## 4 Results and Discussion
### Two-Dimensional Examples  
In the 2D drag minimization problem, the model is set up as shown in Figure 2, which is the visualization of the schematic Figure 1. The length and width of the flow domain \(\Omega\) are 10L and 7L, respectively, and the length and width of the design domain \(\Omega_{s}\) are 2L and L, respectively, with L = 1 (dimensionless). The fixed-point C is used to constrain the generation of solid material, and the structure generated in the first step of optimization is assumed to be the initial body shape. In order to solve the Navier-Stokes equation, the initial values of the boundaries are also needed as definite solution conditions. The far-field condition at the inlet \(\partial\Omega^{in}\) and the boundary \(\Gamma\) is a uniform velocity in the \(X\)-direction of \(\mathfrak{u}_{0}=\text{U}=1\) (dimensionless), and the constraint at the outlet \(\partial\Omega^{out}\) is a normal stress \(p=0\) (dimensionless).  
We consider the external flow with uniform velocity flowing around the isolated body. In this case, the fluidic drag is usually expressed as the boundary integral of the stress vector on the body surface, that is, the integral of the compressive stress and the viscous stress. However, because the shape of the topology optimization structure is unknown in advance, it is not applicable to this problem. In 2012, [PERSON] put forward various objective function formulas for drag minimization and lift maximization, including the volume integral and area integral [50], through approximate transformation. In addition, the volume integration formula provides a better estimation of lift and drag coefficients than the conventional line integration along the streamline body; The volume integral formula is not sensitive to small changes in the mesh generated around the object [51]. Therefore, the objective function we use is in the form of the volume integral.  
In the porous media model, the effect of solid material on the fluid is expressed as an artificial Darcy friction in the momentum equation, so the force \(\mathbf{F}\) on the solid exerted by the fluid should be expressed as the volume integral of--\(\mathbf{f}\) due to the action-reaction law. The expression of \(\mathbf{F}\) can be defined as:  
\[\mathbf{F}=-\int_{\Omega}\mathbf{f}\,d\Omega=\int_{\Omega}\mathbf{\alpha u}\, \,\mathrm{d}\Omega \tag{17}\]  
Thus, fluid dynamic drag \(D\) can be expressed as the X-axis component of Equation (17) as:  
\[J_{1}=D=\int_{\Omega}\mathbf{\alpha u}\,\,\mathrm{d}\Omega \tag{18}\]  
where \(\mathbf{u}\) is the X-axis component of velocity vector \(\mathbf{u}\) and \(J_{1}\) is the objective function 1. Meanwhile, the existing drag formula through surface-stress integration is transformed by the Gaussian formula to the volume integral form as:  
\[\begin{array}{l}\int_{\Gamma}\left[(\
abla\mathbf{u}+\
abla\mathbf{u}^{T})-p \mathbf{I}\right]\,\mathbf{n}\,\,\mathrm{d}\Gamma=\int_{\Omega}\
abla\cdot \left[(\
abla\mathbf{u}+\
abla\mathbf{u}^{T})-p\mathbf{I}\right]\mathrm{d} \Omega\\ =\int_{\Omega}\left(\mathrm{Re}\mathbf{u}\cdot\
abla\mathbf{u}-\mathbf{f} \right)\mathrm{d}\Omega=\int_{\Omega}\left(\mathrm{Re}\mathbf{u}\cdot\
abla \mathbf{u}+\mathbf{\alpha u}\right)\mathrm{d}\Omega\end{array} \tag{19}\]  
and the convection term is retained. Therefore, objective function 2 can be described as:  
\[J_{2}=\int_{\Omega}\left(\mathrm{Re}\mathbf{u}\cdot\
abla\mathbf{u}+\mathbf{ \alpha u}\right)\mathbf{e}\,\,\mathrm{d}\Omega \tag{20}\]  
where \(\mathbf{e}\) is the unit vector in X-direction. Therefore, \(J_{1}\) and \(J_{2}\) are just different expressions of drag, with difference is with or without a convective term. \(J_{2}\) is to characterize drag by adding the convective term on the basis of \(J_{1}\).