Component reliability can be estimated by either statistics-based methods with data or physics-based methods with models. Both types of methods are usually independently applied, making it difficult to estimate the joint probability density of component states, which is a necessity for an accurate system reliability prediction. The objective of this study is to investigate the feasibility of integrating statistics- and physics-based methods for system reliability analysis. The proposed method employs the first-order reliability method (FORM) directly for a component whose reliability is estimated by a physics-based method. For a component whose reliability is estimated by a statistics-based method, the proposed method applies a supervised learning strategy through support vector machines (SVM) to infer a linear limit-state function that reveals the relationship between component states and basic random variables. With the integration of statistics- and physics-based methods, the limit-state functions of all the components in the system will then be available. As a result, it is possible to predict the system reliability accurately with all the limit-state functions obtained from both statistics- and physics-based reliability methods.

## Introduction

System reliability can be numerically measured by the probability that the system performs its intended function without failures. As the system state (safe or failed) is determined by the states of its components and it is hard to predict the system reliability directly, the system reliability is usually estimated based on component states. The accurate system reliability prediction requires the joint probability density of component states [1].

A physical component itself can be considered as a system since it may have multiple failure modes, and the reliability of the physical component is determined by the states of all the component failures. For this reason, we also consider a failure mode as a component and a physical component as a system if it has multiple failure modes.

Since there is rarely a closed-form solution to Eq. (1), many approximation methods have been developed, such as the first-order reliability method (FORM) [4], the second-order reliability method (SORM) [5], the saddlepoint approximation method [6], Monte Carlo simulation [7], and matrix-based system reliability method [8]. Numerous applications of these methods have been reported for many systems, such as mechanical, automation, civil, and communication systems.

*i*th failure mode. Then the probability of system failure is computed by

Although this method is easy to use, it may result in an estimated reliability far smaller than the true value. Without the complete joint probability distribution of component states, it is difficult to evaluate the system reliability accurately, especially when component reliabilities are estimated by statistics- and physics-based methods independently.

Recently, Hu and Du [10,11] proposed a new method that reconstructs component limit-state functions with limited reliability information, making it possible to evaluate system reliability using Monte Carlo simulation. The method is effective for cases where component reliability data are provided with respect to system loads. A proof-of-concept method has also recently been proposed for systems with both in-house and outsourced components [12], where the reliabilities of in-house components are estimated with physics-based methods and those of the outsourced ones with statistics-based methods. The study has shown the feasibility of integrating statistics- and physics-based reliability approaches for special problems. The objective of this work is to further investigate the method proposed in Ref. [12]. For the statistics-based methods, samples of basic variables (loading, material properties, dimensions, etc.) and the component states, either safe or failed, are available. We adopt support vector machines (SVM) to build linear limit-state functions with respect to the basic variables since SVM is one of the best classification methods due to its high efficiency and accuracy. It has also been employed in many studies [13,14]. Then, with the limit-state functions generated by SVM and those from physics-based methods, the system reliability could be accurately estimated.

The scope and assumptions of the new method are as follows: components fail due to excessive loads. For components whose reliability is estimated by a statistics-based method, observations of basic random variables are available. Distributions of all basic random variables are known. The study focuses on series systems although it can be extended to parallel systems.

The proposed method has the following advantages: (1) limit-state functions built from statistical data can be easily integrated with those derived from physics. This helps system designers understand the dependency between component failures and enables them to construct a complete joint distribution of component states; (2) the proposed method does not restrict the number of basic random variables (such as loads) shared by components. Hence, it has a broader application scope than the previously proposed methods [10,11] that can accommodate only one common system load.

We provide a brief review of methodologies used in this work, including SVM and first-order reliability method in Sec. 2. In Sec. 3, the SVM method for building limit-state functions and the procedure of system reliability analysis with the proposed method are introduced. Three examples are discussed in Sec. 4. Conclusions and future work are presented in Sec. 5.

## Methodology Review

In this work, we use FORM for physics-based component reliability analysis and use SVM to construct limit-state functions for failure modes (components) whose reliabilities are estimated by a statistics-based method. Both of the methods are briefly reviewed below.

### Support Vector Machine.

*n*-dimensional row vector. The value of $y$ depends on whether the point belongs to the first class or the second one. In this work, if the point falls in the safe region, then $y=+1$; otherwise, $y=\u22121$. The objective of SVM is to separate the training points into two classes with a hyperplane, as shown in Fig. 1, which is given by

where $\omega $ is a weight vector, and $b$ is the bias. The shaded points passed by these hyperplanes are called support vectors (SVs), and there are no points between these hyperplanes.

According to the Karush–Kuhn–Tucker conditions, only the SVs lead to $\lambda i\u22600$. This means that only the SVs appear in the optimal result.

### First-Order Reliability Method.

First-order reliability method is a physics-based reliability method, which linearizes the limit-state function $g(X)$ at the most probable point using the first-order Taylor expansion. Three steps are involved.

*X*-space) are independent. The original random variables $X=(X1,X2,\u2026,Xn)$ are transformed into standard normal random variables $U=(U1,U2,\u2026,Un)$ in the

*U*-space. The transformation is given by [15]

where $Fi(\u22c5)$ and $\Phi (\u22c5)$ are the cumulative distribution functions of $Xi$ and a standard normal variable, respectively, and $T(\u22c5)$ denotes the transformation function.

## System Reliability Prediction With Combined Statistics- and Physics-Based Methods

The objective of this study is to integrate statistics- and physics-based reliability methods so that the joint probability density function (PDF) of all the component states is available. As discussed previously, the two different types of components (failure modes) are defined as follows:

Type I: Type I components have limit-state functions and their reliabilities can be estimated by physics-based reliability methods, such as FORM.

Type II: Type II components do not have limit-state functions, and their reliabilities are estimated by statistics-based methods.

The main idea of this work is to construct limit-state functions for type II components using testing data. Then with all the available limit-state functions, the system reliability could be estimated.

### Construct a Limit-State Function for a Type II Component.

*X*-to-

*U*transformation, we have a new dataset $(ui,yi)$, $i=1,2,\u2026,m$, where

*j*indicates the

*j*th component of the

*i*th sample point.

With sufficient number of experiments, the probability of failure of the component $pf$ can also be estimated with a statistics-based reliability method. We hence assume that the component reliability is available.

In this study, we use SVM to construct the limit-state function in the form of $GII(U)=\beta +\alpha UT$, in which $\beta $ is known and is given by $\beta =\u2212\Phi \u22121(pf)$. Now the task becomes to find a unit vector $\alpha $ that defines the hyperplane $GII(U)$. This can be done with the following two steps:

### System Reliability Analysis.

We now discuss the system reliability analysis using the proposed integrated statistics- and physics-based reliability method. Assume there are $m$ components (failure modes) and $m\u22652$. The limit-state functions $giI(\u22c5),\u2009(i=1,2,\u2026,m1)$ of $m1$ type I components are available. For the other $m2(m2=m\u2212m1)$ type II components, their observations or training points $(x,y)$ are available.

*U*-space can be written as

*i*th and

*j*th components and is calculated by

According to Eq. (18), we have $\alpha jII=(\omega j/\u2225\omega j\u2225)\u2009(j=m1+1,m1+2,\u2026,m)$. To verify the direction of $\alpha jII$, we first substitute $\alpha jII=(\omega j/\u2225\omega j\u2225)$ into Eq. (18). Since $g(X)<0$ or $G(U<0)$ means a failed state, Eq. (18) should be negative at any sample point with $y=\u22121$. Otherwise, we change the direction of $\alpha jII$ by reversing the signs of all the components in it. The details of doing this are shown in Example 1 in Sec. 4.

Then $Rs$ can be easily evaluated by integrating $\varphi G(v)$ in the safe region $\Omega $, and the probability of system failure is $pfs=1\u2212Rs$.

The proposed method provides a new way to approximate linear limit-state functions for components with only estimated probabilities of failure and limited reliability data. The dependency between components is automatically accommodated in the system covariance matrix. Also, it is computationally efficient due to the linear form of all the limit-state functions.

First-order reliability method is used in this feasibility study, but it is not a necessity of the proposed method. Other reliability methods, such as SORM and SVM-based methods, can also be used. One can choose a method if it satisfies the following two requirements: the method can produce the probability of failure $pf$ so that the reliability index is obtained by $\beta =\u2212\Phi \u22121(pf)$, and a directional vector $\alpha $ is available or can be derived. Both SORM and SVM-based methods satisfy the above requirements.

## Examples

In this section, three examples are presented. To validate the proposed method, we at first assume that the true limit-state functions of type II components exist, but unknown to system designers. Using these true limit-state functions, we can obtain the true system reliability from a physics-based reliability method. To mimic the actual physical testing for type II components, we perform computer-based testing (random sampling) by calling the true limit-state functions and then apply the proposed method.

### Example 1—A Mathematical Problem.

*U*-space is

Twenty-six samples of $X$ are generated and the corresponding values of $y$ are computed using Eq. (26). The simulated results are given in Table 1. Training points are given in both *X*- and *U*-spaces, and $pf2$ is also obtained based on Eq. (26).

$x$ | $u$ | $y$ | |||
---|---|---|---|---|---|

1 | 7.6054 | 24.3070 | −2.9932 | −3.7953 | $\u22121$ |

2 | 9.4886 | 22.5981 | −0.6392 | −4.9346 | $\u22121$ |

3 | 10.0151 | 31.5254 | 0.0189 | 1.0169 | $+1$ |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

25 | 8.7630 | 23.4822 | −1.5462 | −4.3452 | $\u22121$ |

26 | 8.2552 | 22.9919 | −2.1810 | −4.6721 | $\u22121$ |

$x$ | $u$ | $y$ | |||
---|---|---|---|---|---|

1 | 7.6054 | 24.3070 | −2.9932 | −3.7953 | $\u22121$ |

2 | 9.4886 | 22.5981 | −0.6392 | −4.9346 | $\u22121$ |

3 | 10.0151 | 31.5254 | 0.0189 | 1.0169 | $+1$ |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

25 | 8.7630 | 23.4822 | −1.5462 | −4.3452 | $\u22121$ |

26 | 8.2552 | 22.9919 | −2.1810 | −4.6721 | $\u22121$ |

To verify the direction of $\alpha 2II$, we first arbitrarily pick one training point, for instance, $u1=(\u22122.9932,\u22123.7953)$, where a failure ($y=\u22121$) occurs. Then we plug $u1$ into Eq. (29) and obtain $G2II(U)=\u22120.11<0$, which indicates a failure. Thus, the failed state is consistent with the label of $u1$, that is $y1=\u22121$, which means $\alpha 2II$ has a correct direction.

To validate the result, we use $g1I(X)$ and $g2II\u2009(true)(X)$ to calculate the system reliability based on FORM. The result is $1.4523\xd710\u22125$ and is regarded as the true probability of system failure. For comparison, we also compute the system reliability using the independence assumption method. All the results are listed in Table 2. The independence assumption method produces an error of 8.1%. The large error comes from neglecting the strong correlation indicated by $\rho 12=0.8815$. The proposed method has an error of only 0.56%, which shows much higher accuracy.

### Example 2—A Cantilever Beam Problem With Multiple Failure Modes.

As shown in Fig. 3, a cantilever beam is subjected to moments $M1$ and $M2$; forces $Q1$ and $Q2$; and distributed loads denoted by $(qL1,qR1)$ and $(qL2,qR2)$. Among these external loads, $M1$, $M2$, and $Q1$ are random variables. The other random variables are the dimensions of $a1$, $a2$, and $b1$; the yield strength $Sa$; and the allowable shear stress $\tau a$. Thus, there are totally eight basic random variables, as listed in Table 3, in which *N* means Normal Distribution and Log*N* means Lognormal Distribution. For each distribution, the first parameter is the mean value and the second one is the standard deviation. The deterministic parameters are listed in Table 4.

Random variables | Distribution | |
---|---|---|

$X1$ | $M1(N\u22c5m)$ | $N(50\xd7103,2\xd7103)$ |

$X2$ | $M2(N\u22c5m)$ | $N(30\xd7103,2\xd7103)$ |

$X3$ | $a1(m)$ | $N(1.5,0.005)$ |

$X4$ | $a2(m)$ | $N(4.5,0.005)$ |

$X5$ | $Q1(N)$ | $Log\u2009N(65\xd7103,13\xd7103)$ |

$X6$ | $b1(m)$ | $N(0.7,0.005)$ |

$X7$ | $Sa(Pa)$ | $N(62.5\xd7106,106)$ |

$X8$ | $\tau a(Pa)$ | $N(3.6\xd7106,105)$ |

Random variables | Distribution | |
---|---|---|

$X1$ | $M1(N\u22c5m)$ | $N(50\xd7103,2\xd7103)$ |

$X2$ | $M2(N\u22c5m)$ | $N(30\xd7103,2\xd7103)$ |

$X3$ | $a1(m)$ | $N(1.5,0.005)$ |

$X4$ | $a2(m)$ | $N(4.5,0.005)$ |

$X5$ | $Q1(N)$ | $Log\u2009N(65\xd7103,13\xd7103)$ |

$X6$ | $b1(m)$ | $N(0.7,0.005)$ |

$X7$ | $Sa(Pa)$ | $N(62.5\xd7106,106)$ |

$X8$ | $\tau a(Pa)$ | $N(3.6\xd7106,105)$ |

Parameters | Values | |
---|---|---|

1 | $Q2(N)$ | $30\xd7103$ |

2 | $b2(m)$ | $2.5$ |

3 | $qL1(N/m)$ | $30\xd7103$ |

4 | $qL2(N/m)$ | $20\xd7103$ |

5 | $c1(m)$ | $0.25$ |

6 | $c2(m)$ | $1.75$ |

7 | $qR1(N/m)$ | $20\xd7103$ |

8 | $qR2(N/m)$ | $103$ |

9 | $d1(m)$ | $1.25$ |

10 | $d2(m)$ | $4.75$ |

11 | $L\u2009(m)$ | $5.1$ |

12 | $w\u2009(m)$ | $0.204$ |

13 | $h\u2009(m)$ | $0.403$ |

Parameters | Values | |
---|---|---|

1 | $Q2(N)$ | $30\xd7103$ |

2 | $b2(m)$ | $2.5$ |

3 | $qL1(N/m)$ | $30\xd7103$ |

4 | $qL2(N/m)$ | $20\xd7103$ |

5 | $c1(m)$ | $0.25$ |

6 | $c2(m)$ | $1.75$ |

7 | $qR1(N/m)$ | $20\xd7103$ |

8 | $qR2(N/m)$ | $103$ |

9 | $d1(m)$ | $1.25$ |

10 | $d2(m)$ | $4.75$ |

11 | $L\u2009(m)$ | $5.1$ |

12 | $w\u2009(m)$ | $0.204$ |

13 | $h\u2009(m)$ | $0.403$ |

Thus, this failure mode is treated as a type I component.

Thus, this failure mode is also treated as a type I component.

where $\rho 12,\u2009\rho 13$, and $\rho 23$ are the correlation coefficients between $G1I(U)$ and $G2I(U)$; $G1I(U)$ and $G3II(U)$; $G2I(U)$ and $G3II(U)$, respectively. Then, the system reliability is evaluated and is given by $pfs=4.1528\xd710\u22123$.

With all the given limit-state functions $g1I(X)$, $g2I(X)$, and $g3II\u2009(true)(X)$, the true system reliability could be obtained using FORM and is assumed as a benchmark for comparison. Likewise, we also estimate the probability of system failure using the independence assumption method. The results are shown in Table 5, which indicates that the proposed method is more accurate than the independence assumption method.

Proposed method | Independence assumption method | True value | |
---|---|---|---|

$pfs$ | $4.1528\xd710\u22123$ | $5.2442\xd710\u22123$ | $4.1582\xd710\u22123$ |

Error (%) | $0.13$ | $26.12$ | — |

Proposed method | Independence assumption method | True value | |
---|---|---|---|

$pfs$ | $4.1528\xd710\u22123$ | $5.2442\xd710\u22123$ | $4.1582\xd710\u22123$ |

Error (%) | $0.13$ | $26.12$ | — |

### Example 3—A System With Multiple Components.

Figure 4 shows a crank-slider system consisting of four physical components. An external moment is applied to joint *A,* which drives beam AB rotating around *A*. For this example, we only focus on the system reliability when $\theta 2=\pi /2$.

All the random variables known by the system designers are listed in Table 6 and the known deterministic parameters are listed in Table 7. Since $D$ and $\tau a5$ are only known by the spring supplier, they are not listed in Table 6. They are denoted as $X9$ and $X10$. Thus, there are actually ten basic random variables in the system. For FM5, the training points are provided in the form of $(X1,X2,X9,X10)$.

Random variables | Distribution | |
---|---|---|

$X1$ | $M1(N\u22c5m)$ | $N(350,65)$ |

$X2$ | $l1(m)$ | $N(0.3,10\u22124)$ |

$X3$ | $l2(m)$ | $N(0.9,10\u22123)$ |

$X4$ | $b1(m)$ | $N(0.022,5\xd710\u22124)$ |

$X5$ | $h1(m)$ | $N(0.019,5\xd710\u22124)$ |

$X6$ | $b2(m)$ | $N(0.015,5\xd710\u22124)$ |

$X7$ | $h2(m)$ | $N(0.009,5\xd710\u22124)$ |

$X8$ | $d4(m)$ | $N(0.0228,10\u22124)$ |

Random variables | Distribution | |
---|---|---|

$X1$ | $M1(N\u22c5m)$ | $N(350,65)$ |

$X2$ | $l1(m)$ | $N(0.3,10\u22124)$ |

$X3$ | $l2(m)$ | $N(0.9,10\u22123)$ |

$X4$ | $b1(m)$ | $N(0.022,5\xd710\u22124)$ |

$X5$ | $h1(m)$ | $N(0.019,5\xd710\u22124)$ |

$X6$ | $b2(m)$ | $N(0.015,5\xd710\u22124)$ |

$X7$ | $h2(m)$ | $N(0.009,5\xd710\u22124)$ |

$X8$ | $d4(m)$ | $N(0.0228,10\u22124)$ |

No. | Deterministic parameters | Values |
---|---|---|

1 | $E2(Pa)$ | $200\xd7109$ |

2 | $E4(Pa)$ | $200\xd7109$ |

3 | $K$ | $1$ |

4 | $l3(m)$ | $0.96$ |

5 | $l4(m)$ | $0.31$ |

6 | $Sa1(Pa)$ | $400\xd7106$ |

7 | $Sa4(Pa)$ | $460\xd7106$ |

8 | $\delta a4(m)$ | $0.0032$ |

No. | Deterministic parameters | Values |
---|---|---|

1 | $E2(Pa)$ | $200\xd7109$ |

2 | $E4(Pa)$ | $200\xd7109$ |

3 | $K$ | $1$ |

4 | $l3(m)$ | $0.96$ |

5 | $l4(m)$ | $0.31$ |

6 | $Sa1(Pa)$ | $400\xd7106$ |

7 | $Sa4(Pa)$ | $460\xd7106$ |

8 | $\delta a4(m)$ | $0.0032$ |

As discussed in Sec. 4, the five FMs in the system are treated as five components at the system level. The first four FMs with known limit-state functions $giI(X)\u2009(i=1,2,3,4)$ belong to type I components, and FM5 is a type II component since its limit-state function $g5II(X)$ is not available.

Using Eq. (24), the estimated probability of system failure is $pfs=1\u2212Rs=0.0133$.

With all the exactly known limit-state functions $giI(X)\u2009(i=1,2,3,4)$ and $g5II\u2009(true)(X)$, the true system reliability could be directly acquired using FORM. The results from different methods are shown in Table 8. It is concluded that the proposed method performs much better than the independence assumption method.

## Conclusions

This work verifies the feasibility of integrating statistics- and physics-based method for system reliability analysis. It is common that component reliability is estimated by a statistic-based method; with the increasing use of physics-based computational models, it is also possible that component reliability is estimated by a physics-based method. This study deals with the difficulty of obtaining the joint probability density when physics-based methods are used for some components (type I) and statistics-based methods are used for other components (type II).

The physics-based method employed in this study is FORM, which is directly used for type I components whose physics-based limit-state functions are available. For type II components whose physics-based limit-state functions are unknown, with a statistics-based method, reliability experiments are performed. Then their reliabilities are estimated. A supervised learning strategy through SVM is developed to create limit-state functions for type II components. The proposed method makes the limit-state functions of all the components available thereby leading to a multivariate normal probability density function, whose integration in the safe region then produces the system reliability.

This feasibility study makes a number of assumptions, such as the distributions of basic random variables for both types of components are known, the component reliability is calculated by FORM, the safety-failure boundary is linear with respect to basic variables in the standard normal space, and sample points from reliability testing in both safe and failure regions are available. If the data set has a nonlinear pattern, the proposed method can still accommodate such nonlinearity by introducing slack variables to the SVM model so that the linear assumption could be violated slightly. If the nonlinearity is high, SVM methodologies that produce nonlinear models should be employed.

## Funding Data

Division of Civil, Mechanical and Manufacturing Innovation, National Science Foundation (CMMI 1562593).