# Phase-field Topology Optimization of Functionally Graded Lattice Structures

## Introduction

Structural topology optimization (TO) is a numerical method which aims, by means of a density function, at optimally distributing a limited amount of material within a volume, representing the initial geometry of a body undergoing specific loads and displacement boundary conditions. The phase-field based TO, first introduced by Bourdin and Chambolle in [1], is an TO method which employs a phase-field approach to approximate the interface perimeter separating regions of solid material and voids.

## Goals

In this project, we aim at developing an optimization procedure to obtain structures with locally varying material properties, exploiting the possibility of additive manufacturing (AM) technology to distribute material within a 3D printed part in an almost continuous fashion. In particular, in this work we focus on functionally graded lattice structures, defined as structures with variable density and where the density variability is obtained by means of lattice cells having different filling ratio, i.e., the ratio of material and void within a single lattice cell.

## Results

A rigorous mathematical proof of the proposed methodology can be found in [2], whereas the numerical implementation of the method together with numerous numerical examples is presented in [3]. Finally, a complete workflow to convert numerical results to 3D printed structures and its experimental validation are reported in [4].

Fig. 1: Conceptual workflow to convert numerical results into 3D printed structures

Video 1: The video shows the material and void regions together with the continuous density distribution within the material region at each time step of the optimization routine for a three-point bending problem (a.k.a. MBB-beam problem). We can observe how the black-and-white solution converges much faster to a stable configuration while more iterations are needed to optimally re-distribute the density within the material (black) areas.

## References

1. Bourdin, B. and Chambolle, A. (2003), Design-dependent loads in topology optimization, ESAIM Contr. Optim.Calc. Var., 9 19-48.
2. Carraturo, M., Rocca, E., Bonetti, E., Hömberg, D., Reali, A. and Auricchio, F. (2019). Graded-material design based on phase-field and topology optimization. Computational Mechanics64(6), 1589-1600.
3. Auricchio, F., Bonetti, E., Carraturo, M., Hömberg, D., Reali, A. and Rocca, E. (2020). A phase-field-based graded-material topology optimization with stress constraint. Mathematical Models and Methods in Applied Sciences30(08), 1461-1483.
4. Alaimo, G., Carraturo, M., Rocca, E., Reali, A. and Auricchio, F., Functionally Graded Material Design for Plane Stress Structures using Phase Field Method, Proceedings of the II International Conference on Simulation for Additive Manufacturing (Sim_AM 2019), Pavia, Italy, 11-13 September 2019.