Adaptive probability analysis using an enhanced hybrid mean value method

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1 Research Paper Struct Multidisc Optim 28, ) DOI /s Adaptive probability analysis using an enhanced hybrid mean value method B.D. Youn, K.K. Choi, L. Du Abstract This paper proposes an adaptive probability analysis method that can effectively generate the probability distribution of the output performance function by identifying the propagation of input uncertainty to output uncertainty. The method is based on an enhanced hybrid mean value HMV+) analysis in the performance measure approach PMA) for numerical stability and efficiency in search of the most probable point MPP). The HMV+ method improves numerical stability and efficiency especially for highly nonlinear output performance functions by providing steady convergent behavior in the MPP search. The proposed adaptive probability analysis method approximates the MPP locus, and then adaptively refines this locus using an a posteriori error estimator. Using the fact that probability levels can be easily set a priori in PMA, the MPP locus is approximated using the interpolated moving least-squares method. For refinement of the approximated MPP locus, additional probability levels are adaptively determined through an a posteriori error estimator. The adaptive probability analysis method will determine the minimum number of necessary probability levels, while ensuring accuracy of the approximated MPP locus. Several examples are used to show the effectiveness of the proposed adaptive probability analysis method using the enhanced HMV+ method. Nomenclature X Random parameter; X =[X 1,X 2,...,X n ] T x Realization of X; x =[x 1,x 2,...,x n ] T U Independent and standard normal random parameter; U =[U 1,U 2,...,U n ] T u Realization of U; u =[u 1,u 2,...,u n ] T β Probability level GX) Performance function; the design is considered a fail if GX) < 0 ĜX) Approximate performance function of GX) t Parametric coordinate along the arc of an n- dimensional sphere u Most probable point n Normalized steepest-descent direction of the performance function U j Approximate MPP locus h Basis vector for MPP locus approximation a Coefficient vector for MPP locus approximation M Full moment matrix in approximating MPP locus B Half moment matrix in approximating MPP locus W Weight matrix in approximating MPP locus A posteriori error estimator of MPP locus ε MPPL Key words TS a Received: Revised manuscript received: April 2004 Published online: 2004 Springer-Verlag 2004 B.D. Youn, K.K. Choi,L.Du Center for Computer-Aided Design and Department of Mechanical Engineering, College of Engineering, The University of Iowa, Iowa City, IA ybd@ccad.uiowa.edu, kkchoi@ccad.uiowa.edu, liudu@ccad.uiowa.edu 1 Introduction The fact that the decision-making process must be made under various uncertainties underscores the need for more effective output probability analysis methods that can identify the propagation of input uncertainty to output uncertainty. To identify the uncertainty propagation of an engineering system, various tools for output probability analysis have been developed: sampling methods, moment estimation methods, most probable point MPP)- based methods, etc. Due to its generality and simplicity, Monte Carlo simulation Rubinstein 1981) has been extensively used TS a Please add keywords.

2 2 as one of sampling methods for output probability analysis. However, given the need for a large sample size to obtain the probability distribution of the output performance function, this method requires intensive computation and is expensive. Several alternative sampling methods, so called variance reduction methods, have been proposed: an importance sampling method Melchers 1989), an adaptive sampling method Bucher 1988), an adaptive importance sampling method Wu 1994), etc. The basic idea behind these methods is to reduce the variance in the estimate during the simulation procedure by using the probability density function of an importance sampling. Thus, the probability density function must be properly selected to yield accurate estimates for high probability levels with possibly a small number of samples Melchers 1989; Bucher 1988; Wu 1994). It has been pointed out Du and Chen 2001) that the adaptive sampling method should choose a proper probability density function, which is the original density f X x) conditional on the failure domain. However, even though the variance reduction method requires a smaller number of samples than Monte Carlo simulation, the computational requirement is still quite significant for large-scale problems. To alleviate such a high computational burden, several moment estimation methods were proposed using a quadrature formula for numerical integration Evans 1972; Evans and Falkenburg 1976), or using a numerical integration scheme and a design of experiment DOE) D Errico and Zaino 1988; Seo and Kwak 2003). The former approximates statistical moments of the output performance function by directly integrating statistical moments of its Taylor series approximations using the quadrature formula. The latter estimates those moments by employing numerical integration with integration points and their corresponding weights, and performing DOE at those integration points. Even though these methods are accurate for high-order statistical moments, they are still expensive for large-scale applications with a sizeable number of uncertain parameters D Errico and Zaino 1988; Seo and Kwak 2003). Two MPP-based methods were developed Wu and Wirsching 1987; Du and Chen 2001) based on the advanced first-order second-moment method that was introduced by using a rotationally invariant reliability measure Hasofer and Lind 1974): the reliability index approach or G-level approach) and the performance measure approach or P-level approach). In this paper, the performance measure approach PMA) *Tu and Choi 1999; Youn et al. 2001, 2003) or fixed norm approach Lee and Kwak )) is used for output probability analysis, since it is difficult in the reliability index approach RIA) to set proper output performance function levels a priori, whereas the probability levels β i [ 6, 6] can be easily set a priori in PMA. For MPP search, the advanced mean value method Wu and Wirsching 1987) is shown to be either inefficient or unstable for a concave performance function Youn et al. 2001, 2003; Youn and Choi 2003). Thus, a hybrid mean value HMV) method was developed in Youn et al. 2001, 2003) to handle both convex and concave performance functions more effectively. However, it is shown in this paper that the HMV method could fail to converge for highly nonlinear output performance functions. To improve numerical stability and efficiency, an enhanced HMV HMV+) method is proposed in this paper. The HMV+ method is shown to improve numerical stability and efficiency for highly nonlinear output performance functions by providing steady convergent behavior in the MPP search. A probability analysis method was proposed in Du and Chen 2001) by approximating the most probable point MPP) locus, where the MPP locus is approximated using the extrapolated least-squares method, and the number of probability levels are predetermined. An adaptive probability analysis method is proposed in this paper by employing an a posteriori error estimator. That is, the MPP locus is refined by adaptively determining additional probability levels using an a posteriori errorestimator. Thus, this method allows us to use a minimum number of necessary probability levels, while controlling accuracy even for highly nonlinear performance functions. For approximation of the MPP locus of probabilistic structural responses, using the fact that probability levels β i [ 6, 6] can be easily set a priori in PMA, the MPP locus is approximated using interpolated least squares and moving least-squares methods Lancaster and Salkauskas 1986). The comparison between leastsquares and moving least-squares methods is discussed in terms of the rate of convergence. The proposed method is demonstrated using numerical examples. Comparisons are also made to other probability analysis methods available in the literature. 2 Enhanced HMV HMV+) method for output probability analysis Probability analysis provides a complete probability distribution of output performance function by identifying the propagation of input uncertainty to output uncertainty. In this paper, the MPP-based method is developed to carry out the output probabilistic structural analysis, which employs a series of MPP search carried out using the proposed HMV+ method. 2.1 Output probability analysis When an engineering system is subject to a variety of uncertainties, an output probability analysis provides a valuable probability distribution or density) function, or statistical moments of the output performance function for a decision-based design Chen et al. 2000). The probability distribution function is obtained by identifying the propagation of input uncertainty to output

3 3 uncertainty as Madsen et al. 1986; Haldar and Mahadevan 2000) F G g)= f X x)dx 1...dx n 1) where the uncertainty F X x) orf X x)) of input X is propagated to the uncertainty F G g) of output GX). In this paper, the probability distribution function is approximated by employing PMA, in which HMV+ searches the MPP x i corresponding to a given probability F G g i )=β i where gx i )=g i. Using the MPP-based method, the output probability analysis in 1) can be redefined as F G g i )= f X x)dx 1...dx n = GX) g i f X x)dx 1...dx n =Φ β i ), GX) g i 0 i =1,...,NPL 2) where Φ ) is the standard normal distribution and NPL is the number of probability levels. Taking an inverse transformation of F G for PMA, the corresponding output performance function level is computed using g i = F 1 G Φ β i)) = gx i ), i=1,...,npl 3) The MPPs x i are obtained at a series of probability levels using HMV+. Once the MPP locus is obtained, the probability distribution or density) function of output is approximately constructed by associating the probability levels and their output performance functions, as shown in Fig. 1. TS b For many engineering applications, it is desirable to obtain accurate tail end of probability distribution especially for high levels of probability, i.e. 3σ,whichcanbe estimated using an advanced first-order second-moment method Youn and Choi 2003). The advanced first-order second-moment method Hasofer and Lind 1974) is used in this paper with an emphasis on improving numerical efficiency, while maintaining a desirable level of accuracy. A transformationt: X U is introduced to map the original random space X to a standard, uncorrelated normal random space U. If the random variables X i are mutually independent, the transformation is Rackwitz and Fiessler 1978) T : U i =Φ 1 F Xi x i ) ), i=1 n 4) where F Xi is the probability distribution function of X i. To simplify the probability integration in 2), the limit state surface G g i = 0 can be expressed as a first-order approximation. This is referred to as the first-order reliability method FORM) Hasofer and Lind 1974; Madsen et al. 1986; Haldar and Mahadevan 2000), as shown in Fig. 2. UsingFORM,2)or3)canbesolvedbyformulating an optimization with one equality constraint Tu and Choi 1999; Youn et al. 2001, 2003; Lee and Kwak ) to obtain g i = gu i ), if β i > 0, maximize GU) subject to U = β i, for i =1,...,NPL if β i < 0, minimize GU) subject to U = β i, for i =1,...,NPL 5) Fig. 1 MPP-based method for output probability analysis TS b Please check quality of all your figures send higher resolution versions of Figs. 1, 3 5 and 7 10 if possible.

4 4 Fig. 2 First- and second-order reliability methods Once the MPP u i = T x i ) is obtained for a given probability level, the corresponding output performance function level is computed using g i = gu i ). The MPP-based method using PMA is very effective for the output probability analysis since it allows one to set probability levels β i [ 6, 6] a priori, without knowing the output performance function levels. 2.2 Enhanced hybrid mean value HMV+) method Even though the HMV method Youn et al. 2001, 2003) performs well for convex or concave performance functions, it could fail to converge for highly nonlinear output performance functions. To improve numerical stability and efficiency, an enhanced HMV HMV+) method is proposed by revising the previous HMV algorithm. In this new HMV+ algorithm, if the value of the performance function is decreased at the next search point, the performance function is approximated along the arc with constant probability level β ι ) between the current point and next search point to find a new search point where the approximated performance function has the maximum value. Step 1. Set the iteration counter k = 0 and probability level β i with the convergence parameter ε =10 4.Letu 0) HMV+ = 0. Step 2. Calculate the performance function gu k) HMV+ ) and its sensitivity gu k) HMV+ ). Step 3. Check the Karush Kuhn Tucker condition: sgnβ u k) HMV+ i) u k) HMV+ nk) HMV+ 1 ε for k 2, where n is the normalized steepest ascent direction of GU); and sgnβ ι ) is the signum function, such that it is +1 if β ι > 0and 1 ifβ ι < 0. If it is satisfied, then stop. [ ] Step 4. If k 2 and sgnβ i ) gu k) HMV+ ) guk 1) HMV+ ) < 0, interpolate the performance function along the arc region between these two search points and obtain a new search point u k+1) HMV+ by maximizing the approximate performance function. Otherwise, use the HMV method to obtain a new search point u k+1) HMV+.Letk = k +1 and go to Step 2. In the HMV+ method, the arc-interpolation method is employed when the performance function value is decreased at the next search point for β i > 0orincreased at the next search point for β i < 0. Performance function and its sensitivity values at two search points,, are used to interpolate the performance function along the arc region between these two search points. For the interpolation, a parametric coordinate t is introduced as u k) HMV+ and uk 1) HMV+ U = st)u k 1) + tu k) and U = β t, s,t 0 6) where s + t)= tuk 1) u k) + tu k 1) u k)) 2 +1 t2 )β 4 t tu s t)= tuk 1) u k) k 1) u k)) 2 +1 t2 )βt 4 βt 2 7) The sensitivity of the performance function with respect to a parametric coordinate t can be obtained using a chain rule as β 2 t

5 dg dt = G U i U i t = G ) ds U i dt uk 1) + u k) 8) value for β i > 0 or smaller performance function value for β i < 0, needs to be selected. 5 In 8), the sensitivities of performance function are evaluated at two search points u k 1) and u k) as dg u k 1)) = dt g u k 1)) ds ± ) t) dt u k 1) + u k) = t=0 g u k 1)) u k 1) u k) ) βt 2 u k 1) + u k) dg u k)) = g u k)) ds ± ) t) dt dt u k 1) + u k) = t=1 g u k 1)) β 2 ) t u k 1) u k) uk 1) + u k) 2.3 Numerical examples of MPP search for probability level β ι Three numerical examples are used to demonstrate the effectiveness of the HMV+ method for MPP search at probability level β ι : two for numerical efficiency and one for numerical stability. Example 1: nonlinear performance function 1 The first example is from Youn et al. 2001, 2003), which is given as Then, gu k 1) ),gu k) dgu ), k 1) ) dt, and dguk) ) dt are used to interpolate the performance function using a cubic polynomial as Gt)=a 0 + a 1 t + a 2 t 2 + a 3 t 3 9) The next search point u k+1) is obtained where the approximated performance function G is maximum, as u k+1) = st )u k 1) + t u k) at t = t Gt) is maximum, for β i > 0 where Gt) is minimum, for β i < 0 10) Note that st) may not be unique for one value of t, if βt t>1s, t > 0; s, t 4 βt 4 uk 1) u k) ) 2 ), as shown in 7). This could be true when the angle composed of three points u k 1), u 0), and u k) is more than 90,whichcan be expressed mathematically as u k 1) u k) < 0. One of the two s values, that yields greater performance function Fig. 3 Performance function contour and MPP search iterations using the HMV+ method for Example 1 Table 1 Results of MPP search for Example 1 Iteration X 1 X 2 g x k)) dg k 1)/ dt dg k)/ dt s t MPP

6 6 Table 2 Comparison for numerical efficiency of the HMV+ method Method gx ) x NFE/NSE HMV , 3.161) 17/17 Proposed HMV , 3.156) 8/8 MFD , 3.156) 40/6 SLP , 3.155) 18/12 SQP , 3.157) 18/8 GX)= 0.3X 2 1 X 2 + X 2 0.8X ) where X i N1.2, 0.42), X i N1.0, 0.42), and β t =6.0. The performance function contour and MPP search iterations, using the HMV+ method, are shown in Fig. 3. As shown in Fig. 3, the response is highly nonlinear and contains both saddle point and regions with high curvature. For numerical efficiency, the HMV+ method is compared to some general optimization algorithms: a modified feasible direction MFD), a sequential linear programming SLP), and a sequential quadratic programming SQP). It is shown in Table 1 that the proposed HMV+ method converges in seven iterations. Three interpolations are employed at 2 nd,4 th,and6 th iterations. On the other hand, the HMV method and the other general optimization algorithms converge slowly, as shown in Table 2. The last column in Table 2 shows the number of total function evaluations NFE) and sensitivity evaluations NSE). The total number of function and sensitivity evaluations is the least for the HMV+ method. Example 2: nonlinear performance function 2 Another highly nonlinear performance function is from Du et al. 2003), which is given as GX)= 4+X ) 2 X ) 3 X ) 4 + X 2 12) where X 1 N0.0, 1.0), X 2 N0.0, 1.0), and β t =3.0. The performance function contour and MPP search iterations, using the HMV+ method, are shown in Fig. 4. As Fig. 4 Performance function contour and MPP search iterations using the HMV+ method for Example 2 shown in Fig. 4, the response is highly nonlinear with high curvature contours. Table 3 illustrates that the proposed HMV+ method converges in five iterations with two interpolations at the second and fourth iterations. On the other hand, as shown in Table 4, the HMV method, SLP, and SQP fail to find the MPP, while MFD converged very slowly. Thus, the HMV+ method is shown to be numerically stable and efficient in the MPP search. The result of the most probable point of inverse reliability MPPIR) method Du et al. 2003) for the same problem is shown in Table 4, requiring 20 function evaluations. Example 3: nonlinear performance function 3 A vehicle side-impact is employed to perform MPP search for a high-dimensional nonlinear response. The problem isdescribedindetailinyounet al. 2004). The velocity Table 3 Results of MPP search for Example 2 Iteration X 1 X 2 g x k)) dg k 1)/ dt dg k)/ dt s t MPP

7 7 of the door is used as the performance function, which is expressed as GX)= X 3 X X 5 X X 9 X X 9 X X ) Table 6 Comparison for numerical efficiency of the HMV+ method Method gx ) NFEA HMV /59 Proposed HMV /9 MFD /7 SLP /43 SQP /12 where β t =3.0, X i N1.0, 0.05) for i =1to7, X i N0.3, 0.006) for i =8, 9, and X i N0.0, 10.0) for i =10, 11. Table 5 provides the MPP search results of the proposed HMV+ method in X-space. Throughout the MPP search, the variables X 1, X 2, X 4,andX 8 remain unchanged. The HMV+ method converges in eight iterations along with three interpolations at the the second, fourth and seventh iterations. Other methods shown in Table 6 converge to the same MPP, but are not as efficient as the MPP+ method. Figure 5 shows the performance function contour and MPP search iterations, using the HMV+ method, on the X 10 X 11 hyperplane by setting X i = x i for i =1to9. Fig. 5 Performance function contour and MPP search iterations using the HMV+ method for Example 3 Table 4 Comparison for numerical efficiency of the HMV+ method Method gx ) x NFEA HMV Diverged Proposed HMV , 2.679) 6/6 MPPIR , 2.679) 20/0 MFD , 2.679) 55/5 SLP , 2.996) 16/11 SQP , 2.996) 12/4 3 Adaptive probability analysis An adaptive probability analysis method is proposed in this paper, by taking advantage of the proposed HMV+ method, to approximate the probability distribution of the output performance function. To this end, two methods are integrated in this paper: an interpolated instead of extrapolated) MPP locus approximation using the moving least-squares method; and an adaptive method to choose probability levels effectively depending on the nonlinearity of probabilistic output performance functions. In the interpolated method, the approxima- Table 5 Results of MPP search for Example 3 Iter X 3 X 5 X 6 X 7 X 9 X 10 X 11 g x k)) dg k 1 /dt dg k /dt MPP

8 8 Fig. 6 Flow chart of adaptive probability analysis tion is made in the region surrounded by known samples, whereas in the extrapolated method, the approximation is also made for outside this region. The interpolated MPP locus approximation is studied by comparing the least-squares and moving least-squares methods. For the adaptive method, an a posteriori error estimator is defined as the difference between the initial search point on the approximate MPP locus and the MPP point obtained using the HMV+ method at the selected probability level β ι. The MPP locus is refined by adaptively adding more probability levels until the a posteriori error is small enough at all selected probability levels. Thus, the adaptive method is a closed-loop probability analysis approach, as shown in Fig. 6, unlike the open-loop probability analysis approach where all probability levels β i [ 6, 6], i=1 n, needs to be predetermined. Detailed discussions of this proposed method are presented in the following sections. 3.1 MPP locus approximation In the open-loop probability analysis approach, it is difficult to set a proper number of probability levels without knowing the degree of nonlinearity of the performance function. That is, too many probability levels will be expensive, while too few levels may not yield accurate probability analysis. Furthermore, the MPP search at one probability level is performed independently from the other probability level. Therefore, much of the valuable information generated during the MPP search at one probability level is ignored at other probability levels. These shortcomings can be overcome using the interpolated approximation of the MPP locus, which provides good initial search points for MPP search for the next probability level. Error analyses of the MPP locus approximation are carried out for the least-squares and moving least-squares methods Least squares method The MPP locus approximation using the least-squares method can be formulated as NB U j β)= h i β)a ij = h T a j, forj =1,...,NRV 14) i=1 where NB is the number of basis monomials, NRV is the number of random parameters, a j is the coefficient vector for the j th random parameter, and h is the basis vector. Mutually independent monomials are used as basis functions. In this paper, a quadratic polynomial basis is used to approximate the MPP locus. Higher-order polynomial bases are avoided due to incorrect level of oscillation near the boundaries. To obtain the coefficient vector, a residual E LS can be defined as NPL E j LS = NPL I=1 where I=1 U j β I ) U j β I )) 2 = h T a j U j β I ) ) 2 =Haj U j ) T Ha j U j ) 15)

9 9 h 1 β 1 ) h NB β 1 ) H =..... h 1 β NPL ) h NB β NPL ) U j = [ U j β 1 ) U j β NPL ) ] 16) Here, NPL represents the number of probability levels and the subscript LS denotes the least-squares method. The residual E j LS is a positive definite quadratic form and the necessary condition to minimize E j LS is aj E j LS =0 17) By solving 17), the coefficient vector for the j th random parameter is obtained as a j = H T H ) 1 H T U j, j =1,...,NRV 18) Thus, the MPP locus approximation using the leastsquares method can be expressed as U j β)=h T H T H ) 1 H T U j, j =1,...,NRV 19) Moving least-squares method Lancaster and Salkauskas 1986) The MPP locus approximation using the moving leastsquares method is NB U j β)= h i β)a ij β)=h T β)a j β) i=1 for j =1,...,NRV 20) where h is the basis vector and a j β)=[a 1j β),a 2j β),...,a NBj β)] T is the j th coefficient vector, which is a function of the probability level β. Mutually independent monomials are used as basis functions. In this paper, a quadratic polynomial basis is used to approximate the MPP locus, which is found to provide a sufficiently accurate result, since the MPP locus is not highly oscillatory even if the output performance function is highly nonlinear. To obtain the coefficient vector, the residual E MLS can be defined as NPL E j MLS β)= NPL I=1 I=1 [ 2 wβ β I ) U j β I ) U j β I )] = wβ β I ) [ h T a j U j β I ) ] 2 = [Ha j β) U j ] T Wβ)[Ha j β) U j ] 21) where h 1 β 1 ) h NB β 1 ) H =..... h 1 β NPL ) h NB β NPL ) U j = [ U j β 1 ) U j β NPL ) ] wβ β 1 ) wβ β 2 ) 0 Wβ)= ) 0 0 wβ β NPL ) Here, NPL is the number of probability levels, and the subscript MLS denotes the moving least-squares method. Note that the approximation U j β I ) draws closer to U j β I ) at those points β I with a relatively large weight wβ β I ) by introducing a weight inversely proportional to the distance between the sample and interpolation points. The necessary condition to minimize E j MLS is aj E j MLS =0 23) By solving 23) the coefficient vector for the j th random parameter is obtained as a j β)=m 1 β)bβ)u j, j =1,...,NRV 24) where Mβ) is referred to as the moment matrix given by Mβ)=H T Wβ)H and Bβ)=H T Wβ) 25) Thus, MPP locus approximation using the moving leastsquares method is U j β)=h T β)m 1 β)bβ)u j, j =1,...,NRV 26) Since the coefficient vector is a function of the probability level, 24) must be solved at different probability levels of interest, unlike the least-squares method in 18). If the weight matrix is set as the identity matrix, the moving least-squares approximation in 26) becomes the leastsquares approximation in 19). 3.2 Adaptive set of probability levels Using either the least-squares or moving least-squares method, the approximate MPP locus can be refined as the number of probability levels are increased. The adaptive method to add more probability levels is determined using an a posteriori error estimator, defined as ε MPPL = u 0 k u k β2 k ) where u 0 k and u k are the initial search point and MPP, respectively, at the current probability level β k,andε MPPL is the error measure of the MPP locus. CE c Please complete this sentence

10 10 Fig. 7 Adaptive set of probability levels In the proposed probability analysis method, an a posteriori error estimator is used to decide on the appropriateness of the number of probability levels for accuracy of the approximated MPP locus. That is, the MPP locus and its a posteriori error estimator make it possible to add more probability levels adaptively using the bisection method. The adaptive procedure continues to refine the MPP locus by adding number of probability levels until the a posteriori error is small for all probability levels. Detailed procedures involved in the adap- tive probability analysis are illustrated in Fig. 7. First, the MPP search is carried out at the probability levels ±1/2βH, where βh is predetermined, say βh = 6.0. The HMV+ method is used since it can efficiently and accurately perform MPP search for even a high probability level, e.g. βh = 6.0 shown in Example 1. Next, the MPP locus is approximated based on information obtained at probability levels, β = 0, ±3, ±6, as shown in Fig. 7a). Using the bisection method, the next probability levels, β = ±1.5, ±4.5, are selected and the cor-

11 11 responding MPP searches are carried out starting from the initial search point on the approximate MPP locus to find the MPPs at these probability levels. The a posteriori error estimator defined in 27) is then used to check the CE c. This estimator shows that the error of the MPP locus is unacceptable when β< 3.0, as illustrated in Fig. 7a). Next, the second approximated MPP locus is constructed using information at the probability levels β =0, ±1.5, ±3, ±4.5, ±6, as shown in Fig. 7b). At the same time, using information from the approximated MPP locus, MPP searches are carried out at the selected probability levels β = 3.75, 5.25) where the MPP locus is found inaccurate. The adaptive probability analysis constructed the third approximated MPP locus as shown in Fig. 7c). The accuracy is verified by the a posteriori error at all probability levels and the probability analysis is stopped when the accuracy is satisfied. Finally, using the MPP locus, the cumulative distribution function CDF), probability density function PDF), and statistical moments of the response are generated, as shown in Fig. 7d). The adaptive probability analysis method will accurately estimate the tail end of the probability distribution, since the it will tend to add more probability levels at the tail end of probability distribution rather than near its mean due to the a posteriori error. 3.3 Numerical procedure of adaptive probability analysis A numerical procedure illustrated in Fig. 7 for the proposed adaptive probability analysis is presented in this section. Step 1. Set the MPP locus counter to k = 0 and select the highest probability level β H. Select the convergence parameter ε REL for the MPP search, and ε MPPL for MPP locus approximation. Carry out MPP search at ±1/2β H and ±β H. Step 2. Select probability levels using the bisection method, where the initial search point is not close to the MPP, i.e. ε 1 >ε MPPL, and approximate the k +1 th MPP locus. Step 3. Obtain initial search points on the approximate MPP locus at the selected probability levels. Step 4. Carry out MPP searches starting from the initial search points at the selected probability levels. Step 5. Check to see if the a posteriori error estimator satisfies the error criteria ε>ε MPPL.Ifthe convergence is achieved at all probability levels, then stop. Otherwise, go to Step 2. 4 Numerical results and discussion Example 4: side-impact crashworthiness for MPP locus approximation The side-impact crashworthiness used in Example 3 is employed here for MPP locus approximation in the probability analysis. Two responses are used: a mildly nonlinear response abdomen load) and a highly nonlinear response velocity of door). This study also presents a comparison between the least-squares and moving least-squares methods for the MPP locus approximation. As the number of probability levels is increased, the a posteriori error ε MPPL in 27) is measured to observe the rate of convergence in approximating the MPP locus. From Fig. 8, it can be seen that the moving least-squares method shows a faster rate of convergence than the least-squares method in MPP locus approximation. The moving least-squares method is better in reproducing both local and global behaviors in the MPP locus than the least-squares method. As expected, Table 7 Numerical results of adaptive probability analysis No. of levels, I Probability level β i G No. of analyses Total number of analyses 28

12 12 a larger error in the MPP locus is observed for the highly Abdomen Load Vel. of Door nonlinear response, ε <ε MPPL MPPL. Example 5: mathematical example for adaptive probability analysis A performance function for output probability analysis is defined as GX)=X 1 0.3X 1 X 2 0.1X ) where two random parameters are defined as X i N0.0, 0.5), i =1,2,andβ H =6.0. The process of the output probability analysis is depicted in Fig. 7 of Sect. 3. It is observed in Fig. 7 that the adaptive probability analysis refines the MPP locus by adaptively adding probability levels only in the region where the a posteriori error is greater than the allowable amount. The adaptive probability analysis requires ten MPP searches excluding β i = 0) and 28 analyses, as shown in Table 7. For comparison, the exact MPP locus in Fig. 7 is constructed using 32 probability levels. Compared to Monte Carlo simulation with samples, the adaptive probability analysis is shown to yield accurate CDF, PDF, and statistical moments, as shown in Fig. 7d). Numerical efficiency of the adaptive probability analysis is compared to those of Monte Carlo simulation and NESSUS Southwest Research Institute 1996), as shown in Table 8. NESSUS fails to complete the output probability analysis, due to the failure of the MPP search for β<0. For the adaptive probability analysis, different MPP search methods are employed, where the AMV method diverged. On the contrary, the adaptive probability analyses with HMV and HMV+ converged, where the proposed HMV+ method enhances numerical efficiency and stability, compared to all other methods. Example 6: side-impact crashworthiness for adaptive probability analysis The same side-impact application used for Example 4 is considered for both abdomen load and door velocity. β H is set to 4.0 for both responses. The result of output probability analysis for abdomen load is shown in Table 9. Eight probability levels excluding β i =0) are required to complete the analysis. In addition to the probability level at β i = 0, the initial probability levels are selected at β i = ±2 and±4, and MPP searches are carried out starting from the initial point at the origin in U-space. When the next probability levels are added at β i = ±1 and±3, MPP searches are carried out starting from the initial points obtained from the approximated MPP locus. Thus, the MPP searches for β i = ±1 and±3 require fewer analyses than the ones for probability levels β i = ±2and±4, as shown in Table 9. Fig. 8 Errors in MPP locus, ε MPPL Table 8 Results of probability analysis for Example 5 Monte Carlo Adaptive Probability Analysis Method NESSUS simulation AMV HMV HMV+ No. of analyses Diverged Diverged 41 28

13 13 Fig. 9 CDF and statistical moments of abdomen load Fig. 10 CDF and statistical moments of velocity of door Table 9 Numerical results of adaptive probability analysis for abdomen load No. of levels, I Probability level β i G No. of analyses Total number of analyses 27 Table 10 Numerical results of adaptive probability analysis for velocity of door No. of levels, I Probability level β i G No. of analyses Total number of analyses 44 Table 11 Results of probability analysis for Example 6 Monte Carlo Adaptive Probability Analysis Method NESSUS simulation AMV HMV HMV+ Abdomen load Velocity of door Diverged Diverged Diverged 44

14 14 To verify numerical accuracy, CDF and statistical moments obtained using the adaptive probability analysis are compared to those obtained using Monte Carlo simulation with samples, as shown in Fig. 9. Note that a relatively larger error of the adaptive probability analysis occurs near 10% and 90%, which is mainly due to the error of FORM. The second-order reliability method SORM) might be able to reduce these errors. The result of output probability analysis for door velocity is shown in Table 10. Eleven probability levels excluding β i = 0) are required to complete the output probability analysis. The adaptive probability analysis of door velocity requires more probability levels than the abdomen load, since the former response is more nonlinear than the latter. For numerical accuracy the adaptive probability analysis is again compared to Monte Carlo simulation with samples, as shown in Fig. 10. Note that a relatively larger amount of error of the adaptive probability analysis occurs between 50% and 95%, which is again mainly due to the error of FORM. For a given performance function value, the adaptive probability analysis estimates smaller probability than Monte Carlo simulation, since the response is a concave function in this example. As observed in Table 11, the adaptive probability analysis with HMV+ for abdomen load is carried out efficiently, while the same eight probability levels are employed for NESSUS with less efficient result. In general, NESUSS is unable to set the minimum necessary number of probability levels. Moreover, it is found that the HMV+ method makes the adaptive probability analysis more efficient. For the highly nonlinear response of the door velocity, NESSUS and adaptive probability analyses methods with AMV or HMV fail to complete output probability analysis due to the failure of the MPP searches, whereas the proposed adaptive probability analysis method with HMV+ has carried out output probability analysis efficiently. 5 Conclusions This paper proposes an adaptive output probability analysis method that aids in the decision-based design process by efficiently and accurately identifying the propagation of the input uncertainty to the output uncertainty. In addition, the enhanced hybrid mean value HMV+) method is proposed to improve numerical stability and efficiency in the MPP search. Three examples are used to show numerical efficiency and accuracy of the HMV+ method, as compared to the original HMV method and general optimization algorithms. By using MPP locus approximation and an a posteriori error estimator, it is found that the adaptive probability analysis uses the least number of necessary probability levels adaptively, and to perform output probability analysis efficiently. A comparison study between the least-squares and moving least-squares methods shows that the latter converges faster in approximating the MPP locus. Two numerical examples are used to demonstrate the effectiveness of the proposed adaptive probability analysis using the HMV+ method, in terms of numerical efficiency and stability. It has also been found that numerical efficiency in adaptive probability analysis does not depend on the number of random parameters, but on the degree of nonlinearity of the performance function. Acknowledgements Research is partially supported by the Automotive Research Center sponsored by the U.S. Army TARDEC. References Bucher, C.G. 1988: Adaptive Sampling An Iterative Fast Monte Carlo Procedure. 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