Owing to different densities of the respective phases, solid-solid
phase transitions often are accompanied by changes in workpiece size
and shape. In my talk I will address the question of finding an
optimal phase mixture in order to accomplish a desired workpiece
shape.
From mathematical point of view this corresponds to an optimal shape
design problem subject  to a static mechanical equilibrium problem
with phase dependent stiffness tensor, in which the two phases exhibit
different densities leading to different internal stresses. Our goal
is to tackle this problem using a phasefield relaxation. To this end
we first briefly recall previous works regarding phasefield approaches
to topology optimization (e.g.  by Bourdin \& Chambolle, Burger \&
Stainko and Blank, Garcke et al.).
We add  a Ginzburg-Landau term to our cost functional, derive an
adjoint equation for the displacement and choose a gradient
flow dynamics with an articial time variable for our phasefield
variable. We discuss well-posedness results for the resulting system
and  conclude with some numerical results.