## 2 Theory
### Filtering and Parameterizations  
Filtering enables the removal of checkerboard patterns [30; 31] in optimal design results and the computation of layouts independent from mesh refinements [31]. Mesh-independent density filtering is also used [15; 32]. The main concept of this filtering is defining the physical element density to be a weighted average of the neighboring design variables:  
\[\widetilde{\rho_{i}}\ =\ \frac{\sum_{j\in_{e,i}}w\big{(}\mathbf{x}_{j}\big{)}v_{ j}\rho_{j}}{\sum_{j\in_{e,i}}w\big{(}\mathbf{x}_{j}\big{)}v_{j}} \tag{8}\]  
where \(\widetilde{\rho_{i}}\) is the filtered density of element \(i\), \(\rho_{j}\) is the design variable of element \(j\), \(\mathbf{x}_{j}\) is the location of element \(j\), \(v_{j}\) is the volume of element \(j\), and \(N_{e,i}\) is the number of the set of the neighborhood element \(i\) within a certain filter radius \(r\). \(w\big{(}\mathbf{x}_{j}\big{)}\) is a weighting function specified by \(w\big{(}\mathbf{x}_{j}\big{)}=r\ -\ \|\mathbf{x}_{j}\ -\ \mathbf{x}_{\text{c}}\|\). In order to mitigate the gray region between the solid and void regions, a threshold projection filter is applied using a smoothed Heaviside function [25]:  
\[\overline{\widetilde{\rho}}_{e}\ =\ \frac{\tanh(\beta\eta)\ +\ \tanh(\beta( \widetilde{\rho}_{e}-\eta))}{\tanh(\beta\eta)\ +\ \tanh(\beta(1-\eta))} \tag{9}\]where \(\eta\) is the threshold in the projection filter, and \(\beta\) is a variable to control the sharpness of the projection filter; as \(\beta\)\(\rightarrow\)\(\infty\), Equation (9) becomes a discrete Heaviside projection. \(\overline{\overline{\rho}_{e}}\) is a physical design density filtered by the threshold projection filter, which is used to represent the degree of occupancy of the material in element \(e\). The solid isotropic material with penalization (SIMP) method [33] is used to interpolate the [PERSON]'s modulus of each element, yielding the following expression for the interpolation:  
\[E_{e}=\overline{\overline{\rho}_{e}}(E_{1}-E_{0})+E_{0} \tag{10}\]  
where \(E_{1}\) is the [PERSON]'s modulus of the solid phase (the black regions), and \(E_{0}\) is the [PERSON]'s modulus of the void phase (the white regions), which generally have a relation of \(E_{0}=10^{-9}\).\(E_{1}\). The superscript \(p\) is a penalization factor (generally, \(p=3\)). [PERSON] et al. [19] proposed an energy interpolation scheme to mitigate the numerical instability due to the excessive distortion in low stiffness elements, which is inevitable in the topology optimization process. The energy interpolation form of element \(e\) is expressed as:  
\[\phi_{e}(\mathbf{u}_{e})\ =\ [\phi(\alpha_{e}\mathbf{u}_{e})-\phi_{L}(\alpha_{e }\mathbf{u}_{e})+\phi_{L}(\mathbf{u}_{e})]E_{e} \tag{11}\]  
\(\phi\) is the stored elastic energy density function, and \(\phi_{L}\) is the stored elastic energy density under small deformation. \(\alpha_{e}\) is the interpolation factor for element \(e\), which is 1 for solid elements and is 0 for void elements. Therefore, solid elements are treated as nonlinear elements in the numerical analysis (i.e., \(\phi_{e}\ =\ \phi(\mathbf{u}_{e})E_{1}\)), whereas void elements are treated as linear elements (i.e., \(\phi_{e}\ =\ \phi_{L}(\mathbf{u}_{e})E_{0}\)). The interpolation factor \(\alpha_{e}\) is expressed similarly to the threshold projection filter in Equation (9) using a smoothed Heaviside function:  
\[\alpha_{e}\ =\ \frac{\tanh(\beta_{1}\rho_{0})\ +\ \tanh(\beta_{1}\left( \overline{\overline{\rho}_{e}}^{p}-\rho_{0}\right))}{\tanh(\beta_{1}\rho_{0}) \ +\ \tanh(\beta_{1}(1-\rho_{0}))} \tag{12}\]  
where \(\rho_{0}\) is a threshold between 0 and 1. If \(\beta_{1}\)  
If the three points are defined as the target points, as shown in Figure 1, and the optimization process is performed without the slope constraint, the topology optimization process maintains the inherent nonlinearity of the initial structure.