###### Abstract
The adjoint variable method is a popular numerical method that is often used in sensitivity analysis and optimization problems. In this method, the objective function and constraint conditions of the optimization problem are represented as operators and vectors, which are used to calculate the partial derivatives of the design parameters. The main advantage of the adjoint variable method is that its computational complexity is much lower than that of the direct differential method, and it is widely used in engineering optimization and design because of its rapidity in calculation. However, for the nonlinear prediction model, its application range may be limited. The optimization of the racing car frame plays an important role in improving the performance of the car. At present, more research has focused on the optimization of the frame. In recent years, many scholars have conducted some research on frame optimization. [PERSON] et al. [22] considered the strength and mode of the frame under complex road conditions and improved the relevant parameters through optimization, but only the topology optimization was performed for the cockpit, and the mass became heavier. [PERSON] et al. [23] used the optimization method SIMP to optimize the topology of some frame structures, which improved the modal performance. The frame mass was reduced by size optimization, but the design parameters were not determined by sensitivity analysis, so the size optimization was blind. [PERSON] et al. [24] made the mechanical properties of the frame superior through optimization analysis and realized the lightweight of the frame. However, only the topology optimization of the front cabin was performed, and the sensitivity analysis was not used. [PERSON] et al. [25] mainly improved the torsional stiffness through sensitivity analysis, but this method had not been applied in the field of lightweight design. To sum up, in the field of the formula racing car frame, most of the literature only does topology optimization on the local part, and the size optimization does not set the appropriate design variables, which is blindness. In addition, in international journals, many studies have applied sensitivity analysis to other parts of the car, such as [26; 27; 28]. In recent years, it has been difficult to find the application of sensitivity analysis in the field of lightweight frame design in the literature, so this research is essential for frame design. The above papers are compared with the research in this paper, as shown in Table 1. The main contributions of this paper are as follows: (1) Global topology optimization of the frame based on the variable density method. Compared with the local topology optimization, the global topology optimization optimizes the frame of the cockpit and the rear cabin at the same time. The optimization results will retain the material in the area with higher unit density to obtain the trend of the whole vehicle pipe fittings, so that the designed frame structure is more accurate and the anti-deformation performance will be better. (2) The sensitivity analysis is introduced into the field of racing frames. The adjoint variable method is used to analyze the sensitivity of the frame, and the influence factors of each variable are obtained according to the results. The size of the frame is optimized based on the variables with greater influence. Compared with other sensitivity analysis methods, this method provides an effective way to calculate the gradient of design parameters and has a faster calculation speed. It can determine the best design parameters in a shorter time, explore a wider design space, and more effectively optimize the quality and performance of the frame. More importantly, it provides appropriate design parameters for size optimization and improves accuracy, so that the effect of lightweight design will be better. In the second section of this paper, the model establishment of the frame is introduced. The frame structure is simulated and analyzed by Hyper Works software under various working conditions.