Design of graded porous bone-like structures via a multi-material topology optimization approach

Abstract

Graded porous structures combine robustness of porous structures and high stiffness of bulk designs. This study aims to design optimized graded porous bone-like structures through a novel multi-material topology optimization approach, which generalizes the concept of multiple materials. Namely, each material can have not only distinct material property but also a different level of local porosity, or a combination of both, thus allowing the realization of multiple levels of porosity. With separated density and material/porosity fields, we propose two types of multi-porosity local volume constraints to enable graded porosity considering linear and bi-linear material constitutive relations. Through the proposed framework, single- and multi-material structures can be obtained with a natural transition between the bulk and multiple levels of porous regions. We adopt the Bi-value Coding Parameterization (BCP) scheme combined with the Solid Isotropic Material with Penalization (SIMP) method to interpolate the stored energy functions. Through several examples with multiple porosity levels and various material properties, we demonstrate the effectiveness of the proposed framework with two novel constraints to generate optimized multi-material and multi-porosity structures. We further investigate the interactions among material properties, multiple porosity levels, structural stiffness, and robustness. Compared with conventional bulk designs, the optimized bone-like structures with multi-level graded porosity, although less stiff, are found to be more robust, i.e., their structural stiffness is less influenced by the load variations and material deficiency. The resulting graded porous composite designs showcase the capability of the proposed multi-material formulation to optimize the distributions of not only different types of materials but also multiple levels of porosity.

Appendices

Appendix: 1. Sensitivity analysis

Appendix 1 presents sensitivity analysis that is derived for the proposed formulations. We take X as a general notation representing J₁, J₂, J₃ (as shown in Table 1), $G_{G}^{(\ell )}$, G_L1, or G_L2, and the sensitivities with respect to ρ_e and $\xi _{e}^{(k)}$ are given as

$$ \frac{\partial X}{\partial \rho_{e}} = \underset{i \in \mathscr{I}_{i}(r)}{\sum} \left( \frac{\partial X}{\partial \overline{\rho}_{i}} \frac{\partial \overline{\rho}_{i}}{\partial \widetilde{\rho}_{i}} \frac{\partial \widetilde{\rho}_{i}}{\partial \rho_{e}} \right), $$

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$$ \frac{\partial X}{\partial \xi_{e}^{(k)}} = \underset{i \in \mathscr{I}_{i}(r)}{\sum} \left( \sum\limits_{j=1}^{N_{m}} \left( \frac{\partial X}{\partial m_{i}^{(j)}} \frac{\partial m_{i}^{(j)}}{\partial \overline{\xi}_{i}^{(k)}} \right) \frac{\partial \overline{\xi}_{i}^{(k)}}{\partial \widetilde{\xi}_{i}^{(k)}} \frac{\partial \widetilde{\xi}_{i}^{(k)}}{\partial \xi_{e}^{(k)}} \right). $$

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For the first type of objective, J₁, the sensitivities with respect to $\overline {\rho }_{i}$ and $m_{i}^{(j)}$ are

$$ \frac{\partial J_{1}}{\partial \overline{\rho}_{i}}=-(1-\epsilon_{0}) p_{\rho} \overline{\rho}_{i}^{p_{\rho}-1}\left( \sum\limits_{j=1}^{N_{m}}\left( m_{i}^{(j)}\right)^{p_{m}} E^{(j)}\right) u_{i}^{\mathrm{T}} k_{0} u_{i}, $$

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$$ \frac{\partial J_{1}}{\partial m_{i}^{(j)}}=-(1-\epsilon_{0}) \overline{\rho}_{i}^{p_{\rho}}\left( p_{m} \left( m_{i}^{(j)}\right)^{p_{m}-1} E^{(j)}\right) u_{i}^{\mathrm{T}} k_{0} u_{i}, $$

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respectively, and k₀ is the element stiffness matrix with unit Young’s modulus. The sensitivities of J₂ with respect to $\overline {\rho }_{i}$ and $m_{i}^{(j)}$ are

$$ \begin{array}{@{}rcl@{}} \frac{\partial J_{2}}{\partial \overline{\rho}_{i}}&=&-(1-\epsilon_{0}) p_{\rho} \overline{\rho}_{i}^{p_{\rho}-1}\left( \sum\limits_{j=1}^{N_{m}}\left( m_{i}^{(j)}\right)^{p_{m}} E^{(j)}\right)\\ &&\times \sum\limits_{\gamma=1}^{N_{c}} u_{i,\gamma}^{\mathrm{T}} k_{0} u_{i,\gamma}, \end{array} $$

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$$ \begin{array}{@{}rcl@{}} \frac{\partial J_{2}}{\partial m_{i}^{(j)}}&=&-(1-\epsilon_{0}) \overline{\rho}_{i}^{p_{\rho}}\left( p_{m} \left( m_{i}^{(j)}\right)^{p_{m}-1} E^{(j)}\right) \\ &&\times \sum\limits_{\gamma=1}^{N_{c}} u_{i,\gamma}^{\mathrm{T}} k_{0} u_{i,\gamma}, \end{array} $$

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respectively. The sensitivities of J₃ with respect to $\overline {\rho }_{i}$ and $m_{i}^{(j)}$ are

$$ \begin{array}{@{}rcl@{}} \frac{\partial J_{3}}{\partial \overline{\rho}_{i}}&=&-(1-\epsilon_{0}) p_{\rho} \overline{\rho}_{i}^{p_{\rho}-1}\sum\limits_{j=1}^{N_{m}}\left( \left( m_{i}^{(j)}\right)^{p_{m}} W_{i}^{(j)}\right), \\ \frac{\partial J_{3}}{\partial m_{i}^{(j)}}&=&-(1-\epsilon_{0}) \overline{\rho}_{i}^{p_{\rho}}p_{m} \left( m_{i}^{(j)}\right)^{p_{m}-1} W_{i}^{(j)}, \end{array} $$

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respectively. For the global constraint, the sensitivities of $G_{G}^{(\ell )}$ with respect to $\overline {\rho }_{i}$ and $m_{i}^{(j)}$ are

$$ \begin{array}{@{}rcl@{}} \frac{\partial G_{G}^{(\ell)}}{\partial \overline{\rho}_{i}}&=& \sum\limits_{j=1}^{N_{m}} \begin{cases} \frac{A_{i} m_{i}^{(j)}}{{\sum}_{h=1}^{N_{e}} A_{h}} & \text{if } j \in \mathcal{G}^{\ell}\\ 0 & \text{otherwise}, \end{cases} \\ \frac{\partial G_{G}^{(\ell)}}{\partial m_{i}^{(j)}}&=& \begin{cases} \frac{A_{i} \overline{\rho}_{i}}{{\sum}_{h=1}^{N_{e}} A_{h}} & \text{if } j \in \mathcal{G}^{\ell}\\ 0 & \text{otherwise}, \end{cases} \end{array} $$

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respectively. For the local constraint L1, the sensitivities with respect to $\hat {\rho }_{i}$ and $m_{i}^{(j)}$are

$$ \begin{aligned} & \frac{\partial G_{L1}}{\partial \rho_{i}}= \underset{z \in \mathscr{I}_{i}(R)}{\sum} \left( \frac{\partial G_{L1}}{\partial \hat{\rho}_{z}} \frac{\partial \hat{\rho}_{z}}{\partial \overline{\rho}_{i}}\right), \quad \text{where} \\ & \frac{\partial G_{L1}}{\partial \hat{\rho}_{z}} = \frac{1}{\alpha_{z} N_{e}}\left( \frac{1}{N_{e}} \sum\limits_{h=1}^{N_{e}}\left( \frac{\hat{\rho}_{h}}{\alpha_{h}}\right)^{q}\right)^{\frac{1}{q}-1} \left( \frac{\hat{\rho}_{z}}{\alpha_{z}}\right)^{q-1}\\ &\quad\quad\quad\quad\text{ and} \quad \frac{\partial \hat{\rho}_{z}}{\partial \overline{\rho}_{i}} = \frac{A_{z}}{{{\sum}_{h \in \mathscr{I}_{i}(R)} A_{h}}}, \\ &\frac{\partial G_{L1}}{\partial m_{i}^{(j)}}=-\frac{\hat{\rho}_{i} \alpha_{0}^{(j)}}{N_{e} {\alpha_{i}^{2}}}\left( \frac{1}{N_{e}} \sum\limits_{h=1}^{N_{e}} \left( \frac{\hat{\rho}_{h}}{\alpha_{h}}\right)^{q}\right)^{\frac{1}{q}-1} \left( \frac{\hat{\rho}_{i}}{\alpha_{i}}\right)^{q-1}, \end{aligned} $$

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respectively. For the local constraint L2, the sensitivity with respect to $\overline {\rho }_{i}$ is

$$ \begin{aligned} & \frac{\partial G_{L2}}{\partial \overline{\rho}_{i}}= \sum\limits_{j=1}^{N_{m}} \underset{z \in \mathscr{I}_{i}(R^{(j)})}{\sum} \left( \frac{\partial G_{L2}}{\partial \hat{V}_{z}^{(j)}} \frac{\partial \hat{V}_{z}^{(j)}}{\partial \overline{\rho}_{i}}\right), \quad \text{where}\\ & \frac{\partial G_{L2}}{\partial \hat{V}_{z}^{(j)}}=\frac{1}{\alpha_{0}^{(j)} N_{e}}\left( \frac{1}{N_{e}}\sum\limits_{h=1}^{N_{e}} \left( \sum\limits_{j=1}^{N_{m}}\left( \frac{\hat{V}_{h}^{(j)}}{\alpha_{0}^{(j)}}\right)\right)^{q}\right)^{\frac{1}{q}-1} \end{aligned} $$

$$ \begin{aligned} &\qquad\qquad\times\left( \sum\limits_{j=1}^{N_{m}}\left( \frac{\hat{V}_{z}^{(j)}}{\alpha_{0}^{(j)}}\right)\right)^{q-1},\\ & \frac{\partial \hat{V}_{z}^{(j)}}{\partial \overline{\rho}_{i}} = \frac{A_{i} m_{z}^{(j)}}{{{\sum}_{h \in \mathscr{I}_{e}(R^{(j)})} A_{h}}}. \end{aligned} $$

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The sensitivity of G_L2 with respect to $m_{i}^{(j)}$ is

$$ \begin{aligned} & \frac{\partial G_{L2}}{\partial m_{i}^{(j)}}= \underset{z \in \mathscr{I}_{i}(R^{(j)})}{\sum} \left( \frac{\partial G_{L2}}{\partial \hat{V}_{z}^{(j)}} \frac{\partial \hat{V}_{z}^{(j)}}{\partial m_{i}^{(j)}}\right), \\ & \frac{\partial G_{L2}}{\partial \hat{V}_{z}^{(j)}}=\frac{1}{\alpha_{0}^{(j)} N_{e}}\left( \frac{1}{N_{e}}\sum\limits_{h=1}^{N_{e}} \left( \sum\limits_{j=1}^{N_{m}}\left( \frac{\hat{V}_{h}^{(j)}}{\alpha_{0}^{(j)}}\right)\right)^{q}\right)^{\frac{1}{q}-1} \\&\qquad\qquad\times\left( \sum\limits_{j=1}^{N_{m}}\left( \frac{\hat{V}_{z}^{(j)}}{\alpha_{0}^{(j)}}\right)\right)^{q-1}\\ & \frac{\partial \hat{V}_{z}^{(j)}}{\partial m_{i}^{(j)}} = \frac{A_{i} \overline{\rho}_{e}}{{{\sum}_{h \in \mathscr{I}_{i}(R^{(j)})} A_{h}}}. \end{aligned} $$

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The terms $\partial m_{i}^{(j)}/\partial \overline {\xi }_{i}^{(k)}$, ${\partial \overline {\xi }_{i}^{(k)}}/{\partial \widetilde {\xi }_{i}^{(k)}}$ and ${\partial \widetilde {\xi }_{i}^{(k)}}/{\partial \xi _{e}^{(k)}}$ are given as

$$ \begin{aligned} &\frac{\partial m_{i}^{(j)}}{\partial\overline{\xi}_{i}^{(k)}}=\frac{1}{2^{N_{\xi}}}s_{j,k} {\prod}_{t=1 \atop t \neq k}^{N_{\xi}}\left( 1+s_{j,t} \overline{\xi}_{i}^{(t)}\right), \\ &\frac{\partial \overline{\xi}_{i}^{(k)}}{\partial \widetilde{\xi}_{i}^{(k)}}=\frac{\beta_{\xi} (\tanh(\beta_{\xi}(\widetilde{\xi}_{i}^{(k)}-\eta_{\xi}))^{2}-1)}{\tanh(\beta_{\xi}(\eta_{\xi}-1))-\tanh (\beta_{\xi} \eta_{\xi})}, \\&\qquad\qquad\text{and} \quad \frac{\partial \widetilde{\xi}_{i}^{(k)}}{\partial \xi_{e}^{(k)}}=\frac{\omega_{e, i}}{{\sum}_{h \in \mathscr{I}_{e}(r)} \omega_{h, i}}. \end{aligned} $$

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The terms ${\partial \overline {\rho }_{i}}/{\partial \widetilde {\rho }_{i}}$ and ${\partial \widetilde {\rho }_{i}}/{\partial \rho _{e}}$ are given as

$$ \begin{array}{@{}rcl@{}} \frac{\partial \overline{\rho}_{i}}{\partial \widetilde{\rho}_{i}}&=&\frac{\beta_{\rho}(\tanh(\beta_{\rho} (\widetilde{\rho}_{i}-\eta_{\rho}))^{2}-1)}{\tanh(\beta_{\rho}(\eta_{\rho}-1))-\tanh(\beta_{\rho} \eta_{\rho})}, \\ &&\text{and} \quad \frac{\partial \widetilde{\rho}_{i}}{\partial \rho_{e}}=\frac{\omega_{e, i}}{{\sum}_{h \in \mathscr{I}_{e}(r)} \omega_{h, i}}. \end{array} $$

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Appendix: 2. Framework summary

We present the algorithm to show the optimization procedures of the proposed framework. In Algorithm 1, i refers to the i th optimization iteration, $i_{\max \limits }$ is the number of optimization iterations, Δ₀ is the convergence toleration, and Δ is the maximum absolute change of design variables.

Appendix: 3. Nomenclature

A :

Element area

$\alpha _{0}^{(j)}$ :

Prescribed local volume fraction upper bound for the j th material/porosity field

α :

Local volume fraction upper bound

β _ρ :

Heaviside parameter for density variable

β _ξ :

Heaviside parameter for material/porosity variable

$\overline {\boldsymbol {D}}$ :

Tangent stiffness matrix

Δ :

Maximum absolute change of design variables

Δ ₀ :

Convergence toleration

E :

Young’s modulus

E _t :

Tension Young’s modulus

E _c :

Compression Young’s modulus

𝜖 ₀ :

A sufficiently small value

ε :

Strain

$\overline {\boldsymbol {\varepsilon }}$ :

Principal strain

$\mathcal {E}_{\gamma }$ :

The γ th strain condition

F :

External force

G_G(⋅):

Global volume constraint function

G_L1(⋅):

The first local volume constraint function

G_L2(⋅):

The second local volume constraint function

$\mathcal {G}^{(\ell )}$ :

Sets of materials that belong to the ℓ th constraint

$i_{\max \limits }$ :

The number of optimization iterations

${\mathscr{I}}(\cdot )$ :

Influence region

J₁(⋅):

Objective function: compliance under single load case

J₂(⋅):

Objective function: compliance under multiple load cases

J₃(⋅):

Objective function: negative potential energy

$\boldsymbol {\mathcal {K}}_{h}$ :

Admissible displacement space

m ^(j) :

The j th physical material/porosity

N _ξ :

The number of material/porosity design variable

N _e :

The number of elements

N _ℓ :

The number of global volume constraints

N _m :

The number of material/porosity variables

N _c :

The number of multiple load cases

η _ρ :

Heaviside threshold for density variable

η _ξ :

Heaviside threshold for material/porosity variable

p _ρ :

Density penalization parameter

p _m :

Material/porosity penalization parameter

π_h(⋅):

Potential energy function

ρ :

Density design variable

$\widetilde {\boldsymbol {\rho }}$ :

Filtered density variable

$\overline {\boldsymbol {\rho }}$ :

Projected (physical) density field

$\hat {\boldsymbol {\rho }}$ :

Local volume fraction

q :

P-norm parameter

R :

Influence radius

R ^(j) :

Influence radius for the j th material/porosity field (applicable to the L2 constraint)

r :

Filtering radius

s :

Mapping matrix

$\overline {\boldsymbol {\sigma }}$ :

Principal stress

u _h :

Displacement field obtained from the minimization of potential energy

v _h :

Arbitrary admissible displacement field

ν _t :

Poisson’s ratio for tension

ν _c :

Poisson’s ratio for compression

V ₀ :

Global volume fraction

$\hat {\boldsymbol {V}}^{(j)}$ :

The j th material/porosity local volume fraction

W(⋅):

Stored-energy function

$\hat {W}(\cdot )$ :

Interpolated stored energy function

w :

Distance between two elements

ξ :

Material/porosity design variable

$\overline {\boldsymbol {\xi }}^{(k)}$ :

The k th filtered material/porosity variable

$\widetilde {\boldsymbol {\xi }}^{(k)}$ :

The k th projected material/porosity variable

x :

Centroid of element