# An Aggregation-Free Local Volume Fraction Formulation for Topological Design of Porous Structure

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Optimization Problem for Infill Structure

#### 2.1. Formulation for Porous Structure Design

_{e}

_{,i}denotes the neighborhood set of elements that fall inside the ith element’s filter radius r

_{min}. The weighting factor ω

_{ei}is given as

_{e}is parameterized by ${\overline{\rho}}_{e}$ according to the SIMP interpolation [52]:

_{0}is the elastic modulus of solid material and E

_{min}(=10

^{−6}E

_{0}) is a minor stiffness to prevent singularity in finite element analysis. η denotes the power exponent of the Young’s modulus interpolation law, which is typically set as 3.

_{e}

_{,i}is the set of adjacent elements that fall inside the ith element’s circle or sphere within the influence radius r

_{max}.

_{f}represents the global volume faction while ψ is its corresponding threshold. V

_{e}is the eth elemental volume of the solid material. The parameter φ denotes the local volume fraction for all voxels. It should be highlighted that the restriction on overall material usage is unnecessary since the local volume fraction is already specified. Under normal circumstances, the value of ψ needs to be satisfied with the condition ψ ≤ φ when the total volume fraction constraint is active.

#### 2.2. Augmented Lagrangian Method

_{Φ}and N

_{Ψ}are the constants related to number of the local and global volume constraints. The penalization terms Φ

^{(k)}(ρ) and Ψ

^{(k)}(ρ) are defined as follows [47].

^{(k)}can be updated as following:

_{max}are the update factor and the upper bound for γ respectively, with the adoption to ensure numerical stability.

#### 2.3. Sensitivity Analysis

_{i}and k

_{0}are the displacement vector for ith element and local stiffness matrix when physical density ${\overline{\rho}}_{i}=1$.

^{(k)}(ρ) and Ψ

^{(k)}(ρ) with respect to physical density ${\overline{\rho}}_{i}$ can be computed as

_{e}to design variable ρ

_{e}can be written as

#### 2.4. MMA Algorithm

^{(k)}reads [48]:

_{e}. When a positive move-limit m is predefined, it yields:

## 3. Numerical Examples

#### 3.1. Example 1

_{max}= 6, respectively. For comparison purposes, the optimized topologies procured from the proposed method and the p-norm aggregation scheme and the distributions of the $\overline{\overline{\rho}}$ field are drawn in Figure 2, respectively. Additionally, the histograms of physical density $\overline{\rho}$ as well as local volume fraction $\overline{\overline{\rho}}$ are displayed in Figure 3.

#### 3.2. Example 2

#### 3.3. Example 3

_{max}on optimized results are discussed. The global volume fraction was fixed as ψ = 0.45. The upper bound of the local volume fraction ranged from 0.55 to 0.70. Meanwhile two different influence radii r

_{max}= 6 and 12 were adopted. The resulting topologies by various combinations of the parameter φ and r

_{max}are listed in Figure 6. Figure 7 depicts the relationship between compliance, influence radius, and the local volume fraction constraint.

_{max}resulted in the following three effects: larger diameters for holes, fewer holes and stronger rods. As observed from Figure 7, the resultant structure became stiffer as the influence radius r

_{max}and the upper bound of the local volume fraction φ grew. In conclusion, the local volume fraction constraint determined the local porosity, while the influence radius r

_{max}played an important role on the void size of topology. These phenomena can be referenced as the design law for porous structures.

#### 3.4. Example 4

_{max}, a mesh-independence study was performed in Example 4. Four distinct meshes were utilized, including 400 × 200, 800 × 400, 1200 × 600 and 1600 × 800 elements. Two filtering radii and two volume fractions were fixed as r

_{min}= 3, r

_{max}= 6, φ = 0.6 and ψ = 0.45, respectively. The optimized topologies with the resulting compliance, volume fraction are displayed in Figure 8.

#### 3.5. Example 5

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The optimized topologies obtained from: (

**a**) the proposed method: c = 85.780, V

_{f}= 0.546; (

**b**) c = 75.184, V

_{f}= 0.564 as well as the Local volume fraction distribution from: (

**c**) the proposed method: $max\left({\overline{\overline{\rho}}}_{e}\right)=0.598$; (

**d**) the p-norm method: $max\left({\overline{\overline{\rho}}}_{e}\right)=0.779$.

**Figure 3.**Histogram of (

**a**) local volume fraction $\overline{\overline{\rho}}$; (

**b**) physical density $\overline{\rho}$.

**Figure 4.**The optimized topologies by varying the global volume fraction from: (

**a**) ψ = 0.35: c = 104.069, V

_{f}= 0.350, $max\left({\overline{\overline{\rho}}}_{e}\right)=0.595$; (

**b**) ψ = 0.40: c = 94.551, V

_{f}= 0.401, $max\left({\overline{\overline{\rho}}}_{e}\right)=0.596$; (

**c**) ψ = 0.45: c = 89.242, V

_{f}= 0.450, $max\left({\overline{\overline{\rho}}}_{e}\right)=0.595$; (

**d**) ψ = 0.50: c = 87.399, V

_{f}= 0.502, $max\left({\overline{\overline{\rho}}}_{e}\right)=0.598$.

**Figure 5.**Iteration histories of compliance, global and local volume fraction from: (

**a**) ψ = 0.35; (

**b**) ψ = 0.40; (

**c**) ψ = 0.45; (

**d**) ψ = 0.50.

**Figure 6.**The optimized topologies obtained from (

**a**) φ = 0.55, r

_{max}=6: c = 99.577, V

_{f}= 0.449, $max\left({\overline{\overline{\rho}}}_{e}\right)=0.551$; (

**b**) φ = 0.55, r

_{max}= 12: c = 95.983, V

_{f}= 0.450, $max\left({\overline{\overline{\rho}}}_{e}\right)=0.546$; (

**c**) φ = 0.65, r

_{max}=6: c = 86.498, V

_{f}= 0.449, $max\left({\overline{\overline{\rho}}}_{e}\right)=0.645$; (

**d**) φ = 0.65, r

_{max}= 12: c = 82.466, V

_{f}= 450, $max\left({\overline{\overline{\rho}}}_{e}\right)=0.646$; (

**e**) φ = 0.75, r

_{max}=6: c = 78.921, V

_{f}= 0.451, $max\left({\overline{\overline{\rho}}}_{e}\right)=0.745$; (

**f**) φ = 0.75, r

_{max}= 12: c = 76.044, V

_{f}= 0.445, $max\left({\overline{\overline{\rho}}}_{e}\right)=0.746$.

**Figure 8.**The optimized topologies from different meshes: (

**a**) 400 × 200: c = 90.231, V

_{f}= 0.451, $max\left({\overline{\overline{\rho}}}_{e}\right)=0.597$; (

**b**) 800 × 400: c = 88.148, V

_{f}= 0.449, $max\left({\overline{\overline{\rho}}}_{e}\right)=0.599$; (

**c**) 1200 × 600: c = 86.557, V

_{f}= 0.446, $max\left({\overline{\overline{\rho}}}_{e}\right)=0.603$; (

**d**) 1600 × 800: c = 86.208, V

_{f}= 0.448, $max\left({\overline{\overline{\rho}}}_{e}\right)=0.603$.

**Figure 10.**The optimized topologies (

**a**) regardless of the local volume fraction: c = 2954.265, V

_{f}= 0.600, $max\left({\overline{\overline{\rho}}}_{e}\right)=1.000$; (

**b**) by porosity control: c = 4658.439, V

_{f}= 0.595, $max\left({\overline{\overline{\rho}}}_{e}\right)=0.695$.

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**MDPI and ACS Style**

Long, K.; Chen, Z.; Zhang, C.; Yang, X.; Saeed, N.
An Aggregation-Free Local Volume Fraction Formulation for Topological Design of Porous Structure. *Materials* **2021**, *14*, 5726.
https://doi.org/10.3390/ma14195726

**AMA Style**

Long K, Chen Z, Zhang C, Yang X, Saeed N.
An Aggregation-Free Local Volume Fraction Formulation for Topological Design of Porous Structure. *Materials*. 2021; 14(19):5726.
https://doi.org/10.3390/ma14195726

**Chicago/Turabian Style**

Long, Kai, Zhuo Chen, Chengwan Zhang, Xiaoyu Yang, and Nouman Saeed.
2021. "An Aggregation-Free Local Volume Fraction Formulation for Topological Design of Porous Structure" *Materials* 14, no. 19: 5726.
https://doi.org/10.3390/ma14195726