# Designing Bioinspired Composite Structures via Genetic Algorithm and Conditional Variational Autoencoder

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## Abstract

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## 1. Introduction

^{170}legal positions in the game of Go, which has a 19 × 19 grid size with three possibilities at each point [30]. For such massive systems, machine learning (ML) appears to be a suitable solution and has been proposed in the literature, the well-known one being AlphaGo by Alphabet Inc.’s Google DeepMind [31]. ML can build up a hard-to-discern numerical relationship between input and output from large and complex data through learning from past experiences. Therefore, many researchers replace conventional simulation methods with ML to accelerate the development process of various tasks, such as predicting the properties of chemical molecules [32,33] and composites [24,34,35,36,37,38,39]. Furthermore, more recent studies have demonstrated ML’s exceptional capability in solving inverse design problems. That is, through the ML model such as Generative Adversarial Network (GAN) [40,41,42] and Conditional Variational Autoencoder (CVAE) [43,44,45], chemical molecular or composites structure can be obtained using the desired requirement as the input of ML model.

## 2. Materials and Methods

#### 2.1. Composite Design Space, Fabrication, and Finite Element Analysis

^{19}combinations, which is unrealistic in terms of applying brute force algorithms to find optimal designs.

_{c}, i.e., G ≥ G

_{c}[51]. The energy release rate, G, is the energy dissipation with a newly created fracture surface area and is defined as the rate of change in potential energy with a crack area. Thus, G may be regarded as the “driving force” for fracture; G

_{c}is the fracture energy (critical energy release rate) and is a material property independent of the applied loads and the specimen geometry. In this study, since the material ahead of the crack tip is either soft or stiff (Figure 2), the corresponding G

_{c}value is used for a particular design pattern. Accordingly, we introduced a fracture criterion, the ratio of the energy release rate and the critical energy release rate G/G

_{c}, to compare the tendency of crack propagation among different patterns [52,53].

_{1}and U

_{2}are the strain energy before and after the crack extension, and da is the crack extending length (for two-dimensional problems). The units of G are J/m

^{2}.

_{c}= 249 J/m

^{2}and FLX9870-40a with Young’s modulus E = 0.455 MPa and G

_{c}= 414 J/m

^{2}. E and G

_{c}are assumed isotropic within each cell, and G

_{c}is for Mode I only. Those values for single materials were measured in this study using the same experimental setup to ensure consistent results with the composite ones. Our experiments show that PolyJet generally produces strong interface bonding between two different materials, whereas the commonly used extrusion-based 3D printing, i.e., FDM, produces a weak interface and can significantly affect the mechanical performance of multi-material printed structures [56]. Thus, FDM is unsuitable as its specimens almost always fail at the stiff-soft boundary instead of material fracture.

#### 2.2. Experimental Section

**Figure 3.**(

**a**) Experimental setup for the tensile test with DIC. (

**b**) Sample displacement field calculated using DIC. (

**c**) picture of a sample specimen. (

**d**) Dimension of the specimens. The red box is the design area. Unit: mm.

_{c}. We recorded the load from the machine’s readings when the crack started to propagate. This critical load was applied in the FEM model to calculate the strain energy release rate G according to Equation (1). Because the crack had already started to extend at that moment, the energy release rate calculated by FEM was equal to the critical energy release rate G

_{c}.

^{3}) by calculating the area under the load-displacement curve before fracture. Finally, we will compare G/G

_{c}and experimental toughness and verify if G/G

_{c}is a good predictor of toughness.

_{c}.

#### 2.3. Conditional Variational Autoencoder (CVAE)

#### 2.4. Genetic Algorithm (GA)

## 3. Results and Discussion

#### 3.1. Dataset Analysis

_{c}varies significantly from about 0.33 to 1.12, which means the toughness of composite materials is highly tunable with just 12.5% soft material. Note that the G

_{c}value depends on the material ahead of the crack tip (crack propagation direction), i.e., G

_{c}= 249 J/m

^{2}for the stiff material and 414 J/m

^{2}for the soft material. Figure 7 shows the distribution of the fracture criterion and stiffness of the random patterns. We observe an intriguing distribution of the data points in two clusters, which will be discussed later.

#### 3.2. Experimental Results

_{c}, but the material ahead of the crack tip was soft for one pattern and stiff for the other. The objective is threefold: First, to verify the accuracy of FEM simulations using stiffness. Second, to verify whether the material ahead of the tip is indeed critical, as commonly believed. Third, to verify whether G/G

_{c}correlates with toughness, as defined by the area under the load-displacement curve at fracture.

_{c}indeed have similar toughness values, confirming that G/G

_{c}can predict toughness—the higher G/G

_{c}, the lower the experimental toughness. As the fracture criterion G/G

_{c}shows how much deformation the pattern can sustain before the crack starts to propagate, under the condition of similar stiffness, the pattern’s toughness can be predicted by the fracture criterion. A smaller G/G

_{c}means the pattern can undergo larger cross-head displacement and absorb more energy before the crack propagates. Third, we observe no apparent correlation between the toughness and type of material ahead of the crack tip. We will evaluate the exact contribution of the material distribution on the toughness by XAI in Section 3.4.

#### 3.3. CVAE versus GA

_{c}(0.2–1.2) as target properties (Figure 9) for CVAE and GA to design. For each set of target properties, the decoder of CVAE takes in a latent vector randomly sampled from Gaussian noise and the target properties, and then outputs its prediction. As for GA, we set the maximum iteration to 50 and output the design with the best fitness. After the predictions by CVAE and GA, we simulate their designs in FEM to calculate their stiffness and toughness as ground truth. The error of each design is calculated by averaging the error of stiffness and the error of toughness. Figure 10b,d is the scatter plot of the resulting properties distribution of composites designed by CVAE and GA. Although the target properties are uniformly distributed, the results of CVAE and GA are still confined to the range of the original FEM dataset. This is expected as the original dataset is generated randomly and should represent the overall distribution of the composite design space; we can conclude that the inability of CVAE and GA is due to the non-existence of such design outside of the original dataset, as shown in Figure 7. As for the design accuracy, both CVAE and GA perform very well inside the plausible properties range, with errors well below 20% (Figure 10a,c). Moreover, Figure 9b,d shows that the resulting designs by GA distribute somewhat more evenly than CVAE. This suggests that GA slightly outperforms CVAE within the original dataset range and can more accurately design a composite with desired properties. Additionally, two specific design examples of different combinations of stiffness and toughness are provided in Figure 11 to help better understand the evaluation process.

#### 3.4. Discussion with Explainable AI (XAI) and Fracture Mechanics

^{2}. However, the energy release rate increases to 332 J/m

^{2}if the pattern has soft material just ahead of the crack tip, indicating that the soft material placed ahead of the crack tip doubles the crack propagation tendency. Correspondingly, if the resistance to crack propagation (i.e., the fracture toughness) of the soft material is not two-fold that of the stiff material, placing the soft material ahead of the crack tip cannot make the pattern tougher as intuitively expected.

^{2}and 414 J/m

^{2}, respectively. Therefore, the fracture energy of soft material is only ~70% higher than that of stiff material, which means that soft material placed ahead of the crack tip does not help the toughness of the pattern. On the other hand, if soft material is placed behind the crack tip, the energy release rate decreases to 98 J/m

^{2}, a reduction of 47% compared to the case of stiff material around the crack tip. Therefore, to make the pattern tougher by reducing the energy release rate, soft material (of higher G

_{c}) should be preferably placed behind the crack tip instead of ahead of the crack tip unless the soft material’s G

_{c}is orders of magnitude larger than the stiff material’s G

_{c}.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Schematic of the composite design. The dimension of the design space is 13 mm × 13 mm. The displacement-controlled tensile test is conducted in the y-direction. The red box indicates the two elements behind the crack tip.

**Figure 4.**The architecture of the neural network for the CVAE model. The numbers in the parentheses are the hidden units.

**Figure 6.**The neural network’s architecture for predicting the mechanical properties of composite material. The numbers in the parentheses are the hidden units.

**Figure 10.**Comparison of CVAE and GA. (

**a**) Error contour plot of CVAE (

**b**) Scatter plot of properties generated by CVAE. (

**c**) Error contour plot of GA. (

**d**) Scatter plot of properties generated by GA. The unit of K is N/m.

**Figure 11.**Two examples with different target stiffness and toughness values illustrating the design process and the results of GA and CVAE. The unit of K is N/m.

**Figure 12.**Shapley value plot of each element. (

**a**) K by rank. (

**b**) G/G

_{c}by rank. (

**c**) K by value. (

**d**) G/G

_{c}by value. The unit of K is N/m.

**Figure 13.**Scatter plots of the FEM dataset depending on the material behind the crack tip: (

**left panel**): stiff material; (

**right panel**): soft material. The unit of K is N/m.

**Table 1.**Dataset analysis of 120,000 data generated from finite element method. 25%, 50%, and 75% indicate the properties’ first, second, and third quartiles.

K (N/m) | G/G_{c} | G (J/m^{2}) | |
---|---|---|---|

mean | 1337 | 0.735229 | 199.377518 |

std | 25.40 | 0.130645 | 59.964850 |

min | 1210 | 0.329780 | 82.115000 |

25% | 1324 | 0.728642 | 181.958000 |

50% | 1340 | 0.763818 | 190.629000 |

75% | 1353 | 0.804388 | 202.360000 |

max | 1444 | 1.122954 | 464.903000 |

**Table 2.**Comparison of simulated and experimental stiffness K (* soft material ahead of the crack tip).

No. | K (FEM) (N/m) | Experimental Stiffness, K (N/m) | Error (%) | |||
---|---|---|---|---|---|---|

Specimen 1 | Specimen 2 | Specimen 3 | Average | |||

697 | 1329 | 1478 | 1610 | 1396 | 1495 | 11.1 |

10,668 * | 1301 | 1546 | 1555 | 1424 | 1508 | 13.7 |

57,881 | 1375 | 1602 | 1423 | 1440 | 1488 | 8.6 |

49,130 * | 1308 | 1317 | 1365 | 1431 | 1371 | 4.6 |

15,935 | 1277 | 1478 | 1524 | 1362 | 1455 | 12.2 |

99,251 * | 1248 | 1347 | 1457 | 1324 | 1376 | 9.3 |

**Table 3.**Comparison of simulated fracture criterion G/G

_{c}and experimental toughness of 6 design patterns (* soft material ahead of the crack tip).

No. | G/G_{c} | Experimental Toughness, (J/m^{3}) | |||
---|---|---|---|---|---|

Specimen 1 | Specimen 2 | Specimen 3 | Average | ||

697 | 0.373 | 15,381 | 18,899 | 13,000 | 15,760 |

10,668 * | 0.374 | 15,114 | 9949 | 11,355 | 12,139 |

57,881 | 0.695 | 7971 | 9982 | 7836 | 8596 |

49,130 * | 0.693 | 9489 | 8141 | 8164 | 8598 |

15,935 | 1.008 | 9398 | 5734 | 5377 | 6836 |

99,251 * | 1.007 | 6698 | 6543 | 5819 | 6353 |

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**MDPI and ACS Style**

Chiu, Y.-H.; Liao, Y.-H.; Juang, J.-Y. Designing Bioinspired Composite Structures via Genetic Algorithm and Conditional Variational Autoencoder. *Polymers* **2023**, *15*, 281.
https://doi.org/10.3390/polym15020281

**AMA Style**

Chiu Y-H, Liao Y-H, Juang J-Y. Designing Bioinspired Composite Structures via Genetic Algorithm and Conditional Variational Autoencoder. *Polymers*. 2023; 15(2):281.
https://doi.org/10.3390/polym15020281

**Chicago/Turabian Style**

Chiu, Yi-Hung, Ya-Hsuan Liao, and Jia-Yang Juang. 2023. "Designing Bioinspired Composite Structures via Genetic Algorithm and Conditional Variational Autoencoder" *Polymers* 15, no. 2: 281.
https://doi.org/10.3390/polym15020281