In order to improve laminate quality, and reduce part distortion as well as pro-cessing time, many different process optimization tools have been developed in the past. Prior to the development of process simulation tools, cure cycle op-timization involved an expensive and time-consuming experimental trial-and-error processes [180], which was conducted only on rare occasions [181, 182]. The introduction of the first numerical process analysis by Loos and Springer led to an investigation of the effects of different temperature cycles on the laminate qual-ity [157]. Thus, the previously experimentally trial and error approach was hence-forth mostly conducted by means of numerical process analysis tools [81, 183].
With help of numerical tools, Ciriscioli et al. found that the cure cycle recom-mended by the manufacturer is insufficient for thicker parts, and proposed re-duced heating rates on the second ramp to reduce thermal gradients [184]. Kim and Lee introduced cooling and reheating steps to prevent temperature over-shoots in thick autoclave laminates [185]. Chen et al. investigated the effect of moisture upon the optimal temperature path to achieve minimal residual stress [186]. These trial and error findings quickly showed that several process-related criteria have to be weighted against each other as they are contradictory:
For instance, low overall temperatures result in low residual stresses but also in high processing times. Originating from this problem, several optimization strategies were investigated to find the optimal cure cycle for the processing of a given laminate. Two main approaches towards an optimal cure cycle are discussed in literature: The first approach results in a rule-based control of the manufacturing environment (e.g. autoclave) and is mostly referred to as
"ex-pert system" [187]. Here, a set of sensors is located in the most critical area of the part and the temperature and pressure is controlled according to previously determined simple rules, as shown in Figure 2-5. Thus, the final cure cycle is determined "on the fly" during the part production. The work on these "smart"
cure strategies mostly includes an evaluation of process simulation to define the control rules and an initial set of process parameters [188, 189]. In contrast to this rule-based approach, others determined the cure cycle numerically a priori by means of increasingly sophisticated global optimization strategies utilizing process simulation and a multi-objective fitness function evaluation.
Figure 2-5Control of the curing process in a rule-based heuristic system [9].
In an early work by Ciriscioli and Springer one of the first heuristic expert systems was set up, which takes composite temperature, compaction, residual stress and void formation in an autoclave into account [190]. In their work, the rule-based cure strategy was validated by means of cure simulations and ex-periments. However, they encountered difficulties in determining the viscosity and degree of cure online. Furthermore, at times different rules contradicting each other at certain cure states posed a challenge. Other researchers followed in using process simulation to set up a heuristic rule-based system to find optimal cure cycles with minimal residual stress [9, 156]. Sheen was one of the first to use numerical optimization to minimize the temperature gradients in a part by means of finite element simulation [191]. Pillai et al. developed a framework called "local criterion optimization" for the process simulation approach devel-oped by Bogetti and Gillespie to determine an optimized cure cycle with help of a weighted optimization function considering the final degree of cure, max-imum temperature, and residual stress [36, 41, 192]. Subsequent process trend analysis resulted in operating rules for the autoclave and ensured improved part quality as well as reproducibility. Thus, Pillai used numerical multi-objective process optimization to generate a rule-based framework for autoclave
process-ing. In general, the different numerical optimization techniques employed can be divided in three categories:
• Deterministic methods(e.g. Nelder-Mead’s or Powel’s methods) calculate a local maximum or minimum of the fitness function with comparably low computational effort [40, 175, 193–195]. However, as the local extremum found might differ significantly from the global extremum, the quality of the result depends largely on the user-defined initial parameter set.
• On the contrary, stochastic methods (e.g. evolutionary strategy, random walk, simulated annealing, or genetic algorithms) employ random search techniques in the chosen design space [196]. They are more likely to find a global optimum but require more computational effort, which is the reason why they are often used in more recent studies [5, 197–201].
• Neural network algorithms[10, 202, 203] couple a set of input parameters (e.g. cure cycle time, heating rate, dwell temperature) with a set of output parameters (e.g. laminate quality and residual stress) using "hidden units", as shown in Fig. 2-6. These hidden units obtain information from the input units and proceed this information to the output units. The links in-between these units are weighted. The weight values have to be determined by training of the neural network.
Although the process simulation computational effort by means of neural net-works is very small, the training of the weights needs to be extensive and typi-cally requires at least several thousand and possibly up to millions of individual data sets, depending on the difficulty of the problem. As the amount of process simulations for teaching of the neural network alone exceeds the amount of sim-ulations required by the other two methods to find an optimal cure cycle this method is recommended for use in special cases only where a very large number of investigations is required, such as process-interactive model-predictive moni-toring and control or process investigations taking stochastic material variability into account [10, 204].
Independent of the method chosen, all numerical optimization algorithms are inherently reliant on the quality of the fitness function. The multi-objective fitness functionFprojects all assessment parametersJof interest for a given cure cycle, such as laminate quality and processing time, onto one value.
J ⊂Rn∧F: J→R (2-58)
Figure 2-6Functionality of neural networks [10].
The optimal solution is the global minimum or maximum of the fitness function, depending on the formulation of the problem. Within this work, the optimal solution is defined by the minimum of the fitness function:
∃x∈R:x=min(F(J)) (2-59)
A large variety of fitness parameters was used in the presented literature. Ruiz and Trochu developed one of the most diverse fitness functions in literature, which uses the weighted sum of seven different cure cycle assessment parame-ters [5, 146, 155].
F= Xn
k=1
ωk Jk (2-60)
The cure cycle assessment parameters used in this work were: the final extent of cure, the maximum obtained exothermic temperature, the cross-over at the after-gelation point (AGP), the degree of cure gradients, the induced curing stress, the thermal stress, and the processing time. Each individual parameter was expressed as a sigmoid function between zero and unity, with zero indicating
the desired optimal state. In case of the final extent of cure the objective function is written as follows:
Jfc = Afc
Bfc+egfc (2-61)
gfc =−αilast−αmin
αmin
Cfc
αult−αmin
(2-62) whereαilast is the minimal degree of cure apparent in the laminate, andαminand αultthe minimal required extent of cure and the ultimate degree of cure for which the resin is considered fully polymerized. Afc, Bfc and Cfc are coefficients of the sigmoid function. These values are chosen dependent on the process specifi-cations and material in question. For a given set of parameters, an exemplary objective function Jfcis shown in Figure 2-7.
Final degree of cure [-]
0 0.2 0.4 0.6 0.8 1
FitnessvalueJfc[-]
0 0.2 0.4 0.6 0.8
1 αmin 0.8
αult 0.95 Afc 1 Bfc 1 Cfc -3
Figure 2-7 & Table 2-1Sigmoid function and corresponding coefficients for the final extent of cure objective.
The objective function presented in this figure leads to a preferred extent of cure between 0.9 and unity. The impact of this preference on the overall fitness function is defined by the magnitude of the weighting factor ωfc in relation to all other weighting factors (see Eq. 2-60 ). Given that the produced parts are subjected to different requirements, the objective function coefficients as well as the weighing factors have to be set specifically for the target part: In the automobile industry the maximal cycle time restriction is set to several minutes whereas in the aerospace industry the maximal cycle time can be several hours.
Thus, no overall best solution exists and the application of an optimization strategy to determine the cure cycle requires experience in the case-specific set-up of the fitness function.
Finally, the robustness of the optimized cure cycle should also be considered. As the manufacturing process as well as the material initial state are subjected to
small variations, the determined cure cycle should lead to an optimal part even if these imperfections are considered. To account for batch-to-batch variation in the optimization process, Michaud investigated four different optimization techniques capable of finding the global optimum and improved an algorithm based on an evolutionary strategy [196]. In the investigation of 2.54 cm thick laminates, the resulting cure cycle led to a decrease of the amount of inferior parts by almost 80 % compared to optimization techniques neglecting batch-to-batch variation [38].