Representation of Topology Using Homology Groups and its Application to Structural Optimization. 1st Report. Verification of Applicability of Homology Theory in Abstract Problem of Closed Surface.

Abstract

In this paper, application of the homology theory, which is a field of mathematics concerned with topology, to structural optimization is attempted. A homology group, which is one of the Abelian groups, can represent the topology of a structure in a standardized way ; that is, the canonical direct sum decomposition. Therefore, topological information of a structure can be described in the formulation using homology groups. An abstract combination problem of triangular elements is chosen as a numerical example to verify the generality of this method using the homology theory. Several of the triangular elements are combined into a structure. Under the topological constraint condition represented by homology groups, a structure which has the minimum sum of numbers of triangular elements, its sides, and its vertices, is found by genetic algorithm. Furthermore, homology groups are utilized to describe the fitness function in the genetic algorithm as another example.