50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference<br>17th 4 - 7 May 2009, Palm Springs, California AIAA 2009-2251 A Generalized Complementary Intersection Method (CIM) for System Reliability Analysis Pingfeng Wang1, Byeng D. Youn2, and Chao Hu3 Department of Mechanical Engineering, the University of Maryland at College Park College Park, MD, 20742, USA In many years, researchers desire to evaluate system reliability accurately and efficiently. This paper presents a generalized Complementary Intersection Method (CIM) for system reliability analysis for series, parallel, and mixed systems. The CIM provides an innovative way to evaluate system reliability by decomposing the probability of a high-order joint failure event into probabilities of complementary intersection events. However, its application has been limited to a series system only. In this paper, a generalized CIM framework is proposed for system reliability analysis with any system structure (or configuration) (e.g., series, parallel, and mixed). The CIM is generalized for both parallel and mixed system reliability. The System Structure Matrix (SS-Matrix) is proposed to characterize any system structure in a comprehensive manner. The Binary Decision Diagram (BDD) technique is employed to identify system’s mutually exclusive path sets, of which each path set is a series system. On consequence, system reliability with any system structure is decomposed into the probabilities of the mutually exclusive path sets, which can be evaluated using different reliability analysis methods. Three examples are used to demonstrate the uniqueness and effectiveness of the proposed methodology. Nomenclature Ei Eij EiEj Pf Φ P(Ei) fx(x) pfs Gi β = = = = = = = = = = event (or first-order complementary intersection event) of the ith system component complementary intersection event for the ith and jth system components the joint event of the ith and jth system components probability of failure standard Gaussian cumulative distribution function probability of event Ei probability density function probability of system failure function of the ith constraint reliability index I. Introduction T HE importance of reliability has been well conceived by engineers in the last few decades. Considerable advances have been made in field of reliability assessment and Reliability-Based Design Optimization (RBDO) [1-5]. However, the research in system reliability analysis has been stagnant, mainly due to two technical difficulties. First, it is hard to formulate system reliability explicitly for a given system redundancy. Second, even if system reliability is given explicitly, it’s extremely difficult to assess system reliability effectively with numerical methods. Since system reliability prediction is of great importance in aerospace, mechanical, electronic and civil engineering fields, its technical development will have an immediate and major impact on complex engineering system designs. Due to the difficulties, system reliability analysis provides the bounds of system reliability. Ditlevsen proposed the most widely used second-order system reliability bounds method, which gives much tighter bounds compared 1 Graduate Student, pfwang@umd.edu, AIAA Student Member. Assistant Professor, bdyoun@umd.edu, Corresponding Author, AIAA Member. 3 Graduate Student, huchaost@umd.edu 1 American Institute of Aeronautics and Astronautics 2 Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. with the first-order bounds for both series and parallel systems [6]. Other equivalent forms of Ditlevsen’s bounds are given by Thoft-Christensen and Murotsu [7], Karamchandani [8], Xiao and Mahadevan [9] and K. Ramachandran [10]. Recently, Song and Der Kiureghian formulated system reliability to a Linear Programming (LP) problem, referred to as the LP bounds method [11]. The LP bounds method is able to calculate the optimal bounds for system reliability based on available information. However, it is extremely sensitive to accuracy of the given information, which is the probabilities for the first-, second-, or higher-order joint safety events. To assure high accuracy of the LP bounds method for system reliability prediction, the probabilities must be given very accurately. Besides the system reliability bound methods, one of the most popular approaches is the multimodal adaptive importance sampling (AIS) method, which is found satisfactory for the system reliability analysis of large structures [12, 13]. The integration of surrogate model techniques with MCS-based methods can be an alternative approach to system reliability prediction as well [14]. This approach can construct the surrogate model for multiple limit-state functions to represent the joint failure region. This approach is quite practical but accuracy of the method depends on fidelity of the surrogate model. It is normally expensive to build an accurate surrogate model with high confidence. Most recently, Youn and Wang [15] introduced an innovative concept of complementary intersection event and proposed the Complementary Intersection Method (CIM) for system reliability analysis with any system structure (e.g., series, parallel, and mixed). The CIM method provides not only a unique formula for system reliability but also an effective numerical method to evaluate the system reliability with high efficiency and accuracy. The CIM decomposes the probabilities of high-order joint failure events into probabilities of complementary intersection events. For large scale systems, a CI-matrix was proposed to store the probabilities of component safety and complementary intersection events. Then, series system reliability can be evaluated efficiently by any reliability method, such as First-Order Reliability Method (FORM), Second-Order Reliability Method (SORM), Eigenvector Dimension Reduction (EDR) method, or Monte Carlo Simulation (MCS). The accuracy and efficiency of these reliability methods used for the evaluation of the CI-matrix were also compared. However, the application of the CIM has been limited to a series system only. The work presented in this paper is the continuous investigation of the CIM for system reliability analysis. In this work, the CIM is advanced for parallel systems and mixed systems, and a generalized CIM framework is proposed for system reliability analysis with any system structures. The CIM is generalized for both parallel and mixed system reliability. The System Structure Matrix (SS-Matrix) is proposed to characterize any system structure in a comprehensive manner. The Binary Decision Diagram (BDD) technique is employed to identify system’s mutually exclusive path sets, of which each path set is a series system. On consequence, system reliability with any system structure is decomposed into the probabilities of the mutually exclusive path sets, which can be evaluated using different reliability analysis methods. Three examples are used to demonstrate the uniqueness and effectiveness of the proposed methodology. Section II reviews the complementary interaction method (CIM) and CI-matrix. Section III presents system reliability analysis for series, parallel, and mixed system, and the proposed unified system reliability analysis framework will be introduced in Section IV. The proposed method is demonstrated with three examples in Section V. II. Complementary Intersection Method (CIM) This section reviews the Complementary Intersection Method (CIM). First, the complementary intersection (CI) event is defined and the probability decomposition theorem is reviewed. Then, the probability of the second-order joint failure event is defined in terms of the probability of component safety events and complementary intersection events. Second, the CI-matrix is defined to facilitate the system reliability analysis for large-scale problems, which is composed of the probabilities of the component safety and complementary intersection events. A. CI event and probability decomposition theorem This subsection reviews the definition of CI event. This event enables the decomposition of the probability of any second or higher-order events into the probabilities of the CI events. Definition: Complementary Intersection (CI) Event Let an Nth-order CI event denote E12…N ≡ {X | G1⋅ G2 ⋅ …⋅ GN ≤ 0}, where the component safety (or 1st-order CI) event is defined as Ei = {X | Gi ≤ 0, i = 1, 2, …, N}. 2 American Institute of Aeronautics and Astronautics The defined Nth-order CI event is actually composed of N distinct intersections of component events Ei and their complements Ēj in total where i = 1,…, N, j =1,..., N, i ≠ j. For example, for the second order CI event Eij, it is composed of two distinct intersection events, Ē1E2 and E1Ē2. These two events are the intersections of E1 (or E2) and the complementary event of E2 (or E1). Thus, we refer to the defined event as the Complementary Intersection (CI) Event. Theorem: Decomposition of the Probability of an Nth-Order Joint Safety Event With the definition of the CI event, the probability of an Nth-order joint safety event can be decomposed into the probabilities of the CI events as N N N N N 1 m −1 N −1 P ∩ Ei = N −1 ∑ P ( Ei ) − ∑ P ( Eij ) + ∑ P ( Eijk ) + + ( −1) P E + + − 1 P E ( ) ( ) ∑ ijl 12 N i =1; i =1; i =1; i =1 2 i =1 m j = 2; j = 2; j = 2; i< j k =3 i< j<k l =m i < j << l (1) The detailed derivation of Eq. (1) can be found in Ref. [15]. It is noted that each CI event has its own limit state function, which enables the use of any reliability analysis methods. In general, higher-order CI events are expected to be highly nonlinear. Considering the tradeoff between computational efficiency and accuracy, this paper uses the probabilities of the first and second-order CI events in Eq. (1) for system reliability analysis. However, more terms in Eq. (1) can be employed as advanced reliability analysis methods are developed in future. Based on the definition of the CI event, the second-order CI event can be denoted as Eij ≡ {X | Gi ⋅ Gj ≤ 0}. The CI event can be further expressed as Eij = Ēi Ej ∪ Ei Ēj where the component failure events are defined as Ēi = {X | Gi > 0} , Ēj = {X | Gj > 0}. The event Eij is composed of two events: Ei Ēj = {X | Gi ≤ 0 ∩ Gj > 0} and Ēi Ej = {X | Gi > 0 ∩ Gj ≤ 0}. Since the events, Ēi Ej and Ei Ēj, are disjoint, the probability of the CI event Eij can be expressed as P( Eij ) ≡ P( X | Gi ⋅ G j ≤ 0) = P( X | Gi > 0 ∩ G j ≤ 0) + P( X | Gi ≤ 0 ∩ G j > 0) (2) = P( Ei E j ) + P( Ei E j ) Based on the probability theory, the probability of the second-order joint safety event Ei ∩ Ej can be expressed as P( Ei E j ) = P( Ei ) − P( Ei E j ) (3) = P( E j ) − P( Ei E j ) From Eqs. (2) and (3), the probabilities of the second-order joint safety and failure events can be decomposed as 1 P ( Ei E j ) = P ( Ei ) + P ( E j ) − P ( Eij ) (4) 2 1 P( E i E j ) = 1 − P( Ei ) + P( E j ) + P ( Eij ) (5) 2 B. CI-matrix For large-scale systems, the CI events can be conveniently written in the CI-matrix. For instance, if the system includes m components in total, the CI-matrix is defined as P ( E1 ) P ( E12 ) P ( E13 ) P ( E1m ) P ( E2 ) P ( E23 ) P ( E2 m ) (6) CI = P ( E3 ) P ( E3m ) P ( E ) m In the upper triangular CI-matrix, the diagonal elements correspond to the component reliabilities (or probabilities of the first-order CI events) and the element on ith row and jth column corresponds to the probability of the second-order CI event Eij if j < i. The probabilities of the second-order joint safety and failure events in Eqs. (4) and (5) can be evaluated with the probabilities of all component safety and complementary intersection events found from the CI-matrix. The CI-matrix facilitates to evaluate system reliability. The probability of the complementary 3 American Institute of Aeronautics and Astronautics intersection events can be computed using any reliability analysis method, such as MCS, FORM, SORM, EDR method and Stochastic Expansion (SE) methods and so on. III. Generalized CIM for System Reliability Analysis This section will generalize the CIM for system reliability analysis for series, parallel, and mixed systems. Section A will briefly review the CIM for series system reliability analysis from Ref. [15]. The proposed CIM for parallel and mixed system reliability analysis are developed in Sections B and C, respectively. Section D provides a generalized CIM framework for system reliability analysis. A. Series system reliability analysis Although the second-order system reliability bounds method or the LP bounds method can generally give fairly narrow system reliability bounds, system reliability is not unique with an explicit formula. This section introduces an explicit formula for system reliability assessment, which is developed based a mathematical inequality equation. In addition to the second-order reliability bounds method and LP bounds method, this explicit formula can provide an alternative way for system reliability assessment. Considering a structural serial system with m components, the probability of system failure can be expressed as m Pfs = P ∪ Ei i =1 (7) th where pfs represents the probability of system failure and Ēi denotes the failure event of the i component. Based on the well known Boolean bounds in Eq. (8), the first-order system reliability bound is given in Eq. (9), suggested by Ang and Amin [10] and Cornell [11]. m m max( P( Ei )) ≤ P(∪ Ei ) ≤ ∑ P ( Ei ) i =1 i (8) i =1 m max P Ei ≤ Pfs ≤ min ∑ P Ei , 1 i =1 ( ) ( ) (9) However, these methods provide wide bounds of system reliability. Thus, the second-order bounds method proposed by Ditlevsen [14] in Eq. (10) is widely used because it gives quite narrow bounds of system reliability. m i −1 m m P E1 + ∑ max P Ei − ∑ P Ei E j , 0 ≤ Pfs ≤ min ∑ P Ei − ∑ max P Ei E j , 1 j <i i =2 j =1 i=2 i =1 ( ) ( ) ( ) ( ) ( ) (10) where E1 is the event having the largest probability of failure. Since the probabilities of all events are non-negative, the following inequalities must be satisfied as 2 m max( P( Ei )) ≤ i m P( Ei ) ≤ ∑ P( Ei ) ∑ i =1 i =1 (11) Based on Eqs.(10) and (11), the probability of system failure (Pfs) of a serial system failure can be simplified to a unique explicit formula as m Pfs ≅ P E1 + ∑ P Ei − ( ) i=2 i −1 ( ) ∑ P ( E E ) i 2 (12) j j =1 It is proven in Ref.[15] that this approximate probability lies in the second-order bounds in Eq. (10). From Eq. (12), serial system reliability can be assessed as (1 − the probability of system failure) and formulated as m Rs = P( E1E2 Em −1Em ) ≅ P ( E1 ) − ∑ P Ei − i=2 i −1 ( ) ∑ P ( E E ) i j =1 j 2 A, where A ≡ 0, if A > 0 if A ≤ 0 (13) Note that the terms inside the bracket, 〈 〉, should be ignored if it is less than zero and Rs should be set to zero if the approximated one given by Eq. (13) is less than zero. Equation (13) provides an explicit and unique formula for system reliability assessment based on the second-order reliability bounds shown in Eqn.(10) and an inequality equation (11). To the application of large system with multiple failure events, The CI-matrix facilitates to evaluate system reliability. The probability of the CI events can be computed using any reliability analysis method, such as MCS, First-Order Reliability Method (FORM), Second-Order Reliability Method (SORM), Eigenvector Dimension Reduction (EDR) Method or Stochastic Expansion (SE) methods. B. Parallel system reliability analysis 4 American Institute of Aeronautics and Astronautics A parallel system reliability formula can be obtained based on the formula of series system reliability by using the De Morgan’s law. According to the De Morgan’s law, the probability of parallel system failure can be expressed as m m m P ∩ Ei = 1 − P ∩ Ei = 1 − P ∪ Ei i =1 i =1 i =1 (14) ___ where Ei is the ith component failure event. Eq. (14) relates the probability of failure of a parallel system with the probability of series system safety (reliability). If we treat Ei as the ith component failure event in a series system, the right side of Eq. (14) is then the series system reliability. Based on this relationship, the probability of parallel system failure can be obtained from Eq. (13) by treating safe events in the series system as failure events in the parallel system as A, if A > 0 A ≡ (15) i=2 j =1 0, if A ≤ 0 Finally, parallel system reliability can be obtained from Eq. (15) by one minus the probability of system failure as i −1 m P(failure of a parallel system) ≅ P( E1 ) − ∑ P( Ei ) − m Rs _ parallel ≅ P( E1 ) + ∑ P( Ei ) − i=2 ∑ P ( E E ) i i −1 ∑ P ( E E ) i j 2 , j =1 2 j , A, A ≡ 0, if A > 0 if A ≤ 0 (16) C. Mixed system reliability analysis A mixed system may have various system structures (or configurations). There is no unique system reliability formula available for a mixed system. This study develops a systematic procedure for mixed system reliability analysis using the CIM. The developed procedure is introduced below with a mixed system example. Considering a mixed system with N components, the following procedure can be proceeded to carry out system reliability analysis. Step I: Constructing a System Structure Matrix (SS-Matrix) A 3-by-M SS-Matrix can be used to model the system structure in a systematic manner. The first row of the matrix contains a component number, while the second and third rows correspond to the starting and end nodes of the component. Generally, the total number of columns of a SS-Matrix, M, is equal to the total number of system components, N. In the case of complicated system structures, one component may repeatedly appear in between different sets of nodes and, consequently, M could be larger than N, for example a 2-out-of-3 system. E6 ① ⑤ ③ E1 E2 ② E3 E4 ⑥ E5 ④ E7 Figure 1 A mixed-system block diagrams Let us consider a mixed system with 7 components, as shown in Fig. 1. The system structure matrix is a 3×7 matrix. The first column of the system structure matrix, [1, 1, 2]T, indicates that the 1st component connects nodes 1 and 2. The SS-Matrix for the system in Fig. 1 can be constructed as 1 2 3 4 5 6 7 SS- Matrix = 1 2 3 4 5 2 3 2 3 4 5 6 4 5 Step II: Finding system path sets based on the SS-Matrix Based on the SS-Matrix, the Binary Decision Diagram (BDD) technique can be employed to find the mutually exclusive system path sets, of which each path set is a series system. More information on the BDD can be found in references [16] and [17]. 5 American Institute of Aeronautics and Astronautics For the mixed system shown in Fig. 2, the mutually exclusive path sets can be found by using the BDD as { Pathset = E1 E2 E3 E4 E5 , E1 E2 E6 E4 E5 , E1 E2 E3 E7 E5 , E1 E2 E6 E4 E3 E7 E5 } Step III: Calculating system reliability for all mutually exclusive path sets and the overall system Due to the mutual exclusiveness, the mixed system reliability, RM, is the sum of the probabilities of all paths as Np Np Rs _ mixed = Pr ∪ Path i = ∑ Pr ( Path i ) (17) i =1 i =1 where Pathi is the ith mutually exclusive path set obtained by the BDD and Np is the total number of mutually exclusive path sets. For the system in Fig. 2, the system reliability can be calculated as 4 4 Rs _ mixed = Pr ∪ Path i = ∑ Pr ( Path i ) (18) i =1 i =1 = Pr( E1 E2 E3 E4 E5 ) + Pr( E1 E2 E6 E4 E5 ) + Pr( E1 E2 E3 E7 E5 ) + Pr( E1 E2 E6 E4 E3 E7 E5 ) where the probability of each individual path set can be calculated using the series system reliability formula in Eq.(13). D. Generalized CIM framework for system reliability analysis As a series system or a parallel system can be viewed as special case of a mixed system, the proposed CIM with the SS-Matrix and BDD can perform system reliability analysis with any system structures (or configurations) (e.g., series, parallel, and mixed). Figure 2 shows a generalized CIM framework for system reliability. Figure 2 A generalized CIM framework for system reliability analysis IV. Case Studies This section presents three case studies for a series system, a parallel system and a mixed system, respectively, to demonstrate the efficiency and accuracy of the proposed generalized CIM for system reliability. For each case study, the generalized CIM framework is demonstrated in a wide range of system reliability levels and compared with Monte Carlo Simulation (MCS). For series and parallel systems, the results of the generalized CIM framework are also compared with First-Order Bound (FOB) and Second-Order Bound (SOB) methods. The main objective of the examples is to demonstrate the accuracy of the proposed generalized CIM framework for system reliability analysis. So in three examples we focus on a mathematical error produced by a system reliability formulae rather than a numerical error by a numerical method. In order to eliminate the numerical error in system reliability analysis, the Monte Carlo Simulation (MCS) with 1,000,000 sample points is used to obtain the CI matrix. A. Series system example: an internal combustion engine system The following internal combustion engine case study demonstrates the application of the generalized CIM for series system reliability analysis. Five random variables are considered in this example: the cylinder bore b, compression ratio cr, exhaust valve diameter dE, intake valve diameter dI, and the revolutions per minute (rpm) at peak power, ω. All the random variables are assumed to follow normal distribution with statistical information shown in Table 1. More detail information of this example can be found in Ref. [19]. 6 American Institute of Aeronautics and Astronautics Table 1 Statistical information of input random variables for combustion engine Random Variable b (mm) cr (mm) dE (mm) dI ω, (×10-3) Mean ----------- Standard Deviation 0.40 0.15 0.15 0.05 0.25 Distribution Type Normal Normal Normal Normal Normal From a thermodynamic viewpoint, nine component safety events are defined as follows: E1 = {1.2 N c b − 400 ≤ 0} { ( min. bore wall thickness ) 0.5 } E2 = 8V / ( 200π N c ) − b ≤ 0 ( max. engine height ) E3 = {d I + d E − 0.82b ≤ 0} ( valve geometry and structure ) E4 = {0.83d I − d E ≤ 0} ( min. value diameter ratio ) E5 = {d E − 0.89d I ≤ 0} ( max. value diameter ratio ) −5 E6 = 9.428 × 10 4V / (π N c ) (ω /d I 2 ) − 0.6Cs ≤ 0 ( max. Mech/Index ) E7 = {0.045b + cr − 13.2 ≤ 0} ( knock-limit compression ratio ) E8 = {ω − 6.5 ≤ 0} ( max. torque converter rpm ) E9 = {230.5Qηtw − 3.6 × 106 ≤ 0} ( max. fuel economy ) { (19) } where ηtw = 0.85951(1 − cr−0.33 ) − Sv , V = 1.859 × 106 mm3 Q = 43,958 kJ/kg, Cs = 0.44, and N c = 4 In this study, system reliability analyses are performed at the eight different design points as listed in Table 2. These design points are the reliability-based optimum designs using FORM with eight different target component reliability levels from 80% to 99.9%. The results of system reliability analysis at these design points are summarized in Table 3 and also graphically shown in Fig. 3. From the results, it is found that the first-order bounds method gives too wide bounds to be of practical use. Whereas, the second-order bounds method gives tighter bounds. The generalized CIM method provides more accurate results at all reliability levels and its high accuracy is maintained at high reliability levels, which are often encountered in engineering practices. The error in the CIM comes from the probabilities of the third- or higher-order joint failure events. As the reliability level decreases, the effects of the third- or higher-order joint failure events increase. That is why the error increases as the system reliability decreases. Table 2 Eight different design points for system reliability analysis designs points 1 2 3 4 5 6 7 8 b 82.1025 82.3987 82.5511 82.6770 82.8234 82.8750 82.9204 82.9977 Mean values for random variables cr dE dI 35.8039 30.3274 9.3397 36.1754 30.4835 9.3684 36.3630 30.5676 9.3811 36.5187 30.6334 9.3920 36.7006 30.7121 9.4049 36.7655 30.7407 9.4096 36.8222 30.7657 9.4137 36.9197 30.8084 9.4204 7 American Institute of Aeronautics and Astronautics ω 5.2827 5.5983 5.7550 5.8901 5.9498 5.9754 5.9772 5.9795 Table 3 Results of System Reliability Analysis with MCS, FOB using MCS, SOB using MCS, and Generalized CIM using MCS (N=1,000,000) Analysis Method FOB SOB 1 2 System Reliability Level at Each Design 3 4 5 6 7 8 Upper 0.9989 0.9899 0.9745 0.9495 0.8984 0.8742 0.8490 0.7988 Lower 0.9949 0.9506 0.8744 0.7432 0.5367 0.4318 0.3410 0.1513 Upper 0.9949 0.9520 0.8822 0.7741 0.6224 0.5554 0.4987 0.3967 Lower 0.9949 0.9517 0.8798 0.7653 0.5929 0.5190 0.4418 0.3049 CIM 0.9949 0.9518 0.8805 0.7674 0.6026 0.5312 0.4612 0.3371 MCS 0.9949 0.9520 0.8820 0.7731 0.6179 0.5476 0.4871 0.3748 Figure 3 Results of system reliability analysis at eight different reliability levels B. Parallel System Example: A Ten Brittle Bar System The following ten-bar system example demonstrates the application of the generalized CIM framework for parallel systems. As shown in Fig. 4, ten brittle bars are connected in parallel to sustain a load applied at one end. This case study is modified from the example employed in Ref. [20] by increasing the total number of bars from 2 to 10. Ten bars are all brittle with different fracture strains εf, and the strain level for the maximum allowable load should be one of the ten fracture strain levels. At the ith fracture strain level εfi, components with fracture strains below εfi fail, and the overall allowable load is simply the strength summation of components with fracture strains above εfi. Thus, as the maximum allowable load, the system strength RT can be formulated as at ten fracture strain levels εf1,…, εf10, as shown in Eq. (20). 8 American Institute of Aeronautics and Astronautics 10 10 RT = max ∑ R j (ε ) = max ∑ R j (ε fi ) ε j =1 1≤i ≤10 j =i (20) R1 (ε f 1 ) + R2 (ε f 1 ) + R3 (ε f 1 ) + R4 (ε f 1 ) + R5 (ε f 1 ) + R6 (ε f 1 ) + R7 (ε f 1 ) + R8 (ε f 1 ) + R9 (ε f 1 ) + R10 (ε f 1 ), R2 (ε f 2 ) + R3 (ε f 2 ) + R4 (ε f 2 ) + R5 (ε f 2 ) + R6 (ε f 2 ) + R7 (ε f 2 ) = max + R8 (ε f 2 ) + R9 (ε f 2 ) + R10 (ε f 2 ), ...... R10 (ε f 10 ) 1 2 …… 10 F (a) (b) Figure 4 Ten brittle bar parallel system: (a) system structure model; (b) brittle material behavior in parallel system For example, at fracture strain εf2, the 1st brittle bar fails due to fracture and thus does not contribute to the overall system strength, which becomes the strength summation of the surviving nine brittle bars. This scenario can be treated as a component corresponding to fracture strain εf2. There are totally ten components corresponding to ten strain levels, respectively, as shown in Eq.(20). The brittle bar system fails to sustain the load F only if all the ten components fail. This is a parallel system with ten components. The component safety events can be expressed in terms of several random variables. 10 10 j =i j =i Gi = F − ∑ R j (ε fi ) = F − ∑ ( E j Aj ) ⋅ ε fi , 1 ≤ i ≤ 10 (21) where, Rj represents the allowable load that can be sustained by the jth brittle bar; Aj the cross section area of the jth brittle bar; Ej Young’s modulus of the jth brittle bar. Random variables and their random properties are summarized in Table 4. Ten different system reliability levels are used for comparison with ten different loading conditions (F). These loading points are used to validate the generalized CIM method at different reliability levels. Table 5 summarizes the results of system reliability analysis which are graphically summarized in Fig. 5. It can be seen that the first-order bounds are too wide to be of practical use. Whereas, the second-order bounds method gives tighter system reliability bounds compared with the first order bounds method. The generalized CIM method provides more accurate results at all reliability levels and its high accuracy is maintained at high reliability levels, which are often encountered in engineering practices. The error in the CIM comes from the probabilities of the third- or higher-order joint failure events. For a parallel system, as the reliability level decreases, the effects of the third- or higher-order joint failure events also decrease. That is why the error decreases as the system reliability decreases. 9 American Institute of Aeronautics and Astronautics Table 4 Statistical information of input random variables for the ten bar system Random Variable E1-E10 (GPa) A1 (mm2) A2 (mm2) A3 (mm2) A4 (mm2) A5 (mm2) A6 (mm2) A7 (mm2) A8 (mm2) A9 (mm2) A10 (mm2) εf1 εf2 εf3 εf4 εf5 εf6 εf7 εf8 εf9 εf10 F (kN) Mean 200.0 100.0 120.0 140.0 140.0 140.0 150.0 150.0 150.0 200.0 300.0 0.0010 0.0012 0.0018 0.0025 0.0027 0.0030 0.0033 0.0036 0.0040 0.0050 --- Standard Deviation 10.0 5.0 5.0 5.0 10.0 10.0 10.0 15.0 15.0 15.0 25.0 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 0.0010 0.0011 30.0 Distribution Type Gumbel Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Uniform Uniform Uniform Uniform Uniform Uniform Uniform Uniform Uniform Uniform Normal Table 5 Results of System Reliability Analysis with MCS, FOB using MCS, SOB using MCS, and Generalized CIM using MCS (N=1,000,000) Analysis Method System Reliability Level at Each Design 1 2 3 4 5 6 7 8 9 10 Upper 0.4133 0.5639 0.7331 0.9216 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Lower 0.1594 0.2054 0.2507 0.2974 0.3444 0.4395 0.4865 0.5334 0.5803 0.9705 Upper 0.3537 0.467 0.5854 0.7065 0.8293 1.0000 1.0000 1.0000 1.0000 1.0000 Lower 0.3192 0.4062 0.4849 0.5507 0.6068 0.6917 0.7161 0.7459 0.7897 0.9943 CIM 0.3417 0.4456 0.5490 0.6482 0.7388 0.8714 0.9017 0.9069 0.9051 0.9943 MCS 0.3301 0.4272 0.5226 0.6131 0.6961 0.8314 0.8813 0.9192 0.9476 0.9998 FOB SOB 10 American Institute of Aeronautics and Astronautics Figure 6 Results of system reliability analysis at ten different reliability levels C. Mixed System Example: A Cantilever Beam-Bar Mixed System The following cantilever beam-bar system [18] is employed in this study to demonstrate the effectiveness of the generalized CIM framework for mixed system reliability analysis. The system is considered as an ideally elasticplastic cantilever beam supported by an ideally rigid-brittle bar, with a load applied at the midpoint of the beam, as shown in Fig. 6. There are three failure modes and five independent failure events E1 ~ E5. These three failure modes are formed by different combinations of failure events as: • 1st Failure Mode: The fracture of the brittle bar (event E1) occurs, and subsequently the formation of a hinge at the fixed point of the beam (event E2). • 2nd Failure Mode: The formation of a hinge at the fixed point of the beam (event E3) followed by the formation of another hinge at the midpoint of the beam (event E4). • 3rd Failure Mode: The formation of a hinge at the fixed point of the beam (event E3) followed by the fracture of the brittle bar (event E5). The five safety events can be expressed as: E1 = { X , T | 5 X /16 − T ≤ 0} , E2 = { X , L, M | LX − M ≤ 0} , E3 = { X , L, M | 3LX / 8 − M ≤ 0} , E4 = { X , L, M | LX / 3 − M ≤ 0} , E5 = { X , L, M , T | LX − M − 2 LT ≤ 0} Considering these three failure modes, the system success event can be obtained as: ES = ( E1 ∪ E2 ) ∩ { E3 ∪ ( E4 ∩ E5 )} (22) (23) The statistical information of the random input variables is given in Table 6. Ten different system reliability levels are used for comparison with ten different loading conditions (X). Figure 7 shows the system block diagram. Figure 6 A Cantilever beam-bar system 11 American Institute of Aeronautics and Astronautics Table 6 Statistical information of input random variables Random Variable L T M X Mean 5.0 1000 150 --- ① Standard Deviation 0.05 300 30 20 ② Distribution Type Normal Normal Normal Normal ④ ③ Figure 7 System block diagram for mixed system example From the system block diagram shown in Fig. 7, the SS-Matrix for this mixed system can be obtained by its definition as: 1 2 3 4 5 SS-Matrix = 1 1 2 2 3 2 2 4 3 4 With this SS-Matrix, the BDD can be constructed as shown below in Fig. 8 Figure 8 BDD diagram for mixed system example The BDD shown in Fig. 8 indicates the following mutually exclusive system path sets as Pathset = {E1 E3 , E1 E2 E3 , E1 E3 E4 E5 , E1 E2 E3 E4 E5 } The system reliability analysis is carried out with ten different loading conditions (X) using the generalized CIM. The MCS is used for a benchmark solution. Table 7 summarizes the results. It can be found that the CIM gives very accurate mixed system reliability results at all reliability levels. In recent literature, the LP bound method was developed [18] and is able to calculate the optimal bounds for mixed system reliability based on available information. Results regarding this example using the LP bound method can be found in Ref. [18]. Compared with the LP bound method, the generalized CIM not only gives a unique estimate of mixed system reliability, but also provides numerical schemes for system reliability. It is known that the LP bound method can suffer when an approximate LP algorithm is used for over constrained problems. Besides, it is extremely sensitive to the accuracy of the given input information, which are the probabilities for the first-, second-, and high-order joint safety events. 12 American Institute of Aeronautics and Astronautics Table 7 Results of different system reliability analysis methods: (1) MCS, (2) Complementary Interaction Method (CIM) using MCS (N=1,000,000) System Reliability at Each Point Analysis Method 1 2 3 4 5 6 7 8 9 10 CIM 0.3546 0.4251 0.4981 0.5724 0.6444 0.7110 0.7708 0.8666 0.9308 0.9681 MCS 0.3548 0.4252 0.4982 0.5725 0.6445 0.7111 0.7708 0.8667 0.9309 0.9681 V. Conclusion This paper presented a generalized Complementary Intersection Method (CIM) for system reliability analysis for series, parallel, and mixed systems. The CIM decomposes the probability of a high-order joint failure event into probabilities of complementary intersection events, and provides an innovative way to evaluate system reliability. However, its application has been limited to a series system only. In this paper, a generalized CIM framework is proposed for system reliability analysis for a system with any structure or configuration (e.g., series, parallel, and mixed). Within the generalized framework, (i) the CIM was firstly generalized for parallel system reliability analysis with a unique explicit formula, (ii) the System Structure Matrix (SS-Matrix) was proposed to characterize any system structure in a comprehensive manner, and (iii) the Binary Decision Diagram (BDD) technique is employed to identify system’s mutually exclusive path sets, of which each path set is a series system. On consequence, system reliability with any system structure is decomposed into the probabilities of the mutually exclusive path sets, which can be evaluated as series systems using the CIM formula by different reliability analysis methods. The proposed CIM framework is generally applicable for system reliability analysis for different engineering systems, regardless the structure or configuration. The proposed methodology was demonstrated to be effective and efficient by three case studies, one for series systems, one for parallel systems, and one for mixed systems. Acknowledgments The work presented in this paper has been partially supported by US National Science Foundation (NSF) under Grand No. GOALI-07294, U.S. Army TARDEC by the STAS contract (TCN-05122), General Motors under Grant No. TCS02723. References 1 Youn, Byeng D., Choi, K. K., and Du, L., “Enriched Performance Measure Approach (PMA+) for ReliabilityBased Design Optimization,” AIAA Journal, Vol. 43, No. 4, pp. 874-884, 2005. 2 Chen, X., Hasselman, T.K., and Neill, D. 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