A new approach for solving singular systems in topology optimization using Krylov subspace methods

In topology optimization, elements without any contribution to the improvement of the objective function vanish by decrease of density of the design parameter. This easily causes a singular stiffness matrix. To avoid the numerical breakdown caused by this singularity, conventional optimization techniques employ additional procedures. These additional procedures, however, raise some problems. On the other hand, convergence of Krylov subspace methods for singular systems have been studied recently. Through subsequent studies, it has been revealed that the conjugate gradient method (CGM) does not converge to the local optimal solution in some singular systems but in those satisfying certain condition, while the conjugate residual method (CRM) yields converged solutions in any singular systems. In this article, we show that a local optimal solution for topology optimization is obtained by using the CRM and the CGM as a solver of the equilibrium equation in the structural analysis, even if the stiffness matrix becomes singular. Moreover, we prove that the CGM, without any additional procedures, realizes convergence to a local optimal solution in that case. Computer simulation shows that the CGM gives almost the same solutions obtained by the CRM in the case of the two-bar truss problem.