# Fail-Safe Topology Optimization Using Damage Scenario Filtering
## 2 Theory of Fail-Safe Structural Topology Optimization
### Sensitivity Analysis  
In the context of the fail-safe topology optimization problem, there are numerous design variables that are typically addressed through finite element methods. In this process, the restatement of the stiffness matrix and the solution of the multivariate equilibrium equations are required with each analysis iteration. Therefore, using optimization criteria methods proves to be an ideal choice. In this section, by leveraging the principles of simple material interpolation theory, a detailed investigation of optimization criteria algorithms for fail-safe optimization problems is conducted. Originating from the objective function and constraint conditions, the optimization criteria formula for updating design variables, described by the Lagrangian function, is derived. The construction of the Lagrangian function for the optimization problem outlined in the preceding section is as follows:  
\[L=f_{r}(\rho_{e})+\lambda(V-V_{\text{max}})+\sum\limits_{i=1}^{m}\Bigl{(} \lambda_{1}^{i}\Bigr{)}^{\text{T}}\Bigl{(}K^{i}u^{i}-\hat{\mathbf{f}}\Bigr{)}+ \sum\limits_{e=1}^{n}\lambda_{2}^{e}(0-\rho_{e})+\sum\limits_{e=1}^{n}\lambda_ {3}^{e}(\rho_{e}-1) \tag{7}\]  
In Equation (7), \(\lambda,\lambda_{1},\lambda_{2}\) serve as Lagrange multipliers. For the fail-safe structural topology optimization problem, \(\mathbf{\rho}\) is a matrix composed of \(\rho_{e}\). When \(\mathbf{\rho}\) reaches an extremum, the Lagrangian function described above satisfies the Kuhn-Tucker conditions as follows:  
\[\left\{ \begin{array}{l}\frac{\partial L}{\partial\rho_{e}}=\frac{\partial f _{r}}{\partial\rho_{e}}+\lambda\frac{\partial V}{\partial\rho_{e}}+\sum \limits_{i=1}^{m}\bigl{(}\lambda_{1}^{i}\bigr{)}^{\text{T}}\frac{\partial\bigl{(} K^{i}u^{i}\bigr{)}}{\partial\rho_{e}}-\lambda_{2}^{e}+\lambda_{3}^{e}=0\\ V=V_{\text{max}}\\ K^{i}u^{i}=\hat{\mathbf{f}}\\ \lambda_{2}^{e}(0-\rho_{e})=0\\ \lambda_{2}^{e}(\rho_{e}-1)=0\\ \lambda_{2}^{e}>0\ \ \,\ \ \lambda_{3}^{e}>0\ \ \,\ \ e=1,2,\ldots,n\end{array} \right. \tag{8}\]After establishing the mathematical model for the topology optimization of fail-safe structures, similar to standard topology optimization, it is necessary to solve the sensitivity information required when updating the design variables, that is, the sensitivity calculation of the objective function obtained by the KS function with respect to the cell density. According to the chain derivation rule, the partial derivative of the objective function with respect to the unit density is expressed as follows:  
\[\frac{\partial f_{r}}{\partial\rho_{e}}=\frac{\sum\limits_{i=1}^{m}\left(e^{ \gamma f^{(i)}(\rho)}\frac{\partial f^{(i)}(\rho)}{\partial\rho_{e}}\right)}{ \sum\limits_{i=1}^{m}e^{\gamma f^{(i)}(\rho)}} \tag{9}\]  
In Equation (9), for ease of description, assuming \(h^{i}=e^{\gamma f^{(i)}(\rho)}\) and \(H=\sum\limits_{i=1}^{m}e^{\gamma f^{(i)}(\rho)}\), the calculation of the sensitivity information of the volume fraction with respect to the design variables for the constraint conditions is as follows:  
\[\frac{\partial L}{\partial\rho_{e}}=\sum\limits_{i=1}^{m}h^{i}\frac{\frac{ \partial\left(\left(u^{i}\right)^{\mathrm{T}}\left(K^{i}\right)\left(u^{i} \right)\right)}{\partial\rho_{e}}}{H}+\lambda v_{e}+\sum\limits_{i=1}^{m} \left(\lambda_{1}^{i}\right)^{\mathrm{T}}\left(\frac{\partial\left(K^{i} \right)}{\partial\rho_{e}}u^{i}+^{i}\frac{\partial\left(u^{i}\right)}{ \partial\rho_{e}}\right) \tag{10}\]  
In Equation (10), after consolidation, the aforementioned formula can be simplified as  
\[\frac{\partial L}{\partial\rho_{e}}=-\sum\limits_{i=1}^{m}\frac{h^{i}}{H}u \frac{\partial K}{\partial\rho_{e}}u+\lambda v_{e} \tag{11}\]  
In Equation (11), setting the left side of the equation equal to zero yields the following result:  
\[\frac{\sum\limits_{i=1}^{m}\frac{h^{i}}{H}p\left(\rho_{e}^{k}\right)^{p-1}q_{e} }{v_{e}}=\lambda \tag{12}\]  
This represents the deformation energy of the structure under local damage and the proportion of the second local failure scenario among all failure scenarios.