*2.2. Metamodelling*

As has been shown in the introduction, a range of different metamodelling procedures such as kriging and NN have been implemented in the field of SMC processing and SMC material description, each exhibiting individual problem-dependent benefits and drawbacks [8,34]. Jin et al. and Simpson et al. evaluated a range of metamodelling procedures, with Simpson et al. recommending the use of NN when dealing with highly nonlinear or large problems containing many parameters, which (as will be shown) is the case in the proposed procedure [30,32,49]. Furthermore, the use of FF-NN has been shown by Twu and Lee et al. to be beneficial in comparison to alternative approaches for metamodelling in the field of SMC optimisation; thus, this approach is used [30]. The cited work is greatly recommended for a more in-depth description of general procedures in setting up and training FF-NN.

Contrary to prior papers presented in the introduction, which use a small number of non-binary input variables, the initial filling state of each element (which may be completely filled or unfilled, thus, binary) is proposed for defining the geometry and position of the preform and for use as input variables (thus, 1200 binary input variables are used in total). However, this results in a high number of network connections, even without considering the further setup of the FF-NN. Training an FF-NN with a high number of connections may lead to a loss in accuracy for out-of-sample data commonly known as overfitting [50]. This is especially prone when using a limited sample size as is the case in simulation-based optimisation of the SMC process [51].

Using a large number of binary input variables is a known procedure in the field of Optical Character Recognition (OCR), in which neural networks are used to detect printed or handwritten letters in black and white images [49,52,53]. Cybenko and Hornik et al. have shown that an FF-NN with a single hidden layer can, when using sigmoid transfer functions, describe a continuous function to an arbitrary degree of accuracy [30,54,55]. However, the necessary number of nodes in the hidden layer and sample sizes have been part of an intensive debate. While general procedures for finding the most optimal parameters (commonly known as hyperparameters) for an FF-NN, such as the robust design methodology proposed by Taguchi, have been used, these parameters are usually set based on experience and trial and error procedures [56,57].

Furthermore, larger sample sizes are preferred, and in the field of OCR of handwriting, extensive databases have been formed [53]. However, sample size is limited in the discussed application on compression moulding due to the calculation e ffort and general practicability. Therefore, overfitting is assumed as given and may occur regardless of the chosen number of neurons in the hidden layer. However, there are metamodelling approaches that mitigate the e ffects of overfitting, which will be shown to be successful. Hastie et al. propose the use of ensemble modelling approaches, in which the outputs of multiple FF-NN with identical architecture (but which may each exhibit a di fferent form of overfitting e.g., due to di fferences in training procedures) are combined to increase the accuracy of the model as a whole [51,58]. In this paper, bootstrap aggregation ("bagging") is implemented, and the mean of the outputs of 100 FF-NN trained with random starting weights is used to predict the maximum absolute deflection of the plate. Thus, only a limited number of neurons are used, and the focus is set on showing the general applicability of this procedure and its accuracy in prediction.

In total, 36 samples consisting of a unique rectangular preform geometry and position and affiliated maximum absolute deflection were created by simulation procedures shown in Section 2.1 for training of the FF-NN (see Figure 2 for a representation of the preform samples). Definition of the samples was based on the following principles:


**Figure 2.** Geometries of preform samples used in feed forward NN (FF-NN) training. Preform resulting in the lowest maximum absolute deflection of the sample set is outlined.

Thus, limiting the samples to only a single geometry was chosen for reasons of consistency. Including alternative geometries based on these principles may be beneficial for increasing the accuracy of the trained FF-NN. However, this would necessarily increase the number of samples, and including all elements in the sample set may not be always possible (e.g., inclusion of corner elements when using round preform geometries).

The material and processing parameters applied are shown in Tables 1 and 2 and Figure 3. Material data correspond with SMC0400 of Menzolit Srl., Turate, Italy [59].


**Table 1.** Material parameters of SMC0400 [59].

**Table 2.** Process settings [59]. SMC: Sheet Moulding Compounds.


**Figure 3.** Shear rate and temperature-dependent viscosity [59].

Training of the FF-NN is conducted using the "fitnet" function implemented in the MATLAB Deep Learning Toolbox by Levenberg–Marquardt backpropagation, as this has been shown to be the fastest method in training the FF-NN [60–62]. FF-NN architecture and training parameters are summarised in Table 3. For a complete description of Levenberg–Marquardt backpropagation, refer to the cited paper by Hagan and Menhaj [60].

For initial validation of the metamodel, a comparison of the maximum deflection predicted by the metamodel and calculation by FEM for three preforms not used in FF-NN training (Figure 4) are compared. Further validation is conducted on preform geometries obtained using the optimisation procedure.

**Table 3.** Architecture of the neural networks (NN) and training parameters.

 **Figure 4.** Preform geometries used in initial metamodel validation: validation geometry 1 (**a**), validation geometry 2 (**b**), validation geometry 3 (**c**).

(**c**)
