# A differential equation-driven update strategy for density-based topology optimization: implementation with MATLAB codes

## TL;DR

This paper introduces a new method for topology optimization using differential equations, implemented in MATLAB, which can improve optimization performance compared to traditional methods.

## Contribution

A novel differential equation-driven update strategy for density-based topology optimization is introduced, using absolute increments related to the OC method.

## Key findings

- The differential equation-driven update scheme effectively solves density distribution optimization problems.
- The absolute increment format offers a more active optimization process than traditional relative increment formats.
- MATLAB implementations and numerical examples demonstrate the effectiveness of the proposed method.

## Abstract

Differential equation-driven evolution strategies are often associated with boundary-driven topology optimization methods, such as the level set method. However, differential equations can also be utilized effectively in density-based approaches. This paper presents a design update scheme formulated using differential equations to evolve elemental densities in topology optimization. The proposed scheme transforms the differential equation into an absolute increment format, closely related to the optimality criteria (OC) method, which is traditionally implemented in a relative increment format in density-based methods. The relative increment format of the OC method typically ensures an efficient and stable optimization process, whereas the absolute increment format tends to enable a more active and responsive optimization process, potentially leading to optimized results with improved performance. Furthermore, the absolute increment format can be converted into a relative one if needed. This study explores compliance minimization problems for both isotropic composite and single-material cases. Detailed MATLAB implementations for these cases are presented and thoroughly explained. Numerical examples demonstrate that the differential equation-driven update scheme effectively addresses density distribution optimization problems, offering an alternative to classical density methods.