A new rational approach to multi-input multi-output 3D topology optimization

A new 3D topology optimization approach is presented that is based on the singular value decomposition of the input/output transfer matrix of the system. To start with, the input and output vectors, i.e. the acting loads and the quantities of interest for the designer, are chosen and the input-output transfer matrix is derived. Such matrix, say $G\left(p\right)$, depends on the vector of the design variables p (the densities at the element level). The singular value decomposition of $G\left(p\right)$ is the core of the proposed approach. It provides singular values as well as left and right singular vectors. Singular values are shown to uniquely define a few matrix norms that can be conveniently computed and used as goal functions to be minimized. Left and right singular vectors respectively represent the principal input/output pairs of the system whose gain is the associated singular value. Numerical optimization is pursued via the method of moving asymptotes (MMA) [1] that calls for the semi-analytic computations of objective functions and constraints. The results of a few 3D numerical investigations are presented and discussed in much detail. An in-house Matlab code developed for the sake of this paper, and based on the ones presented in [2] and [3], is provided in full as an Appendix to the paper.