# A Novel Shape-Adjustable Surface and Its Applications in Car Design
## 1 Introduction
[20] made Coons surfaces of the second type adjustable under the action of two shape parameters. Analogously, [PERSON] et al. [21], [PERSON] et al. [22] and [PERSON] [23] replaced the blending function in the second type of Coons patch with a second-order trigonometric blending function, a blending function with shape parameter \(\lambda\), and a RBF-Hermit function, respectively, to make the patch shape modifiable. Nevertheless, these two types have their own disadvantages in free-form surface modeling. The first type of Coons cannot be further modified finely, and is only suitable for simple modeling. For the second type, including its various variants' shape parameters, the compatibility conditions of the cross-boundary derivatives are very stringent, and it is also difficult to obtain the data directly in design practice [24]. This represents a primary issue that this study aims to resolve. Automobile modeling comprises not only aesthetic features, but also aerodynamic ability. The traditional method of measuring the air drag coefficient is the wind-tunnel test. In the 1980s, the Motor Industry Research Association from the UK established the MIRA car model, which can be decomposed into more than ten surfaces connected by rounded corners, and used it as a standard model for tunnel testing. However, because of the high cost of the tunnel test, computational fluid dynamics (CFD) simulation is more widely used in the conceptual design stage of vehicle models, in which the virtual CAD model reproduces the basic fluid dynamics features of real cars in CFD simulation software and becomes the major test object to communicate between the designer's scheme and CFD technology. [PERSON] et al. [25] proposed a new boundary representation method of the CAD model, which is more suitable for mesh generation in CFD preprocessing. [PERSON] et al. [26] built a bioinspired CAD car model to take advantage of the efficient movement of boxfish in water due to its streamline features. The shape of this model is unified and integral, and its characteristic lines consist of the intersection of several subsurfaces. In the long-term research of [PERSON] et al. [27; 28; 29], they built various CAD car body models with different complexity to explore the influence of A-pillar and C-pillar of automobiles of different shape (rounded or angular) on driving pitching stability. In their research in 2014 [29], they used the large eddy simulation method on two CAD models with full details; they drew the conclusion that the vortex shed from the A-pillar edge results in a destabilizing tendency while the vortex shed from the C-pillar edge results in a stabilizing tendency. [PERSON] et al. [30] reduced the drag coefficient by nearly 10% by optimizing the angle of departure and the shape of the Rear body diffuser in a racing car CAD model. In the study from [PERSON] et al. [31] on the influence of various eddy viscosity turbulence models on the CFD simulations of a particular racing car type, they also pointed out that the shape of some specific surfaces (i.e., backlight-decklid junction, spoiler base, roof rails, shark-fin) would affect the results of CFD simulation. In the comparison between the large eddy simulation model and the wall-modeled large eddy simulation model from [PERSON] et al. [32], a realistic generic car body CAD model according to the styling of the Audi A4 and the BMW 3 series was adopted. In this research, they pointed out that only the true and accurate CAD models could fully reproduce the real aerodynamic behavior present in automotive geometries. However, this research also held the idea that making use of simplified bodies is the most efficient tool to study fundamental aerodynamic processes and focus on specific features of automotive aerodynamics for the tester. Therefore, in their other studies [33; 34], [PERSON] et al. utilized the CAD model with different degrees of simplification to assess various LES models and to simulate the airflow around the wheels. In the research of [PERSON] et al.