Topology Optimization of Continuum Elastic Structures through the Progressive Directional Selection Method

Resumo

The present work deals with a new approach for Topology Optimization (TO) of two-dimensional
continuum elastic structures through the Progressive Directional Selection (PDS) method. To achieve the best
topology of a structure, a typical goal is to define the best material distribution in a domain, considering an
objective function and mechanical constraints. In general, most of the studies address the compliance minimization
of structures. Numerical methods for TO of continuum structures have been investigated extensively. Most of
those methods are based on finite element analysis, where the design domain is discretized into a fine mesh of
elements. In such a setting, the optimization procedure is to find a structure's topology by determining for every
point in the design domain if there should be material (solid element) or not (void element). To control the process
of inserting or removing finite elements without forgotten the continuum representation, standard algorithms as
Homogenization, Solid Isotropic Microstructure with Penalization (SIMP), and Evolutionary Structural
Optimization (ESO) are applied in many studies. The latter approach is based on the simple concept that the
optimal design can be achieved by gradually removing inefficient material (elements) from the design space. The
ESO algorithms are easy to understand and implement. However, ESO is heuristic, and there is no proof that an
optimum solution can be achieved by element elimination and admission. The original scheme is inefficient once
it needs to find the best solution by comparing several intuitively generated solutions. To avoid this problem but
taking advantage of ESO's simplicity, the PDS method, which is inspired by natural selection observed in biology,
applies a strategy to minimize the strain energy of a discretized analyzed domain with a volume restriction. Based
on the performance criteria adopted for the problem, the selected population is reached through an iterative process
that converges when the optimal topology does not evolve anymore, i.e., there is no change in the final set of
selected elements. An instructional problem that shows the essence of PDS for a rigid body problem and one
example of topology optimization of a classical problem found in the literature are investigated.