Topology Optimization Of Two-Dimensional Elastic Structures Using Boundary Elements And The Topological Derivative

Abstract

Topological Optimization provides a powerful framework to obtain the optimal domain
topology for several engineering problems. The Topological Derivative is a function which
characterizes the sensitivity of a given problem to the change of its topology, like opening a small hole
in a continuum or changing the connectivity of rods in a truss.
A numerical approach for the topological optimization of 2D linear elastic problems using Boundary
Elements is presented in this work. The formulation of the problem is based on recent results which
allow computing the topological derivative from strain and stress results. The Boundary Element
analysis is done using a standard direct formulation. Models are discretized using linear elements and
a periodic distribution of internal points over the domain. The total potential energy is selected as cost
function. The evaluation of the topological derivative at internal points is performed as a postprocessing
procedure. Afterwards, material is removed from the model by deleting the internal points
with the lowest values of the topological derivate. The new geometry is then remeshed using a
weighted Delaunay triangularization algorithm capable of detecting “holes” at those positions where
internal points have been removed. The procedure is repeated until a given stopping criteria is
satisfied.
The proposed strategy proved to be flexible and robust. A number of examples are solved and results
are compared to those available in the literature.