# Optimizing Vehicle Body Cross-Sections Using a Parametric Mathematical Model
## 1 Introduction  
Conceptual design plays a crucial role in determining the performance and structure of automotive bodies. Various studies have explored methods and models that are both easy to modify and computationally efficient. One approach focuses on using technical data from key vehicle connections to enhance crash safety and static/dynamic stiffness [1]. Another proposes a conceptual design tool utilizing optimization algorithms to optimize overall body performance [2]. Structural dynamics models have been developed to predict and optimize body-in-white structures, simplifying models and improving design efficiency [3]. A multidisciplinary approach addressing static loads and crash safety through energy weighting and topology optimization has also been explored [4]. Additionally, innovative methods have been introduced to model the flexibility effects of connected components in automotive bodies [5], addressing limitations of traditional beam theory. While current models often rely on finite element principles, challenges such as extensive data input, long design cycles, slow optimization, and high computational costs remain significant. [PERSON] et al. [6] proposed a more accurate and robust sensitivity calculation method for nonlinear eigenvalue problems through theoretical derivation using the transfer matrix approach, laying new theoretical foundations and computational tools for research and applications in related fields. The method proposed by [PERSON] et al. [7] discretizes non-uniform and curved beams into constant beam segments, even in cases involving non-coincident nodes or skewed angles. This modular approach simplifies complex modeling procedures significantly. By employing unified transfer matrices that account for variouscombinations of shear deformation, rotational inertia, and axial loads, the method provides a new perspective for dynamic modeling. In this paper, based on the body conceptual design of the body performance requirements, using transfer matrix method to establish a meshless fully parameterized simplified body conceptual model, which can be easily modified and quickly calculated, and constructed the body cross-sectional parameters, dimensional parameters on the body stiffness performance parameters of the functional relationship. The cross-sectional parameters and dimensional parameters of the body are constructed as a function of the body stiffness performance parameters. With the goal of the lightest body mass, the geometric parameters of the beam cross-section are optimized by using stochastic gradient genetic algorithm. This paper aims to address the challenge of optimizing the body stiffness performance in structural design, which is a critical design requirement. Traditional methods often involve complex and resource-intensive iterative processes, making them inefficient when adjusting body design parameters. In particular, there are limitations in traditional methods when establishing the relationship between stiffness performance and the body's cross-sectional dimensions and geometric parameters. To solve this problem, this paper proposes a meshless, fully parameterized simplified body conceptual model based on the transfer matrix method. This model allows for rapid calculations and flexible adjustments, enabling modifications to meet various design requirements and significantly improving the efficiency of design optimization. Through this model, the relationship between the body's cross-sectional and dimensional parameters and its stiffness performance is clearly established, enabling fast and accurate design optimization. Compared to existing traditional methods, our model offers significant advantages in terms of calculation time, flexibility, and stiffness prediction accuracy. We will demonstrate, through comparison with current industry standards, how our approach reduces computational complexity while maintaining or improving result accuracy.