## 3 Topology Optimization Model Considering Fatigue Characteristics
### Topology Optimization Model Formulation  
In this paper, the bracket is subjected to cyclic loading in the vertical direction during fixture loading. To satisfy fatigue resistance, it is suggested to convert this fatigue resistance problem into a problem of minimizing the mean compliance of the structure under the 19 extreme operating conditions. Ignoring damping effect, the mathematical formulation of the topology optimization model is as follows:  
\[ \begin{array}{rl}\min f(\rho):&\overset{n}{:}\sum\limits_{i=1}^{n}\alpha_{i} \big{(}\mathbf{F}_{i}^{T}\mathbf{U}_{i}(\omega;\rho)\big{)}^{2}\\ s.t.:&\big{[}\mathbf{K}_{1}(\rho)-\omega^{2}\mathbf{M}_{1}\left(\rho\right) \mathbf{U}_{i}=\mathbf{F}_{i},i=1,2,3\\ &\big{(}\mathbf{K}_{2}(\rho)-\omega^{2}\mathbf{M}_{2}\left(\rho\right)\mathbf{U} _{i}=\mathbf{F}_{i},i=4,8,12,16\\ &\big{(}\mathbf{K}_{3}(\rho)-\omega^{2}\mathbf{M}_{3}\left(\rho\right)\mathbf{ U}_{i}=\mathbf{F}_{i},i=5,9,13,17\\ &\big{(}\mathbf{K}_{4}(\rho)-\omega^{2}\mathbf{M}_{4}\left(\rho\right)\mathbf{ U}_{i}=\mathbf{F}_{i},i=6,10,14,18\\ &\big{(}\mathbf{K}_{5}(\rho)-\omega^{2}\mathbf{M}_{5}\left(\rho\right)\mathbf{ U}_{i}=\mathbf{F}_{i},i=7,11,15,19\\ &\sum\rho_{nd}\omega_{nd}<=V_{0}\\ &0<\delta<\rho_{nd}<1,nd=1,2,\ldots,l\end{array}  \tag{1}\]  
where \(f\) is the objective value function, \(\omega(\omega=20\pi,\text{rad}/\text{s})\) is the prescribed angular frequency, \(i\) is the number of the extreme conditions, \(n\) is the maximum number of extreme condition, \(nd\) is the number of nodes which are discretized by FEM, \(l\) is the maximum number of nodes,\(\alpha_{i}\) is the weighted coefficient, \(\mathbf{K}(\rho)\) is the global stiffness matrix penalized by the SIMP method and \(\mathbf{K}_{1},\mathbf{K}_{2},\ldots,\mathbf{K}_{5}\) mean that there are five kinds of boundary conditions in FEM, \(\mathbf{M}(\rho)\) is the global mass matrix penalized by the SIMP method and \(\mathbf{M}_{1},\mathbf{M}_{2},\ldots,\mathbf{M}_{5}\) mean that there are five kinds of boundary conditions in FEM, \(\mathbf{U}_{i}\) is a vector of node displacements, \(\mathbf{F}_{i}\) is the loading vector, \(V_{0}\) is the allowed fraction of total material volume, \(\delta(\delta=0.001)\) is the lower boundary of the design variable to prevent singularities in the global matrix. Considering that there are six load cases in each extreme condition, it means:  
\[\mathbf{U}_{i}=[\mathbf{U}_{i}^{1}\ \mathbf{U}_{i}^{2}\ \mathbf{U}_{i}^{3}\ \mathbf{U}_{i}^{4}\ \mathbf{U}_{i}^{5}\ \mathbf{U}_{i}^{6}] \tag{2}\]  
\[\mathbf{F}_{i}=[\mathbf{F}_{i}^{1}\ \mathbf{F}_{i}^{2}\ \mathbf{F}_{i}^{3}\ \mathbf{F}_{i}^{4}\ \mathbf{F}_{i}^{5}\ \mathbf{F}_{i}^{6}] \tag{3}\]  
Figure 6: Six kinds of loads in condition 1. The green regions are fixed boundary conditions. The red regions are the location of external loads. (**a**) represents the first load case in condition 1; (**b**) represents the second load case in condition 1; (**c**) represents the third load case in condition 1; (**d**) represents the fourth load case in condition 1; (**e**) represents the fifth load case in condition 1; (**f**) represents the sixth load case in condition 1.  
where superscript 1, 2,..., 6 are indexes which represent load cases in each extreme condition. For each load case, for example  
\[(\mathbf{K}_{1}(\rho)-\omega^{2}\mathbf{M}_{1}(\rho))\mathbf{U}_{1}^{1}=\mathbf{ F}_{1}^{1} \tag{4}\]  
where \(\mathbf{K}_{1}(\rho)-\omega^{2}\mathbf{M}_{1}(\rho)\) needs to be decomposed as \(\mathbf{K}_{1}(\rho)-\omega^{2}\mathbf{M}_{1}(\rho)=\mathbf{LU}\) via LU decomposition. \(\mathbf{L}\) is the lower triangular matrix and \(\mathbf{U}\) is the upper triangular matrix. Because matrix \(\mathbf{K}_{1}\) and \(\mathbf{M}_{1}\) is unchanged for different load cases, if the boundary conditions are the same in FEM, the LU decomposition is only implemented once for every boundary condition and matrix \(\mathbf{L}\) and \(\mathbf{U}\) are saved in the memory.