1 Introduction

Aero-engines, gas turbines, and other blade machines depend heavily on their turbine blades, which are made of Ni-based single crystal superalloy and are cast to survive the harsh service environment of high temperature and high pressure. In order to obtain improved performance in terms of combustion efficiency and thrust-weight ratio, the turbine inlet temperature (1700–2000 K) has far exceeded the melting point (about 1700 K) of the turbine blade [1]. The film cooling method is normally suggested to enhance the high-temperature creep resistance and maintain the normal state of the turbine blade [2]. A coating of air film is created on the surface of the turbine blade by conducting cold air through hundreds of tiny holes placed in the proper positions across the turbine blade. As a result, the blades are safeguarded by being separated from the hot gases.

The key to increasing the aero-engine’s operational efficiency is to correctly machine the film cooling holes of the turbine blade [3], where two accuracy issues need to be resolved: exact dimension and accurate location. The film cooling holes have a deep depth and a relatively narrow radius, often between 0.2 and 0.8 mm, which are typical high-aspect-ratio micro holes, and are therefore challenging to produce using traditional techniques [4]. With the development of EDM and ultrashort pulse laser processing technology, as well as the maturity of related experiments, mathematical modelling, and other process methods [5, 6], the machining error of the micro hole gradually satisfies the accurate dimensional requirements of roundness, taper, contour, and other parameters. However, due to the intricate design and thin-walled structure, the cooling rate of the blade surface varies according to the curvature statues, which influences the casting process of the turbine blade and causes uneven deformation [7]. In other words, the real surface of the blade is not the same as the theoretical surface, and the theoretical position of the film cooling hole is not necessarily on the real blade surface. The position deviation existed. Additionally, the creation of the air film and the direction of material removal depend greatly on the axial direction of the holes. As a result, during the actual machining process, the real surface normal should be considered [8]. The following outcomes may be attained if the film cooling holes are machined based on the theoretical position and axial direction: (1) The holes may not be perforated or the opposing wall may be damaged because of the discrepancy between the real depth and the theoretical depth; (2) the holes next to one other are crossed, which alters how cold air flows; (3) the interference with the holes may cause damage to the cavity ribs, compromising the structural integrity of the blade; (4) there is a chance that the machine tool and the blade will collide or interact, which could lead to production mishaps; and (5) The insulation and cooling effects will be impacted by the misalignment between the air film forming area and the blade surface.

To achieve this, researchers have optimised casting accuracy to the greatest extent possible by planning the parameters in advance with the law of the casting process on the effect of wall thickness and shrinkage of the turbine blade [9], or by analysing the deviation of the cast blade surface and compensating the casting moulds [10], or by performing finite element modelling of the casting process and predicting the casting results by numerical simulation [11]. Although these techniques have mostly succeeded in controlling the casting process and increasing the geometric accuracy of the blade surface, there are still some deviations, especially in the region of the turbine blade with a considerable curvature where tiny deviations are exponentially amplified.

Given the inevitable deviation, a methodical approach has been taken [12]: gathering the point cloud data of the real turbine blade using contact or non-contact measurements (sampling), utilising point cloud registration methods to determine the spatial relationship between the real blade and the theoretical blade (location), utilising reverse engineering to reconstruct the CAD model of the genuine blade (modelling), and finally, mapping the theoretical holes onto the real blade to calculate the real film cooling holes (distribution). Even though numerous high-precision measuring tools (such as CMMs, blue ray scanners) and algorithms are used to obtain accurate point clouds of the blade surface, and useful alignment techniques like principal component analysis (PCA), minimum potential energy (MPE), random sampling consistency (RANSAC), and iterative closest point (ICP) [13, 14] are used to achieve the precise position of the real blade, the transformation process is still rigid [15]. The local deviations still exist. As a result, the projected holes are unable to satisfy the real holes’ design specifications in terms of both position and axial direction.

According to the previous discussion, the inverse modelling method can be proposed as modifying the surface shape of the theoretical blade to resemble as closely as possible the deformed surface of the real blade, and modifying the theoretical film cooling holes to achieve the precise location [16]. Reconstructing the geometric model of the blade is crucial. Currently, the inverse modelling of the blade is guided by the non-uniform rational B-spline (NURBS) fitting approach. By sweeping the cross-section curves and fitting the point cloud data, the blade surface is recreated [17, 18]. To increase the credibility of the point cloud, point cloud filtering techniques like homogeneous filter, statistical outlier removal, and chord-height deviation points are used. These techniques consider the errors, noise, and point deviations at large curvature positions introduced by the machining and subsequent measurements of the real blade [19]. However, the suggested inverse modelling method cannot result in a surface shape mapping between the theoretical blade and the actual blade. Fortunately, free-form deformation (FFD) technology, one of the widely used CAD model deformation techniques, can modify the control points of the created surface, changing the original CAD model in the process [20]. For instance, the use of rigid registration and FFD technology can ensure a perfect match in the blade body and a uniform allowance in the leading and trailing edge area when the precision-forged blade has a small deviation in the blade body area, but a large deviation in the leading and trailing edge area, or even torsional deformation in the blade [21]. This method can successfully finish the recovery and reconstruction of the CAD model of the blade while dealing with the issue of repairing deformed and damaged blades [22, 23]. Although the position matching issue between the real model and the theoretical model of the surface is solved by this method, it is challenging to manage the shape and precision of the leading and trailing edge regions.

This paper proposes an adaptive location method for film cooling holes based on blade design intent. Blade design intent means that the design process of the blade considers the influence of the mean line curve, thickness, and other design parameters of the cross-section curve on the geometric shape. Adaptive location indicates that for each manufactured blade, a personalised distribution of the film cooling holes is recalculated to satisfy the accuracy criteria. Therefore, based on the design principle of the blade cross-section curve, the position and normal direction of the points on the theoretical blade surface are transformed to achieve the precise location of the film cooling holes by the translation and rotation matrix calculated in the free deformation process from the theoretical cross-section curve to the real cross-section curve. The rest of the paper is structured as follows: The important parameters and the CAD modelling technique for the turbine blade are briefly introduced in Sect. 2. Section 3 provides a flexible deformation approach of the blade surface to determine the correspondence between the theoretical blade and the actual blade. The related optimization procedure for the precise position of the holes is provided in Sect. 4. The position and machining of the holes in a turbine blade are discussed in Sect. 5, which serves as proof that the procedure described in this work is valid. In Sect. 6, the conclusions and prospect are covered.

2 Principles of parametric geometric modelling of blades

As a special free-form part, the main area of the blade consists of the leading edge, trailing edge, and convex and concave surfaces. After sweeping the 2D cross-section curves, the NURBS surfaces of the blade are constructed as shown in Fig. 1a. Obviously, the modelling accuracy can be improved by increasing the number of section curves. Therefore, creating precise section curves for the blade CAD model is the main challenge.

Fig. 1
figure 1

CAD models of a the turbine blade and b the cross-section curve

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The mean line is established first in the blade design process in accordance with the demands of aerodynamic performance. Then, the various thicknesses are superimposed over the mean line to create the matching inscribed circles, and the envelope curve is the cross-section curve, as shown in Fig. 1b. The correlation of different regions of the cross-section curve is established based on the mean line curve, which realises the dimension reduction of the structure of the cross-section curve. In particular, the convex and concave curves are symmetrical with respect to the mean line curve, and the points in these curves are:

$$\begin{array}{l}{p}_{{\text{cv}}}=s+r\bullet {n}_{{\text{cv}}}\\ {p}_{{\text{cc}}}=s+r\bullet {n}_{{\text{cc}}}\end{array}$$
(1)

where \(s\) is the discrete point of the mean line curve, \({p}_{{\text{cv}}}\) and \({p}_{{\text{cc}}}\) are the corresponding points of \(s\) in the convex and concave curves, respectively, the distance from these points to \(s\) is \(r\) referring to the radius of the inscribed circle, and \({n}_{{\text{cv}}}\) and \({n}_{{\text{cc}}}\) are the directions from these points to \(s\), which are symmetrical with respect to the tangent \(t\) of s in the mean line. The leading and trailing edge curves are shown as circular arcs at the endpoints of the mean line curve. Such section curves are designed in a forward manner.

Considering that the theoretical model and measuring data of the blade contour are presented in the process of machining the film cooling holes, and it is difficult to directly obtain the mean line curve and thickness distribution of the corresponding cross-section curve, an inverse implement of the design method of the cross-section curve is applied to obtain the mean line curve and thickness distribution of the measured blade. Specifically, the algorithm is as follows: Figuring out the inscribed circles at various points along the section curve, then fitting the centres of each inscribed circles. The fitted curve is the mean line curve, and the diameter of the inscribed circle is assumed to be the thickness of the section curve. As a result, the parametric geometric modelling of the blade for the manufacturing process is achieved.

Next, the positions of the film cooling holes in the turbine blade are introduced, as shown in Fig. 2. In general, the holes are mainly distributed in groups in the path from root to tip of the blade and are located in the leading edge, concave, and convex surfaces. The adaptive method suggested in this research will be used to handle these surfaces because they underwent significant deformation throughout the casting process. The trailing edge surface will not be discussed for the non-existence of the film cooling holes.

Fig. 2
figure 2

The distribution of the film cooling holes

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3 Flexible deformation method of the blade surfaces

Due to the limits of the casting process, deviations in the turbine blades are unavoidable. However, these deviations may only be described as Euclidean distances between contours, which cannot account for the deformation between the theoretical and real blade surfaces. The geometric connection between the points and vectors of the concave and convex curves will be impacted by the deviations applied to deform the section curves directly because the concave and convex curves are symmetric with respect to the mean line curve. The parametric geometric modelling principle of the blade gives the way of superimposing the thickness on the mean line curve and reduces the dimension of the surface structure of the turbine blade. Therefore, it is essential to assess the deformation based on the mean line curve and transform the theoretical cross-section curve in order to match the real cross-section curve. This technique simultaneously increases the computational effectiveness of the algorithm and lowers the computational complexity of the concave and convex curves deformation.

3.1 Free form deformation method

Flexible deformation of the mean line curve using the FFD method is suggested to analyse deformation brought on by casting variations. The mean line curve properly satisfies the FFD method because it is a unidirectional, unclosed free curve. In contrast to FFD, which modifies the NURBS curve’s control points, this article uses the real mean line curve as its target curve and transforms its discrete points to produce the new curve, as shown in Fig. 3. The distance between the corresponding points is defined as the translation, and the variation in the normal direction of the curve where the point is located is defined as the rotation, and the rotation angle is \(\theta\) in the transformation process. \({s}_{{\text{origin}}}\) and \({s}_{{\text{target}}}\) are the discrete points of the original and target curves, respectively, and \(m\) and \(t\) are the normal and tangent directions of the discrete point, respectively.

Fig. 3
figure 3

Deformation of the free-form curve

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The deformation matrix consisting of translations and rotations ensures that discrete points and vectors on the curve remain in the same relative geometric position to the curve as the curve is deformed. It should be noted that the cross-section curves and mean line curves treated in this paper are 2D shapes. Therefore, the translation matrix \({T}_{{\text{warp}}}\) and rotation matrix \({R}_{{\text{warp}}}\) using nonhomogeneous coordinates are represented as:

$$\begin{array}{l}{T}_{{\text{warp}}}={s}_{{\text{target}}}-{s}_{{\text{origin}}}\\ {R}_{{\text{warp}}}=\left[\begin{array}{cc}{\text{cos}}\theta & -{\text{sin}}\theta \\ {\text{sin}}\theta & {\text{cos}}\theta \end{array}\right]\end{array}$$
(2)

3.2 Deformation decomposition of the cross-section curve

With the FFD method for the mean line curve, the deformations implied by the manufacturing deviations in turbine blade can be decomposed. Prior to that, it is necessary to explain and define the discrete points that make up the section curves and mean line curve.

Considering that the concave and convex curves are symmetric with respect to the mean line curve, and the leading edge curve is related to the endpoint of the mean line curve, the mean line curve of the section curve is discretised according to the equal arc length method. The discrete points of the concave and convex curves are the points on those curves that are closest to the discrete points of the mean line curve (see Fig. 4a). The leading edge curve is discretized using the equal arc length approach based on the endpoint of the mean line curve, and the discrete points of the leading edge curve are generated (see Fig. 4b). Based on these explanations, the deformation process of the cross-section curve is illustrated using an arbitrary point on the theoretical mean line curve, where \({s}_{{\text{origin}}}\) is the point in the theoretical mean line curve, \({p}_{{\text{origin}}}\) is the corresponding point in the theoretical section curve (concave, convex or leading edge curves), and \({n}_{{\text{origin}}}\) is the arbitrary vector of \({p}_{{\text{origin}}}\).

Fig. 4
figure 4

The discrete points in a concave and convex curves and b leading edge curve

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Based on the definition of the discrete points, the novel method of the deformation process of the cross-section curve is as follows: Under the initial condition of the rigid alignment of the theoretical and real blades, firstly, the theoretical and real mean line curves will be transformed to the position with the smallest average error (twisting deformation), and then the theoretical mean line curve will be deformed to the real mean line curve implementing the FFD method (warping deformation), and based on these operations, the thickness variation at the corresponding positions of the two curves will be calculated (shrinking deformation), so as to achieve the overall deformation of the theoretical cross-section curves. The \({s}_{{\text{origin}}}\), \({p}_{{\text{origin}}}\), and \({n}_{{\text{origin}}}\) are deformed using the deformation order and the corresponding deformation matrix (see Fig. 5). In this case, the deformations of the discrete points of the concave and convex curves are calculated in the same process, and a similar calculation process is proposed for the leading edge curve.

Fig. 5
figure 5

Deformation process of the cross-section curves

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3.3 Twisting deformation process

After the initial rigid alignment between the theoretical and real blades, the twisting deformation describes the translation and rotation variations between the theoretical and real section curves when the relative deviations are minimal. Therefore, the 2D registration method, such as 2D ICP, is proposed to calculate the rotation matrix \({R}_{{\text{twist}}}\) and translation matrix \({T}_{{\text{twist}}}\) describing the twisting deformation. Additionally, the transformation of twisting deformation is rigid. Thus, the discrete points from the theoretical mean line, concave, convex, and leading edge curves have the same transformation matrices. The relative geometric values after twisting are represented as:

$$\begin{array}{l}{s}_{1}={R}_{{\text{twist}}}\bullet {s}_{{\text{origin}}}+{T}_{{\text{twist}}}\\ {p}_{1}={R}_{{\text{twist}}}\bullet {p}_{{\text{origin}}}+{T}_{{\text{twist}}}\\ {n}_{1}={R}_{{\text{twist}}}\bullet {n}_{{\text{origin}}}\end{array}$$
(3)

where \({s}_{1}\) is the discrete point of the mean line curve after 1 deformation, \({p}_{1}\) is the discrete point of the section curves after 1 deformation, and \({n}_{1}\) is the vector of \({p}_{1}\) after 1 deformation. The twisting transformation is shown in Fig. 5a and b.

3.4 Warping deformation process

Following the twisting transformation, the warping deformation describes how the theoretical and real section curves differ in shape. The twisted mean line and section curves are warped using the FFD method, which is devised in this paper to calculate the translation and rotation of each discrete points. The calculation process of the translation matrix \({T}_{{\text{warp}}}\) and the rotation matrix \({R}_{{\text{warp}}}\) is shown in Sect. 3.1. In the deformation process of discrete points of the concave, convex, and mean line curves, the rotations are calculated through the matching normal direction of the mean line curve. When computing the deformations of the discrete points of the leading edge curve and the end point of the mean line curve, the rotations are determined by the vectors created by the discrete points of the leading edge curve and the endpoint of the mean line curve. Figure 6 illustrates the warping deformation processes from the region to be deformed to the target region at the concave, convex, and leading edge following the twisting process, respectively.

Fig. 6
figure 6

The warping deformation process of the a concave, convex, and b leading edge regions

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Therefore, the relative geometric values after warping are represented as:

$$\begin{array}{l}{s}_{2}={s}_{1}+{T}_{{\text{warp}}}\\ {p}_{2}={R}_{{\text{warp}}}\bullet \left({p}_{1}-{s}_{1}\right)+{T}_{{\text{warp}}}+{s}_{1}\\ {n}_{2}={R}_{{\text{warp}}}\bullet {n}_{1}\end{array}$$
(4)

where \({s}_{2}\) is the discrete point of the mean line curve after 2 deformations, \({p}_{2}\) is the discrete point of the section curves after 2 deformations, and \({n}_{2}\) is the vector of \({p}_{2}\) after 2 deformations. The warping transformation is shown in Fig. 5b and c.

It should be emphasised that each discrete point on the mean line curve experiences a distinct deformation matrix during warping deformation, which must be calculated independently.

3.5 Shrinking deformation process

Since the warping deformation has corrected the deviation of the two mean line curves, the shape of the section curve after warping is similar to the real section curve. The only discrete points of the cross-section curves that still have a deviation are caused by the shrinking deformation. Figure 7 illustrates the concave, convex, and leading edge shrinking deformation processes from the to-be-deformed zone to the target region following the warping process. The rotation is no longer included, and the translation matrix \({T}_{{\text{shrink}}}\) is represented as

Fig. 7
figure 7

The shrinking deformation process of the a concave, convex, and b leading edge regions

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$${T}_{{\text{shrink}}}={p}_{{\text{target}}}-{p}_{2}$$
(5)

Therefore, the relative geometric values after shrinking are represented as

$$\begin{array}{l}{s}_{3}={s}_{2}\\ {p}_{3}={p}_{2}+{T}_{{\text{shrink}}}\\ {n}_{3}={n}_{2}\end{array}$$
(6)

where \({s}_{3}\) is the discrete point of the mean line curve after 3 deformations, \({p}_{3}\) is the discrete point of the section curves after 3 deformations, and \({n}_{3}\) is the vector of \({p}_{3}\) after 3 deformations. The warping transformation is shown in Fig. 5c–d.

Similarly, during shrinking deformation, each discrete point on the mean line curve has a specific deformation matrix that must be calculated separately.

3.6 Deformation of whole blade surfaces

Combining the above twisting, warping, and shrinking deformations, the discrete point \({p}_{{\text{origin}}}\) and vector \({n}_{{\text{origin}}}\) on the theoretical cross-section curve are finally transformed into the following form:

$$\begin{array}{c}{p}_{3}={R}_{{\text{warp}}}\bullet {R}_{{\text{twist}}}\bullet {p}_{{\text{origin}}}-{R}_{{\text{warp}}}\bullet {R}_{{\text{twist}}}\bullet {s}_{{\text{origin}}}+{R}_{{\text{twist}}}\bullet {s}_{{\text{origin}}}+{T}_{{\text{twist}}}+{T}_{{\text{warp}}}+{T}_{{\text{shrink}}}\\ {n}_{3}={R}_{{\text{warp}}}\bullet {R}_{{\text{twist}}}\bullet {n}_{{\text{origin}}}\end{array}$$
(7)

As a result, the translation matrix \(T\) and rotation matrix \(R\) of the flexible deformation process of the theoretical cross-section curve to the real cross-section curve are represented as:

$$\begin{array}{c}T=-{R}_{{\text{warp}}}\bullet {R}_{{\text{twist}}}\bullet {s}_{{\text{origin}}}+{R}_{{\text{twist}}}\bullet {s}_{{\text{origin}}}+{T}_{{\text{twist}}}+{T}_{{\text{warp}}}+{T}_{{\text{shrink}}}\\ R={R}_{{\text{warp}}}\bullet {R}_{{\text{twist}}}\end{array}$$
(8)

where the deformation process is shown in Fig. 5a–d.

Applying this method, the deformation field of the cross-section curve is created by calculating the flexible deformation matrix for each discrete point corresponding to the mean line curve on the concave, convex, and leading edge curves. Finally, the method is extended to all the cross-section curves to complete the spatial transformation of the discrete points of the theoretical blade surface, which constitutes the CAD model of the deformed blade surface and realises the flexible deformation of the whole blade surfaces.

4 Precision location and optimization of holes

4.1 Analysis of deformed holes

Based on the overall flexible deformation data of the blade, the position and axial direction of the theoretical film cooling holes are transformed to determine the position and axial direction of the holes on the real blade surface. The fitting method of the surface parameter domains is used to calculate the deformation of the theoretical holes because the theoretical positions of the film cooling holes are not exactly the same as the discrete points when the transformation matrix of the theoretical blade is calculated.

For blade CAD models, the concave, convex, and leading edge surfaces are generally composed of NURBS surfaces, which are represented with a parameter \(u\) within the section curves and a parameter \(v\) between the curves. Meanwhile, the deformation, including translation matrix \(T\) and rotation matrix \(R\), can be represented as three values \(x, y, {\text{and}} \theta\). Combining the above variables, three fitting functions \(x={f}_{1}\left(u,v\right)\), \(y={f}_{2}\left(u,v\right)\), and \(\theta ={f}_{3}\left(u,v\right)\) are constructed based on the parameters \(\left(u,v\right)\) of the discrete points of the deformed blade surfaces and the deformations \(\left(x,y,\theta \right)\).

After calculating the \(\left(u,v\right)\) parameters of the theoretical holes on the corresponding surfaces and solving these three fitting functions, the deformation of the corresponding holes can be determined, and the associated deformation matrices can be built. Since the points on the blade surface are 3D data and the deformations are 2D data, the translation matrix \({T}_{{\text{hole}}}\) and the rotation matrix \({R}_{hole}\) of the real holes are represented as:

$$\begin{array}{l}{T}_{{\text{hole}}}={\left[x,y,0\right]}^{T}\\ {R}_{{\text{hole}}}=\left[\begin{array}{ccc}{\text{cos}}\theta & -{\text{sin}}\theta & 0\\ {\text{sin}}\theta & {\text{cos}}\theta & 0\\ 0& 0& 1\end{array}\right]\end{array}$$
(9)

Therefore, the relative information of the deformed holes can be obtained as follows:

$$\begin{array}{l}{p}_{{\text{hole}},{\text{def}}}={R}_{{\text{hole}}}\bullet {p}_{{\text{hole}},{\text{origin}}}+{T}_{{\text{hole}}}\\ {n}_{{\text{hole}},{\text{def}}}={R}_{{\text{hole}}}\bullet {n}_{{\text{hole}},{\text{origin}}}\end{array}$$
(10)

where \({p}_{{\text{hole}},{\text{origin}}}\) and \({n}_{{\text{hole}},{\text{origin}}}\) are the position and axial direction of the film cooling holes in the theoretical blade, respectively, and \({p}_{{\text{hole}},{\text{def}}}\) and \({n}_{{\text{hole}},{\text{def}}}\) are the position and axial direction of the film cooling holes in the flexible deformed blade, respectively.

4.2 Optimization and adjustment of holes

There are still small deviations between the positions of the deformed holes and the real blade surfaces, and these deviations may be caused by the non-smoothness of the local surface. Therefore, it is necessary to optimise the position and axial direction of the flexible deformed holes based on the design principle of the film cooling holes.

When designing the film cooling holes, the path from the root to the tip of the blade is typically considered the basic direction, and the holes in this direction will be considered to be the basic group. The direction of the arc length parameter of the cross-section curve will guide the planning of various groups (see Fig. 2). The holes in the same groups have essentially the same sizes, axial directions, and places along the same sweeping curve (see Fig. 8a). However, the axial directions of the holes within the group may not be similar in the leading edge surface of the blade. Therefore, it is necessary to divide subgroups based on the similarity of theoretical axes. The hole axes within the subgroups are essentially consistent (see Fig. 8b). Additionally, the subgroup is considered the fundamental group in the optimisation process.

Fig. 8
figure 8

The similar axial directions a in the same group and b in different subgroups

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Therefore, an optimisation method is developed:

  • Step 1: A low-order virtual curve is fitted for the flexible deformed holes \(\left\{{p}_{{\text{i}}}\right\}\) in the group. The appropriate projection curve is created after projecting the virtual curve onto the real blade surface. The ultimate, exact locations of the real holes are the corresponding holes on the projection curve.

  • Step 2: The Quaternionic interpolation of the axial direction \(\left\{{n}_{{\text{i}}}\right\}\) of the flexible deformed holes in the direction of the arc length parameter of the projection curve is implemented based on the projected positions of the holes. It should be noticed that the axes in the subgroups are proposed to be optimised when existing the subgroups.

Following the aforementioned optimization, the precise position and axial direction of the film cooling holes are obtained, and the adaptive location of the film cooling holes is achieved.

5 Case and discussion

In order to verify the accuracy of the adaptive location method of the film cooling holes proposed in this paper, the following experiment processes are adopted. Firstly, the SHINING 3D OptimScan 5 M Plus-200 is applied to measure the casting turbine blade and create a point cloud of the blade surfaces. This measuring equipment is a high-precision 3D inspection scanner with narrow-band blue light source. The single-shot accuracy is 0.01 mm. The preliminary alignment and position are then achieved by using six points provided by the turbine blade design department as the matching points of ICP algorithm to transform the measured point cloud data with the theoretical blade. Based on this, the adaptive method developed in this paper is used to predict the position and axial direction of the real holes. Finally, the five-axis femtosecond laser processing equipment created by the Xi'an Institute of Optics and Precision Mechanics is then used to process the calculated holes. Figure 9 depicts the entire procedure along with the hardware and software.

Fig. 9
figure 9

Adaptive location and machining framework for film cooling holes

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The blade is approximately 35 × 25 × 8 mm in size, and because it is clamped to a fixture, there are more than 3 million points and more than 6 million triangular meshes of the blade and fixture after 3D scanning. The calculation time of the flexible deformation process of the blade is mainly related to the number of the blade sections, and the algorithms are programmed through the secondary development of Siemens NX 1899. The computation time for a single cross-section is less than 5 min on a PC with an Intel(R) Core(TM) i7-11800H CPU (RWI 4.8 GHz), a 16 GB RAM, an NVIDIA GeForce RTX 3060 Laptop GPU, and a 512 GB SSD. The sampling and fitting of the cross-section curves of the point cloud take about 4 min. The adaptive approach presented in this paper has a running time that is less than a minute. The whole computation time is under 22 min when the deformed blade is rebuilt using 4 cross-sections.

It should be noted that the original coordinate system of the scanned point cloud of the turbine blade is related to the spatial position and state of the 3D scanner; it is difficult to predict the coordinate system of the measured point cloud in advance. The six-point ICP algorithm is used to produce the initial transformation, which is an essential first step. However, when the spatial position of the measured point cloud and the theoretical blade differs significantly from each other and the application of the six-point ICP algorithm is unable to obtain an accurate location, other coarse alignment methods, such as principal component analysis, minimum potential energy, random sampling consensus, and other algorithms, can be considered first. Then, the six points provided by the design department are carried out for subsequent operations.

Different location results are examined in the experiment and contrasted, including (I) the three-plane location method based on fixture, (II) the ICP method of blade surfaces, (III) the ICP method of six points from the design department, and (IV) the adaptive location method in this paper, as shown in Fig. 10. The regions for location calculation are highlighted as yellow. Based on these location methods, the deviations of the real blade and the theoretical blade as well as the distribution of the film cooling holes are compared to verify the accuracy of the adaptive location method of the film cooling holes proposed in this paper.

Fig. 10
figure 10

Regions for location calculation of a I, b II, c III, and d IV methods

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5.1 Deviations of blade surfaces

The related deviation cloud maps can be created for the four location methods mentioned above as shown in Fig. 11. The first three cloud maps among them describe the deviation of the real blade calculated by the corresponding location methods from the theoretical blade, while the last cloud map describes the deviation of the real blade calculated by Method III from the theoretical blade calculated by Method IV. The deviation statistics are shown in Table 1.

Fig. 11
figure 11

Deviation cloud maps of a I, b II, c III, and d IV methods

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Table 1 Deviation statistics of cloud map (mm)
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According to the aforementioned deviation statistics, Method I yields the worst location outcomes in terms of maximum, minimum, average, and standard deviation. The explanation for this is because even if the fixture is perfectly aligned when the blade is manually clamped, the ultimate position of the blade may not be exact due to differing clamping strengths. At the same time, the lack of clamping accuracy amplifies the location mistakes of the blade surfaces. While Methods II and III both employ the ICP algorithm and select distinct model regions, their end results are similar. Method II outperforms Method III somewhat. The maximum and minimum deviations, as well as the standard deviation, of Methods II and III, have all decreased when compared to Method I. Because the distances between the real model located by Methods I–III and the theoretical model are quite close, the positive and negative deviations cancel each other out when computing the average value, leaving Methods I–III with almost the same average values. The Method IV suggested in this paper conducts overall flexible deformation on the blade surfaces and has a reduced average value and a more uniform distribution of deviations. In comparison to other approaches, the average value falls by almost 75%. The standard deviation falls by 41.9% in comparison to Method II, which has the strongest location effect. Therefore, the overall flexible deformation of surfaces in this paper is demonstrated to be effective.

5.2 Distribution of film cooling holes

The theoretical blade is suggested as the reference during the comparison procedure for blade surface deviation. However, the deformed turbine blade during the hole measuring process causes the datum of film cooling holes to shift. Based just on the theoretical position, it is challenging to evaluate the quality of the actual holes. Therefore, it is necessary to detect the geometric information of the holes based on the real location of the film cooling holes in the real turbine blade. It is suggested to use optical detection, CT, and other testing tools. However, there is currently no quantitative method to determine whether real holes still fulfil design specifications even after their reference is lost. As a result, the femtosecond laser is used to machine the real holes predicted by Method IV, and the position and radius are then determined. The real holes are compared with the holes predicted by Methods I–IV. The theoretical holes are projected onto the real blade surfaces to produce the predicted holes from Methods I to III. The deviations of the hole positions are shown in Fig. 12. The horizontal coordinate is the hole number and the vertical coordinate is the hole deviation. The deviation statistics are shown in Table 2. Only those in groups 1 (in the convex surface), 5 (in the leading edge surface), and 9 (in the concave surface) were chosen for comparison because of the sheer volume of film cooling holes that were present.

Fig. 12
figure 12

The position deviations of holes located by different methods

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Table 2 Deviation statistics of hole position (mm)
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The deviation data for film cooling holes shows that the holes computed using Method I have a larger deviation, and the tendency is consistent with the surfaces. The findings of Method IV were noticeably superior to those of Methods II and III for group 1 in the convex surface and group 5 in the leading edge surface. The deviations in group 9 of Methods II–IV were superior to one another because group 9 had a reduced surface deformation. The greatest deviation of the hole position for Method IV, however, fell by nearly 80% when viewed holistically, and the average and standard deviation also decreased by about 70% when compared to Methods I–III. In addition, the minimal deviations for the four approaches have a low absolute value. The effectiveness of Method IV established in this paper for the precise location for the film cooling holes of the turbine blade is demonstrated by all of these comparison results.

In addition to the location and optimization of the holes, the axial direction of the film cooling holes is optimised by Method IV in this paper. The angle between the axial direction of the holes and the normal direction of the blade surface, which was computed by several ways, was examined to demonstrate the correctness of the localization and optimization method. Considering that among Methods I–III, Methods II and III have the smallest deviation in the hole position, and the difference is not significant, Method III is chosen for comparison with Method IV. In Method III, the real holes are created by projecting the theoretical holes to the real surface. The hole axes are not changed throughout the locating process. Therefore, the real axial direction of the film cooling hole is the theoretical axial direction. Based on the aforementioned calculations, the angle between the axial direction of the theoretical hole and the normal direction of the theoretical blade and the angle between the axial direction of the real hole and the normal direction of the real blade calculated by Methods III and IV are obtained, as shown in Fig. 13. The horizontal coordinate is the hole number and the vertical coordinate is the angle between the axial direction and the normal direction.

Fig. 13
figure 13

The angles between axial and normal directions of the theoretical and real blades located by Methods III and IV

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Based on the angle between the axial direction and the normal direction of the theoretical blade, the angle deviations of the real blade located by the Methods III and IV are obtained, as shown in Fig. 14. The horizontal coordinate is the hole number and the vertical coordinate is the angle deviations. Since the positive change of the angle direction cannot be determined, the absolute values are utilised for all the angle deviations, as shown in Table 3.

Fig. 14
figure 14

The angle deviations caused by Methods III and IV

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Table 3 Deviation statistics of the angle between axial and normal directions (°)
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It can be shown that while the angle deviation under the various methods varies little in the concave and convex surfaces, it drastically changes in the leading edge surfaces with large curvatures under Methods III and IV. However, as can also be inferred from the data in Table 3, the overall change of the angle computed by Method IV is relatively steady. Additionally, the angle variation of Method IV is noticeably better than that of Method III at the tip or root regions, such as Points 1–1, 1–9, 5–1, 5–11, 9–1, and 9–9, which further demonstrates the efficacy of the current method in this paper for the precise axial location of the film cooling holes of the deformed turbine blade.

The following conclusions can be reached from the experimental data and statistical analysis:

  1. 1.

    Compared with the ICP method of the blade surfaces (i.e., Method II), the overall flexible deformation of the theoretical model using the method in this paper produced a more homogeneous distribution of deviations of the surfaces, with a decrease of about 75% in the mean value and 41.9% in the standard deviation.

  2. 2.

    When compared with the real film cooling holes after adopting the ICP method of the blade surfaces (i.e., Method II) or the ICP method of six points (i.e., Method III), the maximum deviation of the position of the film cooling holes after adopting the method in this paper has decreased by about 80%, and the mean value and standard deviation have also decreased by about 70%. Also, the angle between the axial direction of the film cooling holes and the normal direction of the blade surface is more consistent with the distribution of the angle on the theoretical blade.

6 Conclusions

An adaptive location method for the film cooling holes is provided based on the design intent of the blade in order to address the impact of the casting deviation of the turbine blade on the location datum of the holes. The free transformation matrix from the theoretical cross-section curve to the real cross-section curve is constructed to deform the blade surfaces flexibly, and the accurate location of the position and axial direction of the film cooling holes is satisfied. This is done by decomposing the casting deviations of the cross-section curves into the twisting, warping, and shrinking deformations based on the theoretical mean line curve. The main conclusions are the following:

  1. 1.

    The implementation of the overall flexible deformation method of the blade significantly improves location accuracy of the blade surface when compared to the rigid transformation method (ICP) of the overall blade surfaces, with an average error reduction of about 75% and a standard deviation reduction of 41.9%. The technique can be applied to the adaptive machining of various deformed blade surfaces in addition to the deformation and reconstruction of the casting turbine blade.

  2. 2.

    When compared to the position of the film cooling holes after using the rigidity transformation method (ICP) of the overall blade surfaces, the maximum deviation of the position of the film cooling holes by applying the method of this paper is reduced by about 80%, and the mean and standard deviation are also reduced by about 70%. The distribution of the angle between the axial direction and the normal direction is more consistent with the angle on the theoretical blade. So the improvement of the location accuracy of the film cooling holes is significant, and the final deviation meets the machining requirements.

Finally, there are still a few issues. In addition to changing the blade surface, the casting process also causes variations in the turbine blade’s internal cavity. These deviations influence how the film cooling holes inside the blade interact with one another and vary the depths of the holes, both of which require more study.