Topology optimization for two-dimensional structures using boundary cycles in homology theory

Abstract

Various prominent methods for structural topology optimization have been developed, especially since the late 1980s when the homogenization method was proposed. However, knowledge in the topology of mathematics was nearly not applied in structural optimization. With the application of the homology theory in the topology, this study proposed an original method of optimizing the topology of a structure consisting of elastic triangular plates sustaining a static load. The total volume of the structure was minimized under the constraint that the strain energy density in all elements is equal to a prescribed value. Although apparent design variables are the material density of triangular plates, the coefficients of boundary cycles of 3-simplexes are utilized as auxiliary variables that express inner forces in the plates. The arbitrary values of the auxiliary variables always satisfy the equilibrium among inner forces in triangular plates and do not generate useless elements with no stress throughout the optimization process. Simultaneous linear equations for stress analysis need not be solved repeatedly because the volume minimization process is concurrently involved in the analysis. The number of variables can be reduced to less than that of triangular plates with eigenvalue decomposition. The validity of the proposed method was verified by applying it to several numerical examples.