Multi-Scale Topology Optimization of Discrete Lattice Structures

1.583 Final Project
Amira Abdel-Rahman
PhD Student
Center for Bits and Atoms (CBA)

Background

Digital Material

Discrete set of parts with relative positions and orientations; where global geometry is determined from local constraints, reversibly join parts with dissimilar properties to build heterogeneous functional systems.

* Kenneth C Cheung and Neil Gershenfeld. Reversibly assembled cellular composite materials.science, 341(6151):1219–1221, 2013.
* Benjamin Jenett, Christopher Cameron, Filippos Tourlomousis, Alfonso Parra Rubio,Megan Ochalek, and Neil Gershenfeld. Discretely assembled mechanical metamaterials.Science Advances, 6(47):eabc9943, 2020.

Applications

Toyota Lattice Racing Car (TLDR)

NASA Wing (MADCAT)

Test Case Study: Morphing Wing

Project Overview

Inverse Design Process

Microstructure Optimization

Compliant Mechanism Design of Trusses

\[ \begin{aligned} & \underset{\rho^e}{\text{minimize}} & & V(\rho^e)=\sum_{e \in \Omega } \rho^e \upsilon^e \\ & \text{subject to} & & K (\rho^e)d-F=0 \\ & & & g= L_i^T d\leq d_{max,i} \ \forall i \in 1,..,m \\ & & & \rho^e_{min} \leq \rho^e \ \forall e \\ & \end{aligned} \]

Using the adjoint method

The gradient of objective function: \[ f= \sum_{e \in \Omega } \rho^e \upsilon^e \\ \frac{\delta f}{\delta \rho_e}=\sum_{e \in \Omega } \upsilon^e \] The gradient of each constraint $i \ \forall (1,..,m)$: \[ \frac{\delta g_A}{\delta \rho_e}= - \lambda_i^T \frac{\delta K(\rho_e)}{\delta \rho_e } d \\ \]

2D Initial Results

3D Initial Compliant Mechanisms Results

Results: Microstructures after Manual Cleanup

Homogenization

From nodes & edges to voxels to elasticity tensor

\[ {\mathbf{C}}^H=\frac{1}{\left|{\Omega}_m\right|}{\int}_{\Omega_m}\mathbf{C}\left(\varepsilon \left({\mathbf{u}}_m^0\right)-\varepsilon \left({\mathbf{u}}_m\right)\right)\left(\varepsilon \left({\mathbf{u}}_{\mathrm{m}}^0\right)-\varepsilon \left({\mathbf{u}}_m\right)\right)\mathrm{d}{\Omega}_m \] \[ {\int}_{\Omega_m}\mathbf{C}\varepsilon \left({\mathbf{u}}_{\boldsymbol{m}}\right)\varepsilon \left({\boldsymbol{\nu}}_m\right)\mathrm{d}{\Omega}_m={\int}_{\Omega_m}\mathbf{C}\varepsilon \left({\mathbf{u}}_m^0\right)\varepsilon \left({\boldsymbol{\nu}}_m\right)\mathrm{d}{\Omega}_m,\kern1.25em \forall {\boldsymbol{\nu}}_m\mathbf{\in}{H}_{per}\left({\Omega}_m,{\mathbb{R}}^d\right) \] * Dong G, Tang Y, Zhao Y. A 149 Line Homogenization Code for Three-Dimensional Cellular Materials Written in matlab. ASME. J. Eng. Mater. Technol.

Homogenization Results

Youngs Modulus

Lattice	E11	E22	E33
Cube	6.9e7	6.9e7	6.9e7
Cuboct	9.85e7	9.85e7	9.85e7
Auxetic	2.584e8	1.608e8	1.608e8
Chiral	1.12e8	1.14e8	1.14e8

Shear Modulus

Lattice	G23	G31	G12
Cube	1.45e6	1.45e6	1.45e6
Cuboct	5.04e7	5.04e7	5.04e7
Auxetic	3.34e7	6.78e7	3.34e7
Chiral	5.45e7	6.4e7	5.45e7

Poisson Ratio

Lattice	v12	v13	v23	v21	v31	v32
Cube	0.056	0.056	0.056	0.056	0.056	0.056
Cuboct	0.294	0.294	0.294	0.294	0.294	0.294
Auxetic	0.063	0.063	0.376	0.039	0.039	0.376
Chiral	0.2	0.2	0.43	0.21	0.21	0.43

Macrostructure Optimization

Multi-Material Topology Optimization

The materials distribution is determined by the local volume fraction fields, α i (i = 1, … , p) \[ \boldsymbol{A}^{h} := \left\{ \alpha^{h} \in \left\{\alpha_{i}^{h} \in \boldsymbol{V}^{h}\left(\Omega^{h}\right)\right\}_{i\in \{1, \ldots, p\}} \left| \begin{array}{l} \\\\\\ \end{array} \right.\right. \] \[ \left. \begin{array}{ll} \sum_{i=1_{}}^{p} \alpha_{i}^{h} = 1, & \\ \int_{\Omega^{h}_{}} \alpha_{i}^{h} \ dx = \Lambda_{i} |\Omega^h|,\quad & i=1, \ldots, p \\ \boldsymbol{l}_{i}^{h} \leqslant \alpha_{i}^{h} \leqslant \boldsymbol{u}_{i}^{h}, \quad & i=1, \ldots, p \end{array} \right\} \] * Rouhollah Tavakoli and Seyyed Mohammad Mohseni. Alternating active-phase algorithmfor multimaterial topology optimization problems: a 115-line matlab implementation.Structural and Multidisciplinary Optimization, 49(4):621–642, 2014.