# A Novel Reliability Analysis Approach under Multiple Failure Modes Using an Adaptive MGRP Model
## 3 Reliability Analysis Approach under Multiple Failure Modes
### Adaptive Multiple Response Gaussian Process
The mean and variance of the predicted value can be expressed as follows:  
\[\left\{\begin{array}{l}\mu_{\hat{\mathbf{\mathsf{y}}}}(x^{*})=h^{m}(x^{*})\mathbf{ \mathsf{B}}^{m}+r(x^{*})^{\mathsf{T}}\mathbf{\mathsf{R}}^{-1}(\mathbf{\mathsf{Y}}-\mathbf{ \mathsf{h}}^{m}\mathbf{\mathsf{B}}^  
nificantly improved by the RMQGS method, the accuracy of the model still cannot meet the requirements due to its limited numbers. Therefore, the adaptability of the MRGP model needs to be enhanced to improve its accuracy and efficiency. Generally, U-function, H-function, and EFF-function are used to search for the best update iteration sample points from the candidate sample points of MC simulation and then update the initial MRGP model until it meets the accuracy conditions. However, the three learning functions have different screening conditions for sample points. If they are used alone, the efficiency is low, and if they are used in combination, the complementary effects are difficult to achieve. To avoid that, two novel learning functions are defined according to the iteration form of the MRGP model.  
To increase high-quality sample points, when the number of iterations is odd, the probability value of the U-function is defined as the condition for judging the sign of the maximum possible wrongly predicted response value. The mathematical expression is given as follows:  
\[\Theta_{U}(x)=\Phi\bigg{(}\bigg{|}\frac{\mu_{g}(x)}{\sigma_{g}(x)}\bigg{|} \bigg{)} \tag{12}\]  
where \(\mu_{g}(x)\) and \(\sigma_{g}(x)\) denote the predicted mean and standard deviation of the MRGP model, and \(\Phi(\bullet)\) is the cumulative distribution function for a standard normally distributed variable. Then, the sample points that need to be updated can be provided as follows:  
\[\mathbf{x}^{\Theta}=\arg\min_{x\in S_{\text{MC}}}\Theta_{U}(x) \tag{13}\]  
where \(S_{\text{MC}}\) is the sample repository generated by MC simulation. When the number of iterations is even, a novel learning function is proposed to screen the sample points closest to the limit state surface, which is expressed as follows:  
\[\Upsilon_{U}(x)=\varphi\bigg{(}\bigg{|}\frac{\mu_{g}(x)}{\sigma_{g}(x)}\bigg{|} \bigg{)}/\big{|}\mu_{g}(x)\big{|} \tag{14}\]  
where \(\varphi(\bullet)\) is the probability density function for a standard normally distributed variable. Then, the updated sample points can be given by the following:  
\[x^{\text{r.}}=\arg\max_{x\in S_{\text{MC}}}\Upsilon_{U}(x) \tag{15}\]  
Due to the results of Equations (13) and (15) being probability values, the probability evaluation and termination criterion should be made to ensure sample points are updated. The variation coefficient of the estimated failure probability is calculated according to MC simulation, and its expression is provided as follows:  
\[\text{Cov}\Big{(}\hat{P}_{f}\Big{)}=\sqrt{\frac{1-\hat{P}_{f}}{(N_{\text{MC}} -1)\hat{P}_{f}}} \tag{16}\]  
where \(\hat{P}_{f}\) is the estimated failure probability. It is considered that the estimation is unacceptable when \(\text{Cov}\Big{(}\hat{P}_{f}\Big{)}\geq 5\%\), and then the sample repository needs to be expanded. Otherwise, it is considered acceptable, and the number of sample points is sufficient. Further, the termination criterion of the adaptive learning procedure can be expressed as follows:  
\[\xi=\frac{N_{\text{MC}}-\sum\limits_{j=1}^{N_{\text{MC}}}\Theta_{U}(x)}{N_{ \text{MC}}\hat{P}_{f}} \tag{17}\]  
where \(N_{\text{MC}}\) is the number of sample points by MC, when \(\xi\leq 5\%\), the iteration terminates. Otherwise, some new sample points are added.