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.

本研究提出了一种有效的 Moore-Penrose 伪逆近似方法，当对于矩阵，与特征值零相关的特征向量（在此称为零特征向量）是预先已知的。该方法将计算成本降低了几个数量级。近似是通过添加小幅度对角矩阵来正则化矩阵并在其正则求逆后与投影矩阵相乘来执行的。投影去除了零特征向量的分量。获得良好近似的条件是小幅度矩阵的幅度应足够小于矩阵的最小非零特征值。当矩阵是无支撑弹性问题（弹性体不受支撑的问题）中的刚度矩阵时，零特征向量表示刚体运动。该方法应用于鲁棒的无支撑拓扑优化，显示出其优异的准确性和效率。发现观察到的计算时间与刚度矩阵的大小成正比。此外，对细网格问题进行稳健的拓扑优化会产生具有生物特征的结构。