Descripción
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 \emph{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 artificial time variable for our phasefield
variable. We discuss well-posedness results for the resulting
system and conclude with some numerical results.