Structural Optimization under Topological Constraint Represented by Homology Groups. (Topological Constraint on One-Dimensional Complex by Use of Zero- and One-Dimensional Homology Groups).

Abstract

Topology of any one-dimensional complex can be represented by zero- and one-dimensional homology groups, which are isomorphic to the direct sum of additive groups. In this paper, a method is proposed to impose constraint on the topology of a frame treated as a one-dimensional complex by use of homology groups in the field of structural optimization. As the numerical examples, the total strain energy of the frame is minimized under topological constraints and constant weight. Useless members are eliminated from a ground structure by use of genetic algorithm. Any number of additive groups can be freely set up as a topological constraint because of generalized inverse matrices, and a rule of coding in the genetic algorithm is prescribed so that all strings(corresponding to chromosomes in biological systems)generated in the optimization process could satisfy the topological constraints. As the result it is found that loops in the topology of the optimum structure adjoin each other. The proposed method is also applied to the topology optimization of a square, flat panel board fixed on a rigid wall and loaded vertically on points distant from the wall.