Early Quality Prediction of Complex Double-Walled Hollow Turbine Blades Based on Improved Whale Optimization Algorithm

Dong, Yiwei; Gong, Yuhan; Bo, Xu; Tan, Zhiyong

doi:10.1115/1.4066855

Graphical Abstract Figure

Abstract

The precision in forming complex double-walled hollow turbine blades significantly influences their cooling efficiency, making the selection of appropriate casting process parameters critical for achieving fine-casting blade formation. However, the high cost associated with real blade casting necessitates strategies to enhance product formation rates and mitigate cost losses stemming from the overshoot phenomenon. We propose a machine learning (ML) data-driven framework leveraging an enhanced whale optimization algorithm (WOA) to estimate product formation under diverse process conditions to address this challenge. Complex double-walled hollow turbine blades serve as a representative case within our proposed framework. We constructed a database using simulation data, employed feature engineering to identify crucial features and streamline inputs, and utilized a whale optimization algorithm-back-propagation neural network (WOA-BP) as the foundational ML model. To enhance WOA-BP’s performance, we introduce an optimization algorithm, the improved chaos whale optimization-back-propagation (ICWOA-BP), incorporating cubic chaotic mapping adaptation. Experimental evaluation of ICWOA-BP demonstrated an average mean absolute error of 0.001995 mm, reflecting a 36.21% reduction in prediction error compared to conventional models, as well as two well-known optimization algorithms (particle swarm optimization (PSO), quantum-based avian navigation optimizer algorithm (QANA)). Consequently, ICWOA-BP emerges as an effective tool for early prediction of dimensional quality in complex double-walled hollow turbine blades.

1 Introduction

The aero-engine is the most critical part of an airplane, and its performance largely depends on the turbine inlet temperature. In the case of limited material temperature resistance, proposing a new blade cooling structure represented by thin-walled, double-walled blades is of great significance for improving engine performance. Currently, research on investment casting of turbine blades has reached a relatively advanced stage, with investment casting being extensively applied in the high-precision molding of intricate structures. Nonetheless, accurately predicting the intricate casting of turbine blades presents a formidable challenge. This challenge arises not only from the process conditions during casting [1], but also from the complex internal structural deformation and external contour. Given the prohibitively high cost of casting double-walled turbine blades, there is a pressing demand for novel methodologies to forecast casting formation. In this work, the modeling of the accuracy prediction of complex structures inside a double-walled turbine blade is used as a case study. Figure 1 shows the complex cooling structure explored.

Fig. 1

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Double-walled turbine blade

Numerical simulation of the casting process with fluid-thermal-solid coupling is widely used to predict the thermal deformation of various materials [2,3]. Numerical simulation of the casting process is the basis for high-quality, short-cycle, and low-cost manufacturing of new-generation net-forming precision-cast turbine blades. However, due to the long computational cycles required for numerical simulation and the poor access to data sets, it is unsuitable for rapid, real-time evaluation. Therefore, more efficient prediction methods need to be constructed. In today’s era of Industry 4.0 and digitization of manufacturing [4], new communication technologies combined with state-of-the-art algorithms in the field of machine learning (ML) and deep learning (DL) offer a wide range of applications for data-driven smart manufacturing [5–7]. ML and DL methods are used to automate data-driven quality analytics by enabling them, and their application in this field is known as quality prediction [8,9].

Various types of sensors monitor the casting environment, and ML methods utilize the numerical simulation data of fluid-thermal-solid coupling for casting defect prediction [10–12], as well as prediction of product dimensions and quality [13–15].

To further reduce the cost of time for a large number of repetitive simulations, small samples of simulation results are used to train neural network models to quickly predict geometric deviations [16] and stress–strain of products under different parametric conditions [17]. Meanwhile, ML can be used in the additive manufacturing stage [18], to predict the quality of porosity [19,20], tensile strength [21], surface roughness [22], microstructural defects [23], and molding quality of the product [24–26].

At present, researchers globally try to use intelligent algorithms for regression prediction modeling of the cold-effect design of a turbine blade [27], and regression prediction modeling based on traditional neural networks provides a new solution to the above problems and has achieved specific research results [28,29]. However, researchers have not yet carried out relevant work using intelligent algorithms in the blade casting link, and the simple use of traditional neural networks still has certain deficiencies in the prediction accuracy and endpoint hit rate. To solve the above problems, our research group combined the improved whale swarm optimization algorithm to improve the traditional neural network. The whale optimization algorithm (WOA), as a new swarm intelligent optimization algorithm, still has some deficiencies but has the potential for development. Thus, it is crucial to improve this algorithm and expand its application areas [30,31].

In this paper, an ML-based method for predicting the early dimensional accuracy of the typical structure of a complex double-walled hollow turbine blade is proposed. The framework and key algorithms of the method are investigated, and the effectiveness of the method is demonstrated through a verification using a complex double-walled hollow turbine blade. Figure 2 illustrates the overall technical route of this paper. The main contributions of this paper are as follows:

A data-driven framework for predicting the early dimensional accuracy of typical structures of a complex double-walled hollow turbine blade is proposed;
The performance of different ML algorithms is investigated, and the results show that the ICWOA algorithm exhibited better prediction accuracy;
The potential relationship between the casting conditions with different process parameters and the deformation of the typical structure of a complex double-walled hollow turbine blade is revealed through feature engineering techniques.

Fig. 2

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Research for early mass prediction of complex double-walled turbine blades

2 Whale Optimization Algorithm

Whale optimization Algorithm simulates the humpback whale bubble-net attacking method, which is more straightforward to operate and converges faster than other optimization algorithms. The standard WOA mainly relies on the coefficient vector, $A$ ⁠, to select the path to search for the prey and uses the probability, $P$ ⁠, to determine the final predation mechanism. The computational flow of the standard WOA is shown in Fig. 3.

Fig. 3

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WOA algorithm flow

WOA includes two mechanisms, shrinking encircling and spiraling updating, to model the unique bubble-net hunting strategy of humpback whales; in addition, humpback whales also randomly search for prey. The following section briefly introduces the mathematical model of WOA.

2.1 Encircling Prey.

Humpback whales can identify the location of their prey and surround them. The location of the true optimal solution in the target space is not a priori. Moreover, WOA assumes that the optimal solution for the current population is the location of the prey, denoted by

X^{*}

⁠, and that after

X^{*}

has been determined, the other individuals move toward

X^{*}

in the following manner:

D = | C \cdot X^{*} - X (t) |

(1)

X (t + 1) = X^{*} - A \cdot D

(2)

where

t

is the number of generations of the current iteration,

A

and

C

are the coefficients,

X^{*} (t)

is the position vector of the current optimal solution (i.e., the location of the prey), and

X (t)

is the position vector of the current solution.

X^{*}

needs to be updated each time a better solution appears during each iteration, and

A

and

C

are calculated as follows:

A = 2 α \cdot r_{1} - α

(3)

C = 2 \cdot r_{2}

(4)

where

α

is a coefficient that decreases linearly with iterative generations from 2 to 0,

α = 2 - \frac{2 t}{T_{max}}

⁠, t is the current number of iterations,

T_{max}

is the maximum number of iterations, and

r_{1}

and

r_{2}

are random number between [0,1].

2.2 Bubble-Net Attacking Method.

WOA includes two movement mechanisms to simulate the bubble-net hunting strategy of the humpback whale population. The first is a shrinking circle encirclement, which linearly decreases the value of

α

in Eq. ; the another is a spiral update, which is mathematically modeled as follows:

X (t + 1) = D^{^{'}} \cdot e^{b k} \cdot \cos (2 π k) + X^{*} (t)

(5)

D^{^{'}} = | X^{*} (t) - X (t) |

(6)

where

D^{^{'}}

is the distance between the current search individual and the current optimal solution,

b

is the spiral shape parameter, and

k

is a [

- 1, 1

] uniformly distributed random number.

Since two types of predation behaviors exist in the process of approaching prey, WOA chooses either bubble-net predation or contraction encirclement, according to the probability

p

⁠. The position update formula is as follows:

X (t + 1) = {\begin{matrix} X^{*} (t) - A \cdot D & p \leq 0.5 \\ D^{^{'}} \cdot e^{b k} \cdot \cos (2 π k) + X^{*} (t) & p \geq 0.5 \end{matrix}

(7)

where

p

is the probability of the predation mechanism and the random number with the value range [0,1]. As the number of iterations t increases, the parameter

A

and the convergence factor

α

gradually decrease, and if

| A | < 1

⁠, each whale gradually surrounds the current optimal solution, which belongs to the local optimal-seeking stage in WOA.

2.3 Searching for Prey.

WOA simulates the behavior of humpback whale pods searching for prey locations in addition to their approach to prey, as expressed by the following equation:

D = | C \cdot X_{r a n d} - X (t) |

(8)

X (t + 1) = X_{r a n d} - A \cdot D

(9)

where

X_{r a n d}

is a random selection of individuals in the current population, and in this movement mode, humpback whales search randomly according to each other’s position. WOA uses the numerical magnitude of

| A |

to determine whether the population of whales is conducting a shrinking circle encirclement or searching for prey; when

| A | < 1

⁠, the population conducts a shrinking circle encirclement, and when

| A | \geq 1

⁠, the population conducts a search for prey.

3 Improvements to Whale Optimization Algorithm

Although WOA has good optimization-seeking capability, the basic WOA still has the disadvantages of low solution accuracy, slow convergence, and a tendency to fall into the local optimum. In the population intelligence optimization algorithm, randomly generated arrays are required; however, the results are often poor. Since chaotic sequences are nonlinear, ergodic, and unpredictable, random arrays can be replaced by chaotic mappings. To overcome these shortcomings, we improve the WOA in three aspects: population initialization, position update strategy, and prevention of falling into the local optimum. The improved algorithm is ICWOA, and the overall flowchart is shown in Fig. 4.

Fig. 4

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ICWOA algorithm flow

3.1 Cubic Map.

The standard Cubic mapping [32] function can be expressed as shown in Eq.

x_{n + 1} = b x_{n}^{3} - c x_{n}

(10)

Here

b

and

c

are chaotic influence factors, and the range of cubic mapping is different for different values of

b

and

c

⁠. Generally, when

c \in (2.3, 3)

⁠, the sequence generated by cubic mapping is chaotic. In addition,

x_{n} \in (- 2, 2)

when

b = 1

⁠, and

x_{n} \in (- 1, 1)

when

b = 4

⁠. The maximum Lyapunov exponent of 16 commonly used 1D chaotic mappings was calculated and analyzed in the literature [33], and the cubic mapping expression is shown in Eq.

x_{n + 1} = ρ x_{n} (1 - x_{n}^{2})

(11)

where

x_{n} \in (0, 1)

⁠;

ρ

is the control parameter, and the chaotic property of the cubic mapping has a strong relationship with the value of parameter

ρ

⁠. From Ref. [33], the chaotic property of cubic mapping is known to be similar to the maximum Lyapunov exponent of the logistic mapping and tent mapping, and it is better than 1D mappings such as sine, circle, Singer, and Kent mapping. An improved cubic mapping with a maximum Lyapunov exponent of 1.0984 was proposed in the literature [34], which is better than the Lyapunov exponent of 0.6930 for the logistic mapping.

In this paper, the distribution curves of the values taken after $10^{3}$ iterations of the cubic chaos mapping are given. The initial value of $x_{n} \in (0, 1)$ ⁠, and the cubic mapping has good chaos traversal when $x_{0} = 0.3$ and $ρ = 2.59$ ⁠. The simulation results are shown in Fig. 5.

Fig. 5

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Cubic map

3.2 Improved Nonlinear Convergence Factor.

Since the convergence factor for linear variation does not well regulate the global search ability and local exploitation ability, a nonlinear convergence factor [35–37] is proposed in this paper as follows:

a = a_{f i r s t} - (a_{f i r s t} - a_{f i n a l}) \sin (μ \frac{t}{max_i t e r} π + φ)

(12)

where

a_{f i r s t}

and

a_{f i n a l}

are the initial and final values of the control parameters, respectively;

max_i t e r

is the maximum number of iterations;

t

is the current number of iterations;

μ

and

φ

are the regulation parameters.

3.3 Adaptive Weighting Strategy and Stochastic Difference Method Variation Strategy.

WOA is prone to falling into the local optimum and early convergence in late local development. To enable the algorithm to maintain the population diversity and to jump out of local optimum in time, we propose an adaptive weight strategy [38,39] and random difference variation strategy [40,41].

The mathematical expressions of the adaptive weighting strategy are as follows:

ω = 1 - \frac{e^{\frac{t}{max_i t e r}} - 1}{e - 1}, X (t + 1) = ω \cdot X^{*} (t) - A \cdot D

(13)

X (t + 1) = ω \cdot X^{*} (t) + D \cdot e^{b k} \cdot \cos (2 π k)

(14)

The stochastic difference variance strategy is as follows:

X (t + 1) = r_{1} \cdot [X^{*} (t) - X (t)] + r_{2} \cdot [X^{^{'}} (t) - X (t)]

(15)

where

r_{1}

and

r_{2}

are both random numbers of [0,1];

X^{^{'}} (t)

is a randomly selected individual in the population.

Each individual must go through the encircling predation, spiral update, and prey search stages, and the adaptive weighting strategy is used to update the position when the individual performs encircling predation or spiral update. Afterward, the individual needs to update it again by the random difference variation strategy to take the optimal position before and after its change, which speeds up the convergence of the population and effectively prevents the population from falling into the local optimum. The population works in concert by these two strategies, giving the algorithm a better optimality-finding effect.

4 Data Processing of Critical Parts of a Double-Walled Blade

For data-driven prediction algorithms, the quality of the dataset is critical. In this study, the deformation of structures such as impact holes with different aperture sizes, rotary section, trailing-edge cutback, and pin fin were the key structures affecting the cooling efficiency of the blade [42–44]. From the perspective of double-wall blade casting, among the effects of different process parameters on the deformation of single-crystal nickel-based high-temperature alloys during the solidification stage, the pouring temperature [45], preheating temperature, and drawing speed had a great influence on the deformation of casting complex structures of the blade. Therefore, we processed the data for the numerical simulation casting deformation of these structures under different process parameters and used them as the initial input features for the prediction model.

4.1 Impact-Hole Error Statistics.

Since the procast software exports the model as an stereolithography (STL) point cloud file, the point cloud was extracted locally for the impact-hole area, a circle was fitted to the 3D discrete points, with known coordinates of the three-dimensional spatial discrete points (⁠ $x_{i}, y_{i}, z_{i}$ ⁠), and a spatial circle was constructed so that the spatial points were as close as possible to the fitted spatial circle.

First, all the discrete points were on a plane as much as possible, and the plane equation can be expressed as

a x + b y + c z - 1 = 0

(16)

The above can be written in matrix form as

M A = L_{1}

(17)

where

M = {[\begin{matrix} x_{1} & x_{2} & \dots & x_{n} \\ y_{1} & y_{2} & \dots & y_{n} \\ z_{1} & z_{2} & \dots & z_{n} \end{matrix}]}^{T}, A = {(a, b, c)}^{T}, L_{1} = {(1, 1, 1)}^{T}

Calculation of the error can be done using the Euclidean paradigm: $‖ M A - L_{1} ‖ = \sqrt{\sum_{i = 1}^{n} {(M A - L_{1})}^{2}}$ ⁠.

This is a hyper-deterministic equation to solve for $A = {(M^{T} M)}^{- 1} M^{T} L_{1}$ ⁠, which is the normal vector of the plane, according to the method of least squares.

Assuming that all discrete points are on the circle, the middle of the line connecting any two points had to pass through the center of the circle. Let the center of the circle be $C$ (⁠ $x_{0}$ ⁠, $y_{0}$ ⁠, $z_{0}$ ⁠). We took two points, $P_{1}$ (⁠ $x_{1}$ ⁠, $y_{1}$ ⁠, $z_{1}$ ⁠) and $P_{2}$ (⁠ $x_{2}$ ⁠, $y_{2}$ ⁠, $z_{2}$ ⁠). Then, the ${vector}_{1}$ of the line connecting $P_{1}$ and $P_{2}$ is expressed as (⁠ $x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1}$ ⁠), and the coordinates $P_{12}$ of the midpoint of the line connecting $P_{1}$ and $P_{2}$ are $(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}, \frac{z_{1} + z_{2}}{2})$ ⁠.

The center of the circle

C

was connected to

P_{12}

with

{vector}_{2}

as

(\frac{x_{1} + x_{2}}{2} - x_{0}, \frac{y_{1} + y_{2}}{2} - y_{0}, \frac{z_{1} + z_{2}}{2} - z_{0})

⁠. For

P_{1}

and

P_{2}

to be on the circle,

{vector}_{1} * {vector}_{2} = 0

had to be satisfied, i.e.,

\begin{aligned} (x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1}) \\ \cdot (\frac{x_{1} + x_{2}}{2} - x_{0}, \frac{y_{1} + y_{2}}{2} - y_{0}, \frac{z_{1} + z_{2}}{2} - z_{0}) \end{aligned}

(18)

Thus, it was possible to obtain

Δ x_{12} \cdot x_{0} + Δ y_{12} \cdot y_{0} + Δ z_{12} \cdot z_{0} - l_{1} = 0

(19)

where

Δ x_{12} = x_{2} - x_{1}

⁠,

Δ y_{12} = y_{2} - y_{1}

⁠,

Δ z_{12} = z_{2} - z_{1},

l_{1} = \frac{x_{2}^{2} + y_{2}^{2} + z_{2}^{2} - x_{1}^{2} - y_{1}^{2} - z_{1}^{2}}{2}

⁠.

All points were on the circle; then, we had

\begin{aligned} B & = [\begin{matrix} Δ x_{12} & Δ y_{12} & Δ z_{12} \\ Δ x_{13} & Δ y_{13} & Δ z_{13} \\ \dots & \dots & \dots \\ Δ x_{1 n} & Δ y_{1 n} & Δ z_{1 n} \\ Δ x_{23} & Δ y_{23} & Δ z_{23} \\ Δ x_{24} & Δ y_{24} & Δ z_{24} \\ \dots & \dots & \dots \\ Δ x_{2 n} & Δ y_{2 n} & Δ z_{2 n} \\ \dots & \dots & \dots \\ Δ x_{(n - 1) n} & Δ y_{(n - 1) n} & Δ z_{(n - 1) n} \end{matrix}] \\ C & = [\begin{matrix} x_{0} \\ y_{0} \\ z_{0} \end{matrix}], L_{2} = [\begin{matrix} l_{1} \\ l_{2} \\ \dots \\ l_{(n - 1) n / 2} \end{matrix}] \Rightarrow BC = L_{2} \end{aligned}

(20)

where the above equation is also an overdetermined equation. Calculation of the error can be done using the Euclidean paradigm:

‖ B C - L_{2} ‖ = \sqrt{\sum_{i = 1}^{n} {(B C - L_{2})}^{2}}

⁠. Since the center of the circle

C

had to be in the plane of the aforementioned control, it satisfied

a x_{0} + b y_{0} + c z_{0} - 1 = 0

⁠, i.e.,

A^{T} C = 1

(21)

where

A

is the normal vector of the plane, which was found by

A = {(M^{T} M)}^{- 1} M^{T} L_{1}

⁠. Therefore, it could be solved by constructing an optimization problem under the constraints of Eq.

f (C) = {‖ BC - L_{2} ‖}^{2} + λ (AC - 1)

(22)

where

λ

is the Lagrangian multiplier. When we took the derivative concerning

C

with respect to

λ

and made the derivative value 0, the transformation yielded

\begin{aligned} [\begin{matrix} B^{T} B & A \\ A^{T} & 0 \end{matrix}] \cdot [\begin{matrix} C \\ λ \end{matrix}] = [\begin{matrix} B^{T} L_{2} \\ 1 \end{matrix}] \\ \Rightarrow [\begin{matrix} C \\ λ \end{matrix}] = {[\begin{matrix} B^{T} B & A \\ A^{T} & 0 \end{matrix}]}^{- 1} [\begin{matrix} B^{T} L_{2} \\ 1 \end{matrix}] \end{aligned}

(23)

The radius of a circle could be determined from the average of the distances from all points to the center of the circle

\bar{r} = \frac{1}{N} \sum_{i = 1}^{N} \sqrt{{(x_{i} - x_{0})}^{2} + {(y_{i} - y_{0})}^{2} + {(z_{i} - z_{0})}^{2}}

(24)

Therefore, using the STL point cloud data, we constructed a spatial circle so that the spatial points were as close as possible to the fitted spatial circle, and according to the least squares method, we could derive the normal vector of the plane. According to the point coordinates, to find the center of the spatial circle and, at the same time, get the fitted radius, we coded and solved the derivation process of the equation of the fitted circle curve using matlab, as shown in Fig. 6, to obtain the simulation mode with l 1.0-mm, 1.2-mm, 1.3-mm, and 1.5-mm impact-hole circle fitting.

Fig. 6

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Impact-hole circle fitting: $(a)$ impact hole (1.0 mm), $(b)$ impact hole (1.2 mm), $(c)$ impact hole (1.3 mm), and $(d)$ impact hole (1.5 mm)

Figure 7 shows the error probability density function (PDF) of the variation amount of four different orifice impact holes. The dynamic leaf impact-hole apertures were 1.0 mm, 1.2 mm, 1.3 mm, and 1.5 mm, and the variation of 1.0-mm impact-hole aperture was 0.10–0.15 mm; the 1.2-mm impact-hole aperture variation was 0.06–0.20 mm; the 1.3-mm impact-hole aperture variation was 0.11–0.21 mm, and the 1.5-mm impact-hole aperture variation was 0.16–0.30 mm.

Fig. 7

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Four types of impact-hole aperture variation

Although the statistical mean and standard deviation of the overall geometric deviation parameters on the impact holes of different bore diameters differed, their probability density functions were close to the Gaussian distribution, and the statistical results are shown in Fig. 8. The mean value of the 1.0-mm impact holes was 0.12142, with a standard deviation of 0.00919. The mean value of the 1.2-mm impact holes was 0.13033, with a standard deviation of 0.02939. The mean value of the 1.3-mm impact holes was 0.16109, with a standard deviation of 0.02097. Further, the mean value of 1.5-mm impact holes was 0.21790, with a standard deviation of 0.02985. The statistical mean value of the variation of different impact-hole diameters gradually increased, and the standard deviation exhibited a decreasing trend when the hole diameter was 1.3 mm, and it started to increase again when the hole diameter was 1.5 mm. The variation of the impact-hole diameter directly affected the cooling efficiency in its working condition.

Fig. 8

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Statistical results for different impact holes

4.2 Rotary Section Error Statistics.

Figure 9 shows the error PDF of the radius of curvature of the rotary section. The original design model, chamfer, of the front chamber rotary section of the moving leaf was 0.4–0.8 mm. Because the model shrinks irregularly after the simulation casting, it does not form a complete chamfer angle, so the radius of curvature was used here instead of the chamfer to indicate the change law. Moreover, the radius of curvature of the chamfer of the front chamber rotary section increased after the shrinkage deformation, the deformation of the front chamber rotary section area was larger, and the radius of curvature was mainly concentrated in 8–9.5 mm. The original design model chamfer of the dynamic lobe middle chamber rotary section was 0.4 mm, the radius of curvature of the chamfer of the front chamber rotary section increased after shrinkage deformation, and the radius of curvature was concentrated in 8.5–10 mm.

Fig. 9

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Curvature of rotary section

The statistical mean and standard deviation of the geometric deviation parameters of the front chamber rotary section and the middle chamber rotary section were different, and the statistical results are shown in Fig. 10. The mean value of the anterior chamber was 8.99706, with a standard deviation of 0.69427; the mean value of the middle chamber was 9.29326, with a standard deviation of 0.62595. The mean value of the middle chamber rotary section was higher than that of the front chamber rotary section. Still, the standard deviation of the middle chamber rotary section was lower than that of the front chamber rotary section, revealing that the curvature variation of the middle chamber rotary section was more concentrated.

Fig. 10

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Statistical results of the front and middle chamber rotary section

4.3 Trailing-Edge Cutback and Pin Fin Error Statistics.

Figure 11 shows the error PDF of the trailing-edge cutback and pin fin. Here, the maximum spacing of the trailing-edge cutback position was taken to indicate its variation law. The spacing size of the trailing-edge cutback in the original design model was 0.4 mm, and the spacing increased at the trailing-edge cutback after shrinkage deformation, mainly concentrated in 0.1–0.18 mm. A Gaussian distribution for the trailing-edge cutback was fitted to the distribution, where the mean was 0.14839, and the standard deviation was 0.0302. The diameter of the original design model of the pin fin was 1.2 mm, and here, the same circle fitting as the impact hole was used to fit the cylindrical cross section of the pin fin to obtain the diameter of the pin fin after casting deformation. The specific method is introduced in Sec. 4.1, so we will not repeat it here. The radius of the pin fin after shrinkage deformation changed in the range of 0.02–0.20 mm, which was mainly concentrated in 0.04–0.16 mm. A Gaussian distribution for pin fin was fitted to the distribution, where the mean was 0.09754, and the standard deviation was 0.03934.

Fig. 11

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Curvature of trailing-edge cutback and pin fin

After processing the initial feature input data, and considering the volatility of the investment casting process, as well as the fact that the deformation deviation follows the characteristics of a normal distribution, the initial data are normalized to the database using the Zscore method, as shown below:

z = \frac{x - μ}{σ}

(25)

where

x

is the original data,

μ

is the mean of

x

⁠, and

σ

is the standard deviation. In this way, the bias index was scaled to zero mean and unit variance.

5 Comparison of Prediction Model Results

The dataset with normalization done in Sec. 4 is used as input to train the early neural network model. To illustrate the performance of the ICWOA algorithm, in addition to comparing the ICWOA algorithm with the WOA algorithm, the ICWOA algorithm is also compared with the PSO algorithm. To better compare the differences between the different algorithms, each algorithm has the same setup conditions, where the population size and the maximum number of iterations are set to 30 and 1000, respectively. Figure 12 depicts the critical structure prediction modeling framework.

Fig. 12

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Modeling process of critical structure prediction modeling with ICWOA

Convergence performance is one of the important indicators of optimization algorithms. To evaluate the ICWOA algorithm, we show the convergence curves of different thickness prediction models in Fig. 13. It can be seen that the ICWOA-BP model converged faster than the WOA-BP model, and at the same time, the prediction error was better. This indicates that the WOA algorithm may fall into local optimal solutions. In addition, the fast iteration speed was also proved to be another advantage of ICWOA. From these results, it can be concluded that ICWOA outperforms WOA in global optimization for the same amount of data.

Fig. 13

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Convergence curves of the ICWOA and WOA in different test sets

A detailed comparison between ICWOA-BP, WOA-BP, and PSO-BP predicted deformations and simulation results is shown in Fig. 14, and it is obvious that the WOA-BP prediction accuracy was significantly improved after the improvement. From Figs. 14(b)–14(d), WOA-BP as well as PSO-BP were caught in the phenomenon of the local optimal solution, and large deviations appeared locally. In contrast, the optimized ICWOA-BP did not have a large deviation point, and the maximal difference between the actual value and the predicted value was stabilized at $\pm 0.005 mm$ under the data of the deformation of four different impact holes, except that the improved ICWOA-BP performed slightly better than PSO-BP. In addition, the fast iteration speed was another advantage of ICWOA. In the validation set, the average MAE of the optimized ICWOA-BP model was $1.955 \times 10^{- 3}$ ⁠, a 36.21% reduction in the error compared with the original model WOA-BP. The result indicates that the prediction accuracy of the ICWOA model is satisfactory.

Fig. 14

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Predicted deformation bias of ICWOA, WOA, and PSO in different ML models: $(a)$ impact hole (1.0 mm), $(b)$ impact hole (1.2 mm), $(c)$ impact hole (1.3 mm), $(d)$ impact hole (1.5 mm), $(e)$ front chamber rotary section, $(f)$ middle chamber rotary section, $(g)$ trailing edge cutback, and $(h)$ pin fin

To further demonstrate the advantages and usefulness of the algorithms, the ICWOA is compared with the quantum-based avian navigation optimizer algorithm (QANA) [46]. Inspired by the navigation mechanism of migratory birds, Zamani et al. proposed a robust and scalable differential evolutionary algorithm. QANA is very competitive state-of-the-art the swarm intelligence (SI) and the differential evolution (DE) algorithms and was awarded the best theory paper in Engineering Applications of Artificial Intelligence from 2020 to 2022. It is also proven to be accurately applicable to solving real-world engineering optimization problems. The QANA algorithm is due to the WOA algorithm in terms of accuracy and convergence speed [47], so it is compared with the ICWOA algorithm to verify the optimization effect. A detailed comparison between ICWOA-BP and QANA-BP predicted deformations and simulation results is shown in Fig. 15. In some feature dimensions, such as Figs. 15(a), 15(c), and 15(d), the QANA algorithm and the ICWOA algorithm do not differ much in their predictions. In other feature dimensions, the ICWOA algorithm is more stable and has a higher prediction accuracy, and the QANA algorithm appears to have a large deviation.

Fig. 15

View large Download slide

ICWOA and QANA comparison: $(a)$ impact hole (1.0 mm), $(b)$ impact hole (1.2 mm), $(c)$ impact hole (1.3 mm), $(d)$ impact hole (1.5 mm), $(e)$ front chamber rotary section, $(f)$ middle chamber rotary section, $(g)$ trailing edge cutback, and $(h)$ pin fin

In addition, the improved ICWOA model and the comparison of MAE results of different models are shown in Table 1. It can be seen that there are some effects of prediction bias in the data with different features, but the prediction accuracy of the ICWOA-BP model is the highest among all the models, which is improved by about 22.05% on average compared with PSO-BP, and the performance of ICWOA-BP is better than PSO-BP. Meanwhile, the average MAE improved by 14.29% compared to QANA-BP, and ICWOA-BP was superior to QANA-BP.

Table 1

Performance evaluation of the ML models

	MAE in testing phase
Key parts of the blade	BP	PSO-BP	WOA-BP	ICWOA-BP	QANA-BP
Impact hole (1.0 mm)	5.696 × 10⁻³	1.217 × 10⁻³	1.976 × 10⁻³	1.124 × 10⁻³	1.249 × 10⁻³
Impact hole (1.2 mm)	6.162 × 10⁻³	2.954 × 10⁻³	3.078 × 10⁻³	1.791 × 10⁻³	2.083 × 10⁻³
Impact hole (1.3 mm)	6.420 × 10⁻³	2.479 × 10⁻³	2.726 × 10⁻³	1.406 × 10⁻³	1.616 × 10⁻³
Impact hole (1.5 mm)	6.507 × 10⁻³	2.511 × 10⁻³	3.229 × 10⁻³	2.353 × 10⁻³	2.801 × 10⁻³
Front chamber rotary section	6.485 × 10⁻³	2.177 × 10⁻³	3.856 × 10⁻³	1.940 × 10⁻³	2.180 × 10⁻³
Middle chamber rotary section	7.462 × 10⁻³	3.658 × 10⁻³	3.214 × 10⁻³	2.567 × 10⁻³	3.209 × 10⁻³
Trailing-edge cutback	6.629 × 10⁻³	2.211 × 10⁻³	3.056 × 10⁻³	2.131 × 10⁻³	2.449 × 10⁻³
Pin fin	9.789 × 10⁻³	2.870 × 10⁻³	3.387 × 10⁻³	2.331 × 10⁻³	2.664 × 10⁻³
Average	6.894 × 10⁻³	2.510 × 10⁻³	3.065 × 10⁻³	1.955 × 10⁻³	2.281 × 10⁻³

	MAE in testing phase
Key parts of the blade	BP	PSO-BP	WOA-BP	ICWOA-BP	QANA-BP
Impact hole (1.0 mm)	5.696 × 10⁻³	1.217 × 10⁻³	1.976 × 10⁻³	1.124 × 10⁻³	1.249 × 10⁻³
Impact hole (1.2 mm)	6.162 × 10⁻³	2.954 × 10⁻³	3.078 × 10⁻³	1.791 × 10⁻³	2.083 × 10⁻³
Impact hole (1.3 mm)	6.420 × 10⁻³	2.479 × 10⁻³	2.726 × 10⁻³	1.406 × 10⁻³	1.616 × 10⁻³
Impact hole (1.5 mm)	6.507 × 10⁻³	2.511 × 10⁻³	3.229 × 10⁻³	2.353 × 10⁻³	2.801 × 10⁻³
Front chamber rotary section	6.485 × 10⁻³	2.177 × 10⁻³	3.856 × 10⁻³	1.940 × 10⁻³	2.180 × 10⁻³
Middle chamber rotary section	7.462 × 10⁻³	3.658 × 10⁻³	3.214 × 10⁻³	2.567 × 10⁻³	3.209 × 10⁻³
Trailing-edge cutback	6.629 × 10⁻³	2.211 × 10⁻³	3.056 × 10⁻³	2.131 × 10⁻³	2.449 × 10⁻³
Pin fin	9.789 × 10⁻³	2.870 × 10⁻³	3.387 × 10⁻³	2.331 × 10⁻³	2.664 × 10⁻³
Average	6.894 × 10⁻³	2.510 × 10⁻³	3.065 × 10⁻³	1.955 × 10⁻³	2.281 × 10⁻³

6 Conclusion

In this paper, a data-driven framework based on ML techniques was developed for estimating and screening early products. A complex double-walled hollow turbine blade was analyzed as a typical case of the proposed framework. Owing to the extremely high cost of real blade casting, this paper adopts WOA-BP neural network as the basic model of the machine learning algorithm, and proposes a cubic chaos mapping adaptive optimization algorithm ICWOA-BP based on the compilation of the database using the simulation data, and adopts the characteristic structure of the complex double-wall hollow turbine blade as the initial input, and simplifies the data processing. The main findings of this paper are as follows:

A data-driven framework for early dimensional accuracy prediction of complex double-walled hollow turbine blades is proposed, and the effectiveness of the method is illustrated with an example of a critical part of a complex double-walled hollow turbine blade;
We found that six feature dimensions provided the best performance. The selection of features in different critical areas had a significant impact on the performance of the predictive model;
The optimized WOA method was used for the hyperparameter optimization of BP, and the prediction results in the small dataset could still maintain good prediction accuracy, showing good improvement. In the validation set, the average MAE of the optimized ICWOA-BP model was $1.955 \times 10^{- 3}$ ⁠, reducing the error by 36.21% compared with the original model. The prediction errors of the improved model on the test set were affected by different feature data, the average improvement was about 22.09% compared with PSO-BP, and the performance of ICWOA-BP was better than that of PSO-BP. Meanwhile, the average MAE improved by 14.29% compared to QANA-BP, and ICWOA-BP was superior to QANA-BP.

Acknowledgment

This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 52475491 and 51705440), in part by the Natural Science Foundation of Fujian Province, China (Grant No. 2019J01044), in part by the Aeronautical Science Foundation of China (Grant Nos. 20170368001 and 20230003068002), in part by the Foundation of State Key Laboratory of Intelligent Manufacturing Equipment and Technology (Grant No. IMETKF2024013), in part by the CAS-Fujian STS Project (Grant No. 2022T3071), and in part by the National Science and Technology Major Project (Grant Nos. J2019-II-0022-0043 and J2019-VII-0013-0153).

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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Early Quality Prediction of Complex Double-Walled Hollow Turbine Blades Based on Improved Whale Optimization Algorithm

Abstract

1 Introduction

2 Whale Optimization Algorithm

2.1 Encircling Prey.

2.2 Bubble-Net Attacking Method.

2.3 Searching for Prey.

3 Improvements to Whale Optimization Algorithm

3.1 Cubic Map.

3.2 Improved Nonlinear Convergence Factor.

3.3 Adaptive Weighting Strategy and Stochastic Difference Method Variation Strategy.

4 Data Processing of Critical Parts of a Double-Walled Blade

4.1 Impact-Hole Error Statistics.

4.2 Rotary Section Error Statistics.

4.3 Trailing-Edge Cutback and Pin Fin Error Statistics.

5 Comparison of Prediction Model Results

6 Conclusion

Acknowledgment

Declaration of Competing Interest

Data Availability Statement

References

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Early Quality Prediction of Complex Double-Walled Hollow Turbine Blades Based on Improved Whale Optimization Algorithm

Abstract

1 Introduction

2 Whale Optimization Algorithm

2.1 Encircling Prey.

2.2 Bubble-Net Attacking Method.

2.3 Searching for Prey.

3 Improvements to Whale Optimization Algorithm

3.1 Cubic Map.

3.2 Improved Nonlinear Convergence Factor.

3.3 Adaptive Weighting Strategy and Stochastic Difference Method Variation Strategy.

4 Data Processing of Critical Parts of a Double-Walled Blade

4.1 Impact-Hole Error Statistics.

4.2 Rotary Section Error Statistics.

4.3 Trailing-Edge Cutback and Pin Fin Error Statistics.

5 Comparison of Prediction Model Results

6 Conclusion

Acknowledgment

Declaration of Competing Interest

Data Availability Statement

References

Get Email Alerts

Cited By

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