## 1 Introduction  
The shapes of engineering objects are significant design factors that affect their functionality, particularly their aerodynamic aspects. For example, engineers have developed the shapes of ground vehicles for many years to reduce aerodynamic drag and increase fuel efficiency [1; 2; 3; 4; 5]. They have also attempted to reduce wind noise and increase the driving stability at high-speed operating conditions via the modification of a vehicle's body shape [6; 7; 8; 9]. Not only this, but designing the system's proper configuration also becomes important for an aircraft because the shape significantly affects its performance, such as aerodynamic drag, lift, pitching moment, and stability margin [10; 11; 12; 13]. Optimizing the shape of an object to achieve performance improvements is an iterative process that cannot be performed in a single attempt but that instead requires the initial design to be modified several times. One of the most widely used shape design methods is to parameterize the given shape (frequently called the baseline design or baseline model) by using representative variables relating to the configuration, such as length, width, or angle of the baseline model [14; 15; 16]. However, this shape parameterization is so simple that it cannot be used to explore a wide range of designs in the shape optimization process. Instead, representing the shape of the baseline model based on B-spline curves [17; 18; 19] and non-uniform rational B-splines (NURBS) [20; 21; 22] and searching for the optimal shape by using the control points of these entities is a better method that many researchers have been using in optimization problems [8; 23; 24; 25; 26; 27; 28]. Although this approach can search for a larger variety of shapes than the representative variables-based parameterization method, shape exploration is still limited and generating global and radical shape changes from the baseline design requires numerous co-constrained control points. The Free Form Deformation (FFD) [29; 30] and mesh deformation using radial basis function [31; 32; 33] are other shape deformation methods that vary the baseline design based on the control points, but these methods not only require many design parameters in finding the optimal shapes, but unreasonable optimal shapes are sometimes given when the shape optimization is performed using these methods [34]. To overcome the disadvantages of the previous shape design methods, [PERSON] et al. [35] propose an efficient shape design method called the design-by-morphing method. In their study, configurations of the multiple baseline models are represented in the functional space with spectral basis functions, and the coefficients of the spectral basis functions for new objects are computed to conduct morphing. When the design-by-morphing method is applied to generate the optimal head and tail shapes of a high-speed train, the authors show the usefulness of the method by creating the optimal high-speed train whose aerodynamic performance is enhanced by approximately 13.5% compared with the best-performing baseline model. Later, [PERSON] and [PERSON] [36] invent a multi-resolution morphing method based on the design-by-morphing method. In their method, the coefficients of the spectral basis functions of the baseline models are divided into low- and high-frequency groups, where the coefficients in the low-frequency group are likely to define the baseline models' abstract shapes, while those in the high-frequency group are associated to the local geometric features of the models; then, they create a new design by computing new spectral coefficientsin which the low-frequency coefficients are obtained by calculating the weighted sum of the low-frequency coefficients of the baseline models and the high-frequency coefficients are obtained by assigning the high-frequency coefficients of the best-performing baseline model.