A digital twin for smart manufacturing of structural composites by liquid moulding

Abstract

In this work, the authors present a digital twin (DT) to analyse the manufacturing process of structural composites by using resin transfer moulding (RTM). During RTM, a dry textile preform is impregnated with a polymer resin injected in a closed mould. RTM is one of the most used production methods for high-performance structural composites. The DT is focused on detecting in-homogeneous resin flow produced by race-tracking channels that divert resin flow to the outlet gates of the mould, producing dry spots and lack of impregnation. The DT core contains two surrogate models based on encoder/decoder deep learning architectures, providing the fast/accurate response necessary for interrogation during manufacturing. The first surrogate acts as the disturbance detector, providing the on-the-fly representation of the fabric permeability with the only information gathered by a set of five pressure sensors distributed over the mould surface. The second offers real-time representation of a set of quantities of interests (QoI): namely, the flow progress and the pressure field inside the mould. Training of both surrogates was performed with synthetic data generated by high-fidelity multi-physics simulations of the flow progress in a porous preform by following Darcy’s law. Errors in the pressure field predictions of the surrogates are lower than 1$\%$ with consultation time <50 ms, enabling encapsulation in the digital twin. The DT performance was evaluated by comparing the response against a set of RTM experiments for different race-tracking scenarios. The two most relevant novelties of the work are the use of the concept of the instantiated DT, which provides information on the current state of the process from data provided by a network of distributed sensors, and that this DT has been trained exclusively with synthetic data from multiphysics simulations while evaluated against experimental data from injection tests.

Appendix A: Composite loss function

Training is carried out by minimizing the loss function defined as

$$\begin{aligned} \mathcal {L} = \frac{1}{N}\sum _{i=1}^N \rho (r_i), \end{aligned}$$

(A1)

where $r_i$ stands for the residual (difference between the ground truth and the model prediction) and $\rho (r_i)$ the error norm. To improve the accuracy of the model, it is necessary to encourage the identification of the regions in the model with large residuals. Inspired by recent results [1, 5, 12, 13], an error loss based on a wing shape [12, 13] that is of $C^1$ differentiability class was introduced. Mathematically, $\rho _{n-log}$

$$\begin{aligned} \rho _{n-log} (r_i) =\left\{ \begin{array}{ll} \omega \, C_1 |{r_i} |^n &{} \mathrm{{if}}\; |{r_i} |< \delta _1 \\ \omega \left( \ln (1+|{r_i} |)-C_2\right) &{} \textrm{otherwise,} \end{array} \right. \end{aligned}$$

(A2)

with $\delta _1$ the residual threshold, n an even natural number and $C_1=1/(n \delta _1^{n-1}(1+\delta _1))$ and $C_2=\ln (1+\delta _1)-C_1\delta _1^n$ parameters to ensure $C^1$ continuity at $r_i=\delta _1$. The $n=4$ potential is introduced to identify differentiability at $r_i=0$, while the natural logarithm is used to penalise the presence of large outliers. The factor $\omega$ is used to increase the influence of residuals. The factors $\delta _1=10^{-3}$ and $\omega =2.0$ were selected in for optimum results [15] for optimum results, see Fig. 17 .

Fig. 17

Error norms $\rho (r_i)$ functions used for the encoder-decoder training: a) $\delta _1=10^{-3}$ and $\omega =2.5,2.0,3.0$, b) $\omega =2.5$ and $\delta _1=10^{-4},10^{-3},10^{-2}$