A simple and efficient pseudo-inverse approximation for real symmetric matrices and applications to elasticity problems. (1st October 2021) - Fukada, Yoshiki
Highlights: The computation cost is reduced by several orders of magnitudes. The computation consists of a regular inverse, a projection and projector construction. Applicable when the zero's eigenvectors of the matrix are known. It enables structural analyses for floating elastic bodies without inertia modeling. The proof and an error analysis are also presented. Abstract: This study proposes an efficient approximation method for the Moore–Penrose pseudo-inverse when, for a matrix, the eigenvectors associated to the eigenvalues zero (referred to herein as zeros eigenvectors) are known in advance. The method reduces the computational cost by several orders of magnitude. The approximation is performed by the addition of a small-amplitude diagonal matrix to regularise the matrix and multiplication with a projection matrix after its regular inversion. The projection removes the components of the zeros eigenvectors. The condition for obtaining a good approximation, is that the amplitude of the small-amplitude matrix should be sufficiently smaller than the smallest non-zero eigenvalue of the matrix. When the matrix is a stiffness matrix in a support-free elasticity problem (a problem whereby the elastic body is unsupported), the zeros eigenvectors indicate rigid-body motions. The method was applied to robust support-free topology optimization, revealing its excellent accuracy and efficiency. The observed computational time was found to be proportional to the size of the stiffness matrix. Furthermore, conducting robust topology optimization for fine-mesh problems resulted in structures that exhibited biological features.