## 2 Reliability Analysis and Reliability-Based Design Optimization  
### Reliability Analysis Using the First-Order Reliability Method  
The time-invariant reliability problem can be characterized by an \(n\)-vector of random variables \(\mathbf{x}=\left(x_{1},x_{2},x_{3},\ldots,x_{n}\right)^{\mathsf{T}}\) and the subset \(\Omega\) of their outcome space, which defines the failure events of the limit-state function \(g(\mathbf{x})\). The failure probability \(P_{f}\) is given by the \(n\)-fold integral  
\[P_{f}=P(\mathbf{x}\in\{\Omega\equiv g(\mathbf{x})\leq 0\})=\int_{g(\mathbf{x}) \leq 0}k(\mathbf{x})d\mathbf{x} \tag{1}\]  
where \(k(\mathbf{x})\) is the joint probability density function (PDF) of \(\mathbf{x}\). By transforming the random variables into the standard normal space \((\mathbf{u}=\mathrm{T}(\mathbf{x}))\), the failure probability integral can be written as  
\[P_{f}=\int_{G(\mathbf{u})\leq 0}\varphi_{n}(\mathbf{u})d\mathbf{u} \tag{2}\]  
where \(G(\mathbf{u})\) is the limit-state function transformed from the original space into the standard normal space and \(\varphi_{n}(\cdot)\) is the \(n\)-variate standard normal density function for the random vector. The component reliability problem utilizes a reliability method to compute the failure probability. The FORM linearizes the limit-state function \(G(\mathbf{u})\) at a point \(\mathbf{u}^{*}\), which is obtained by solving a constraint optimization problem as  
\[\underset{\mathbf{u}*}{\arg\min}\ \{\|\mathbf{u}\|\ |G(\mathbf{u})=0\ \} \tag{3}\]  
The point \(\mathbf{u}^{*}\) indicates a location in the linearized limit-state function, in which the distance from the origin in the standard normal space is the minimum. This point is commonly known as the \"design point\" or \"most probable point\" (MPP). The linearized limit-state function is written as  
\[G(\mathbf{u})=\
abla G(\mathbf{u}^{*})(\mathbf{u}-\mathbf{u}^{*}) \tag{4}\]  
where \(\
abla G(\mathbf{u}^{*})=\left(\partial_{u_{1}^{*}}G,\ \partial_{u_{2}^{*}}G,\ldots, \partial_{u_{n}^{*}}G\right)\) denotes the gradient row vector. The linearization represents the failure domain \(G(\mathbf{u})\leq 0\), that is, the half-space defined by \(\beta-\alpha\mathbf{u}\leq 0\). \(\beta=\alpha\mathbf{u}^{*}\) is the reliability index and \(\mathbf{\alpha}=-\frac{\
abla G(\mathbf{u}^{*})}{\|\
abla G(\mathbf{u}^{*})\|}\) denotes the normalized negative vector at the design point (refer to Figure 1). This representation leads to  
\[G(\mathbf{u})=\|\
abla G(\mathbf{u}^{*})\|(\beta-\alpha u) \tag{5}\]The probability content of the half-space, which is defined by the reliability index, gives the approximation of the failure probability as  
\[P_{f}\approx\Phi(-\beta) \tag{6}\]  
where \(\Phi(\cdot)\) denotes the cumulative distribution function (CDF) of the standard normal distribution. It should be noted that there are various algorithms for approximation of the design point in Equation (3) and the reliability index [34,35]. The accuracy of the FORM depends upon how well the actual limit-state surface is represented by the linear approximation. For nonlinear limit-state functions having a larger number of random variables, the FORM may not identify accurate design points or encounter a convergence issue while searching for the most probable points for complex systems. The nonlinearity of the limit-state surface is caused by various factors, such as the nonlinear relationship between random variables and the transformation of random variables from the original space to standard normal space. The second-order reliability method (SORM) [13,36,37] can improve the assessment given by FORM by including information about the curvature. The SORM approximates the limit-state surface by using a hyper-paraboloid with tangent hyperplane and main curvatures at the design point. Furthermore, [PERSON] and [PERSON] [38] showed that a limit-state function might have multiple design points. An optimization algorithm utilized to solve Equation (3) may converge to a local design point that results in a gross approximation error.