International Journal of Aerospace Engineering

Volume 2015, Article ID 431824, 11 pages

http://dx.doi.org/10.1155/2015/431824

## A Reconstruction Algorithm for Blade Surface Based on Less Measured Points

College of Mechanical Science and Engineering, Nanling Campus, Jilin University, Changchun 130025, China

Received 5 July 2015; Revised 14 October 2015; Accepted 19 October 2015

Academic Editor: Roger L. Davis

Copyright © 2015 Jia Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A reconstruction algorithm for blade surface from less measured points of section curves is given based on B-spline surface interpolation. The less measured points are divided into different segments by the key geometric points and throat points which are defined according to design concepts. The segmentations are performed by different fitting algorithms with consideration of curvature continuity as their boundary condition to avoid flow disturbance. Finally, a high-quality reconstruction surface model is obtained by using the B-spline curve meshes constructed by paired points. The advantage of this algorithm is the simplicity and effectivity reconstruction of blade surface to ensure the aerodynamic performance. Moreover, the obtained paired points can be regarded as measured points to measure and reconstruct the blade surface directly. Experimental results show that the reconstruction blade surface is suitable for precisely representing blade, evaluating machining accuracy, and analyzing machining allowance.

#### 1. Introduction

Blades with free-form surface are widely used in aviation industry. Generally, the blade cannot be finished in once machining, and the measurement and reconstruction are needed repeatedly until the deviation between reconstruction model of semifinished blades and the designed ones meets the requirements. Besides that, the blade design and damaged blade repairing also need the reconstructed model from the existing ones. Therefore, the reverse technology about blade surface has received extensive attention recently in both research and industrial areas [1–3]. For reconstruction of blades, the measured points are usually obtained from the coordinate measuring machine and then transformed into a 3D model by using surface reconstruction algorithms for further work [4].

The algorithms [5] for surface reconstruction proposed in literature can be divided into two classes: triangular mesh algorithms and parametric surface algorithms. In general, the purpose of triangular mesh method is to transform the measured points data to a mesh surface. First, a triangular mesh is generated as a seed. And then by an optimized algorithm for regional growth a smooth surface passes through the optimized mesh which is reconstructed [6]. The triangulation method for measured points has a low efficiency in the calculation process [7]. And the triangular mesh surface is imprecise and easily influenced by noise for reverse engineering [8–11]. However, parametric surface can be more efficient and accurate to represent the reconstruction models, which are widely used in many practical applications.

Parametric surface algorithms mainly include quadratic surface fitting, B-spline surfaces, nonuniform rational B-spline surfaces (NURBS), lofted surface fitting, and sweep surface fitting [12–16]. When parametric surfaces are used to represent the object, the use of B-spline surface is popular due to the controllability and high accuracy. Dan and Lancheng [17] proposed a new parametric NURBS surface reconstruction algorithm. They selected measured points as control points to construct a NURBS surface; then all measured points were used to modify the surface by least squares minimization. Weiss et al. [18] proposed the surface fitting method, the cloud data is firstly fitted for different patch surfaces, and then the patch surfaces are fitted by an appropriate surface. Yin [19] proposed a similar NURBS surface algorithm which used the chosen points as the control points to construct initial surfaces and then modified them by boundary conditions. An automatic reconstruction of B-spline surface was proposed by Lin et al. [20]. The generated surface is smooth enough to meet the machining requirements, but the main drawback of the generated surface is that it usually does not coincide with the original points. Gálvez and Iglesias [21] proposed a NURBS surface reconstruction algorithm based on a PSO approach in which no pre-/postprocessing is required. The method could obtain all relevant surface data that is very time consuming. As mentioned above, all of the existing algorithms take a significant amount of time or have low accuracy, since they need to parameterize the measured points and then apply various fitting methods to construct surfaces based on minimization conditions. Moreover, the works cannot be directly used to reconstruct surfaces for blades. These algorithms were applied to reconstruct blade surfaces without considering the specific design requirements; the reconstructed surface model could not satisfy the aerodynamic performance of blades.

Usually, a blade model is reconstructed through two-dimensional section curves. The section curves are stacked to build the three-dimensional surface model through a skinning operation. Ma and Kruth [22] provided a NURBS curve and surface fitting algorithm. For the first time, the weights of control points were determined by singular value decomposition and then the control points were calculated by a least squares minimization. Abbas et al. [23] proposed a constructive method to generate a B-spline curve under certain boundary conditions. The method is not used to construct a complete curve but to achieve local control. Yoo [24] introduced a base surface algorithm by creating a smooth implicit surface from a sequence of CT image data and then the reconstruction surface is constructed by using the base surface. A B-spline surface fitting algorithm introduced by Yuwen et al. [25] directly extracted section curves and created a B-spline surface by skinning the section curves. Wu et al. [26] developed an adaptive slicing method for cloud data. The point cloud was segmented to a series of layers whose thickness was defined by shape-error tolerance. Zhang et al. [27] proposed a similar slicing method that used an iterative algorithm to acquire specific shape-error tolerance. Hsiao and Chen [28] proposed a surface reconstruction method based on the feature curves. The main point of the method is the consideration of surface patches stitching. To achieve an accurate reconstructed surface model, the section curves must be recognized as accurately as possible. A surface reconstruction method based on the imperfect points was proposed by Li et al. [29]. First, interpolation B-spline surface is generated according to the given points and then generating a new B-spline surface after removing some points. Finally, the influence is analyzed by comparing the two surfaces. These surface reconstruction methods usually require uniform and intensive enough points to reconstruct the curve and obtain accurate reconstruction surfaces [30]. However, in actual machining process of blades, it is usually to measure a small amount of points, in order to reduce the measuring time and improve the machining efficiency [31]. The above methods are not applicable to reconstruct a smooth blade surface based on less measured points.

To obtain a smooth reconstruction surface model for blade according to the given measured points, a novel method which considers the influence of design parameter on aerodynamic performance for blade is introduced in this paper. The proposed method has two major advantages: (1) it is efficient to compute the section curves for surface reconstruction since the processing reserves the design characteristic enough to ensure aerodynamic performance of blade; (2) the method generates rectangular net from less measured points for reconstructing B-spline surface quickly and guides the choice of the paired points as measured points effectively.

This paper is organized as follows. Section 2 details the algorithm for reconstructing the blade surface model. Section 3 presents several experimental results. Some concluding remarks are drawn in Section 4.

#### 2. The Proposed Reconstruction Method

This section details the B-spline surface reconstruction algorithm. A B-spline interpolation surface algorithm is proposed to obtain the blade surface model based on section curves fitting algorithm. In the first step, based on the coordinates of the measured points, a series of section curves is created covering the measured points. In the second step, the surface model is created. The detailed reconstruction process is shown in Figure 1.