# Novel Artificial Neural Network Aided Structural Topology Optimization
## 2 Theoretical Background
### Solid Isotropic Material Penalty Method  
The most well-known solid isotropic material punishment (SIMP) method is adopted in this study. The density is taken as the design variable. It is gradually adjusted through iterative processes to obtain optimized configuration. Via this method, the constraint factor of the optimized structure is set from 0 to 1. The objective function and constraint conditions of the SIMP method for the minimum compliance problem are given in Equation (1). \[\left\{ \begin{array}{l}\min_{x}:c(x)=\mathbf{U}^{\mathrm{T}}\mathbf{K}\mathbf{U}=\sum \limits_{i=1}^{N}{(x_{e})^{p}\mathbf{u}_{e}}^{\mathrm{T}}k_{0}\mathbf{u}_{e}\\ \mathrm{s.t.}\left\{\begin{array}{c}\frac{V(\mathbf{x})}{V_{0}}=f\\ \mathbf{K}\mathbf{U}=\mathbf{F}\\ \mathbf{x}\in\chi=\left\{\mathbf{x}\in\mathbb{R}^{n},\ 0<\!x_{\min}\leq x\leq 1 \right\}\end{array}\right.\end{array} \right. \tag{1}\]where \(F\) and \(U\) denote load and displacement vectors, respectively; \(K\) is the total stiffness matrix; \(x_{e}\) is the design variable; \(p\) is the penalty factor, taken as 3 in this study; \(c\) is the overall compliance of the structure; \(x\) is the vector composed of all design variables; \(V_{0}\) is the volume of the initial structure; \(f\) is the percentage of the final total volume optimized by the design to the initial volume, namely, the target volume fraction; \(n\) is the total number of units in the design domain; \(x_{\min}\) is the lower limit of the minimum density, usually taken as a non-zero minimum to prevent singular matrices in finite element computations. The problem can be solved using several different optimization methods, such as the optimal criteria method, the sequential linear programming method, and the moving asymptotic line method. However, the optimal criterion method is used in this paper. The density update criteria are given in Equation (2). \[\left\{ \begin{matrix}\max(x_{\min},x_{e}-m),&x_{e}B_{e}^{\eta}\leq\max(x_{ \min},x_{e}-m)\\ x_{e}B_{e}^{\eta}&\max(x_{\min},x_{e}-m)\leq x_{e}B_{e}^{\eta}\leq\min(1,x_{e }+m)\\ \min(1,x_{e}+m)&\min(1,x_{e}+m)\leq x_{e}B_{e}^{\eta}\end{matrix} \right. \tag{2}\]  
where \(m\) denotes a positive number, limiting the range of variable changes for each iteration; \(\eta\) is the numerical damping coefficient taken as \(1/2\) in this study; \(B_{e}\) is the density update factor, obtained from the sensitivity of each iteration, given in Equation (3). \[B_{e}=\frac{-\frac{\partial c}{\partial x_{e}}}{\lambda\frac{\partial V}{ \partial x_{e}}} \tag{3}\]  
where the Lagrange multiplier \(\lambda\) can be obtained by dichotomy. The sensitivity of the objective function is given in Equation (4). \[\frac{\partial c}{\partial x_{e}}=-p(x_{e})^{p-1}\mathbf{u_{e}}^{\mathrm{T}}\mathbf{k }_{0}\mathbf{u_{e}} \tag{4}\]  
It is assumed that the derivative of the design variable with respect to volume is 1. In the optimization process, mesh dependency and a checkerboard pattern might appear. It is indicated that different grid resolutions used for optimization lead to different results. When the number of grids is increased, voids might appear in the final configuration because of grid dependency. That is, different numbers of grids might result in different topology optimization. To prevent this situation, unit density is defined as a weighted average parameter for adjacent units. The filtered unit density is written in Equations (5) and (6). \[\widetilde{x}_{e}=\frac{1}{\sum\limits_{i\in N_{e}}H_{ei}}\sum\limits_{i\in N _{e}}H_{ei}x_{i} \tag{5}\]  
\[H_{ei}=\max(0,r_{\min}-\Delta(e,i)) \tag{6}\]  
where \(N_{e}\) denotes the distance from unit \(e\), \(\Delta(e,i)\) is smaller than the filtering radius \(r_{\min}\) of all elements; \(H_{ei}\) is the weight factor. The weight of elements near element \(e\) is negatively correlated with the distance between two elements. The sensitivity of the ith fine-grained unit could be modified as in Equation (7).