# On barrier and modified barrier multigrid methods for 3d topology optimization

## TL;DR

This paper introduces a novel multigrid-based approach for large-scale 3D topology optimization problems, demonstrating significant computational efficiency improvements over traditional methods.

## Contribution

The paper develops a PBM algorithm utilizing a special Hessian structure and multigrid preconditioning, offering a new efficient solution technique for large 3D topology optimization problems.

## Key findings

- PBM outperforms OC and IP methods in computation time
- Multigrid preconditioning effectively reduces linear system complexity
- Proposed method handles large-scale problems efficiently

## Abstract

One of the challenges encountered in optimization of mechanical structures, in particular in what is known as topology optimization, is the size of the problems, which can easily involve millions of variables. A basic example is the minimum compliance formulation of the variable thickness sheet (VTS) problem, which is equivalent to a convex problem. We propose to solve the VTS problem by the Penalty-Barrier Multiplier (PBM) method, introduced by R.\ Polyak and later studied by Ben-Tal and Zibulevsky and others. The most computationally expensive part of the algorithm is the solution of linear systems arising from the Newton method used to minimize a generalized augmented Lagrangian. We use a special structure of the Hessian of this Lagrangian to reduce the size of the linear system and to convert it to a form suitable for a standard multigrid method. This converted system is solved approximately by a multigrid preconditioned MINRES method. The proposed PBM algorithm is compared with the optimality criteria (OC) method and an interior point (IP) method, both using a similar iterative solver setup. We apply all three methods to different loading scenarios. In our experiments, the PBM method clearly outperforms the other methods in terms of computation time required to achieve a certain degree of accuracy.