# Large Deformation Modeling of Wing-Like Structures Based on Support Vector Regression
## 1 Introduction  
With the development of modern, high-performance aircrafts, structure flexibility is getting larger and larger to meet the requirements of high flight performance. Geometric nonlinearities must be considered in the aircrafts' design. Large deformation subjected to the aerodynamic forces during flight leads to essential effects on stiffness characteristics and aerodynamic configuration, which obviously affect stability, dynamic response and flight performance [1; 2; 3; 4; 5]. Accurate calculation on large deformation is of great significance in modern aircrafts' design [6]. Full-order models, like the finite element method (FEM) in physical degrees of freedom (DoFs), often carry a high computational burden, especially for complex wing-like structures [7; 8]. With geometric nonlinearities, the linear modal techniques cannot be used directly to dramatically simplify the structural model. High computational burden gradually becomes the key factor restricting large deformation simulation, especially in aeroelastic or optimization analysis. A reduced-order model (ROM) based on a modal form with test cases is an attractive theory for large deformation economic calculation. Several authors have applied the approach in static or dynamic response simulation in the recent past [9; 10; 11; 12; 13; 14; 15; 16; 17]. A review of ROM methods for geometric nonlinear structures was presented by [PERSON] et al. [18]. ROM methods considering geometricnonlinearities require a nonlinear stiffness expression given in advance. [PERSON] [9] used quadratic and cubic terms of the basis function to describe nonlinear stiffness. For determination of stiffness coefficients in ROM, two approaches can be chosen. The first one, which is called the direct method, is intrusive, as it explicitly expresses the nonlinear stiffness coefficients [18]. But, the stiffness coefficients are not available in most commercial FEM source. For the complex model, the approach should be impractical [19]. The second one, which is called the indirect method, is non-intrusive, and is the most used method in ROM construction. In this approach, a series of load cases is specified, and the corresponding deformations are calculated by a nonlinear FEM software package, like MSC.Nastran or Abaqus. The nonlinear stiffness coefficients can be solved by a least squares regression analysis through test cases, including load cases and deformations. The accuracy of ROM depends on the choice of test cases [13,20]. Load vector in test cases must ensure that the desired nonlinear effects are excited in the whole procedure [15,21]. Due to the requirement and characteristics proposed, the construction of ROM needs lots of test cases to ensure the calculation accuracy. Even the calculation procedure of ROM is much faster than the full-order model, but a lot of time will be spent in obtaining test cases. Support vector regression (SVR) is a powerful nonlinear regression approach, which belongs to the support vector machine (SVM) [22,23]. SVR introduces a nonlinear kernel to model the nonlinear function of multiple variables and uses insensitive loss function to accept small deviations from objective function. Compared to other regression techniques, SVR is based on the structural risk minimization principle [24] so that it matches the available training sample reasonably well, and generalizes well to the new testing dataset [25,26]. In this regard, SVR has better generalization capacity and is also suitable for small sample size regression. [PERSON] et al. pointed out that SVR performs relatively better with less samples and more features [27]. With the least square regression approach used in ROM described before, empirical risk minimization is employed to force the regression model to converge to the sample target value as much as possible, without consideration of the structural characteristics of the regression model [25]. Traditional ROM needs lots of datasets to meet the accuracy requirements.