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Analysis of structure optimization problem of dual-phase materials

For demonstrating the potential of our framework for the structure optimization of multiphase materials in terms of a target property, a simple sample problem is considered. The sample problem is the structure optimization of artificial dual-phase steels composed of the soft phase (ferrite) and hard phase (martensite). Examples of microstructures are shown in Fig.3. The prepared dual-phase microstructures can be divided into four major categories: laminated microstructures, microstructures composed of rectangle- and ellipse-shaped martensite/ferrite grains, and random microstructures. The size of microstructure images is (128times 128~mathrm {pixels}) and the total number of prepared microstructures is 3824. As an example of a target material property, the fracture strain was selected since fracture behavior is strongly related to the geometry of the two phases. The fracture strain is the elongation of materials at break. As shown in Methodology, the fracture strains for each category were calculated on the basis of the GTN fracture model18,19. Figure4 illustrates the relationship between martensite volume fraction and fracture strain for each category. This shows that laminated microstructures have a relatively high fracture strain. Also, microstructures with a lower martensite volume fraction (higher ferrite volume fraction) possess a higher fracture strain.

Examples of artificial dual-phase microstructures used for training. Black and white pixels respectively correspond to the hard phase (martensite) and soft phase (ferrite). The size of microstructure images is (128times 128) pixels. The dataset can be divided into four major categories. (a) Laminated microstructures. This category only has completely laminated microstructures. (b) Microstructures composed of rectangular martensite grains. This category includes partially laminated structures, such as these shown in the lower left panel. (c) Microstructures composed of elliptical martensite grains. (d) The random microstructures.

Relationship between martensite volume fraction and fracture strain, and examples of microstructures. (a) Plot showing correspondence between martensite volume fraction and fracture strain. (b) Examples of microstructures. Their martensite volume fractions and fracture strains are shown in the plot.

Microstructures generated by the machine learning framework trained by several datasets. (a) Examples of microsturctuers generated for several fracture strains by the network trained using All dataset. (b) Each column corresponds to the microstructures obtained by the models trained using all microstructures, only the random microstructures, only the microstructures composed of ellipse-shaped martensite grains, or only the microstructures composed of rectangle-shaped martensite grains. However, the Rectangle dataset is limited to include only the microstructures whose martensite volume fraction is between 20% and 30%. The given fracture strains are 0.1, 0.3, 0.7, and 0.9 for the All, Random, and Ellipse datasets, and 0.05, 0.1, 0.2, and 0.3 for the Rectangle dataset.

To show the applicability of our framework, we prepared several datasets: all microstructures (All), only random microstructures (Random), only microstructures composed of ellipse-shaped martensite grains (Ellipse), and only microstructures composed of rectangle-shaped martensite grains (Rectangle). Then, we trained VQVAE and PixelCNN using these datasets. The Rectangle dataset is limited to include only the microstructures whose martensite volume fraction is between 20% and 30% to consider the case in which martensite grains are located separately from each other.

Figure5a shows examples of microstructures generated for several fracture strains using the network trained by All dataset. Figure5b summarizes the trend of the microstructures obtained by the networks trained using the above datasets with gradually increasing fracture strain. For the All, Random, and Ellipse datasets, we can see the trend that martensite grains become smaller and thinner as the target fracture strain increases. Since the larger area fraction of the soft phase (ferrite) contributes to the realization of higher elongation as we can see in Fig.4, this result is reasonable. In addition, it should be noted that the laminated structure corresponding to the highest fracture strain ((text {FS}=0.9)) was generated only for the All case in which the laminated structures are included in the training dataset. Additionally, in the case of the controlled martensite volume fraction of the input microstructures (Rectangle), the martensite grains tend to arrange more uniformly as the given fracture strain increases.

Generated microstructures and trend of martensite volume fraction. (a) Microstructures generated at fixed tensile strength and several fracture strains. The tensile strength is set as 700 MPa. The given FSs are 0.1, 0.3, 0.4, 0.5, 0.7, and 0.9. (b) Trend of martensite volume fraction relative to the change in fracture strain. For each fracture strain, the martensite volume fractions of 3000 microstructures generated corresponding to the fracture strain and fixed tensile strength ((700 mathrm {MPa})) were calculated. The black lines and green triangles in the boxes denote median and mean values, respectively.

From these results, we can conclude that there are at least two different strategies for the realization of a higher fracture strain: one is to decrease the size of martensite grains and also to arrange them uniformly, and the other to alternatively make a completely laminated composite structure27. The fact that the laminated structures never appear without providing the laminated structures in the training dataset implies that there exists an impenetrable wall for a simple optimization process, such as a gradient descent algorithm used to train neural networks, to figure out the robustness of laminated structures from the other structures.

Next, the tensile strength is given in addition to the fracture strain as another label for PixelCNN for considering the balance between strength and fracture strain (ductility). In this case, all microstructure data are used for training. The microstructures are generated at the fixed tensile strength of (700 mathrm {MPa}). The generated microstructures are shown in Fig.6a. The laminated structures seem to be dominantly selected as the target fracture strain increases. The trend that martensite grains become smaller and thinner is not seen when the tensile strength is fixed.

In addition, the martensite volume fractions were calculated for 3000 microstructures generated corresponding to several fracture strains. The tensile strength was fixed at (700 mathrm {MPa}) again. The box plot of the trend of the martensite volume fraction relative to the change in fracture strain is shown in Fig.6b. The martensite volume fraction decreases as the given fracture strain increases. At the same time, the martensite volume fraction approaches a constant value. For the realization of a higher ductility without decreasing the tensile strength, the shape of martensite grains approaches that of the laminated structures as the martensite volume fraction decreases. This result implies that laminated structures can achieve a higher tensile strength with a smaller martensite volume fraction. As a result, the laminated structures can be considered as the optimized structures with respect to the shape of martensite grains for the realization of a higher ductility without decreasing their strength. The laminated structures were actually reported to exhibit improved combinations of strength and ductility27.

Correspondence between the target fracture strains given as inputs and the actual fracture strains. For each target fracture strain, 30 microstructures were generated. Then, fracture strains are calculated using the physical model18,19. (a) Plot of relationship. (b) Box plot of relationship. The black lines and green triangles in the boxes denote median and mean values, respectively. (c) Microstructures whose fracture strains are smaller than 20% of the target fracture strains. The values above the panels denote the given target fracture strains (left) and actual fracture strains (right).

To validate the effectiveness of the present framework, fracture strains are calculated using the physical model18,19 for each microstructure obtained using the framework. In this case, the network trained by giving only fracture strain as the target property is used. Figure7a,b show the correspondence between the target fracture strains for generated microstructures and the actual calculated fracture strains. Also, the coefficient of determination was 0.672. It is clear that our framework captures well the general trend of microstructures relative to the fracture strain. However, it should be noted also that there exist several microstructures whose actual fracture strains are far less than the target strains. Figure7c shows the typical microstructures whose fracture strains are smaller than 20% of the target fracture strains. Additionally, the coefficient of determination for the data without data points corresponding to the microstructures shown in Fig.7c was 0.76. All of them are partially incomplete laminated structures. This can be understood as follows. Although laminated structures has a potential to realize higher fracture strains as shown in Fig.4, this is true only when the microstructures are completely laminated. Even when one martensite layer has a tiny hole, the gap between martensite grains becomes the hot spot that induces much earlier rupture. Thus, the box plot shown in Fig.7b is understood to show decreasing values as a result of an attempt to completely laminate the structures to realize the given target fracture strain. This indicates that the framework recognizes the structures shown in Fig.7c to be structurally close to completely laminated structures even though they have far less fracture strains than the completely laminated structures.

As a consequence, these results illustrate that our framework provides a powerful tool for the optimization of material microstructures in terms of target properties, or at least for capturing the trend of microstructures in terms of the change in target property in various cases.

The above results of the generation of material structures corresponding to the target fracture strain indicate that our framework captures the implicit correlation between the material microstructures and the fracture strain. However, generally, it is difficult to interpret implicit knowledge captured by machine learning methods. For that reason, we cannot hastily conclude that machine learning can understand this problem and acquire meaningful knowledge for material design similarly to humans or that it just obtains physically meaningless problem-specific knowledge. Usually, human researchers attain the background physics by noting a part or behavior that will affect a target property during numerous trial-and-error experiments. Generally, this process is time-consuming. Accordingly, approaching implicit knowledge obtained by machine learning methods could be beneficial for achieving a more efficient way to extract general knowledge for material design. Thus, we discuss how to approach the physical background behind the implicit knowledge captured by our framework. In particular, we investigate whether the machine learning framework can identify a part of material microstructures that strongly affects a target property in a similar way human experts can predict on the basis of their experiences.

To identify a critical part of microstructures, we consider calculating a derivative of material microstructures with respect to the fracture strain. This is based on the assumption that human experts unconsciously consider the sensitivity of material microstructures to a slight change in target property. Accordingly, the following variable (Delta) is defined as the derivative:

$$begin{aligned} Delta :=frac{partial D(mathbb {E}_{P(theta |epsilon _f, M_r)}[ theta ])}{partial epsilon _f}, end{aligned}$$

(3)

where (mathbb {E}_{P(theta |epsilon _f, M_r)}[ theta ]) is the expectation of a spatial arrangement of fundamental structures (theta) according to (P(theta |epsilon _f, M_r)), which is the probability distribution captured by PixelCNN. Here, (M_r) and (epsilon _f) are the reference microstructure under consideration and the calculated fracture strain for the microstructure, respectively. In other words, (mathbb {E}_{P(theta |epsilon _f, M_r)}[ theta ]) is the deterministic function of (epsilon _f) and (M_r). In addition, D is the CNN-based deterministic decoder function; hence, (Delta) has the same pixel size of the input microstructure images.

If the machine learning framework correctly captures the physical correlation between the geometry of the material microstructures and the fracture strain, (Delta) is expected to correspond to the areas in (M_r) that highly affects the determination of the fracture strain even without giving the physical mechanism itself. For numerical calculation, (Delta) is approximated as

$$begin{aligned} Delta thickapprox {D(mathbb {E}_{P(theta |epsilon _f+Delta epsilon _f, M_r)}[ theta ])-D(mathbb {E}_{P(theta |epsilon _f, M_r)}[ theta ])}/Delta epsilon _f, end{aligned}$$

(4)

where (Delta epsilon _f) is the gap of the fracture strain, which is set as 0.01 in this paper. Because it is difficult to compare quantitatively the distribution of this variable with the critical microstructure distributions obtained from the physical model, in this paper, we only discuss the location of crucial parts. Thus, the denominator (Delta epsilon _f) is ignored for the calculation of (Delta) in the rest of this paper.

Comparison of derivatives of microstructures with respect to the fracture strain obtained using the machine learning framework with the distributions of void volume fractions calculated on the baisis of physical model. (a)(d) Comparisons for several microstructures. The left, middle, and right column correspond to the reference microstructures, the void distributions obtained using the physical model, and the derivative obtained by the machine learning framework, respectively.

Figure8 shows the comparison of the parts of microstructures critically affecting the determination of the fracture strain obtained by the physical model and our machine learning framework. In the case of the results from machine learning, the absolute values of (Delta) defined in Eq.(3) for each pixel are shown as colormaps. On the other hand, because the fracture behavior is formulated as damage and void-growth processes in the physical model, the void distribution in a critical state directly shows the critical points for the determination of fracture strain. Thus, in the case of the physical model, the calculated void distribution in a critical state is shown in Fig.8. The details of the physical model and the experiment for the determination of some parameters are given in Methodology. For ease of comparison, the ranges of visualized values are changed for each image, while the relative relationships among values of each colormap are kept. Thus, we compare the results qualitatively in terms of the distribution of areas having relatively high values in the next paragraph.

Figure8a,b illustrate the crucial parts of the microstructures composed of relatively long and narrow rectangle-shaped martensite grains. We can see an acceptable agreement between the results of the physical and machine learning methods in terms of the overall distribution of crucial areas which are shown in red in the colormaps of Fig.8. In addition, Fig.8c,d show the parts that critically influence the fracture behavior in the microstructures composed of similarly shaped martensite grains. As an important difference between them, in Fig.8c, the rectangle-shaped martensite grains are irregularly arranged and some martensite grains are close to each other, which might critically affect the fracture behavior, whereas in Fig.8d, circular martensite grains are almost regularly arranged. About Fig.8c, the machine learning framework seems to capture the crucial parts that are predicted by the physical model. As mentioned above, the distributions seem to be dominantly affected by the martensite grains being close to each other. In other words, the short-range interactions among a small number of martensite grains are dominant for the determination of the fracture strain in this case. Also, in Fig.8d, both the physical model and the machine learning framework can predict that the crucial parts are uniformly distributed in square areas.

On the other hand, the physical model also predicts the influence of long-range interactions among martensite grains on fracture behavior, which can be seen in Fig.8c,d as a bandlike distribution. However, the bandlike distribution resulting from the long-range interactions does not seem to be captured by the machine learning framework owing to the characteristic of PixelCNN. Because a global stochastic relationship among the fundamental elements is factorized as a product of stochastic local interactions in PixelCNN as defined in Eq.(1), the extent of interaction exponentially decreases as distance increases. Therefore, the long-range interactions are difficult to be captured by PixelCNN. The discussion of the limitation of PixelCNN in capturing long-range interactions and a remedy for this limitation can be found in28. Figure9 illustrates a sample case showing that the relatively long-range interactions are important for the dertermination of fracture strain. In this case, the determination of the part that critically affects the fracture behavior seems to be difficult using the framework based on PixelCNN.

Sample case showing that a relatively long-range interactions among martensite grains are important for the determination of fracture strain.

For incompletely laminated structures such as that shown in Fig.8a, the martensite layers are expanded to achieve a higher fracture strain even though increasing the martensite volume fraction basically contributes to the decrease in the fracture strain, as shown in Fig.4. Similarly, we can see in Fig.8c that the martensite grains tended to expand to fill the hot spots between them. Additionally, as mentioned above, even though completely laminated structures are structurally similar to incompletely laminated structures, the fracture strains of completely laminated structures are much higher than those of incompletely laminated structures. Thus, eliminating tiny holes that could be causes of hot spots and reaching ({ completely}) laminated structures markedly improve their fracture strains. Altogether, these results imply that the framework recognizes the potential of laminated structures to achieve a higher fracture strain in a similar way that human researchers reach an intuition on completely laminated structures as a result of the consideration of reducing the occurrence of hot spots.

From the above results, we can conclude that our framework can identify the areas that critically affect a target property without human prior knowledge when the local topology of microstructures is dominant for the target property. This implies that machine learning designed consistent with metallurgists process of thinking can approach the background or the meaning of the implicitly extracted knowledge in a similar way that humans acquire an empirical knowledge.

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Identification of microstructures critically affecting material properties using machine learning framework based on metallurgists' thinking process |...