σ
∈
H1loc(Ω;R2)
while the primal variables are
un-controllable and possibly non-unique.
The corresponding nonlinear mixed finite element
method is introduced, analysed, 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 with the new mixed
discretisation schemes is intended for
uniform and adaptive mesh-refinements.