*2.3. Optimisation*

The optimisation of a performance metric of an SMC-based component can be defined as a multivariable optimisation problem (MVO). The mathematical description of an MVO is [63–65]:

$$\min(f(\mathbf{x})), \mathbf{x} \in \mathcal{S} \tag{9}$$

where *f* : *Rn* → *R* is the objective function and *x* = (*<sup>x</sup>*1, *x*2, ... , *xn*)*<sup>T</sup>* is the decision vector belonging to the nonempty feasible region *S* ⊂ *Rn* [63]. In this case, maximum deflection of the plate in z-direction *Dz* is treated as the objective function, which is defined by the maximum absolute deflection of all nodes *N* observed in this direction (compare Figure 5; notation derived from Islam et al.) [66]:

$$f(\mathbf{x}) = \max \left| (D\_z)\_i \right|, \ i = 1, 2, 3, \dots \\ N. \tag{10}$$

**Figure 5.** Exemplary deflection result of structural simulation. In this case, the maximum absolute deflection *Dz* is 89.89 mm.

The components of the decision vector shown in Table 4 correspond with the input variables of the metamodel described previously (initial filling state of each element, which can only be completely filled or empty).



A challenge in solving MVO is detecting the global minimum of *f*, for which evolutionary algorithms (EA) such as genetic algorithms (GA) have been used successfully [67–69]. These algorithms may include multiple different operators such as crossover and mutation. Here, a non-standard mutation procedure is implemented, which is based on evaluation of the metamodel (Section 2.2).

The setup of the developed optimisation routine is shown in Figure 6. Using an iterative method presented in the following, a preform geometry and position that minimises the objective function is sought.

#### (1) Mutation of preform and evaluation of objective function:

Starting elements (which can be a single element or a group of adjacent elements) are mutated iteratively by applying the eight subsequent procedures summarised in Table 5. During these procedures, the preform geometry is increased (or decreased) by the R adjacent, randomly chosen elements (R initially being one) in the specified direction. Decrease procedures are initialised after the minimum preform size (5% of the part surface area coverage) is reached.

After each procedure, the resulting plate deformation of the new preform geometry is evaluated using the metamodel. Mutation is retained if a decrease of the objective function is predicted. To decrease the likelihood in reaching a local minimum (thus not being the most optimal, global solution for the optimisation), R is increased to three if no decrease in deflection is reached during 10 iterations. The mutation is terminated after a total of 125 iterations.

**Figure 6.** Flow chart of the optimisation procedure.

**Table 5.** Mutation procedures conducted in each optimisation iteration.


#### (2) Evaluation of boundary conditions:

In prior conducted studies, problem-dependent constraints and constraint handling techniques had to be implemented [25,31]. As the description of the preform in the presented approach is based on the discretisation of the geometry also used in process and structural simulation, typical constraints such as limiting the preform to the inside of the mould are not necessary.

Two problem-independent constraints (e.g., independent of the part geometry) are implemented, with which the typical processing defects and limitations of the compression moulding process are addressed:


As starting elements, five evenly spaced elements contained inside the FF-NN training sample preform resulting in the smallest maximum deflection are used, as an optimal solution is presumed in this area (see Figure 2 and Figure 9). One of the following two results are expected after successfully running the optimisation procedure:


#### **3. Results and Discussion**

#### *3.1. Metamodel Validation*

In Figure 7, maximum absolute deflections attained by the metamodel and FEM for the validation geometries are compared. Standard deviations and outliers of the 100 individual FF-NN outputs of which the metamodel is composed are also shown. Maximum plate deflections predicted by the metamodel differ by 2.67% (validation geometry 1), 0.26% (validation geometry 2), and 0.82% (validation geometry 3) from values obtained by FEM, respectively. Therefore, plate deflections are predicted accurately by the metamodel.

**Figure 7.** Comparison of plate deflections for the three preform validation geometries obtained by the metamodel and FEM.

As expected, the individual FF-NN included in the metamodel exhibit a high spread in outputs, exceeding 50% of the metamodel output value (e.g., total spread in predicted deflections for validation geometry 3: 48.47 mm), which is attributed to overfitting during the training process (see Section 2.2). No significant influence of the preform position on the spread of the individual FF-NN outputs can be detected. Potentials for decrease in spread include increasing the sample set size and implementing NN validation procedures; however, these would significantly increase the computational effort.

To further evaluate the decrease in plate deflection from 85.32 mm to 81.29 mm which results from decreasing the charge distance from the clamping location, fibre orientations are compared. Fibre orientation tensor component axx, which is visualised in Figure 8, describes the probability of fibre orientation in the x-axis direction [5]. For a decrease in the charge distance from clamping location results, a decrease of this tensor component in the left half of the plate is observed, which reduces the local flexural modulus and therefore the overall bending stiffness of the plate [71]. Although validation geometries used are symmetric relative to the principal axis of the plate, fibre orientations calculated by FEM are not symmetric relative to this axis, which may result from the non-symmetry and coarseness of the mesh used [72].

**Figure 8.** Comparison of fibre orientation in the x-direction for the three preform validation geometries obtained by process simulation. (**a**) = Validation geometry 1; (**b**) = Validation geometry 2; (**c**) = Validation geometry 3.

#### *3.2. Preform Optimisation*

The starting elements of the performed preform optimisations and resulting preform geometries are shown in Figure 9. Preform geometries resulting from adjacent starting points show a similarity in size and geometry; however, these are not identical in any case. Convergence of the objective function is presented in Figure 10. However, it has to be clear that the early generations do not represent valid solutions, as the minimum preform size (Section 2.3, constraint 2b) is only reached during the final generations.

**Figure 9.** Initial starting elements (left) and resulting optimised preform geometries (right) of the performed optimisations. (**a**) Starting element 1, (**b**) Starting element 2, (**c**) Starting element 3, (**d**) Starting element 4, and (**e**) Starting element 5.

**Figure 10.** Convergence of objective function (maximum plate deflection in z-direction) during preform optimisation.

Although 125 iterations were conducted in each case, no decrease in the objective function value or further change in preform geometry is detected for any starting element from 25 generations onwards. The run time was under five minutes respectively. One can see a high decrease in maximum deflection during the initial iterations, with the tapering off of the attained decrease going further. The minima of the objective function range from 75.3 (Starting element 4) to 81.3 mm (Starting element 1) (Figure 9). Similar to the validation geometries, comparison of the metamodel output with FEM results again confirm accurate prediction by the metamodel, with deviations ranging from 0.46% (Starting element 4) to 2.14% (Starting element 1) (Figure 11).

**Figure 11.** Comparison of deflections obtained by metamodel and FEM for optimised preforms.

As different preforms are achieved depending on the starting element and only one global minima is presumed to exist, optimised preforms represent the local minima of the MVO. Further comparison of the obtained values for the objective function (Starting element 5) with maximum deflection of the sample from which the starting elements were initially taken (Figure 2) show that the algorithm was not capable in reaching this more optimal solution (in comparison, the highest deflection of all the samples was 114.9 mm). This sample is in contact with the full length of the left plate edge, representing the highest achievable flow length while fulfilling the minimum mould coverage defined in the optimisation. While reaching the geometry of the mentioned sample may be possible when strongly increasing the number of conducted iterations, the function of the optimisation algorithm is restricted while approaching it due to it having the minimum mould coverage (5%) for conducting a valid mutation step. The sharp edges of the optimised geometries are a result of the use of S3 elements, and these could be combatted by increasing the element count, adding additional constraints to the optimisation procedure, or using alternative process simulation approaches. Additionally, an additional constraint for avoiding two preforms from forming should be implemented (as is the case in preform (b)), as these may lead to weld lines and should be avoided.
