Adaptive Location Method for Film Cooling Holes Based on The Design Intent of The Turbine Blade

Adaptive Location Method for Film Cooling Holes Based on The Design Intent of The Turbine Blade | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Adaptive Location Method for Film Cooling Holes Based on The Design Intent of The Turbine Blade Yaohua Hou, Jing Wang, Jiawei Mei, Hualong Zhao This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3219667/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 21 Mar, 2024 Read the published version in The International Journal of Advanced Manufacturing Technology → Version 1 posted 5 You are reading this latest preprint version Abstract Due to the inevitable deviation of the casting process, the dimensional error of the turbine blade is introduced. As a result, the location datum of the film cooling holes is changed, which has an impact on the machining accuracy. The majority of pertinent studies concentrate on the rigid location approach for the entire blade, which results in a modest relative position error of the blade surface but still fails to give the exact position and axial direction of the film cooling holes of the deformed blade. In this paper, the entire deformation of the blade cross-section curve is divided into a number of deformation combinations of the mean line curve based on the construction method of the blade design intent. The exact location of the film cooling holes in the turbine blade with deviation is therefore efficiently solved by a flexible deformation of the blade that optimizes the position and axial direction of the holes. The verification demonstrates that the novel method can significantly reduce both the contour deviation of the blade surface and the location issue of the film cooling holes. After machining experiments, the maximum position deviation of the holes is reduced by approximately 80% compared to the rigid location method of the entire blade, and the average value and standard deviation are also decreased by about 70%. turbine blade film cooling holes adaptive location blade deformation Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 1. Introduction Aero-engines, gas turbines, and other blade machines depend heavily on their turbine blades, which are made of Ni-base 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, 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. 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 optimized 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 ][ 10 ], or by analysing the deviation of the cast blade surface and compensating the casting moulds [ 11 ], or by performing finite element modelling of the casting process and predicting the casting results by numerical simulation [ 12 ][ 13 ]. 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 [ 14 ]: gathering the point cloud data of the real turbine blade using contact or non-contact measurements (sampling), utilizing point cloud registration methods to determine the spatial relationship between the real blade and the theoretical blade (location), utilizing 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, etc.) 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) [ 15 ][ 16 ] are used to achieve the precise position of the real blade, the transformation process is still rigid [ 17 ]. 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 [ 18 ]. 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 [ 19 ][ 20 ]. In addition, the characteristics of the theoretical model of the blade are considered, where the cross-section curve is fitted using a variety of arc curves in order to meet the tangent requirements of the cross-section curves. The fact that fewer blades satisfy the circular fit feature is a serious restriction of this approach [ 21 ][ 22 ]. 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 [ 23 ]. 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 [ 24 ]. 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 [ 25 ]. 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 [ 26 ][ 27 ]. 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 Section 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 Section 4 . The position and machining of the holes in a turbine blade are discussed in Section 5 , which serves as proof that the procedure described in this work is valid. In Section 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, convex and concave surfaces. After sweeping the 2D cross-section curves, the NURBS surfaces of the blade are constructed as shown in Fig. 1 .a. 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. 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. 1 .b. 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 $$\begin{array}{l}{p}_{cv}=s+r\bullet {n}_{cv}\\ {p}_{cc}=s+r\bullet {n}_{cc}\end{array}$$ 1 where \(s\) is the discrete point of the mean line curve, \({p}_{cv}\) and \({p}_{cc}\) are the corresponding points of \(s\) in the convex and concave curves, respectively, and the distance from these points to \(s\) is \(r\) referring to the radius of the inscribed circle, \({n}_{cv}\) and \({n}_{cc}\) 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 is 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. 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 parametric geometric modelling principle of the blade gives the way of superimposing the thickness on the mean line curve, 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. 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 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. \({s}_{origin}\) and \({s}_{target}\) are the discrete points of the original and target curves, respectively, \(m\) and \(t\) are the normal and tangent directions of the discrete point, respectively. 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 lien curves treated in this paper are 2D shapes. Therefore, the translation matrix \({T}_{warp}\) and rotation matrix \({R}_{warp}\) using nonhomogeneous coordinates is represented as $$\begin{array}{l}{T}_{warp}={s}_{target}-{s}_{origin}\\ {R}_{warp}=\left[\begin{array}{cc}\text{cos}\theta & -\text{sin}\theta \\ \text{sin}\theta & \text{cos}\theta \end{array}\right]\end{array}$$ 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. 4 .a). 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. 4 .b). 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 \({s}_{origin}\) is the point in the theoretical mean line curve, \({p}_{origin}\) is the corresponding point in the theoretical section curve (concave, convex or leading edge curves), and \({n}_{origin}\) is the arbitrary vector of \({p}_{origin}\) . 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 \({s}_{origin}\) , \({p}_{origin}\) and \({n}_{origin}\) 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. 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 minimum. Therefore, the 2D registration method, such as 2D ICP, is proposed to calculate the rotation matrix \({R}_{twist}\) and translation matrix \({T}_{twist}\) 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 $$\begin{array}{l}{s}_{1}={R}_{twist}\bullet {s}_{origin}+{T}_{twist}\\ {p}_{1}={R}_{twist}\bullet {p}_{origin}+{T}_{twist}\\ {n}_{1}={R}_{twist}\bullet {n}_{origin}\end{array}$$ 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, \({n}_{1}\) is the vector of \({p}_{1}\) after 1 deformation. The twisting transformation is shown in Fig. 5 .a→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 \({T}_{warp}\) and the rotation matrix \({R}_{warp}\) is shown in Section 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. Therefore, the relative geometric values after warping are represented as $$\begin{array}{l}{s}_{2}={s}_{1}+{T}_{warp}\\ {p}_{2}={R}_{warp}\bullet \left({p}_{1}-{s}_{1}\right)+{T}_{warp}+{s}_{1}\\ {n}_{2}={R}_{warp}\bullet {n}_{1}\end{array}$$ 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, \({n}_{2}\) is the vector of \({p}_{2}\) after 2 deformations. The warping transformation is shown in Fig. 5 .b→c. It should be emphasized 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 \({T}_{shrink}\) is represented as $${T}_{shrink}={p}_{target}-{p}_{2}$$ 5 Therefore, the relative geometric values after shrinking are represented as $$\begin{array}{l}{s}_{3}={s}_{2}\\ {p}_{3}={p}_{2}+{T}_{shrink}\\ {n}_{3}={n}_{2}\end{array}$$ 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, \({n}_{3}\) is the vector of \({p}_{3}\) after 3 deformations. The warping transformation is shown in Fig. 5 .c→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 \({p}_{origin}\) and vector \({n}_{origin}\) on the theoretical cross-section curve are finally transformed into the following form $${p}_{3}={R}_{warp}\bullet {R}_{twist}\bullet {p}_{origin}-{R}_{warp}\bullet {R}_{twist}\bullet {s}_{origin}+{R}_{twist}\bullet {s}_{origin}+{T}_{twist}+{T}_{warp}+{T}_{shrink}{n}_{3}={R}_{warp}\bullet {R}_{twist}\bullet {n}_{origin}$$ 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 $$T=-{R}_{warp}\bullet {R}_{twist}\bullet {s}_{origin}+{R}_{twist}\bullet {s}_{origin}+{T}_{twist}+{T}_{warp}+{T}_{shrink}R={R}_{warp}\bullet {R}_{twist}$$ 8 where the deformation process is shown in Fig. 5 .a→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 is 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 \(x,y,\theta\) . 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 \({T}_{hole}\) and the rotation matrix \({R}_{hole}\) of the real holes are represented as $$\begin{array}{l}{T}_{hole}={\left[x,y,0\right]}^{T}\\ {R}_{hole}=\left[\begin{array}{ccc}\text{cos}\theta & -\text{sin}\theta & 0\\ \text{sin}\theta & \text{cos}\theta & 0\\ 0& 0& 1\end{array}\right]\end{array}$$ 9 Therefore, the relative information of the deformed holes can be obtained as follows $$\begin{array}{l}{p}_{hole,def}={R}_{hole}\bullet {p}_{hole,origin}+{T}_{hole}\\ {n}_{hole,def}={R}_{hole}\bullet {n}_{hole,origin}\end{array}$$ 10 where \({p}_{hole,origin}\) and \({n}_{hole,origin}\) are the position and axial direction of the film cooling holes in the theoretical blade, respectively, \({p}_{hole,def}\) and \({n}_{hole,def}\) 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. Therefore, an optimisation method is developed: Step 1: A low-order virtual curve is fitted for the flexible deformed holes \(\left\{{p}_{i}\right\}\) 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 \(\left\{{n}_{i}\right\}\) 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. 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 Tianyuan SHINING 3D OptimScan 5M Plus-200 Blu-ray 3D scanner measures the casting turbine blade and creates a point cloud of the blade surfaces. 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 8 depicts the entire procedure along with the hardware and software. 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 minutes on a PC with an Intel(R) Core(TM) i7-11800H CPU (RWI 4.8GHz), a 16GB RAM, an NVIDIA GeForce RTX 3060 Laptop GPU, and a 512GB SSD. The sampling and fitting of the cross-section curves of the point cloud take about 4 minutes. The adaptive approach presented in this paper has a running time that is less than a minute. The whole computation time is under 22 minutes 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. 9 . 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. 1. Deviations of blade surfaces The related deviation cloud diagrams can be created for the four location methods mentioned above as shown in Fig. 10 . The first three cloud diagrams among them describe the deviation of the real blade calculated by the corresponding location methods from the theoretical blade, while the last cloud diagram 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 . Table 1 Deviation statistics of cloud map (mm) Location Methods Maximum Minimum Average Standard Deviation I 0.297 -0.450 -0.084 0.194 II 0.027 -0.275 -0.083 0.043 III 0.095 -0.329 -0.080 0.069 IV 0.124 -0.149 -0.020 0.025 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 Method I-III and the theoretical model are quite close, the positive and negative deviations cancel each other out when computing the average value, leaving Method 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. 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 Method I-IV. The theoretical holes are projected onto the real blade surfaces to produce the predicted holes from Method I-III. The deviations of the hole positions are shown in Fig. 11 . 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. Table 2 Deviation statistics of hole position (mm) Location Methods Maximum Minimum Average Standard Deviation I 0.242 0.005 0.099 0.062 II 0.150 0.002 0.061 0.043 III 0.162 0.004 0.056 0.040 IV 0.033 0.003 0.019 0.009 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 Method II and III for group 1 in the convex surface and group 5 in the leading edge surface. The deviations in group 9 of Method 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 Method 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. 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 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. 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%, so that 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. Declarations Funding This work was supported by Natural Science Basic Research Program of Shaanxi Province (No. 2022JQ-473); Young Talent Fund of Xi'an Association for Science and Technology (No. 095920221309); National Natural Science Foundation of China (No. 52205438). Competing Interests The authors declare no competing interests. Author Contributions All authors contributed to the study conception and design. Investigation, methodology, software, material preparation, data collection, visualization and analysis were performed by Yaohua Hou, Jing Wang and Jiawei Mei. Resources, supervision and project administration were performed by Jing Wang and Hualong Zhao. The first draft of the manuscript was written by Yaohua Hou and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. References Zhang Jingzhou Z, Shengchang W, Chunhua T, Xiaoming (2020) Recent advances in film cooling enhancement: A review. Chin J Aeronaut 33:1119–1136 Chen Pingting S, Wei L, Xueying R, Jing J, Hongde (2021) Numerical study of using dean vortices to enhance the film cooling performance for fan shaped hole. Int J Therm Sci, 165 Montomoli Francesco M, Michela S, Simone M, Francesco (2012) Geometrical Uncertainty and Film Cooling: Fillet Radii. J Turbomach, 134(1) Cheng Yu-qi, Cheng LWen-longJ et al (2021) A novel cooling hole inspection method for turbine blade using 3D reconstruction of stereo vision. Meas Sci Technol 33(1):015018 Li ZY, Wei XT, Guo YB, Sealy MP (2015) State-Of-Art, Challenges, and Outlook on Manufacturing of Cooling Holes for Turbine Blades. Mach Sci Technol, 19(3) Zhao Wanqin Yu (2018) Zhishui. Self-cleaning effect in high quality percussion ablating of cooling hole by picosecond ultra-short pulse laser. Opt Lasers Eng, 105 Dong YW, Li XL, Zhao Qi Y, Jun, Dao Ming (2016). Modeling of shrinkage during investment casting of thin-walled hollow turbine blades. J Mater Process Tech, 244 Li L, Li B, Ruting Z, Zhangfeng X, Xiang W (2023) Geometric parameters measurement for the cooling holes of turbine blade based on microscopic image sequence topographical reconstruction. Measurement 210:112562 Tian Guo-liang, Kun Bu, Zhao Dan-qing, Zhang Ya-li, Fei Q (2018) Zhang Xian-dong, Ren Shuai-jun. A shrinkage prediction method of investment casting based on geometric parameters. The International Journal of Advanced Manufacturing Technology, 96(1) Wu Haihua L, Dichen C, Xu Dongyang (2010) Rapid casting of turbine blades with abnormal film cooling holes using integral ceramic casting molds. Int J Adv Manuf Technol 50:13–19 Zhang DH, Song JR, Ling LJ (2010) Cavity optimization for investment casting die of turbine blade based on reverse engineering. Int J Adv Manuf Technol 48:9–12 Dong Yiwei L, Xiaolin Z, Qi L, Xiaoji, Dou Yangqing (2017) Geometrical modeling to improve the accuracy of drilled cooling holes on turbine blades. Int J Adv Manuf Technol 93:9–12 Yang L, Chai LH, Liang YF et al (2015) Numerical simulation and experimental verification of gravity and centrifugal investment casting low pressure turbine blades for high Nb–TiAl alloy. Intermetallics 66:149–155 Jian XXue-ChengW, Si-Meng Z, Wang-Sheng MJie-YuZ (2023) Adaptive Drilling of Film Cooling Holes of Turbine Vanes Based on Registration of Point Clouds. IEEE Trans Industr Inf. 10.1109/TII.2023.3254664 Cheng Xu L, Zhongwei Z, Kai S (2017) An automatic and robust point cloud registration framework based on view-invariant local feature descriptors and transformation consistency verification. Opt Lasers Eng 98:37–45 Zijie Wu, Yaonan W, Yang M, Qing Z, He X, Haotian Wu, Mingtao F (2022) Mian Ajmal. Multiview Point Cloud Registration Based on Minimum Potential Energy for Free-Form Blade Measurement. IEEE Trans Instrum Meas 71:1–14 He Wantao L, Zhongwei G, Yanyan C, Xu Z, Kai S, Yusheng (2018) A robust and accurate automated registration method for turbine blade precision metrology. Int J Adv Manuf Technol 97:9–12 Mehran Mahboubkhah M, Aliakbari C, Burvill An investigation on measurement accuracy of digitizing methods in turbine blade reverse engineering. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture,2018,232(9) Hou Yaohua Z, Dinghua M, Jiawei Z, Ying L, Ming Geometric modelling of thin-walled blade based on compensation method of machining error and design intent. J Manuf Process ,2019,44. Khameneifar, Farbod, Feng Hsi-Yung (2014). Airfoil profile reconstruction under the uncertainty of inspection data points. The International Journal of Advanced Manufacturing Technology, 71(1–4) Kamran Mohaghegh, Mohammad H, Sadeghi A, Abdullah R, Boutorabi (2010) Improvement of reverse-engineered turbine blades using construction geometry. Int J Adv Manuf Technol 49:5–8 Kamran Mohaghegh, Mohammad H, Sadeghi A, Abdullah (2007) Abdullah. Reverse engineering of turbine blades based on design intent. Int J Adv Manuf Technol 32:9–10 Yang F, Zhenhua C, Yuepeng C, Shujuan D, Chunming D, Shaopeng N, Wei Z Wen Shiping. A Robotic Polishing Trajectory Planning Method Combining Reverse Engineering and Finite Element Mesh Technology for Aero-Engine Turbine Blade TBCs. Journal of Thermal Spray Technology,2022,31(7). Li, Yongqing, Ni Jun. Constraints Based Nonrigid Registration for 2D Blade Profile Reconstruction in Reverse Engineering. J Comput Inf Sci Eng ,2009,9(3). Zhang Yun C, Zhi-Tong N, Tao Reverse modeling strategy of aero-engine blade based on design intent. Int J Adv Manuf Technol ,2015,81(9–12). Rong Yu Xu, Jinting S, Yuwen A surface reconstruction strategy based on deformable template for repairing damaged turbine blades. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering,2014,228(12) Li L, Ying LCongboT, Yanbin Du (2017) An integrated approach of reverse engineering aided remanufacturing process for worn components. Rob Comput Integr Manuf, 48 Cite Share Download PDF Status: Published Journal Publication published 21 Mar, 2024 Read the published version in The International Journal of Advanced Manufacturing Technology → Version 1 posted Editorial decision: Major Revisions Needed 25 Sep, 2023 Reviewers agreed at journal 04 Aug, 2023 Reviewers invited by journal 04 Aug, 2023 Editor assigned by journal 03 Aug, 2023 First submitted to journal 02 Aug, 2023 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Hou","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAzklEQVRIiWNgGAWjYBACAwh1gIGBvYFkLTwHwBQpWiQSiNRizt5jJvGj4k5i/8zHxx5/YLCTZ2A/i1+nZc8ZM8meM88SZ9xOSzc4wJBs2MCTl4DfYTdyzCR42w4nNtwGMg4wMCcwSPAYENQi+ReoZf7NMyAt9cRpkQbZsuEGD0jLYSK0nDlWbC1z5rDxxjNAv5wxOG7YxpNDQMvx5o0331Qclp13/PCxBxUV1fL87GfwawECFgkogw0cTWyE1AMB8weEllEwCkbBKBgFWAAAiJxIifhP9N4AAAAASUVORK5CYII=","orcid":"https://orcid.org/0009-0004-4703-7422","institution":"Xi'an Institute of Optics and Precision Mechanics of CAS: Chinese Academy of Sciences Xi'an Institute of Optics and Precision Mechanics","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Yaohua","middleName":"","lastName":"Hou","suffix":""},{"id":224059030,"identity":"76f2881b-d82f-43bf-a206-b4207b1f248e","order_by":1,"name":"Jing 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07:51:02","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3219667/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3219667/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s00170-024-13456-4","type":"published","date":"2024-03-21T15:02:50+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":41377433,"identity":"ad0f34a3-8561-4f61-9f03-089d1967854d","added_by":"auto","created_at":"2023-08-10 14:58:21","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":108991,"visible":true,"origin":"","legend":"\u003cp\u003eCAD models of (a) the turbine blade (b) the cross-section curve.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-3219667/v1/f63b78501913899ff16e8181.png"},{"id":41376113,"identity":"87583099-c20b-4911-a34e-6e8f7beeb198","added_by":"auto","created_at":"2023-08-10 14:50:21","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":99498,"visible":true,"origin":"","legend":"\u003cp\u003eThe distribution of the film cooling holes.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-3219667/v1/ca4748553eac26068a8bd4c5.png"},{"id":41376116,"identity":"30f0fd6e-bf81-4864-bb7d-622f75d10990","added_by":"auto","created_at":"2023-08-10 14:50:21","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":16845,"visible":true,"origin":"","legend":"\u003cp\u003eDeformation of the free form curve.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-3219667/v1/a0f968c794616a7abe54fd03.png"},{"id":41376118,"identity":"9e3641d2-a2fb-4bc9-9e8b-45c73149befb","added_by":"auto","created_at":"2023-08-10 14:50:21","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":39246,"visible":true,"origin":"","legend":"\u003cp\u003eThe discrete points in (a) concave and convex curves, and (b) leading edge curve.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-3219667/v1/2f41e350751f708df52428be.png"},{"id":41376117,"identity":"399b8413-74ed-47a2-8f45-64fb3470eb73","added_by":"auto","created_at":"2023-08-10 14:50:21","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":102011,"visible":true,"origin":"","legend":"\u003cp\u003eDeformation process of the cross-section curves.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-3219667/v1/397bb990605e6be141f9f68f.png"},{"id":41376115,"identity":"9fd47601-e86b-46c5-a22c-91b680d79d2a","added_by":"auto","created_at":"2023-08-10 14:50:21","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":38645,"visible":true,"origin":"","legend":"\u003cp\u003eThe warping deformation process of the (a) concave, convex and (b) leading edge regions.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-3219667/v1/28a391e2dcad3f5d314bbe0f.png"},{"id":41376120,"identity":"5a2942d0-647a-4d92-846e-6147afc8d8a1","added_by":"auto","created_at":"2023-08-10 14:50:21","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":40357,"visible":true,"origin":"","legend":"\u003cp\u003eThe shrinking deformation process of the (a) concave, convex and (b) leading edge regions.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-3219667/v1/ea7720d1a69d22c98656852f.png"},{"id":41377435,"identity":"7631f75f-d6e1-4890-81c6-3158fe664a4c","added_by":"auto","created_at":"2023-08-10 14:58:21","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":306064,"visible":true,"origin":"","legend":"\u003cp\u003eAdaptive location and machining framework for film cooling holes.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-3219667/v1/baba4e6d6e7b1df3103225f1.png"},{"id":41377434,"identity":"29151691-f2ff-458b-aebb-49b012b52835","added_by":"auto","created_at":"2023-08-10 14:58:21","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":76653,"visible":true,"origin":"","legend":"\u003cp\u003eRegions for location calculation of (a) I, (b) II, (c) III and (d) IV methods.\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-3219667/v1/d1a046745f47f122b33b629f.png"},{"id":41376121,"identity":"a6c0ee06-86c6-4bab-85d5-b700de462b90","added_by":"auto","created_at":"2023-08-10 14:50:21","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":562749,"visible":true,"origin":"","legend":"\u003cp\u003eDeviation cloud maps of (a) I, (b) II, (c) III and (d) IV methods.\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-3219667/v1/6f2d8ae41ea01408c5608b48.png"},{"id":41376122,"identity":"9f1d4ed5-a609-4a8a-bc07-5c3e6237e231","added_by":"auto","created_at":"2023-08-10 14:50:21","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":205605,"visible":true,"origin":"","legend":"\u003cp\u003eThe position deviations of holes located by different methods.\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-3219667/v1/3ff8fa77b1f7630b62faad20.png"},{"id":53403722,"identity":"98ccb796-0314-4b83-b57a-b7404e313866","added_by":"auto","created_at":"2024-03-25 15:14:03","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1902772,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3219667/v1/0c58d741-00a9-47b4-8192-fa2b779444ed.pdf"}],"financialInterests":"","formattedTitle":"Adaptive Location Method for Film Cooling Holes Based on The Design Intent of The Turbine Blade","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eAero-engines, gas turbines, and other blade machines depend heavily on their turbine blades, which are made of Ni-base 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\u0026ndash;2000 K) has far exceeded the melting point (about 1700 K) of the turbine blade [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. The film cooling method is normally suggested to enhance the high-temperature creep resistance and maintain the normal state of the turbine blade [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. 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.\u003c/p\u003e \u003cp\u003eThe key to increasing the aero-engine's operational efficiency is to correctly machine the film cooling holes of the turbine blade [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e], 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, are therefore challenging to produce using traditional techniques [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. 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 [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e][\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e], 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 [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e]. 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 [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. 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. 5. The insulation and cooling effects will be impacted by the misalignment between the air film forming area and the blade surface.\u003c/p\u003e \u003cp\u003eTo achieve this, researchers have optimized 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 [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e][\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e], or by analysing the deviation of the cast blade surface and compensating the casting moulds [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e], or by performing finite element modelling of the casting process and predicting the casting results by numerical simulation [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e][\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]. 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.\u003c/p\u003e \u003cp\u003eGiven the inevitable deviation, a methodical approach has been taken [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]: gathering the point cloud data of the real turbine blade using contact or non-contact measurements (sampling), utilizing point cloud registration methods to determine the spatial relationship between the real blade and the theoretical blade (location), utilizing 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, etc.) 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) [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e][\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e] are used to achieve the precise position of the real blade, the transformation process is still rigid [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e]. 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.\u003c/p\u003e \u003cp\u003eAccording 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 [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]. 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 [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e][\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e]. In addition, the characteristics of the theoretical model of the blade are considered, where the cross-section curve is fitted using a variety of arc curves in order to meet the tangent requirements of the cross-section curves. The fact that fewer blades satisfy the circular fit feature is a serious restriction of this approach [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e][\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e]. 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 [\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]. 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 [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e]. 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 [\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]. 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 [\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e][\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e]. 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.\u003c/p\u003e \u003cp\u003eThis 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 Section \u003cspan refid=\"Sec2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. Section \u003cspan refid=\"Sec3\" class=\"InternalRef\"\u003e3\u003c/span\u003e 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 Section \u003cspan refid=\"Sec10\" class=\"InternalRef\"\u003e4\u003c/span\u003e. The position and machining of the holes in a turbine blade are discussed in Section \u003cspan refid=\"Sec13\" class=\"InternalRef\"\u003e5\u003c/span\u003e, which serves as proof that the procedure described in this work is valid. In Section \u003cspan refid=\"Sec16\" class=\"InternalRef\"\u003e6\u003c/span\u003e, the conclusions and prospect are covered.\u003c/p\u003e"},{"header":"2. Principles of parametric geometric modelling of blades","content":"\u003cp\u003eAs a special free-form part, the main area of the blade consists of the leading edge, trailing edge, convex and concave surfaces. After sweeping the 2D cross-section curves, the NURBS surfaces of the blade are constructed as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.a. 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.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe 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.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.b. 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\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\begin{array}{l}{p}_{cv}=s+r\\bullet {n}_{cv}\\\\ {p}_{cc}=s+r\\bullet {n}_{cc}\\end{array}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(s\\)\u003c/span\u003e\u003c/span\u003e is the discrete point of the mean line curve, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({p}_{cv}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({p}_{cc}\\)\u003c/span\u003e\u003c/span\u003e are the corresponding points of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(s\\)\u003c/span\u003e\u003c/span\u003e in the convex and concave curves, respectively, and the distance from these points to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(s\\)\u003c/span\u003e\u003c/span\u003e is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(r\\)\u003c/span\u003e\u003c/span\u003e referring to the radius of the inscribed circle, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({n}_{cv}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({n}_{cc}\\)\u003c/span\u003e\u003c/span\u003e are the directions from these points to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(s\\)\u003c/span\u003e\u003c/span\u003e, which are symmetrical with respect to the tangent \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(t\\)\u003c/span\u003e\u003c/span\u003e 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.\u003c/p\u003e \u003cp\u003eConsidering that the theoretical model and measuring data of the blade contour is 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.\u003c/p\u003e \u003cp\u003eNext, the positions of the film cooling holes in the turbine blade are introduced, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. 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.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"3. Flexible deformation method of the blade surfaces","content":"\u003cp\u003eDue 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 parametric geometric modelling principle of the blade gives the way of superimposing the thickness on the mean line curve, 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.\u003c/p\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Free form deformation method\u003c/h2\u003e \u003cp\u003eFlexible 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 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.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. 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 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\theta\\)\u003c/span\u003e\u003c/span\u003e in the transformation process. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({s}_{origin}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({s}_{target}\\)\u003c/span\u003e\u003c/span\u003e are the discrete points of the original and target curves, respectively, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(m\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(t\\)\u003c/span\u003e\u003c/span\u003e are the normal and tangent directions of the discrete point, respectively.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe 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 lien curves treated in this paper are 2D shapes. Therefore, the translation matrix \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({T}_{warp}\\)\u003c/span\u003e\u003c/span\u003e and rotation matrix \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({R}_{warp}\\)\u003c/span\u003e\u003c/span\u003e using nonhomogeneous coordinates is represented as\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\begin{array}{l}{T}_{warp}={s}_{target}-{s}_{origin}\\\\ {R}_{warp}=\\left[\\begin{array}{cc}\\text{cos}\\theta \u0026amp; -\\text{sin}\\theta \\\\ \\text{sin}\\theta \u0026amp; \\text{cos}\\theta \\end{array}\\right]\\end{array}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Deformation decomposition of the cross-section curve\u003c/h2\u003e \u003cp\u003eWith 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.\u003c/p\u003e \u003cp\u003eConsidering 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.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.a). 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.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.b). 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 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({s}_{origin}\\)\u003c/span\u003e\u003c/span\u003e is the point in the theoretical mean line curve, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({p}_{origin}\\)\u003c/span\u003e\u003c/span\u003e is the corresponding point in the theoretical section curve (concave, convex or leading edge curves), and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({n}_{origin}\\)\u003c/span\u003e\u003c/span\u003e is the arbitrary vector of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({p}_{origin}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eBased 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 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({s}_{origin}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({p}_{origin}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({n}_{origin}\\)\u003c/span\u003e\u003c/span\u003e are deformed using the deformation order and the corresponding deformation matrix (see Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e). 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.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Twisting deformation process\u003c/h2\u003e \u003cp\u003eAfter 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 minimum. Therefore, the 2D registration method, such as 2D ICP, is proposed to calculate the rotation matrix \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({R}_{twist}\\)\u003c/span\u003e\u003c/span\u003e and translation matrix \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({T}_{twist}\\)\u003c/span\u003e\u003c/span\u003e 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\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\begin{array}{l}{s}_{1}={R}_{twist}\\bullet {s}_{origin}+{T}_{twist}\\\\ {p}_{1}={R}_{twist}\\bullet {p}_{origin}+{T}_{twist}\\\\ {n}_{1}={R}_{twist}\\bullet {n}_{origin}\\end{array}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({s}_{1}\\)\u003c/span\u003e\u003c/span\u003e is the discrete point of the mean line curve after 1 deformation, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({p}_{1}\\)\u003c/span\u003e\u003c/span\u003e is the discrete point of the section curves after 1 deformation, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({n}_{1}\\)\u003c/span\u003e\u003c/span\u003e is the vector of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({p}_{1}\\)\u003c/span\u003e\u003c/span\u003e after 1 deformation. The twisting transformation is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.a\u0026rarr;b.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e3.4 Warping deformation process\u003c/h2\u003e \u003cp\u003eFollowing 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 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({T}_{warp}\\)\u003c/span\u003e\u003c/span\u003e and the rotation matrix \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({R}_{warp}\\)\u003c/span\u003e\u003c/span\u003e is shown in Section \u003cspan refid=\"Sec4\" class=\"InternalRef\"\u003e3.1\u003c/span\u003e. 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\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e 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.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTherefore, the relative geometric values after warping are represented as\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\begin{array}{l}{s}_{2}={s}_{1}+{T}_{warp}\\\\ {p}_{2}={R}_{warp}\\bullet \\left({p}_{1}-{s}_{1}\\right)+{T}_{warp}+{s}_{1}\\\\ {n}_{2}={R}_{warp}\\bullet {n}_{1}\\end{array}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({s}_{2}\\)\u003c/span\u003e\u003c/span\u003e is the discrete point of the mean line curve after 2 deformations, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({p}_{2}\\)\u003c/span\u003e\u003c/span\u003e is the discrete point of the section curves after 2 deformations, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({n}_{2}\\)\u003c/span\u003e\u003c/span\u003e is the vector of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({p}_{2}\\)\u003c/span\u003e\u003c/span\u003e after 2 deformations. The warping transformation is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.b\u0026rarr;c.\u003c/p\u003e \u003cp\u003eIt should be emphasized that each discrete point on the mean line curve experiences a distinct deformation matrix during warping deformation, which must be calculated independently.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e3.5 Shrinking deformation process\u003c/h2\u003e \u003cp\u003eSince 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\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e 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 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({T}_{shrink}\\)\u003c/span\u003e\u003c/span\u003e is represented as\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$${T}_{shrink}={p}_{target}-{p}_{2}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTherefore, the relative geometric values after shrinking are represented as\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$\\begin{array}{l}{s}_{3}={s}_{2}\\\\ {p}_{3}={p}_{2}+{T}_{shrink}\\\\ {n}_{3}={n}_{2}\\end{array}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({s}_{3}\\)\u003c/span\u003e\u003c/span\u003e is the discrete point of the mean line curve after 3 deformations, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({p}_{3}\\)\u003c/span\u003e\u003c/span\u003e is the discrete point of the section curves after 3 deformations, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({n}_{3}\\)\u003c/span\u003e\u003c/span\u003e is the vector of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({p}_{3}\\)\u003c/span\u003e\u003c/span\u003e after 3 deformations. The warping transformation is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.c\u0026rarr;d.\u003c/p\u003e \u003cp\u003eSimilarly, during shrinking deformation, each discrete point on the mean line curve has a specific deformation matrix that must be calculated separately.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e3.6 Deformation of whole blade surfaces\u003c/h2\u003e \u003cp\u003eCombining the above twisting, warping and shrinking deformations, the discrete point \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({p}_{origin}\\)\u003c/span\u003e\u003c/span\u003e and vector \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({n}_{origin}\\)\u003c/span\u003e\u003c/span\u003e on the theoretical cross-section curve are finally transformed into the following form\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$${p}_{3}={R}_{warp}\\bullet {R}_{twist}\\bullet {p}_{origin}-{R}_{warp}\\bullet {R}_{twist}\\bullet {s}_{origin}+{R}_{twist}\\bullet {s}_{origin}+{T}_{twist}+{T}_{warp}+{T}_{shrink}{n}_{3}={R}_{warp}\\bullet {R}_{twist}\\bullet {n}_{origin}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAs a result, the translation matrix \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(T\\)\u003c/span\u003e\u003c/span\u003e and rotation matrix \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(R\\)\u003c/span\u003e\u003c/span\u003e of the flexible deformation process of the theoretical cross-section curve to the real cross-section curve are represented as\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$$T=-{R}_{warp}\\bullet {R}_{twist}\\bullet {s}_{origin}+{R}_{twist}\\bullet {s}_{origin}+{T}_{twist}+{T}_{warp}+{T}_{shrink}R={R}_{warp}\\bullet {R}_{twist}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere the deformation process is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.a\u0026rarr;d.\u003c/p\u003e \u003cp\u003eApplying 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.\u003c/p\u003e \u003c/div\u003e"},{"header":"4. Precision location and optimization of holes","content":"\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Analysis of deformed holes\u003c/h2\u003e \u003cp\u003eBased on the overall flexible deformation data of the blade, the position and axial direction of the theoretical film cooling holes is 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.\u003c/p\u003e \u003cp\u003eFor blade CAD models, the concave, convex and leading edge surfaces are generally composed of NURBS surfaces, which are represented with a parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(u\\)\u003c/span\u003e\u003c/span\u003e within the section curves and a parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(v\\)\u003c/span\u003e\u003c/span\u003e between the curves. Meanwhile, the deformation, including translation matrix \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(T\\)\u003c/span\u003e\u003c/span\u003e and rotation matrix \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(R\\)\u003c/span\u003e\u003c/span\u003e, can be represented as three values \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(x,y,\\theta\\)\u003c/span\u003e\u003c/span\u003e. Combining the above variables, three fitting functions \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(x={f}_{1}\\left(u,v\\right)\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(y={f}_{2}\\left(u,v\\right)\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\theta ={f}_{3}\\left(u,v\\right)\\)\u003c/span\u003e\u003c/span\u003e are constructed based on the parameters \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left(u,v\\right)\\)\u003c/span\u003e\u003c/span\u003e of the discrete points of the deformed blade surfaces and the deformations \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left(x,y,\\theta \\right)\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eAfter calculating the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left(u,v\\right)\\)\u003c/span\u003e\u003c/span\u003e 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 \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({T}_{hole}\\)\u003c/span\u003e\u003c/span\u003e and the rotation matrix \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({R}_{hole}\\)\u003c/span\u003e\u003c/span\u003e of the real holes are represented as\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$$\\begin{array}{l}{T}_{hole}={\\left[x,y,0\\right]}^{T}\\\\ {R}_{hole}=\\left[\\begin{array}{ccc}\\text{cos}\\theta \u0026amp; -\\text{sin}\\theta \u0026amp; 0\\\\ \\text{sin}\\theta \u0026amp; \\text{cos}\\theta \u0026amp; 0\\\\ 0\u0026amp; 0\u0026amp; 1\\end{array}\\right]\\end{array}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eTherefore, the relative information of the deformed holes can be obtained as follows\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e\n$$\\begin{array}{l}{p}_{hole,def}={R}_{hole}\\bullet {p}_{hole,origin}+{T}_{hole}\\\\ {n}_{hole,def}={R}_{hole}\\bullet {n}_{hole,origin}\\end{array}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({p}_{hole,origin}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({n}_{hole,origin}\\)\u003c/span\u003e\u003c/span\u003e are the position and axial direction of the film cooling holes in the theoretical blade, respectively, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({p}_{hole,def}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({n}_{hole,def}\\)\u003c/span\u003e\u003c/span\u003e are the position and axial direction of the film cooling holes in the flexible deformed blade, respectively.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Optimization and adjustment of holes\u003c/h2\u003e \u003cp\u003eThere 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.\u003c/p\u003e \u003cp\u003eWhen 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.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). The holes in the same groups have essentially the same sizes, axial directions, and places along the same sweeping curve. Therefore, an optimisation method is developed:\u003c/p\u003e \u003cp\u003eStep 1: A low-order virtual curve is fitted for the flexible deformed holes \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left\\{{p}_{i}\\right\\}\\)\u003c/span\u003e\u003c/span\u003e 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.\u003c/p\u003e \u003cp\u003eStep 2: The Quaternionic interpolation of the axial direction \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left\\{{n}_{i}\\right\\}\\)\u003c/span\u003e\u003c/span\u003e 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.\u003c/p\u003e \u003cp\u003eFollowing 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.\u003c/p\u003e \u003c/div\u003e"},{"header":"5. Case and discussion","content":"\u003cp\u003eIn 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 Tianyuan SHINING 3D OptimScan 5M Plus-200 Blu-ray 3D scanner measures the casting turbine blade and creates a point cloud of the blade surfaces. 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\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e depicts the entire procedure along with the hardware and software.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe blade is approximately 35\u0026times;25\u0026times;8 mm in size, and because it is clamped to a fixture, there are more than 3\u0026nbsp;million points and more than 6\u0026nbsp;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 minutes on a PC with an Intel(R) Core(TM) i7-11800H CPU (RWI 4.8GHz), a 16GB RAM, an NVIDIA GeForce RTX 3060 Laptop GPU, and a 512GB SSD. The sampling and fitting of the cross-section curves of the point cloud take about 4 minutes. The adaptive approach presented in this paper has a running time that is less than a minute. The whole computation time is under 22 minutes when the deformed blade is rebuilt using 4 cross sections.\u003c/p\u003e \u003cp\u003eIt 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.\u003c/p\u003e \u003cp\u003eDifferent 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.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e. 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.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e\n\u003ch3\u003e1. Deviations of blade surfaces\u003c/h3\u003e\n\u003cp\u003eThe related deviation cloud diagrams can be created for the four location methods mentioned above as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e. The first three cloud diagrams among them describe the deviation of the real blade calculated by the corresponding location methods from the theoretical blade, while the last cloud diagram 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\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eDeviation statistics of cloud map (mm)\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLocation Methods\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMaximum\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMinimum\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eAverage\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eStandard Deviation\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eI\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.297\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e-0.450\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-0.084\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.194\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eII\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.027\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e-0.275\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-0.083\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.043\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIII\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.095\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e-0.329\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-0.080\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.069\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIV\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.124\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e-0.149\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e-0.020\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.025\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eAccording 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 Method I-III and the theoretical model are quite close, the positive and negative deviations cancel each other out when computing the average value, leaving Method 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.\u003c/p\u003e\n\u003ch3\u003e2. Distribution of film cooling holes\u003c/h3\u003e\n\u003cp\u003eThe 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 Method I-IV. The theoretical holes are projected onto the real blade surfaces to produce the predicted holes from Method I-III. The deviations of the hole positions are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e. The horizontal coordinate is the hole number and the vertical coordinate is the hole deviation. The deviation statistics are shown in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. 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.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eDeviation statistics of hole position (mm)\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLocation Methods\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMaximum\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMinimum\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eAverage\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eStandard Deviation\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eI\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.242\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.005\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.099\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.062\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eII\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.150\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.002\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.061\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.043\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIII\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.162\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.004\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.056\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.040\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eIV\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.033\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.003\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.019\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.009\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe 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 Method II and III for group 1 in the convex surface and group 5 in the leading edge surface. The deviations in group 9 of Method 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 Method 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.\u003c/p\u003e"},{"header":"6. Conclusions","content":"\u003cp\u003eAn 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:\u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eThe 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.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eWhen 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%, so that the improvement of the location accuracy of the film cooling holes is significant, and the final deviation meets the machining requirements.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e \u003cp\u003eFinally, 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.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eFunding\u003c/h2\u003e\n\u003cp\u003eThis work was supported by Natural Science Basic Research Program of Shaanxi Province (No. 2022JQ-473); Young Talent Fund of Xi'an Association for Science and Technology (No. 095920221309); National Natural Science Foundation of China (No. 52205438).\u003c/p\u003e\n\u003ch2\u003eCompeting Interests\u003c/h2\u003e\n\u003cp\u003eThe authors declare no competing interests.\u003c/p\u003e\n\u003ch2\u003eAuthor Contributions\u003c/h2\u003e\n\u003cp\u003eAll authors contributed to the study conception and design. Investigation, methodology, software, material preparation, data collection, visualization and analysis were performed by Yaohua Hou, Jing Wang and Jiawei Mei. Resources, supervision and project administration were performed by Jing Wang and Hualong Zhao. The first draft of the manuscript was written by Yaohua Hou and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eZhang Jingzhou Z, Shengchang W, Chunhua T, Xiaoming (2020) Recent advances in film cooling enhancement: A review. Chin J Aeronaut 33:1119\u0026ndash;1136\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eChen Pingting S, Wei L, Xueying R, Jing J, Hongde (2021) Numerical study of using dean vortices to enhance the film cooling performance for fan shaped hole. 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Rob Comput Integr Manuf, 48\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"the-international-journal-of-advanced-manufacturing-technology","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jamt","sideBox":"Learn more about [The International Journal of Advanced Manufacturing Technology](https://www.springer.com/journal/170)","snPcode":"170","submissionUrl":"https://submission.nature.com/new-submission/170/3","title":"The International Journal of Advanced Manufacturing Technology","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"turbine blade, film cooling holes, adaptive location, blade deformation","lastPublishedDoi":"10.21203/rs.3.rs-3219667/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3219667/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eDue to the inevitable deviation of the casting process, the dimensional error of the turbine blade is introduced. As a result, the location datum of the film cooling holes is changed, which has an impact on the machining accuracy. The majority of pertinent studies concentrate on the rigid location approach for the entire blade, which results in a modest relative position error of the blade surface but still fails to give the exact position and axial direction of the film cooling holes of the deformed blade. In this paper, the entire deformation of the blade cross-section curve is divided into a number of deformation combinations of the mean line curve based on the construction method of the blade design intent. The exact location of the film cooling holes in the turbine blade with deviation is therefore efficiently solved by a flexible deformation of the blade that optimizes the position and axial direction of the holes. The verification demonstrates that the novel method can significantly reduce both the contour deviation of the blade surface and the location issue of the film cooling holes. After machining experiments, the maximum position deviation of the holes is reduced by approximately 80% compared to the rigid location method of the entire blade, and the average value and standard deviation are also decreased by about 70%.\u003c/p\u003e","manuscriptTitle":"Adaptive Location Method for Film Cooling Holes Based on The Design Intent of The Turbine Blade","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-08-10 14:50:16","doi":"10.21203/rs.3.rs-3219667/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Major Revisions Needed","date":"2023-09-25T07:49:21+00:00","index":"","fulltext":""},{"type":"reviewerAgreed","content":"","date":"2023-08-05T00:19:18+00:00","index":0,"fulltext":""},{"type":"reviewersInvited","content":"","date":"2023-08-04T12:00:42+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2023-08-03T04:30:17+00:00","index":"","fulltext":""},{"type":"submitted","content":"The International Journal of Advanced Manufacturing Technology","date":"2023-08-02T06:55:17+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"the-international-journal-of-advanced-manufacturing-technology","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jamt","sideBox":"Learn more about [The International Journal of Advanced Manufacturing Technology](https://www.springer.com/journal/170)","snPcode":"170","submissionUrl":"https://submission.nature.com/new-submission/170/3","title":"The International Journal of Advanced Manufacturing Technology","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"abec77de-3a3f-4371-a7bb-5369711257c6","owner":[],"postedDate":"August 10th, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2024-03-25T15:07:56+00:00","versionOfRecord":{"articleIdentity":"rs-3219667","link":"https://doi.org/10.1007/s00170-024-13456-4","journal":{"identity":"the-international-journal-of-advanced-manufacturing-technology","isVorOnly":false,"title":"The International Journal of Advanced Manufacturing Technology"},"publishedOn":"2024-03-21 15:02:50","publishedOnDateReadable":"March 21st, 2024"},"versionCreatedAt":"2023-08-10 14:50:16","video":"","vorDoi":"10.1007/s00170-024-13456-4","vorDoiUrl":"https://doi.org/10.1007/s00170-024-13456-4","workflowStages":[]},"version":"v1","identity":"rs-3219667","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3219667","identity":"rs-3219667","version":["v1"]},"buildId":"cBFmMYwuxLRRLfASyISRj","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}