$$\def \SO {\operatorname{SO}} \def \R {\mathbb{R}} \def \Z {\mathbb{Z}} \def \Ker {\operatorname{Ker}} \def \Im {\operatorname{Im}} \def \notimplies {\nRightarrow}$$

# Shape Optimization for Additive Manufacturing

G. Allaire, P. Geoffroy, O. Pantz, K. Trabelsi

# Motivation

## Compliance minimization

?
$f$

## 3D printed shape

Otimization Pb
$$-\nabla\cdot A e(u)=f$$
$$-\nabla\cdot A^* e(u)=\theta f$$

## Sequence $\Omega_\varepsilon$

$\varepsilon=$16.00

## A 1+2 steps program

Step 0 - Computing $A^*$
$\omega=(0,1)^2$-periodic

## Correctors

$$w_{ij}:=\textrm{argmin} \int_{\Omega} A(e_{ij}+e(w)):(e_{ij}+e(w))$$ $e_{ij}=e_i\otimes e_j$
Compression along $e_1$

$w_{11}$
Compression along $e_2$

$w_{22}$
Shearing

$w_{12}$

## Hooke's law

$$A^*_{ijkl}:=\int_{\Omega} A(e_{ij}+e(w_{ij})):(e_{kl}+e(w_{kl}))$$
$m_1$
$m_2$
$\small A^*(m_1,m_2)$

# Compliance

$$\min_{\alpha,m} \left\{J^*(\theta,A^*):=\int_\Omega A^*e(u^*):e(u^*)\,dx\right\}$$
$$\int_D \theta\,dx \leq V$$

# Stress Formulation

$$\inf_{m,\sigma} J^*(m,\alpha) = \inf_{\alpha,m,\sigma} \int_D (A^*)^{-1}(m,\alpha) \sigma\cdot \sigma \,dx.$$ $$\left\{ \begin{array}{ll} \nabla\cdot\sigma=0&\text{in } D\\ \sigma n = g&\text{on }\Gamma_N \end{array} \right.$$

# Step 2 : Dehomogenization

## Grid Map $\varphi$

$D \varphi^{-1}(Y)$-periodic shape
$$\xrightarrow[\quad\quad]{\varphi}$$

$Y$-periodic shape $\omega$

## Definition of the shape sequence $\Omega_\varepsilon$

$$\Omega_\varepsilon(\varphi,m)=\left\{x\in D~:~\cssId{phi}{\varphi}(x) \in \cssId{vepsilon}{\varepsilon} \omega(\cssId{m}{m}(x)) \right\}.$$

## Computation of the grid Map $\varphi$

$D\varphi^{-1}=$
$a=$ Rotation field of the cells of angle $\alpha$
$r=$ Dilatation of the cells

## Conformity Condition

$$\nabla^T r = \nabla\alpha$$

## Optimization Steps

1. Homogenization
2. Add conformity constaint $\Delta\alpha=0$
3. Compute dilatation and grid map

## Main features

Post-treatment of the homogenization
Find $\Omega_\varepsilon$ converging toward the optimal composite
$\varepsilon$=Length scale of the fine details
One UNIQUE computation for ALL $\varepsilon$