Multi-scale topology optimization for stiffness and de-homogenization using implicit geometry modeling

Abstract

In this article, we demonstrate the state-of-the-art of multi-scale topology optimization for 3D structural design. Many structures designed for additive manufacturing consist of a solid shell surrounding repeated microstructures, so-called infill material. We demonstrate the performance of different types of infill microstructures, such as isotropic truss or plate lattice structures and show that the best results can be obtained using spatially varying and oriented orthotropic microstructures. Furthermore, we demonstrate how implicit geometry modeling using nTop platform can help to interpret these multi-scale designs as single-scale manufacturable designs (de-homogenization). More importantly, we demonstrate the small difference in performance between these multi-scale and single-scale designs through extensive numerical testing. The presented method is at least 3 orders of magnitude more efficient compared to standard density-based topology optimization, allowing for high-resolution 3D structures to be obtained on a standard workstation PC.

Appendices

Appendix 1. Universal isotropy index

The universal isotropy index M^U is obtained as (Ranganathan and Ostoja-Starzewski 2008),

$$ M^{U} = 5\frac{G^{V}}{G^{R}}+\frac{K^{V}}{K^{R}}-6, $$

(A1.1)

here the superscripts V and R indicate the Voigt and Reuss measurements of the shear modulus G and bulk modulus K. These measurements can be obtained as (Hill 1952),

$$ \begin{aligned} K^{V} & = \frac{\tilde{E}_{1111}}{3} + \frac{2\tilde{E}_{1122}}{3},\\ G^{V} & = \frac{\tilde{E}_{1111}}{5} - \frac{\tilde{E}_{1122}}{5} + \frac{3\tilde{E}_{1212}}{5},\\ K^{R} &= \Big(3\tilde{S}_{1111}+6\tilde{S}_{1122}\Big)^{-1}, \\ G^{R} &= \Big(\frac{4\tilde{S}_{1111}}{5} - \frac{4\tilde{S}_{1122}}{5} + \frac{12\tilde{S}_{1212}}{5}\Big)^{-1}, \end{aligned} $$

(A1.2)

with compliance tensor S = E^− 1 and,

$$ \begin{aligned} \tilde{E}_{1212} & = \frac{{E}_{1212}+{E}_{1313}+{E}_{2323}}{3},\\ \tilde{S}_{1111} & = \frac{{S}_{1111}+{S}_{2222}+{S}_{3333}}{3},\\ \tilde{S}_{1122} & = \frac{{S}_{1122}+{S}_{1133}+{S}_{2233}}{3},\\ \tilde{S}_{1212} & = \frac{{S}_{1212}+{S}_{1313}+{S}_{2323}}{3}. \end{aligned} $$

(A1.3)

Appendix 2. Mapping functions ϕ from spatially varying orientation

In this section we summarize the methodology presented in Groen et al. (2020) to obtain smooth and continuous mapping functions $\boldsymbol {\phi } = \left \{\phi _{1}, \phi _{2}, \phi _{3}\right \}$ from 3 smooth and continuous orthogonal vector fields $\left \{\boldsymbol {n}^{1},\boldsymbol {n}^{2},\boldsymbol {n}^{3}\right \}$. The theory to extract these smooth and continuous vector fields from the optimized microstructure orientation is discussed in more detail in Groen et al. (2020).

From (8) and (9) it follows that if a lamination direction has no width, i.e., w_i = 0, it is not important how the corresponding mapping function ϕ_i looks. Furthermore, if a microstructure is solid, i.e., ρ = 1, then all mapping functions at that point can be relaxed. Hence, we require only an accurate description of ϕ_i in Ω_i,

$$ \begin{aligned} \boldsymbol{x} \in {\varOmega}_{i} &\quad \text{if }\quad {w}_{i}(\boldsymbol{x}) > 0.01 \text{ and } \rho(\boldsymbol{x}) < 0.99. \end{aligned} $$

(A2.1)

To solve for ϕ_i we solve the following least-squares problem,

$$ \begin{aligned} \displaystyle \min_{\phi_{i}(\boldsymbol{x})} & : \mathcal{I}(\phi_{i}(\boldsymbol{x})) = \frac{1}{2}{\int}_{\varOmega} {\alpha^{i}_{1}}(\boldsymbol{x})\left \Vert \nabla \phi_{i}(\boldsymbol{x}) - {\boldsymbol{n}}^{i}(\boldsymbol{x}) \right \Vert^{2} \text{d}\varOmega,& & \\ \textrm{s.t.} & : {\alpha^{i}_{2}}(\boldsymbol{x}) \nabla \phi_{i} (\boldsymbol{x})\cdot {\boldsymbol{t}}^{i,1}(\boldsymbol{x}) = 0, & & \\ \textrm{s.t.} & : {\alpha^{i}_{2}}(\boldsymbol{x}) \nabla \phi_{i} (\boldsymbol{x})\cdot {\boldsymbol{t}}^{i,2}(\boldsymbol{x}) = 0. & & \end{aligned} $$

(A2.2)

Here t^i,1 and t^i,2 span a plane tangent to nⁱ, i.e., for n¹ we have t^1,1 = n² and t^1,2 = n³. The domain is split into three parts, which dictate the weights on the objective ${\alpha ^{i}_{1}}$ and the weights on the constraints ${\alpha ^{i}_{2}}$ that allow us to relax ϕ_i outside Ω_i,

$$ \begin{aligned} {\alpha^{i}_{1}}(\boldsymbol{x}) &= \begin{cases} 0.01 & \quad \text{if }\quad {w}_{i}(\boldsymbol{x}) < 0.01, \\ 0.10 & \quad \text{if } \quad \rho(\boldsymbol{x})>0.99, \\ 1.00 & \quad \text{if } \quad \boldsymbol{x}\in {\varOmega}_{i},\\ \end{cases} \\ {\alpha^{i}_{2}}(\boldsymbol{x})& = \begin{cases} 0.00 & \quad \text{if }\quad {w}_{i}(\boldsymbol{x})< 0.01, \\ 0.00 & \quad \text{if } \quad \rho(\boldsymbol{x}) >0.99, \\ 1.00 & \quad \text{if } \quad \boldsymbol{x}\in {\varOmega}_{i}.\\ \end{cases} \end{aligned} $$

(A2.3)

Numerically, the above-mentioned problem can be solved using a finite element approach on a regular grid. Where the grid can be of similar resolution as $\mathcal {T}^{c}$. Furthermore, the constraints are enforced in an augmented setting using a penalty parameter γ_ϕ = 1000. We can impose an average unit-cell spacing ε. To do so, we define the periodicity scaling parameter P_i based on the average lattice spacing in the domain of interest $\tilde {\varOmega }_{i}$,

$$ {P}_{i} = \frac{2\pi}{\epsilon}\frac {{\int}_{\tilde{\varOmega}_{i}} \text{d}\tilde{\varOmega}_{i}}{{\int}_{\tilde{\varOmega}_{i}} ||\nabla \phi_{i}(\boldsymbol{x})|| \text{d}\tilde{\varOmega}_{i}}. $$

(A2.4)

From the mapping function ϕ_i we can identify the local distance between each plate λ_i using,

$$ \lambda_{i}(\boldsymbol{x}) = \frac{2\pi}{P_{i}\lVert \nabla \phi_{i}(\boldsymbol{x}) \rVert} . $$

(A2.5)

This local spacing can be used to get a description of the actual feature size of the geometry f_i for each lamination direction i.

$$ f_{i}(\boldsymbol{x}) = {w}_{i}(\boldsymbol{x}) \lambda_{i}(\boldsymbol{x}). $$

(A2.6)

This description can then be used to modify the width w_i to add a minimum feature size f_min to avoid very thin plates. Another option can be to impose a uniform feature size f_i in the entire domain, as can be seen in Fig. 7c.