3.1. Optimization Objective
The nonlinear second-order consolidation differential Navier–Stokes equation is used to predict the performance of axial flow fans. Thus, the fan optimization mathematical model is a complex nonlinear problem with constraints and multiple peak values. In this study, the optimization variables are six parameters that determine the spanwise distortion laws of the fan. The optimization objectives are maximizations of the total pressure coefficient and maximum efficiency, marked as
ψOPT and
ηOPT. The flow coefficient
is in the range of 0.24–0.45 The mathematical model of the optimization process can be expressed as a search:
X = (
θ0,
θ1,
A,
β0,
β1,
B) with the constraint
, so that:
where
represents the number of constraints,
C is the constraint limit value, m is the number of working conditions,
n is the number of objective functions,
is the linear weighted average of the
kth objective function, and
represents the weighting factor of the
kth objective function under the
ith working condition [
22].
3.3. Dendrite Net
Dendrite Net (DD) is an essential machine learning algorithm that is similar to support vector machines (SVMs) or multi-layer perceptron (MLP). It transforms the fan system to a Taylor expansion, through which we can identify the importance of the system parameters. A graphical illustration of the learning process is shown in
Figure 8, where
X denotes the input space of the DD,
Al is the input of the
lth DD module and output of the (
l-1)
th DD module, and
W l,l−1 is the weight matrix from the (
l-1)
th module. Additionally, ◦ is the Hadamard product, L expresses the number of modules, and
Y denotes the output space of DD.
The forward-propagation of the DD and liner modules is denoted as:
The error-backpropagation of DD and linear modules is:
The weight adjustment of DD is:
where
and
Y are the DD’s outputs and labels, respectively,
m denotes the number of training samples in one batch, and the learning rate
α can either be adapted with epochs or fixed to a small number based on heuristics [
26].
Employing the six blade parameters generated by LHD, the maximum total pressure coefficient
ψOPT and maximum efficiency
ηOPT are obtained by numerical simulation. The initial sample space used to train the DD network is obtained from the LHD method (shown in
Table 3).
A total of 40 groups are collected in the initial sample space. In order to predict the ψOPT and ηOPT, the DD is trained twice.
Each column in the sample space is first normalized. The normalized equation is as follows [
27]:
where
is column i of the normalized sample space,
is the column i of the original sample space,
is the maximum value in
, and
is the minimum values in
.
The normalized impeller structural parameters are then set as input variables, and the corresponding impeller performance parameters are set as output variables. The effect of each blade parameter on fan performance is a complex nonlinear problem. In order to clearly explore the influence of each parameter on fan performance parameters, two Dendrite Nets with six modules are trained to predict the ψOPT and ηOPT.
Figure 9 shows the variation in DD prediction total pressure and efficiency error with iteration steps (Is). When the number of iteration steps (Is) exceeds 1.5 × 10
6, the error of prediction value (e) of
ψOPT and
ηOPT is reduced to −4.7 × 10
−4 and −6.3 × 10
−6, respectively. The prediction results are in complete agreement with the sample space, indicating that the calculation is convergent.
It is well known that the relationship between the inputs and outputs of the system can be expressed by the accumulation of the trigonometric function or polynomial. This study uses DD as a tool to transform the complex system into a polynomial. The relation spectrum shown in
Figure 10 expresses the impact of the blade angle parameters on
ψOPT and
ηOPT, and the impact contains independent and interaction effects in different orders. The relation spectrum can be read by way of a checklist. “Position of items” corresponds to “Items” in
Table 4. The relationship between blade angle parameters and
ψOPT /
ηOPT can be expressed as:
According to the relation spectrum [
28], the values of curve camber
A and curve camber
B have the most significant influence on the total pressure coefficient
ψOPT.
ψOPT is positively correlated to curve camber
A but negatively correlated to curve camber
B. As for the efficiency
ηOPT, the most influential parameters are the blade angles in span 0 and span 1 (
β0,
β1). The
ηOPT is positively correlated with
β0 and negatively correlated with
β1.
3.4. Optimization Algorithm
As there are two independent objectives in the optimization process of an axial fan, no single solution is globally optimal. For this reason, a multi-objective evolutionary algorithm is required to return a set of promising solutions. This study uses the non dominated sorting genetic algorithms with elite strategy (NSGA-II) to solve the global optimization problem [
29]. We take the blade parameters (
β0,
β1,
θ0,
θ1,
A,
B) of the fan as optimization variables and the fan performance parameters (
ψOPT,
ηOPT) as the optimization objective. The two polynomials, trained to predict the fan performance parameters (
ψOPT,
ηOPT) in DD, are taken as the fitness function. The NSGA-II settings are shown in
Table 5.
The constraint range of the optimization variables should generally be set based on the inherent parameters of the prototype scheme. However, according to the relationship spectrum, the performance parameters are sensitive to some blade parameters. In order to speed up the optimization process, the constrained range of the most critical parameters is modified based on the relation spectrum. The constrained range of blade parameters that are beneficial for the
ψOPT and
ηOPT, such as curve camber
A and blade angles
β0, are higher than those of the prototype. In contrast, the unfavorable blade parameters, such as curve camber
B and blade angles
β1, are lower than those of the prototype. The constraint range of the blade parameters is provided in
Table 6.
We use the polynomial trained by DD to predict the fan performance parameters, which save computation time in the 3D numerical simulation. Thus, the population of the optimization algorithm can be appropriately increased to search the global optimization result. The population size is set at 500, and the algorithm evolution last for 200 generations. The convergence factor
is defined to determine the convergence of the Pareto front as:
where
is the
kth dimension objective of the
ith individual along the Pareto front in the
tth generation population. The convergence factor
α represents the average Euclidean distance between the leading-edge individual and the previous generation.
Figure 11 shows the change in
α with the evolution generation. As illustrated,
α decreases rapidly in the first 100 generations, indicating that the objectives gradually converge in the process of crossover, mutation, and migration. After generation 130, the evolution slows down, and the
α is reduced to 0.001.
Figure 12 shows the Pareto front shape of the 50th, 100th, 150th, and 200th generations after transformation. As the initial population is randomly generated, the distribution position of the population at the 50th generation is relatively disorderly and scattered. When the evolution generation reaches 100, the Pareto front shape initially becomes an approximate arc. Then, the population distribution of the 150th generation moves to the upper right, and the spacing between individuals decreases. Finally, the distance between individuals further reduces in the 200th generation. As the Pareto front shape barely changes compared with the 150th generation, it can be assumed that the optimization process has converged.
The efficiency of the Pareto front ranges from 52.5% to 80.35%, and the pressure coefficient ranges from 0.38 to 0.56. A specific weight factor is generally needed to balance the two objectives to select the final scheme. As there is no apparent bias between pressure and efficiency in this optimization, the equal weight factors are adopted for the normalized efficiency and pressure coefficient. A scheme in the middle of the Pareto front is selected as the mean optimal scheme (MOS). Meanwhile, the maximum pressure scheme (MPS) and the maximum efficiency scheme (MES) are employed to analyze the internal flow variation compared with the original scheme (ORI).