Surrogate Model Based on Data-Driven Model Reduction for Inelastic Behavior of Composite Microstructure

Abstract

On the microscale, most composite materials are composed of heterogeneous materials comprising two or more different phases, such as matrices and inclusions. In addition, composite materials may exhibit high variability, depending on the material and amount of material used. Hence, the effect of microstructure on the macroscopic structural analysis of composite materials must be considered. Computational homogenization can be used to describe an effective constitutive model for heterogeneous composites at the microscopic level. However, a significant computational cost may be incurred owing to the iterative procedure when considering the inelastic behavior of composite materials. Hence, an efficient data-driven model reduction technique, i.e., a deep-learned surrogate model, is proposed. The key idea of the proposed framework is twofold: (1) Data-driven unsupervised model reduction for efficiently managing high-dimensional data from the microstructure and for extracting those features, and (2) the construction of parameterized constitutive models with inelastic behavior by connecting macro- and microscopic levels. Each aspect leverages a variational autoencoder and a gated recurrent unit, which are state-of-the-art components for deep learning. To demonstrate the efficiency and accuracy of the proposed model, the proposed model is applied to a two-dimensional microstructure problem involving inelastic behavior. Consequently, it is discovered that the present surrogate model can provide improved computational efficiency and accuracy within a prescribed parametric space.

Appendices

Appendix 1

A. Network Training History: Two-Dimensional Microstructure Under Uniaxial Tension

The network hyperparameters of the VAE and GRU were manually adjusted to train each network. First, VAEs, which extract features of stress fields for each component $({\sigma }_{11}$, ${\sigma }_{22}$, ${\sigma }_{12}$) and phase (matrix, fiber), were trained by a learning rate of $1\times {10}^{-5}$ and a batch size of 256. Also, epoch, known as an iteration of the training network, was set to be 10,000 and 20,000 when training the matrix region and the fiber region, respectively. Then, GRUs, which connect the macro strain and stress field of each component, were trained by a learning rate of $1\times {10}^{-4}$, batch size of 1024, and epoch of 5000. Convergence histories in the training and validation of different components and material phases are shown in Figs.

Fig. 17

Training history of VAEs in matrix of two-dimensional microstructure under uniaxial tension

17,

Fig. 18

Training history of VAEs in fiber of two-dimensional microstructure under uniaxial tension

18 and

Fig. 19

Training history of GRUs for two-dimensional microstructure under uniaxial tension

19. The loss per epoch was evaluated by the Eq. 13, and MSE described in Sect. 3 for VAE and GRU, respectively.

It is noted that the training histories undergoes stable decaying and convergence with respect to the training progresses. As a result, each network provided benign training characteristics without overfitting. It can be seen by the value of R² as a network training evaluation in Sect. 5.1.1 as well.

B. Network Training History: Two-Dimensional Microstructure with Interphase Zone Under Uniaxial Tension

The hyperparameters related to the network training of VAE and GRU are manually tuned, as summarized in Table 4. The training histories for each network (trained by applying the corresponding hyperparameters as summarized in Table 4) are shown in Figs.

Fig. 20

Training history of VAEs in matrix of two-dimensional microstructure with interphase zone

20,

Fig. 21

Training history of VAEs in interphase zone of two-dimensional microstructure with interphase zone

21,

Fig. 22

Training history of VAEs in particle of two-dimensional microstructure with interphase zone

22 and

Fig. 23

Training history of GRUs for two-dimensional microstructure with interphase zone

23. Likewise, in the previous example, entire networks were trained well with stable convergence characteristics.

Table 4 Training parameter of each network