# Sketch-Guided Topology Optimization with Enhanced Diversity for Innovative Structural Design
## 3 Method
### TO Process with Neural-Encoded Density Function  
The mathematical formulation of solving the aforementioned design problem with the SIMP method is given in Equation (1). The optimizable variables of the design problem are the material density values of the discrete elements, \(\mathbf{\rho}=\{\rho_{e}\}\), where \(\rho_{e}\) is the material density of the e-th element within the value range [0, 1]. \(\rho_{e}=1\) represents that this element has solid material and \(\rho_{e}=0\) represents that this element is void. The design objective is to minimize the total structural compliance \(C\).  
For a specific design problem to be solved, the user-defined hyper-parameters include the applied force \(\mathbf{F}\), the prescribed volume \(V^{*}\), the element stiffness matrix for solid material element \(\mathbf{K}_{e}^{0}\), the material's Young's modulus \(E_{0}\), the element volume \(v_{e}\), and the penalization exponent \(p\). In Equation (1), the first constraint, \(\mathbf{K}(\mathbf{\rho})\mathbf{u}=\mathbf{F}\), is the static equilibrium equation, where \(\mathbf{K}(\cdot)\) is the finite element stiffness matrix as a function of the element densities \(\mathbf{\rho}\) assembled from the element stiffness matrix \(\mathbf{K}_{e}(\mathbf{\rho})\) computed by Equation (2). \(\mathbf{u}\) is the displacement field computed from the static equilibrium equation. The second constraint represents the volume constraint.  
\[\min_{\mathbf{\rho}} C=\mathbf{F}^{\top}\mathbf{u}=\mathbf{u}^{\top}\mathbf{K}(\mathbf{\rho})\mathbf{u}\] (1) subject to \[K(\mathbf{\rho})\mathbf{u}=\mathbf{F}\] \[\sum_{e}\rho_{e}v_{e}\leq V^{*}\]  
Figure 1: Overview of proposed multi-solution TO framework with human controllability.  
\[K(\rho)=[K_{e}(\rho)]=[K_{e}^{0}E_{0}\rho_{e}^{p}] \tag{2}\]  
In the SIMP method, the element densities are continuous within the range [0, 1], and are pushed to 0 or 1 with increasing penalty value (typically initialized as \(p=1\) and gradually increased), thereby generating an approximately binary topology design solution. The element densities are optimized on the basis of an FEA of structural compliance conducted on fixed sample grids using domain-specific methods such as Optimality Criteria (OC) or Method of Moving Asymptotes (MMAs). This approach limits the resolution of the solution and requires an analytical sensitivity analysis. This study adopts the neural-encoded density function proposed by [PERSON] and [PERSON] and \[J_{e} = u_{e}^{\top}K_{e}u_{e} \tag{6}\] \[J = \sum_{e}\rho_{e}^{p}J_{e} \tag{7}\]  
where \(e\) is the analysis element, \(J_{e}\) is the compliance of element \(e\), and \(J\) is the total compliance.  
In the experiment of this study, an FEA is conducted with one sample point located at the center of element. When higher FEA accuracy is required, it is straightforward to use multiple sample points within an element, taking advantage of a continuous density function. The number and distribution of Fourier frequencies have a significant influence on design solutions; this will be analyzed and used to enhance solution diversity in Section 4.