## 2 Topology Optimization Method of Car-Body Structure
### Variable-Density Method  
The variable-density method was developed from the homogenization method, which deals with the intermediate density. It is one of the mainstream topology optimization methods based on finite elements. The optimization criteria adopted by the variable density method have the characteristics of a fast convergence speed, few iterations, and small computation, which is the focus of current structural topology optimization methods. Unlike the homogenization method, the variable-density method mainly uses material description for topology optimization. Upon introducing a reverie material, the material density is between 0 and 1, where 0 represents the hollowed-out state, 1 represents the solid state, and a density between 0 and 1 represents the point between the hollowed-out and solid states. The material's physical characteristics and the element's relative density depend on the interpolation function. Considering the relative density of each unit as a design variable, the number of design variables can be significantly reduced, and the computation is also reduced accordingly.  
The design domain of the variable-density method is discretized into a finite element set defined by the element set \(\mathrm{N_{X}}=\{1,2,\ldots,|N_{X}|\}\) in the x-direction and the element set \(N_{Y}=\{1,2,\ldots,|N_{Y}|\}\) in the y-direction. The element density is taken as the design variable:  
\[(0\leq\rho_{(x,y)}\leq 1,(x,y)\in N_{X}\times N_{Y}) \tag{1}\]  
According to the optimality criterion, the element stiffness is effectively controlled. Then, the overall stiffness of the structure can be reasonably regulated, and the materials can be redistributed within the design domain. Thus, the topology optimization structure with the optimum structural stiffness and material distribution can be obtained [24].  
\[E=\rho_{e}E_{0} \tag{2}\]  
\[E=\frac{\rho_{e}}{1+q(1-\rho_{e})}E_{0} \tag{3}\]  
Solid isotropic material with penalization (SIMP) is a commonly used density-stiffness interpolation model, a common technique in topology optimization problems. The model assumes that the material density is constant within the cell and takes it as a design variable. In order to simplify the calculation and improve the efficiency, the material properties are simulated by the exponential function of the cell density. The SIMP method introduces relative density \(\rho_{e}\) and penalty factor \(P\). When \(0\leq\rho_{e}\leq 1\), the element density is limited by penalty factor \(P\), so the structural elements' density is as close as possible to 0 or 1. If the element density is 0, the material can be deleted; if the element density is 1, the material should be filled. For the SIMP interpolation model, the larger the penalty factor is, the better it is. When the penalty factor takes different values, the penalty effect is also different. The element density can be expressed as follows.  
\[\varphi(x_{i})=x_{i}^{P},x_{i}\in[x_{\min},1],i=1,2,3\ldots,x_{n} \tag{4}\]  
The relation between element density and elastic modulus is \[E(x_{i})=E_{\min}+\varphi(x_{i})(E-E_{\min}),i=1,2,3\ldots,n \tag{5}\]  
where \(E(x_{i})\) is the elastic modulus of the element, \(E_{\min}\) is the elastic modulus of the low-strength material element, and \(x_{i}\) represents the relative density of each element. To ensure the stability of numerical calculation, usually, \(E_{\min}=E/1000\) and \(0<E_{\min}\leq E(x_{i})\leq E\). The general optimization mathematical model of structural topology optimization can be formulated as follows.  
\[\begin{array}{ll}\text{Minimize}:&f(X)=f(x_{1},x_{2},\ldots,x_{n})\\ \text{Subjectro}:&g_{j}(X)\leq 0\\ &j=1,2,3,\ldots,\text{m}\end{array} \tag{6}\]  
where \(n\) represents the number of design variables, \(X\) is the optimization design variable, \(f(X)\) is the optimization objective function, \(g(X)\) represents the design response requiring constraints, and \(j\) is the number of constraint equations.