Assessment of penalty parameters in density-based topology optimization

Resumo

The present paper discusses two penalty parameters — the penalty parameter of intermediate densities
(PPID) and the penalty parameter of the density field gradient (PPDFG) — in the context of density-based topology
optimization problems considering a Solid Isotropic Microstructure with Penalization (SIMP). Both parameters
aim to control the influence of functional terms added to the cost functional to regularize the optimization problem.
PPID aims to the reduction of intermediate densities regions and, consequently, finding out a better definition of
the material contour. PPDFG, in its turn, has as its purpose to mitigate the checkerboard phenomenon.
Displacements, strains, and stresses are obtained via the Finite Element Method, with relative densities defined at
element nodes. The mathematical solution of the optimization problem is carried out taking into account the
Augmented Lagrangean Method. The numerical experiment planning consisted of performing a series of
combinations between the penalty parameters and different finite element meshes, to check their influence on the
checkerboard phenomenon, mesh dependence, intermediate densities regions extension, structural topology, and
values of the cost functional. In particular, the optimization problem aims to minimize the compliance of the
structure with a constraint on the material volume. From the results obtained in this work, it is possible to verify
that PPGDF can mitigate or inhibit the mesh dependence, avoid coarse contours, and the occurrence of the
checkerboard phenomenon — conversely, for high values, extensive intermediate densities regions appear, causing
a sharp increase in the structure compliance. In addition, PPGDF has a greater influence than PPID, which, for the
optimization problem presented, exerted little variation in the compliance of the structure.