# Topology Optimization with Bilevel Knapsack: An Efficient 51 Lines MATLAB Code

## TL;DR

This paper introduces a highly efficient, simple 51-line MATLAB code for topology optimization that produces high-quality designs quickly, with extensions for various boundary conditions and 3D, suitable for learners.

## Contribution

The paper provides a minimalistic, easy-to-understand MATLAB implementation for topology optimization that outperforms existing codes in speed and design quality.

## Key findings

- The code achieves faster average computation times.
- It produces void-solid designs with minimal checkerboard patterns.
- Extensions enable application to different boundary conditions and 3D designs.

## Abstract

This paper presents an efficient 51 lines Matlab code to solve topology optimization problems. By the fact that the presented code is based on an hard 0-1 optimization method that handles the integer part of the optimization in a simple fashion and in sub-polynomial time, it has been possible to not only reduce the number of lines to 51 without sacrificing any readability, but also to obtain void-solid designs with close to none checkerboard patterns with improved efficiency. The numerical results in the paper show that the proposed method has the best average times compared to several codes available in literature. We also present extensions to different boundary conditions and to three dimensional designs. The code can be used by students and the newcomers in topology optimization because of its simplicity and readability. The 51 lines Matlab code and the presented extensions can be downloaded on the webpage https://github.com/vlatorre847/TOSSE.