SPP 2256 - Subproject: Multiscale structure in compliance minimization

Basic data for this project

Description

It is a classic engineering problem to identify which geometry of an elastic material best supports a load while consuming a minimum material amount. Typically, the optimal geometry would involve an infinitely fine microstructure with infinitely many, infinitely fine holes. To avoid dealing with the microstructure, the optimization problem is typically relaxed (meaning that microstructured regions are simply replaced by a non-microstructured material with same macroscopic elastic properties, adapting the optimization problem correspondingly) or strongly regularized (meaning that one adds something like production costs to the optimization problem, which will prevent structures with too many or too fine holes). In contrast, in this project we are interested in the case of very weak regularization, in which fine structures and coarsening phenomena over multiple scales will occur (that is, in some regions the structure will be very fine and in others quite coarse). Our aim is to better understand these structures via variational methods. Thus, rather than modelling and understanding the behaviour or response of an existing material, we here consider a material design problem that results in a complex structure.