TY - RPRT

T1 - On some fundamental properties of structural topology optimization problems

AU - Stolpe, Mathias

PY - 2009

Y1 - 2009

N2 - We study some fundamental mathematical properties of classical structural topology optimization problems. Either compliance is minimized with an upper bound on the volume of the structure, or volume is minimized with an upper bound on the compliance. The design variables are either continuous or 0--1. We show, by examples which can be solved by hand calculations, that the optimal solutions in general are not unique and possibly do not have an active volume constraint. These observations have immediate consequences on the theoretical convergence properties of penalization approaches. Furthermore, we illustrate that the optimal solutions to the considered problems in general are not symmetric even if the design domain, the external loads, and the boundary conditions are symmetric around an axis.
The presented examples can be used as teaching material in graduate and undergraduate courses on structural topology optimization.

AB - We study some fundamental mathematical properties of classical structural topology optimization problems. Either compliance is minimized with an upper bound on the volume of the structure, or volume is minimized with an upper bound on the compliance. The design variables are either continuous or 0--1. We show, by examples which can be solved by hand calculations, that the optimal solutions in general are not unique and possibly do not have an active volume constraint. These observations have immediate consequences on the theoretical convergence properties of penalization approaches. Furthermore, we illustrate that the optimal solutions to the considered problems in general are not symmetric even if the design domain, the external loads, and the boundary conditions are symmetric around an axis.
The presented examples can be used as teaching material in graduate and undergraduate courses on structural topology optimization.

KW - Minimum compliance optimization

KW - Topology optimization

KW - Minimum volume optimization

M3 - Report

T3 - DCAMM Report

BT - On some fundamental properties of structural topology optimization problems

ER -