A New Approach for Solving Singular Systems in Topology Optimization Using Krylov Subspace Methods

TL;DR

This paper introduces a novel approach using Krylov subspace methods, specifically Conjugate Residual and Conjugate Gradient Methods, to effectively solve singular systems in topology optimization, ensuring convergence to local optima.

Contribution

It demonstrates that CGM and CRM can find local optimal solutions even when the stiffness matrix is singular, supported by theoretical proof and simulation results.

Findings

CGM converges to a local optimal solution with singular matrices.

CRM and CGM produce identical solutions in the tested problem.

The approach ensures solution stability in topology optimization with singular systems.

Abstract

In topology optimization, the design parameter with no contribution to the objective function vanishes. This causes the stiffness matrix to become singular. We show that a local optimal solution is obtained by Conjugate Residual Method and Conjugate Gradient Method even if the stiffness matrix becomes singular. We prove that CGMconverges to a local optimal solution in that case. Computer simulation shows that CGM gives the same solutions obtained by CRM in case of a cantilever beam problem.