Quasi-Newton corrections for compliance and natural frequency topology optimization problems

Sequential quadratic programming and interior point methods are effective general purpose optimization algorithms that sometimes exhibit poor performance in topology optimization applications. We postulate that this performance behavior stems from the frequent failure of BFGS updates for common topology optimization problem formulations that reduces the effectiveness of the resulting Hessian approximations. To address this issue, we propose a computationally efficient correction technique that utilizes problem-specific second-order derivative information to remove negative curvature contributions and achieve better overall optimizer performance. The proposed technique is tested on compliance and natural frequency topology optimization problem sets and compared against the quasi-Newton optimizers ParOpt, IPOPT, and SNOPT with the BFGS and SR1 Hessian approximations, as well as MMA. The results show that the optimizer with the proposed correction produces optimized designs in fewer function evaluations with either lower or competitive objective and discreteness metric values compared to other quasi-Newton based optimizers. Finally, scalability of the implementation is demonstrated by solving large-scale problems that have up to 96.9 million degrees of freedom with adaptively refined meshes.