# Topology optimization of unsaturated flows in multi-material porous media: application to a simple diaper model

## TL;DR

This paper develops a topology optimization method for maximizing liquid absorption in multi-material porous media, demonstrated on a simplified diaper model with significant improvements over the reference design.

## Contribution

It introduces a density-based multi-material topology optimization approach for unsaturated flow in porous media, applicable to complex, nonlinear transient problems.

## Key findings

- Achieved up to 45% increase in absorbed liquid compared to reference design.
- Developed sensitivity analysis enabling first-order optimization for nonlinear problems.
- Validated the approach on two- and three-material configurations.

## Abstract

We present a mathematical approach to optimize the material distribution for fluid transport in unsaturated porous media. Our benchmark problem is a simplified diaper model as an exemplary liquid absorber. Our model has up to three materials with vastly different properties, which in the reference configuration are arranged in parallel layers. Here, swelling is neglected and the geometry of a swollen diaper is assumed and treated as a porous medium of high porosity. The imbibition process is then simulated by solving an unsaturated flow problem based on Richards' equation. Our aim is to maximize the amount of absorbed liquid by redistributing the materials. To this end, a density based multi-material topology optimization (based on the well known SIMP model) is performed. A sensitivity analysis for the nonlinear transient problem is provided, which enables the application of first order optimization solvers. We perform two- and three-material optimization and discuss several variants of the problem setting. We present designs with up to 45% more absorbed liquid compared to the reference configuration.