International Journal of Aerospace Engineering

Volume 2017 (2017), Article ID 1874729, 15 pages

https://doi.org/10.1155/2017/1874729

## Three-Dimensional CST Parameterization Method Applied in Aircraft Aeroelastic Analysis

Shaanxi Aerospace Flight Vehicle Design Key Laboratory, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China

Correspondence should be addressed to Hua Su

Received 3 March 2017; Accepted 18 June 2017; Published 4 October 2017

Academic Editor: Kenneth M. Sobel

Copyright © 2017 Hua Su 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

Class/shape transformation (CST) method has advantages of adjustable design variables and powerful parametric geometric shape design ability and has been widely used in aerodynamic design and optimization processes. Three-dimensional CST is an extension for complex aircraft and can generate diverse three-dimensional aircraft and the corresponding mesh automatically and quickly. This paper proposes a parametric structural modeling method based on gridding feature extraction from the aerodynamic mesh generated by the three-dimensional CST method. This novel method can create parametric structural model for fuselage and wing and keep the coordination between the aerodynamic mesh and the structural mesh. Based on the generated aerodynamic model and structural model, an automatic process for aeroelastic modeling and solving is presented with the panel method for aerodynamic solver and NASTRAN for structural solver. A reusable launch vehicle (RLV) is used to illustrate the process for aeroelastic modeling and solving. The result shows that this method can generate aeroelastic model for diverse complex three-dimensional aircraft automatically and reduce the difficulty of aeroelastic analysis dramatically. It provides an effective approach to make use of the aeroelastic analysis at the conceptual design phase for modern aircraft.

#### 1. Introduction

In the wake of requirements for high lift-drag ratio aerodynamic shape and light-weight structure, the aeroelastic phenomena caused by interaction between fluid and structure have a growing influence on the integrated performance of modern aircraft [1, 2]. Aeroelastic analysis becomes an important process for modern aircraft design [3, 4]. Especially in the conceptual design phase, main performance of an aircraft is determined during this phase and, therefore, how to carry out the aeroelastic analysis quickly and stably to improve the integrated performance and the design rationality of the aircraft scheme will help a lot in the following aircraft design.

Aeroelastic analysis is given attention by lots of researchers [5]. Many tools such as ZAERO [6], ENSAERO [7], and NeoCASS [8] are developed to perform aeroelastic analysis based on the frequency domain analysis method and time domain analysis method and have been widely applied on high aspect ratio wing, unmanned aerial vehicle, and hypersonic aircraft. Although there exist some mature aeroelastic analysis and solving method, the aerodynamic modeling and structural modeling are still a complex and time-consuming process. First, the parametric geometric shape and modeling tools should work together closely and automatically; second, the aerodynamic solver and structural solver should coordinate with each other to ensure that the aeroelastic analysis procedure is executed consistently and accurately. In the conceptual design phase, the design scheme of aircraft usually needs constant modification to improve performance. The size parameters and structural topology of the scheme need to change frequently. It is difficult for the traditional CAD-based geometry modeling method to satisfy the need of rapid geometry iteration and large range modification. Aeroelastic analysis applied in the conceptual design phase faces the following problems:(1)The aerodynamic model and structural model established in many aeroelastic literatures were complicated; the sizing of the aerodynamic shape and structural layout is difficult and time-consuming, which makes it hard to meet the demand for rapid modification at the conceptual design phase.(2)Because of the differences between aerodynamic modeling and structural modeling in mechanism, the aerodynamic mesh and structural mesh are not compatible. The data conversion is used additionally to interchange aerodynamic force and structural deformation between the aerodynamic model and the structural model. Extra computation overhead is needed, and some errors are introduced inevitably.(3)Quantity and position of the structural inner elements, such as beam, bulkhead, spar, and rib, are hard to be modified using the traditional CAD-based geometry modeling method. The structural layout is vested and cannot obtain the optimal scheme by topology optimization.

These problems hinder the application of aeroelastic analysis in the conceptual design phase. More effective aerodynamic modeling and structural modeling method should be studied to simplify and improve the aeroelastic modeling and analysis process. The CST method is an analytical method developed by Kulfan [9, 10]. It combines a class function representing a specific class of shapes and a shape function defining the deviation from the class function. These provide an efficient shape parameterization ability on complex geometry using fewer design variables. There are some comparisons with other shape parameterization methods [11, 12], showing that the CST method has advantages in smoothness, mathematical efficiency, fitting performances, and Intuitiveness. CST has been widely used to parameterize and optimize two-dimensional airfoil [13, 14] and simple 3D aircraft [15, 16]. Liu [17, 18] presents a multiblock CST method to model the hypersonic aircraft with complex shape, which can join adjacent surfaces smoothly and retain the good properties of the CST method. Leal [19] proposes an aerostructural optimization method for determining in a preliminary manner morphing wing configurations that provide benefits during various disparate flight conditions with CST parameterization.

The authors in [20, 21] proposed a universal three-dimensional CST method for geometry modeling of complex aircraft. It generates complex three-dimensional geometric shape to support various aircraft aerodynamic shape modeling, which gives a simple and effective way to aerodynamic optimization. This novel three-dimensional CST method is extended in aeroelastic analysis of complex aircraft in this article. A novel parametric structural modeling method is proposed based on gridding feature extraction from the aerodynamic mesh generated by the three-dimensional CST method. An automatic and effective way for aeroelastic modeling and analysis is also established. The structure of this article is as follows. First, the basic principle of the three-dimensional CST method is introduced briefly; then the aircraft characteristic components library is established, including fuselage, wing, and empennage; on this basis, the structural modeling method is presented in detail, and the aeroelastic modeling and analysis process is constructed; finally, a static aeroelastic analysis example is used to verify the proposed aeroelastic modeling and analysis process.

#### 2. Three-Dimensional CST Method

CST method has an efficient and brief shape description for the two-dimensional airfoils and simple three-dimensional geometries. This section gives an improvement to expand the original two-dimensional CST method to three dimensions. A universal three-dimensional CST method is proposed to provide adjustable parameterization ability to complex three-dimensional geometry.

##### 2.1. Basic Cross Section Definition

Most of the aircraft geometry can be described by innumerable cross sections along the axial direction. Combining an analytical function (the class function) with a parametric curve (the shape function), the basic cross section can be defined by the B-spline CST method [22] as follows:where . is the lateral coordinate. is the normalized lateral coordinate. is the total length of the basic cross section in the lateral direction. is the normal height in the lateral position. is the eccentric distance in the lateral position. and are the class function and the shape function defined in CST method. and are control parameters of the class function. For the symmetrical section, is equal to . The class function is defined as follows:

The basic cross section is assembled by the upper and the inverted lower . If we set to 0, ignore the shape function, and change the control parameters of the class function simultaneously, different sections can be generated. Various cross section shapes can be generated by changing control parameters of the class function.

##### 2.2. B-Spline Shape Function

The B-spline function can adjust the range of influence by selecting different orders of sub-Bernstein polynomials. It has better local control ability and computation efficiency than Bernstein polynomials. So it has been chosen as the shape function of the CST method in the basic cross section definition. The B-spline function is combined by the massive low-order Bezier curves to approximate the high-order curve. It is defined as follows:where is the weight factor. order B-spine basic function is defined as the piecewise polynomials at node vector . is the B-spline basic function defined at , which can be obtained by recurrence formula De Boor-Cox.

The basic cross section defined with B-spline function is as follows:

##### 2.3. Three-Dimensional CST Method

The three-dimensional geometry can be considered as a series of original cross sections arranged parallel along the axial direction. By defining proper cross sections using the method mentioned above, we can get an analytical surface to express geometric shape. In order to introduce an analytical axial rule, the coefficient is replaced by the same B-spline CST definition in formula (5). So the two-dimensional CST function can be expanded to a three-dimensional version, which is shown as follows:where . is the axial coordinate. is the normalized axial coordinate. is the total length of the generated surface in the axial direction. Bringing formula (6) to formula (5), we have the definition of the three-dimensional analytical surface.

Formula (8) is the analytical expression of the extensional three-dimensional surface, where is the third-dimensional coordinate value along the direction based on the two-dimensional normalized coordinate and . * and * are the CST’s class functions with control parameter , , and . * and * are the basic functions of the B-spline, which constitute the CST’s shape function. is the discrete control weight factor in the up and low surfaces. Lower case and are the orders of B-spline function, which are expressed as the numbers of lateral and axial control points. The total points of the geometric surface are . More control points mean more design parameters and better parametric geometric shape design ability. When = 0 and = 0, the discrete control weight factor is equal to 1, the shape function is also equal to 1, and then entire geometric surface is controlled by the control factors , , , and of the class function. is the eccentric distance in normal position, and the default value is 0.

The geometric surface defined in formula (8) can be described as the third-dimensional coordinate value calculated by the two-dimensional mesh points dispersed as the lateral and axial control points. and are the normalized coordinates defined in . We also need to define the profile in the lateral direction with respect to the axial coordinate . The profile is expressed the same as the B-spline CST method in formula (5). is the discrete control weight factor in the lateral coordinate. From the above definition, we obtain the three-dimensional analytical surface, which can be transformed into global Cartesian coordinate as formula (10).where , . The outward boundary box is used to control the size of the geometric surface, and then we get the full design parameters of the entire geometric surface, which includes the following: Axial and lateral length: Vertical height: Sectional section control factor: Vertical section control factor: Lateral section control factor: Surface weight factor: Lateral weight factor:

, , , are the size parameters of the geometric surface. *, *, , , , are the control parameters of the class function. and are the discrete control weight factors of the shape function. When and are equal to 1, the number of total parameters is constant and also minimum. When the and are expressed as matrix, some control points with the same size of the matrix elements are located in the geometric surface to improve the complexity of the geometric shape. So the size of design parameters could be adjusted dynamically. With these weight factors, the three-dimensional CST method has more flexible parametric geometric shape design ability.

##### 2.4. Basic Geometric Shape

The basic geometric shape is defined by the upper geometric surface and the inverted lower geometric surface. Figure 1 gives some geometric shapes generated by the above three-dimensional CST method with different control parameters. upp, low, and are the matrix generated by of the upper surface, of the lower surface, and of the lateral coordinate. The shapes in the same lines are generated with the same value of and , but with different , *, *, and . The shapes in the same columns are generated with the same values , , , and , but with different , , and upp. The modified shapes in the third line are generated with different sizes of control weight factors of the upper surface compared with the shapes in the second line.