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Dimensional deviation is a prerequisite for improving the manufacturing process of parts with free-form surfaces, for example, the reverse adjustment of the die cavity of turbine blade. Influenced by random noise of the manufacturing process, dimensional variation is inevitable for batch parts. Therefore, it may cause unacceptable error to estimate batch parts dimensional deviation by a single sample. Meanwhile, the optimum sample size for estimating dimensional deviation is difficult to determine. To overcome this problem, a practical method for estimating of dimensional deviation of parts with free-form surface is proposed. Firstly, displacements of the discrete points on part surface are employed to represent dimensional error of the part. Estimating dimensional deviation of parts is actually to estimate the simultaneous confidence intervals of the discrete point displacements. Secondly, Bonferroni simultaneous confidence intervals are adopted to estimate the confidence intervals of the part dimensional deviation. Moreover, the accuracy of the estimation will be continuously improved by increasing samples. Consequently, a practical dimensional deviation estimation method is presented. Finally, a compressor blade is adopted to illustrate the proposed method. The percentage of estimation error of the blade dimensional deviation that is less than 0.05mm, which is the estimation error limit, of all the 100 times sampling experiments is 99%, exceeding the given confidence level of 95%, while the percentage of the existing method is below the confidence level, only 87%.

With the development of technology and the improvement of manufacturing level, free-form surfaces are more and more applied in automobile, aerospace, and shipbuilding industries. Unlike regular geometric features, free-form surfaces can provide excellent mechanical properties, optical characteristics, fluid characteristics, and so on. Geometric shapes of parts with free-form surfaces are often closely related to parts’ design performance [

Dimensional deviation plays an important role in the manufacturing process of free-form surface parts. As the comprehensive influence of the system error on parts’ dimension, dimensional deviation is the essential prerequisite for evaluating or improving the manufacturing process. Taking the molded parts as an example, part dimensional deviation is the basis of the reverse adjustment of the mold cavity [

In order to improve the efficiency of inspection, researchers have made great efforts to reduce inspection samples in the field of general industrial products. Khalifa et al. [

Through the above literature analysis, it can be seen that, in the field of sampling inspection, scholars have done a lot of research work on sampling methods, such as resampling [

Based on mathematical statistics, in order to estimate dimensional deviation of free-form surface parts, the following two problems need to be solved:

How to quantify the dimension and deformation of free-form surfaces

How to construct the pivot quantity of free-form surfaces to estimate its dimensional deviation.

This work tries to give a solution to the above two problems and propose a practical dimensional deviation estimation method for free-form surface parts. The organization of this paper is as follows: in Section

In order to inspect free-form surface parts and infer its dimensional deviation, it is necessary first to quantify the free-form surface dimensions to construct statistics. However, free-form surfaces are more complex than the regular surfaces. In contrast to regular parts, dimensions of parts with free-form surfaces is difficult to be described by a single number.

Discrete points are a set of points with a specific relative position for describing the geometry of the surface. It can be seen from the surface design process that the discrete points are the basis of the surface design. Thus the coordinate value of the discrete points can be used to describe the dimensions of the free-form surfaces. In this work, a matrix composed of coordinates of discrete points is proposed to quantify the dimensions of free-form surfaces.

A free-form surface which is denoted as

Free-form surface and its discrete points.

The sketch map of deformation of the free-form surface

Deformation of free-form surface.

Due to the influence of the systematic error and random noise in manufacturing process, the sketch map of deformation of free-form surfaces is usually as shown in Figure

Sketch map of dimensional variation of free-form surfaces.

In order to facilitate analysis and description, four definitions are given as follows.

In this section, a quantitative description method for the dimension and deformation of surface parts is proposed, which provides the basis of constructing the statistics and sampling for surface parts.

The aim of this study is to narrow the dimensional deviation confidence interval of the batch free-form surface parts to the given precision according to the samples. The accuracy of sampling is usually measured by the absolute error or relative error in a certain probability and the probability here is called the confidence level.

Assume that the batch surface parts contain

From the above analysis, the estimation of dimensional deviation of surface parts is essentially to estimate the simultaneous confidence intervals of the displacement of all the discrete points on parts surface. Therefore, the estimation of dimensional deviation of surface parts can be transformed into multivariate statistical analysis and multivariate sampling problem.

In order to accomplish the accurate estimation of the dimensional deviation of surface parts, the estimation method for dimensional deviation interval of surface parts is given in the first part of this section. A novel sequential sampling method is proposed in the second part of this section.

Assume that the batch free-form surface parts contain

The average of the displacement at point

The variance of displacement at point

Affected by system errors and random noises, surface parts inevitably deform, which results in inaccurate dimensions or even out of tolerance. For constant systematic errors, the quality characteristics of the batch parts are regarded as approximately normal distribution [

Assume that the average displacement of the samples at point

Displacement variance at point

Then the following can be derived from the theory of mathematical statistics:

The following can be obtained according to the quantile definition of

Furthermore, the dimensional deviation interval with confidence level of

However, a surface is described by a set of discrete points. Only obtaining the displacement interval of a single point is far from enough. Therefore, the simultaneous confidence intervals in (

Let the displacement of

It can be seen from (

Therefore, it is necessary to control the number of discrete points in (

Through the inspection data of selected samples, displacement variance of all the discrete points can be obtained. The points with local maximum displacement variance can be regarded as the key points and used to reflect free-form surface dimensional variation at the neighborhood of those points. Furthermore, the Bonferroni simultaneous confidence intervals of the displacement of the key points can be constructed. Finally, dimensional deviation confidence interval of surface parts can be obtained.

According to combination theory, there are

Randomly select two parts as samples and measure their dimensions.

Calculate the displacement variance of all of the discrete points,

Select the key points with local maximum displacement variance,

Calculate the estimation error of the displacement of the key points. If the estimation error is less than the given error limit for all of the key points, stop sampling. Surface dimension deviation can be accurately estimated by the samples. Otherwise, continue to select one parts as sample and proceed to Step

After obtaining the optimal samples, dimensional deviation of the free-form surface parts can be easily calculated by comparing the difference between the average dimensions of the samples and the parts design dimensions. The flow diagram of the proposed methodology is shown in Figure

Flow diagram of the proposed methodology.

In this section, a set of random numbers that generated by Monte Carlo method are taken as the process parameters to simulate the effect of random noise on manufacturing process of free-form surface parts. Taking the deformed parts obtained by simulation as the object of study, the proposed method is analyzed and discussed.

A blade with complex surface with dimension of about 120 × 90 × 70

Blade CAD model.

Structure of section curve.

Discrete points on blade surface.

The blade is fabricated by injection molding. Its material is glass fiber reinforced PA66 which is produced by BASF SE. The material trademark is A3WG3. There are two main factors, the mold and the process parameters, which affect the dimensional accuracy of blade. When the mold cavity is determined, the dimensional accuracy of the blade is mainly determined by the process parameters [

Process parameters and their values.

process parameter | value | distribution value | |
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A: | Mold temperature/°C | 80 | |

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B: | Melt temperature/°C | 290 | |

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C: | Injection time/s | 3 | |

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D: | Holding pressure/Mpa | 25 | |

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E: | Holding time/s | 15 | |

The blade finite element model including 46447 nodes and 246919 tetrahedron elements for blade injection molding is shown in Figure

The finite element model for blade injection molding.

Blade mesh model

Mold CAD model

Mold mesh model

100 sets of process parameters which generated by Monte Carlo method are used as boundary conditions for blade injection molding simulation. Simulation result and blade dimensional deviation of the first set of parameters are shown in Figure

Blade deformation.

The starting point for calculating displacement is as shown in Figure

Deformation statistical histogram of point

Statistical histogram of displacement at point P2.

Maximum point of variance of deformation.

According to the steps in Figure

Surface fitted by blade variance of deformation.

Displacements estimation error (absolute value) of the four key points

Displacements estimation error of the four points.

It can be seen in Figure

Estimation error of blade dimensional deviation, using this 9 samples, is shown in Figure

Estimation error of blade dimensional deviation.

In order to further illustrate the effectiveness of the proposed algorithm, the sampling experiment is repeated 100 times in accordance with the steps in Figure

Sample size of 100 sampling experiments.

Maximum estimation error for 100 sampling experiments.

As can be seen in Figure

In this study, displacements of the discrete points on free-form surface are employed to represent dimensional error; simultaneous confidence intervals of the discrete point displacements are employed to represent dimension deviation of the surface. These two measures are first presented. In order to demonstrate the necessity of doing these, an existing mean value estimation method is adopted to estimate the blade dimension deviation in this section. Estimation results will be compared with the proposed method.

Equation (

Obviously, only one point’s displacement can be estimated by (

Sample size of 100 comparative experiments.

As can be seen in Figure

Comparison of the estimated results of two methods.

The proposed method

Method of estimate single point

Displacements estimation error (absolute value) of the four key points

Displacements estimation error of the four points of the failed experiment.

As can be seen in Figure

Aiming to estimate the dimensional deviation of free-form surface parts, a practical method is proposed in this paper. The main conclusions of this study are as follows:

Displacements of the discrete points on free-form surface are employed to represent dimensional error. Estimating parts’ dimensional deviation is actually to estimate the simultaneous confidence intervals of the discrete point displacements.

Points with local maximum displacement variance are regarded as the key points and used to reflect free-form surface dimensional variation at the neighborhood of those points. Furthermore, Bonferroni simultaneous confidence intervals of the key points are used to estimate the confidence interval of the dimensional deviation of free-form surface parts. By increasing samples, the estimation accuracy can be improved. As a whole, displacements estimation error of key points decrease as the sample increase.

The proposed method and an classical mean value estimating algorithm are employed to estimate the dimensional deviation of a blade. The percentage of estimation error of the blade dimensional deviation that is less than 0.05mm, which is the estimation error limit, of all the 100 times sampling experiments is 99%, exceeding the given confidence level of 95%, while the percentage of the existing method is below the confidence level, only 87%. Sampling experiments indicated that estimating dimension deviation of free-form surface parts with the inspection information of a single point will lead to unacceptable estimation error. In contrast, the proposed method has high estimation accuracy and confidence.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The authors wish to express their gratitude to the finance support of National Natural Science Foundation of China (Grant No. 51475374).