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Efficient estimation of structural reliability for problems with uncertain intervals
Computers and Structures 80 (2002) 1103–1112 www.elsevier.com/locate/compstruc
Efficient estimation of structural reliability for problems with uncertain intervals Ravi C. Penmetsa, Ramana V. Grandhi
*
Department of Mechanical and Materials Engineering Wright State University, 3640 Colonel Glenn Hwy, Dayton, OH 45435-0001, USA Accepted 13 March 2002
Abstract The uncertain parameters in engineering design may appear as random or intervals based on the information available from the past history or physical experiments. When the random variables are provided with probability distribution functions, suitable methods have to be used in computing the failure probability. Similarly, when there are only bounds or ranges on some of the uncertain variables, failure analysis methods are needed to consider the complete range of uncertain intervals. When dealing with the combination of both random distributions and interval variables, the computational cost involved in estimating the reliability of the structure increases exponentially. Every combination of interval values requires one probabilistic based analysis, which is by itself an expensive procedure. Therefore, to make the problem tractable, use of function approximations is presented in this paper, which reduces the number of actual FEA/computational fluid dynamics (CFD) simulations. The proposed method is demonstrated with structural examples. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction In this paper, a method to deal with the intervals without assigning any probability distribution to them is presented. In order to explore the entire domain represented by the interval variables, high quality approximations are used to reduce the computational effort considerably. The proposed method is an innovative combination of the existing reliability analysis techniques and modern approximation concepts. Several numerical examples are provided in this paper, which demonstrate the applicability of the method to wide a range of problems. While estimating the reliability of structures subject to nondeterministic variables, the uncertainty could be due to randomness in the parameters or due to the parameters that are defined as intervals (tolerances). When the uncertain variables are intervals, interval analysis technique can be used to obtain the bounds on the response.
*
Corresponding author. Fax: +1-937-775-5147. E-mail addresses: [email protected] (R.C. Penmetsa), [email protected] (R.V. Grandhi).
These bounds can later be used in the estimation of the reliability of the structure. Reliability analysis is usually performed when a structure is subjected to random loads, geometric and/or material properties. The ultimate goal in reliability analysis is to accurately and efficiently determine the failure probability of a multi-functional structure. However, in most cases uncertainty does not always imply randomness, and methods to account for a combination of random variables and interval variables have to be developed. Interval analysis techniques typically explore all the combinations that arise from the lower to upper bounds of the interval variables and determine the bounds on the system response. In reliability analysis, the exploration of all the combinations of variables involved is not practical due to the cost involved for each reliability analysis. Therefore, methods that explore the entire range of the interval variables to estimate the failure probability has to be developed. A typical example where the interval analysis is required would be a buckling problem, where the governing equations are nonlinear and the state of the system can be multi-valued. The solution to this problem would highly depend on the initial conditions, where the
0045-7949/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 2 ) 0 0 0 6 9 - X
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uncertainties in the initial conditions manifested by manufacturing anomalies such as initial imperfections are available as the intervals. The solution of the response is bounded within certain limits. Similarly, in aircraft design, the integration of controls, structural and aerodynamics disciplines can be highly nonlinear; they depend on the aircraft operating conditions at that instant, and they are uncertain due to atmospheric turbulence, gust, wind-shear effects, etc. As a result, the dynamic response is uncertain. Reliability analysis involves the evaluation of the multi-dimensional integration.
Pf ¼
Z
fx ðX Þ dX
ð1Þ
X
where fx ðX Þ denotes the joint probability density function (PDF) of the vector of basic random variables X ¼ ðx1 ; x2 ; . . . ; xn ÞT representing random quantities (available as distributions) such as loads, geometry, material properties, and boundary conditions. X is the failure region modeled by the limit-state function or performance function gðX Þ. The failure region is defined by gðX Þ 6 0 and the probability of structural failure is defined by Pf . The above multi-dimensional integration can be solved analytically only for a very limited number of cases. Numerical methods such as the Monte Carlo simulation can generally be performed to evaluate the integration, but it is extremely computer intensive due to complex physical simulations such as computational fluid dynamics (CFD) or finite element methods (FEM). Therefore, more accurate surrogate representations of the limit states have been developed in the past for conducting an efficient reliability analysis [1]. In order to replace the above stated multi-dimensional integration using approximations, an expansion point or the most probable failure point (MPP) is identified in fast probability integration methods. This point has the highest probability of failure in the n-dimensional space. MPP is reached iteratively starting from the mean values of the random variables. The solution procedure uses an optimization procedure and needs several function simulations. Hence, the high quality representation of the limit-state function using the two-point adaptive nonlinear approximation (TANA2) [2,3] makes the search process quite practical for analysis intensive approaches such as CFD, FEM and electromagnetics. Reliability analysis methods have become powerful tools in dealing with uncertain variables in design of engineering systems. These methods [1] deal with uncertain variables which are usually available in various distributions like normal, Lognormal, Weibull, etc. There are not many developments when the uncertain variables are available just as intervals or bounds. In these cases, only the upper and lower limit of a particular variable can be
available and assigning any particular distribution to it would be impractical. Because the characteristics are different, these two types of uncertainties should be treated separately in the reliability analysis. Therefore, in this paper a methodology for dealing with uncertain variables where some are available as intervals and the others are available as distributions is presented. When uncertain variables are available as intervals, every point within the interval has some unknown probability of occurrence. These intervals are usually referred to as uncertainty due to ignorance. Therefore, one cannot assign a particular weight to one value over the other within the interval. Instead, all the possible combinations of these uncertain intervals must be considered in the analysis. When some of the input variables are available as intervals, the MPP is not a unique point. For every combination of values of the interval variables, there would be a new MPP (obtained by using the variables with distributions), which would effect the failure probability estimation. Since the available failure probability estimation methods are dependent on the accurate estimation of MPP, this multiple MPP would result in an inaccurate estimation of the failure probability. Hu and Chen [4] have considered structural failure prediction using the fuzzy uncertainties. The failure of the structure was modeled by a fuzzy event, and described by a membership function. In their paper, the fuzziness in the failure surface is from the limiting value of the response that defines the limit-state surface and not from the fuzziness in the input variables. Braibant et al. [5] presented nondeterministic possibilistic approaches for structural analysis and optimization. They have used built fuzzy response equations and solved them using numerical techniques. Difficulty arises when solving the discretized interval equilibrium equations. In order to improve the efficiency of solution algorithms involving fuzziness, the Hansen Algorithm, Neumann Approximate Vertex Solution, and Vertex Solution [7] were introduced in the paper [6]. The Vertex Solution is considered the most robust but the computation cost could be too large to be applicable when the number of uncertain parameters is very large. The Hansen Algorithm is one of the most popular explicit direct algorithms. For the Neumann Approximate Vertex Solution, a linear approximation is used for the stiffness matrix and load vector with respect to uncertain parameters. The number of numerical simulations for Neumann is 2n, where n is the number of uncertain variables. The ‘‘Vertex Solution’’ requires 2n number of simulations. For the Hansen Algorithm, two modifications were introduced to limit the occurrence of unbounded solutions in the basic algorithm. These methods are applicable for implicit problems whose response behaves in a linear fashion. However, in most engineering problems, the response is
R.C. Penmetsa, R.V. Grandhi / Computers and Structures 80 (2002) 1103–1112
highly nonlinear and a linear approximation at the central values would lead to erroneous results. Similarly, the cost involved in the vertex method is exponential and it is not a viable solution to large-scale engineering problems. Moreover, in the above discussed work they have considered only interval or fuzzy input. There were no provisions to include variables for which the probability distribution information is available. When working with fuzzy input variables, the PDF can be converted to a fuzzy membership function by dividing the PDF with the maximum value of the density function. However, this results in a loss of information that would not be acceptable in many cases. Therefore, a technique to incorporate both random and interval uncertainties, is presented in the current paper. An alternative method to deal with the interval uncertain variables is to assume a uniform distribution for the variables. This would not give any information about the effect of having higher probability density towards the upper or lower limit of the interval variable. Moreover, when uniform distribution is assumed for the uncertain variable, additional information is added which would compromise the accuracy of the result. Therefore, the interval uncertainty has to be handled using interval analysis [9–12] in order to determine the bounds on the failure probability.
2. Interval arithmetic Interval arithmetic defines the rules for a set of numbers, which are closed real intervals rather than single valued. Assuming two interval numbers that are closed real intervals that include any real number ‘‘a’’ and ‘‘b’’ including the end points ‘‘a1 ’’ and ‘‘a2 ’’ for ‘‘a’’, and ‘‘b1 ’’ and ‘‘b2 ’’ for ‘‘b’’.
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Using interval extension principle the range of the response function Y can be determined as follows: Y ð½ 1; 1Þ ¼ ½ 1; 1 ½ 1; 1 ½ 1; 1 ¼ ½ 1; 1 ½ 1; 1 ¼ ½ 2; 2 The actual solution for this problem should be Y ð½ 1; 1Þ ¼ ½0; 2. Clearly, there is an overestimation due to the occurrence of the variable x more than once in the equation. The interval arithmetic definition does not accommodate correlation. Therefore, these issues have to be handled with care when using interval analysis.
3. Limitations of interval analysis From the definition of interval arithmetic operations it can clearly be seen that the evaluation of the range of any rational function would give rise to the exact value of the interval values if the variable occurs only once in the function expression. However, if the variable occurs more than once there is no mechanism in the interval arithmetic to estimate the response bounds without serious overestimation of the predicted bound. Thus, the width of an interval may grow with the number of intervals and the number of times they might occur in the limit-state function. In the proposed method of multi-point approximation (MPA), evaluating the weighting function and the local approximations separately have dealt with this overestimation problem. The use of this technique has reduced the overestimation considerably, however the results that are obtained are conservative and would enclose the actual bound.
4. Multi-point approximation a ¼ ½a1 ; a2 ¼ faja1 6 a 6 a2 g b ¼ ½b1 ; b2 ¼ fbjb1 6 b 6 b2 g The basic operations of addition, subtraction, multiplication and division are as follows: a þ b ¼ ½a1 þ b1 ; a2 þ b2 a b ¼ ½a1 b2 ; a2 b1 ab ¼ ½Minða2 b1 ; a1 b2 Þ; Maxða1 b1 ; a2 b2 Þ a=b ¼ að1=bÞ 1=b ¼ ½1=b2 ; 1=b1 for 0 62 b
The MPA can be regarded as the connection of many local approximations. With function and sensitivity information already available at a series of points, one local approximation is built at each point. All local approximations are then integrated into a MPA by the use of a weighting function. The weighting functions are selected such that the approximation reproduces function and gradient information at the known data points. The MPA can be written using the following general formulation: Fe ðX Þ ¼
2.1. Example on interval analysis Consider a response Y ðxÞ ¼ x2 x, and the range of x ¼ ½ 1; 1.
K X
Wk ðX Þ Fek ðX Þ
ð2Þ
k¼1
where Fek ðXÞ is a two-point local approximation, k is the number of local approximations, and Wk is a weighting function that adjusts the contribution of Fek ðXÞ to Fe ðXÞ
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in Eq. (2). The evaluation of this weighting function involves the selection of a blending function and a power index ‘‘m’’. The procedural details for evaluating the weighting function are discussed in [8]. The weighting function is given by the equation / ðXÞ Wk ðXÞ ¼ PK k j¼1 /j ðXÞ
ð3Þ
where /k ðXÞ is the blending function. The blending function used in this paper is given by: /k ðXÞ ¼
1 hk
ð4Þ
where hk ¼
n X ðxi xi;k Þ2
!m
i¼1
This blending function combines the local approximations into one MPA. The local approximations considered in this paper are TANA2 and they are of the type, g~ðX Þ ¼ gðX2 Þ þ
1 p n X ogðX2 Þ xi;2 i pi ðxi xpi;2i Þ ox p i i i¼1
n 1 X ðxpi xpi;2i Þ2 þ e 2 i¼1 i
ð5Þ
where X2 is the expansion point for the approximation. This equation is a second-order Taylor series expansion in terms of the intervening variables yi ðyi ¼ xpi i Þ, in which the Hessian matrix has only diagonal elements of the same value e. Therefore, this approximation does not need the calculation of the second-order derivatives. Further details of this approximation are available in Refs. [2,3]. Naturally, the accuracy of a local approximation is one of the primary factors on which the quality of MPA is dependent. TANA2 were used as local approximations to construct the MPA of each limit-state function. The TANA2 can capture the information of the limitstate accurately in the vicinity of the data points. MPA retains the information for each failure surface without increasing the computational effort. The MPA adaptively adjusts itself to behave as a local approximation when a design point is closer to one of the data points. Function and gradient values of this MPA correspond directly with their exact counterparts at the points where the local approximations were generated [8].
5. Proposed uncertainty analysis technique In this paper, the uncertainty information is classified as intervals and probability distributions. When the in-
put variables are available as random variables, the failure probability can be estimated using the methods available in the literature. The proposed method involves the estimation of the MPP using the technique presented by Wang and Grandhi [3]. Once the MPP is estimated for the given limit-state function, with the interval variables set to central values, then the domain around the MPP (this is important due to the variation in the interval variables) is approximated using MPA. The algorithm details are given as follows: 1. Estimate the MPP for the limit-state function by setting the values of the interval variables at the central value. MPP search is an optimization technique and it generates data points during the intermediate iterations. These points are stored along with their function and the gradients information. 2. Once the MPP is available for one configuration of the interval variables, an approximate limit-state function is constructed. The design of experiments (DOEs) technique generates data points in the design space represented by the uncertain variables. All the uncertain variables are used to sample the data using DOEs. The random variables are assigned an interval that would move each of the variables by two standard deviations on either side of the MPP. The total number of simulations required is as follows: Number of random variables: M Number of interval variables: N Data points from MPP search: P For the above case, DOEs involves M þ N dimensions to sample. For the first iteration, M þ N points from the DOEs can be used along with P data points to construct the MPA. Therefore, for the first iteration M þ N þ P simulations are required. Thereafter, one additional simulation is required for every data point added until convergence. The new points are selected from the design space that was not sampled in the initial step. 3. During this process, the random variables are sampled within one standard deviation from the MPP. This sampling would result in design points around the MPP, which has been already estimated. This would account for the shift in MPP that may arise due to the presence of interval variables. 4. At each of the design points, the function value and the gradients are evaluated. When the gradients with respect to some uncertain variables are not available, they are estimated using numerical techniques. The availability of function value and gradients are a prerequisite for the approximations discussed in this paper. In the absence of the gradients, a response surface model may be a suitable choice. 5. With the function values and gradients available at all the data points, local approximations are con-
R.C. Penmetsa, R.V. Grandhi / Computers and Structures 80 (2002) 1103–1112
structed at the data points. The local approximations used in this research work are TANA2. Once the local approximations are available, they are combined into one MPA [8], which uses blending functions to combine the local approximations. 6. This MPA is used in conjunction with Monte Carlo simulation to determine the failure probability of the structure. For variables with probability distribution their values are generated by a random number generator, however, for the variables with intervals, the function is evaluated using the interval arithmetic. The failure probability interval is obtained by considering the lower and upper limits of the function value obtained from the MPA.
5.1. Implementation issues While using MPA for estimating the bounds of failure probability, overestimation of the bounds is an important issue. This overestimation of the bounds due to the multiple existence of the same variable in the function is avoided partially by combining the terms in each local TANA2 as shown below g~ðX Þ ¼ a þ
n X
bðxpi i þ ci Þ2
ð6Þ
i¼1
where the terms a, bi , and ci are obtained by reordering Eq. (5). P From Eq. (5), the term 12 e ðxpi i xpi;2i Þ2 can be expanded and the expanded form of the equation would be as shown below:
i n x1 p 1 X i;2 pi pi i
ðxpi i Þ þ e ðx2pi þ x2p i;2 2xi xi;2 Þ pi 2 i¼1 i
ð7Þ
Now the xpi i terms can be combined as follows: g~ðX Þ ¼ gðX2 Þ
þ
n X i¼1
1 p n n X ogðX2 Þ xi;2 i pi 1 X ðxi;2 Þ þ e ðx2pi Þ pi oxi 2 i¼1 i;2 i¼1 1 p ogðX2 Þ xi;2 i pi 1 i ðxi Þ þ eðx2p
2xpi i xpi;2i Þ oxi 2 i pi
!
ð8Þ From the above equation the required constants can be obtained as: 2 ! p og xi;2 1 pi 2 og xi;2i
2xpi;2i þ exi;2
; oxi pi 2 e oxi pi i¼1 p 2 og xi;2i and ci ¼
2xpi;2i e oxi pi
a ¼ gðX2 Þ þ e b¼ ; 2
n X
Therefore, there is a slight overestimation of the bound when the interval arithmetic is applied on each approximation. In order to reduce the overestimation, the weighting function Wk ðX Þ in Eq. (2) is estimated at the central value of the bound and this would tend to underestimate the bound. A combination of the weighting function evaluated at the central value and the local approximations evaluated using the bounds, resulted in a conservative estimation of the response bound. Fig. 1 pictorially demonstrates the proposed algorithm for estimating the failure probability interval.
6. Numerical examples This method can produce a bound on the probability of failure by considering variables with distributions and intervals simultaneously. The interval of failure probability obtained by using MPA of the failure surface is compared with the actual failure probability interval. The error is estimated for both the upper and lower limits individually. While performing the reliability analysis, the values of the interval variables are set to central values. 6.1. Cantilever beam A cantilever beam shown in Fig. 2 subjected to a tip load P is considered in this example. The limit state is defined as tip displacement greater than 0.15 in. (failure). G1 ðX Þ ¼
1 p n n X X ogðX2 Þ xi;2 i pi ogðX2 Þ ðxi;2 Þ þ g~ðX Þ ¼ gðX2 Þ
p ox oxi i i i¼1 i¼1
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4PL3
0:15 6 0:0 Ebh3
where L, b, and h are the length, width and height of the beam with mean values of 30, 0.8359 and 2.5093 in., respectively and the Young’s Modulus, E, is 107 psi. The length, width and height of the beam are considered as the random variables, and the standard deviations are, rL ¼ 3:0 in., rb ¼ 0:08 in. and rh ¼ 0:25 in., respectively. Both L and h are considered as log-normally distributed and b is considered as a normal distribution. The load P is available as an interval instead of a constant value. The interval considered for P is [60, 100] lb. Therefore, for every possible value of this load, there would be a corresponding failure probability and using the proposed method the bound of this failure probability is estimated. Initially, a reliability analysis was performed with a load of 80 lb., which resulted in an MPP and six intermediate design points during the MPP search. After obtaining the MPP, DOEs was used to obtain seven additional points distributed around the MPP in the additional dimensions introduced by the interval variables. The local TANA2 were constructed using these
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Fig. 1. Proposed algorithm details.
Fig. 2. Cantilever beam.
design points and they are blended into one MPA. This MPA was used to determine the bounds on the failure probability. The failure probabilities for the designs vary from 0.004963 to 0.08116 when the load is considered to vary as an interval. The design had higher failure probability for a load of 100 lb. and it was the lowest for a load of 60 lb. at the tip. The obtained results were obvious from the equation of the limit-state function. However, when dealing with implicit nonlinear limit-state functions and multiple interval variables, the minimum values of the interval variables do not always result in a minimum failure probability. The variation of failure probability with the variation in loading is a nonlinear function as shown in Fig. 3. Therefore, a linear approximation at
Fig. 3. Failure probability vs. tip load.
the MPP for the random variables and central values for the interval variables would result in an inaccurate estimation of the failure probability. As shown in Table 1 the failure probability bound was [0.005134, 0.08192] using MPA and it was [0.004963, 0.08116] using the actual function. The results show that there is a great variation in the failure probability and the proposed method is capable of capturing the variation accurately and efficiently. The obtained results are after a convergence in the failure probability and also
R.C. Penmetsa, R.V. Grandhi / Computers and Structures 80 (2002) 1103–1112 Table 1 Cantilever beam results
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is set to the central value and the random variables are used to obtain the failure probability. The MPA values for the current example were conservative and they followed the true trend for the failure probability.
Limit-state function
Failure probability bound
% Difference Lower limit
Upper limit
Monte Carlo MPA
[0.0049, 0.0811] 200,000 simulations [0.0051, 0.0819] 13 simulations
–
–
6.2. Ten-bar truss structure
3.44
0.92
The failure probability bounds of the 10-bar truss, shown in Fig. 5, were calculated in this example. The cross-sectional areas of all the ten truss members are lognormally distributed random variables with 2.5 in. mean value and 0.5 standard deviation. The Young’s modulus is 107 psi and the forces applied are P1 ¼ P2 ¼ 105 lb, as shown in Fig. 4. The limit-state considered in this example is the displacement limit on the maximum displacement of the tip of the truss structure which should be less than 1.8 in. The lengths L1, L2, L3, L4 and L5 (as shown in Fig. 5) are considered uncertain in the interval [350, 370].
after adding new design points from the unselected domains in the Design of Experiment technique. Seven additional design points were considered along with the six intermediate design points to obtain the results, which were obtained using 100,000 Monte Carlo simulations on the approximate model. The proposed method required 13 actual function evaluations and at each function evaluation the gradients are available analytically. However, using the combinatorial approach for the interval analysis to explore the entire domain results in 200,000 exact function evaluations provided 100,000 are required for each reliability analysis of lower and upper bounds. Moreover, this does not consider the possibility of the upper or lower limit of the response lying within the interval of the uncertain variables. Fig. 4 shows the variation of the probability of failure with the uncertainty introduced by the interval loading. As the uncertainty in loading increases, the failure probability interval varies as a nonlinear function of uncertainty. The value of failure probability for the load of 80 lb. is the crisp value when the interval variable
Fig. 5. Ten-bar truss.
Fig. 4. Failure probability variation with uncertainty in loading.
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Table 2 Ten-bar truss results Limit-state function
Failure probability bound
% Difference
MPA For Li ¼ 35000 Monte Carlo
[0.00015, 0.0162]
–
MPA For Li ¼ 36000 Monte Carlo MPA For Li ¼ 37000 Monte Carlo MPA
G1 ðX Þ ¼
0.00039 100,000 simulations 0.00043 204 simulations 0.00229 100,000 simulations 0.00242 204 simulations 0.0133 100,000 simulations 0.0142 204 simulations
)9.30
)5.67
)6.76
Dtip
1:0 6 0:0 1:8
Table 2 compares the results obtained using the Monte Carlo simulation on the actual limit-state and MPA. Since this is a problem with an implicit limit-state function, TANA2 are constructed at the six data points obtained in the process of searching for the MPP and the additional 28 points obtained from the DOEs technique. These local approximations are blended together into one MPA. The construction of TANA2 required the evaluation of gradients at the design points. The gradients of the random variables (cross-sectional areas) were available analytically from the finite element software; however, the gradients of the interval variables were numerically evaluated. The five interval variables would lead to 32 (25 ) different settings for a combinatorial estimation of the bounds of the failure probability. Therefore, in this problem the accuracy of the bound estimated using the MPA is demonstrated by comparing the results of failure probability for different interval settings. Three different settings are explored; one where the interval variables are assigned the central values and the other two where the interval variables are assigned the lower and upper limits respectively. For the case where the values of Li s are considered to be 360 in., which are the central values, the MPA produced an estimate that had 5.6% error. For the interval variable setting of 350 in., the error in the estimation was 9.3%. Moreover, for the case where the interval variables were assigned 370 in. the error was 6.76%. This error is introduced by the approximate function; however, the final result of the interval bound for failure probability
would be a conservative estimate because of the slight overestimation due to the use of interval arithmetic. In our experience, the difference in the prediction of failure probability using this method was found to be less than 10%. In this example, a total of 34 function evaluations are required at the data points. Moreover, since all the gradients are not available analytically, the gradients of the function due to the lengths were obtained numerically. Therefore, 170 additional simulations were required for obtaining the gradients. However, the number of exact simulations is considerably less compared to the Monte Carlo simulation, which would require one reliability analysis (around 100,000) for each of the 32 combinations of the uncertain interval variables. Therefore, by using the proposed method the bounds on failure probability can be obtained efficiently. From this example, it can be seen that the lower/ upper limit of the interval variables would not result in the lower/higher limit of the failure probability. This is because the tip displacement for the truss would be maximum for a particular combination of the interval variables, which is different from the lower or upper limit settings for all the variables. Due to this reason it can be seen that all the possible combinations of the interval variables have to be explored in order to obtain the bounds on the failure probability. However, the use of interval analysis and high quality approximations would result in a conservative estimate of the bounds of failure probability with considerably less computational effort. 6.3. Wing structure example In this example a wing with three spars and five ribs is considered. Fig. 6 shows the wing with the loading at each of its nodes. The ribs and spars are modeled using shear elements, the posts are modeled as rod elements
Fig. 6. Box wing example.
R.C. Penmetsa, R.V. Grandhi / Computers and Structures 80 (2002) 1103–1112
1111
and the skins are modeled as membrane elements. The thicknesses of chord wise skins are physically linked to have the same thickness, this is assumed to be a random variable. The skin thickness is normally distributed with a mean of 1.5 in. and a standard deviation of 0.15 in. Each spar is linked to have the same thickness and the same is done with the ribs. This linking resulted in three random variables for spars and five random variables for the ribs. Each of these eight random variables are normally distributed with a mean of 0.5 in. and a standard deviation of 0.05 in. The cross-sectional area of the posts is assumed to be deterministic. The linking of the variables resulted in 12 random variables and the loading is considered to be available as an interval. As shown in Fig. 6, the load is applied on a set of nodes. The load on the leading edge has one interval value, the load on the trailing edge has one interval, and the intermediate nodes have a certain interval loading. The limit-state considered in this example is the tip displacement >1.0 in. represents a failure.
along with the interval variables to obtain the expanded domain of interest. M þ N þ P (¼ 19) data points are used in the initial analysis. Then one data point is added in every iteration until there is a convergence in the failure probability values. Eighteen iterations were required for the convergence of the failure probability, therefore a total of 37 exact simulations were required to obtain the bounds using the proposed method. However, additional 111 (37 data points 3 variables) simulations were required to determine the gradients of the tip displacement with respect to the load. Considering the low error (<6%) in the results obtained and the number of exact simulations required for obtaining them, it can be seen that the proposed method is capable of producing bounds for the failure probability efficiently and accurately.
Dtip 1:0 6 0:0 ðsafeÞ
A methodology was developed for a general uncertain analysis problem. Both random distributions and intervals of input variables are handled simultaneously with considerable accuracy and efficiency. The use of MPA has enabled modeling of the failure region in multiple dimensions in order to produce accurate results efficiently. The local TANA2 capture the nonlinearity of the failure surface accurately near the expansion point and the comparison point. The proposed method is capable of handling a wide range of random variable distributions. It can also be used for nonlinear problems with more than a dozen random variables. The interval uncertainty can also be handled by using the Fuzzy definition for the uncertain variables that are given as intervals. When dealing with fuzzy numbers, the current estimation would be estimation for a zero degree of possibility. For different degrees of possibility, the bounds of failure probability can be obtained.
Table 3 shows the comparison between the bounds of failure probability estimated by first using Monte Carlo Simulation on exact limit-state function and later on MPA of the limit-state. This example is a special case where the effect of the intervals of loading is known in advance and two Monte Carlo simulations are sufficient to obtain the bounds on the failure probability. One simulation with the loads L1 and L2 set equal to maximum and L3 set to minimum, this would give the maximum deflection, and the other is when the loads L1 and L2 are set to minimum and L3 are set to maximum, this would result in minimum displacement. Each Monte Carlo simulation bound was obtained by running 100,000 actual function evaluations for each of the interval variable settings (one is the lower limit and the other is upper limit). A total of 200,000 exact simulations were performed. The MPP search technique has converged in eight iterations, which gives eight data points. Then perturbing the MPP in each dimension by two standard deviations expands the domain around MPP. This is used
Table 3 Wing tip displacement results Limit-state function
Failure probability bound
% Difference Lower Limit
Upper Limit
Monte Carlo MPA
[0.00119, 0.0422] 200,000 simulations [0.00126, 0.0434] 156 simulations
–
–
)5.80
)1.98
7. Summary
Acknowledgements This research work has been sponsored by the Air Force Office of Scientific Research under contract F4962-00-1-0377. The support for the Graduate Research Assistant has been provided by the Dayton Area Graduate Studies Institute (DAGSI).
References [1] Penmetsa RC, Zhou L, Grandhi RV. Adaptation of fast fourier transformations to estimate the structural failure probability. 41st AIAA/ASME/ASCE/AHS/ASC
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Structures, Structural Dynamics, and Materials Conference, Atlanta, Georgia, April 3–6, AIAA-2000-1718, 2000. Grandhi RV, Wang LP. Improved two-point function approximations for design optimization. AIAA J 1995; 33(9):1720–7. Wang LP, Grandhi RV. Safety index calculation using intervening variables for structural reliability analysis. Comput Struct 1996;59(6):1139–48. Hu Y, Chen B. Fuzzy reliability analysis of structures by using the method of fuzzy optimization. Safety Reliab ASME 1996;2:187–92, OMAE. Briabant V, Oudshoorn A, Boyer C, Delcroix F. Nondeterministic possibilistic approaches for structural analysis and optimal design. AIAA J 1999;37(10):1298–303. Hansen ER. Global optimization using interval analysis. In: Pure and applied mathematics series. New York: Marcel Dekker; 1992.
[7] Dong WM, Wong FS. Fuzzy weighted averages and implementation of the extension principle. Fuzzy Sets Systems 1987;21:183–99. [8] Xu S, Grandhi RV. Multi-point approximation for reducing the response surface model development cost in optimization. Proceedings of the 1st ASMO UK/ISSMO Conference on Engineering Design Optimization, Ilkley, West Yorkshire, UK, July 8–9, 1999. p. 381–8. [9] Moore RE. Interval analysis. Englewood Cliff, NJ: Prentice-Hall; 1996. [10] Moore RE. Methods and applications of interval analysis. Philadelphia: SIAM; 1979. [11] Rao SS, Berke L. Analysis of uncertain structural systems using interval analysis. AIAA-96-1459-CP, 1996. [12] Shin YS, Grandhi RV. A global structural optimization technique using an interval method. J Struct Multidiscip Opt 2001;22(5):351–63.
Efficient estimation of structural reliability for problems with uncertain intervals Ravi C. Penmetsa, Ramana V. Grandhi
*
Department of Mechanical and Materials Engineering Wright State University, 3640 Colonel Glenn Hwy, Dayton, OH 45435-0001, USA Accepted 13 March 2002
Abstract The uncertain parameters in engineering design may appear as random or intervals based on the information available from the past history or physical experiments. When the random variables are provided with probability distribution functions, suitable methods have to be used in computing the failure probability. Similarly, when there are only bounds or ranges on some of the uncertain variables, failure analysis methods are needed to consider the complete range of uncertain intervals. When dealing with the combination of both random distributions and interval variables, the computational cost involved in estimating the reliability of the structure increases exponentially. Every combination of interval values requires one probabilistic based analysis, which is by itself an expensive procedure. Therefore, to make the problem tractable, use of function approximations is presented in this paper, which reduces the number of actual FEA/computational fluid dynamics (CFD) simulations. The proposed method is demonstrated with structural examples. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction In this paper, a method to deal with the intervals without assigning any probability distribution to them is presented. In order to explore the entire domain represented by the interval variables, high quality approximations are used to reduce the computational effort considerably. The proposed method is an innovative combination of the existing reliability analysis techniques and modern approximation concepts. Several numerical examples are provided in this paper, which demonstrate the applicability of the method to wide a range of problems. While estimating the reliability of structures subject to nondeterministic variables, the uncertainty could be due to randomness in the parameters or due to the parameters that are defined as intervals (tolerances). When the uncertain variables are intervals, interval analysis technique can be used to obtain the bounds on the response.
*
Corresponding author. Fax: +1-937-775-5147. E-mail addresses: [email protected] (R.C. Penmetsa), [email protected] (R.V. Grandhi).
These bounds can later be used in the estimation of the reliability of the structure. Reliability analysis is usually performed when a structure is subjected to random loads, geometric and/or material properties. The ultimate goal in reliability analysis is to accurately and efficiently determine the failure probability of a multi-functional structure. However, in most cases uncertainty does not always imply randomness, and methods to account for a combination of random variables and interval variables have to be developed. Interval analysis techniques typically explore all the combinations that arise from the lower to upper bounds of the interval variables and determine the bounds on the system response. In reliability analysis, the exploration of all the combinations of variables involved is not practical due to the cost involved for each reliability analysis. Therefore, methods that explore the entire range of the interval variables to estimate the failure probability has to be developed. A typical example where the interval analysis is required would be a buckling problem, where the governing equations are nonlinear and the state of the system can be multi-valued. The solution to this problem would highly depend on the initial conditions, where the
0045-7949/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 2 ) 0 0 0 6 9 - X
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uncertainties in the initial conditions manifested by manufacturing anomalies such as initial imperfections are available as the intervals. The solution of the response is bounded within certain limits. Similarly, in aircraft design, the integration of controls, structural and aerodynamics disciplines can be highly nonlinear; they depend on the aircraft operating conditions at that instant, and they are uncertain due to atmospheric turbulence, gust, wind-shear effects, etc. As a result, the dynamic response is uncertain. Reliability analysis involves the evaluation of the multi-dimensional integration.
Pf ¼
Z
fx ðX Þ dX
ð1Þ
X
where fx ðX Þ denotes the joint probability density function (PDF) of the vector of basic random variables X ¼ ðx1 ; x2 ; . . . ; xn ÞT representing random quantities (available as distributions) such as loads, geometry, material properties, and boundary conditions. X is the failure region modeled by the limit-state function or performance function gðX Þ. The failure region is defined by gðX Þ 6 0 and the probability of structural failure is defined by Pf . The above multi-dimensional integration can be solved analytically only for a very limited number of cases. Numerical methods such as the Monte Carlo simulation can generally be performed to evaluate the integration, but it is extremely computer intensive due to complex physical simulations such as computational fluid dynamics (CFD) or finite element methods (FEM). Therefore, more accurate surrogate representations of the limit states have been developed in the past for conducting an efficient reliability analysis [1]. In order to replace the above stated multi-dimensional integration using approximations, an expansion point or the most probable failure point (MPP) is identified in fast probability integration methods. This point has the highest probability of failure in the n-dimensional space. MPP is reached iteratively starting from the mean values of the random variables. The solution procedure uses an optimization procedure and needs several function simulations. Hence, the high quality representation of the limit-state function using the two-point adaptive nonlinear approximation (TANA2) [2,3] makes the search process quite practical for analysis intensive approaches such as CFD, FEM and electromagnetics. Reliability analysis methods have become powerful tools in dealing with uncertain variables in design of engineering systems. These methods [1] deal with uncertain variables which are usually available in various distributions like normal, Lognormal, Weibull, etc. There are not many developments when the uncertain variables are available just as intervals or bounds. In these cases, only the upper and lower limit of a particular variable can be
available and assigning any particular distribution to it would be impractical. Because the characteristics are different, these two types of uncertainties should be treated separately in the reliability analysis. Therefore, in this paper a methodology for dealing with uncertain variables where some are available as intervals and the others are available as distributions is presented. When uncertain variables are available as intervals, every point within the interval has some unknown probability of occurrence. These intervals are usually referred to as uncertainty due to ignorance. Therefore, one cannot assign a particular weight to one value over the other within the interval. Instead, all the possible combinations of these uncertain intervals must be considered in the analysis. When some of the input variables are available as intervals, the MPP is not a unique point. For every combination of values of the interval variables, there would be a new MPP (obtained by using the variables with distributions), which would effect the failure probability estimation. Since the available failure probability estimation methods are dependent on the accurate estimation of MPP, this multiple MPP would result in an inaccurate estimation of the failure probability. Hu and Chen [4] have considered structural failure prediction using the fuzzy uncertainties. The failure of the structure was modeled by a fuzzy event, and described by a membership function. In their paper, the fuzziness in the failure surface is from the limiting value of the response that defines the limit-state surface and not from the fuzziness in the input variables. Braibant et al. [5] presented nondeterministic possibilistic approaches for structural analysis and optimization. They have used built fuzzy response equations and solved them using numerical techniques. Difficulty arises when solving the discretized interval equilibrium equations. In order to improve the efficiency of solution algorithms involving fuzziness, the Hansen Algorithm, Neumann Approximate Vertex Solution, and Vertex Solution [7] were introduced in the paper [6]. The Vertex Solution is considered the most robust but the computation cost could be too large to be applicable when the number of uncertain parameters is very large. The Hansen Algorithm is one of the most popular explicit direct algorithms. For the Neumann Approximate Vertex Solution, a linear approximation is used for the stiffness matrix and load vector with respect to uncertain parameters. The number of numerical simulations for Neumann is 2n, where n is the number of uncertain variables. The ‘‘Vertex Solution’’ requires 2n number of simulations. For the Hansen Algorithm, two modifications were introduced to limit the occurrence of unbounded solutions in the basic algorithm. These methods are applicable for implicit problems whose response behaves in a linear fashion. However, in most engineering problems, the response is
R.C. Penmetsa, R.V. Grandhi / Computers and Structures 80 (2002) 1103–1112
highly nonlinear and a linear approximation at the central values would lead to erroneous results. Similarly, the cost involved in the vertex method is exponential and it is not a viable solution to large-scale engineering problems. Moreover, in the above discussed work they have considered only interval or fuzzy input. There were no provisions to include variables for which the probability distribution information is available. When working with fuzzy input variables, the PDF can be converted to a fuzzy membership function by dividing the PDF with the maximum value of the density function. However, this results in a loss of information that would not be acceptable in many cases. Therefore, a technique to incorporate both random and interval uncertainties, is presented in the current paper. An alternative method to deal with the interval uncertain variables is to assume a uniform distribution for the variables. This would not give any information about the effect of having higher probability density towards the upper or lower limit of the interval variable. Moreover, when uniform distribution is assumed for the uncertain variable, additional information is added which would compromise the accuracy of the result. Therefore, the interval uncertainty has to be handled using interval analysis [9–12] in order to determine the bounds on the failure probability.
2. Interval arithmetic Interval arithmetic defines the rules for a set of numbers, which are closed real intervals rather than single valued. Assuming two interval numbers that are closed real intervals that include any real number ‘‘a’’ and ‘‘b’’ including the end points ‘‘a1 ’’ and ‘‘a2 ’’ for ‘‘a’’, and ‘‘b1 ’’ and ‘‘b2 ’’ for ‘‘b’’.
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Using interval extension principle the range of the response function Y can be determined as follows: Y ð½ 1; 1Þ ¼ ½ 1; 1 ½ 1; 1 ½ 1; 1 ¼ ½ 1; 1 ½ 1; 1 ¼ ½ 2; 2 The actual solution for this problem should be Y ð½ 1; 1Þ ¼ ½0; 2. Clearly, there is an overestimation due to the occurrence of the variable x more than once in the equation. The interval arithmetic definition does not accommodate correlation. Therefore, these issues have to be handled with care when using interval analysis.
3. Limitations of interval analysis From the definition of interval arithmetic operations it can clearly be seen that the evaluation of the range of any rational function would give rise to the exact value of the interval values if the variable occurs only once in the function expression. However, if the variable occurs more than once there is no mechanism in the interval arithmetic to estimate the response bounds without serious overestimation of the predicted bound. Thus, the width of an interval may grow with the number of intervals and the number of times they might occur in the limit-state function. In the proposed method of multi-point approximation (MPA), evaluating the weighting function and the local approximations separately have dealt with this overestimation problem. The use of this technique has reduced the overestimation considerably, however the results that are obtained are conservative and would enclose the actual bound.
4. Multi-point approximation a ¼ ½a1 ; a2 ¼ faja1 6 a 6 a2 g b ¼ ½b1 ; b2 ¼ fbjb1 6 b 6 b2 g The basic operations of addition, subtraction, multiplication and division are as follows: a þ b ¼ ½a1 þ b1 ; a2 þ b2 a b ¼ ½a1 b2 ; a2 b1 ab ¼ ½Minða2 b1 ; a1 b2 Þ; Maxða1 b1 ; a2 b2 Þ a=b ¼ að1=bÞ 1=b ¼ ½1=b2 ; 1=b1 for 0 62 b
The MPA can be regarded as the connection of many local approximations. With function and sensitivity information already available at a series of points, one local approximation is built at each point. All local approximations are then integrated into a MPA by the use of a weighting function. The weighting functions are selected such that the approximation reproduces function and gradient information at the known data points. The MPA can be written using the following general formulation: Fe ðX Þ ¼
2.1. Example on interval analysis Consider a response Y ðxÞ ¼ x2 x, and the range of x ¼ ½ 1; 1.
K X
Wk ðX Þ Fek ðX Þ
ð2Þ
k¼1
where Fek ðXÞ is a two-point local approximation, k is the number of local approximations, and Wk is a weighting function that adjusts the contribution of Fek ðXÞ to Fe ðXÞ
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in Eq. (2). The evaluation of this weighting function involves the selection of a blending function and a power index ‘‘m’’. The procedural details for evaluating the weighting function are discussed in [8]. The weighting function is given by the equation / ðXÞ Wk ðXÞ ¼ PK k j¼1 /j ðXÞ
ð3Þ
where /k ðXÞ is the blending function. The blending function used in this paper is given by: /k ðXÞ ¼
1 hk
ð4Þ
where hk ¼
n X ðxi xi;k Þ2
!m
i¼1
This blending function combines the local approximations into one MPA. The local approximations considered in this paper are TANA2 and they are of the type, g~ðX Þ ¼ gðX2 Þ þ
1 p n X ogðX2 Þ xi;2 i pi ðxi xpi;2i Þ ox p i i i¼1
n 1 X ðxpi xpi;2i Þ2 þ e 2 i¼1 i
ð5Þ
where X2 is the expansion point for the approximation. This equation is a second-order Taylor series expansion in terms of the intervening variables yi ðyi ¼ xpi i Þ, in which the Hessian matrix has only diagonal elements of the same value e. Therefore, this approximation does not need the calculation of the second-order derivatives. Further details of this approximation are available in Refs. [2,3]. Naturally, the accuracy of a local approximation is one of the primary factors on which the quality of MPA is dependent. TANA2 were used as local approximations to construct the MPA of each limit-state function. The TANA2 can capture the information of the limitstate accurately in the vicinity of the data points. MPA retains the information for each failure surface without increasing the computational effort. The MPA adaptively adjusts itself to behave as a local approximation when a design point is closer to one of the data points. Function and gradient values of this MPA correspond directly with their exact counterparts at the points where the local approximations were generated [8].
5. Proposed uncertainty analysis technique In this paper, the uncertainty information is classified as intervals and probability distributions. When the in-
put variables are available as random variables, the failure probability can be estimated using the methods available in the literature. The proposed method involves the estimation of the MPP using the technique presented by Wang and Grandhi [3]. Once the MPP is estimated for the given limit-state function, with the interval variables set to central values, then the domain around the MPP (this is important due to the variation in the interval variables) is approximated using MPA. The algorithm details are given as follows: 1. Estimate the MPP for the limit-state function by setting the values of the interval variables at the central value. MPP search is an optimization technique and it generates data points during the intermediate iterations. These points are stored along with their function and the gradients information. 2. Once the MPP is available for one configuration of the interval variables, an approximate limit-state function is constructed. The design of experiments (DOEs) technique generates data points in the design space represented by the uncertain variables. All the uncertain variables are used to sample the data using DOEs. The random variables are assigned an interval that would move each of the variables by two standard deviations on either side of the MPP. The total number of simulations required is as follows: Number of random variables: M Number of interval variables: N Data points from MPP search: P For the above case, DOEs involves M þ N dimensions to sample. For the first iteration, M þ N points from the DOEs can be used along with P data points to construct the MPA. Therefore, for the first iteration M þ N þ P simulations are required. Thereafter, one additional simulation is required for every data point added until convergence. The new points are selected from the design space that was not sampled in the initial step. 3. During this process, the random variables are sampled within one standard deviation from the MPP. This sampling would result in design points around the MPP, which has been already estimated. This would account for the shift in MPP that may arise due to the presence of interval variables. 4. At each of the design points, the function value and the gradients are evaluated. When the gradients with respect to some uncertain variables are not available, they are estimated using numerical techniques. The availability of function value and gradients are a prerequisite for the approximations discussed in this paper. In the absence of the gradients, a response surface model may be a suitable choice. 5. With the function values and gradients available at all the data points, local approximations are con-
R.C. Penmetsa, R.V. Grandhi / Computers and Structures 80 (2002) 1103–1112
structed at the data points. The local approximations used in this research work are TANA2. Once the local approximations are available, they are combined into one MPA [8], which uses blending functions to combine the local approximations. 6. This MPA is used in conjunction with Monte Carlo simulation to determine the failure probability of the structure. For variables with probability distribution their values are generated by a random number generator, however, for the variables with intervals, the function is evaluated using the interval arithmetic. The failure probability interval is obtained by considering the lower and upper limits of the function value obtained from the MPA.
5.1. Implementation issues While using MPA for estimating the bounds of failure probability, overestimation of the bounds is an important issue. This overestimation of the bounds due to the multiple existence of the same variable in the function is avoided partially by combining the terms in each local TANA2 as shown below g~ðX Þ ¼ a þ
n X
bðxpi i þ ci Þ2
ð6Þ
i¼1
where the terms a, bi , and ci are obtained by reordering Eq. (5). P From Eq. (5), the term 12 e ðxpi i xpi;2i Þ2 can be expanded and the expanded form of the equation would be as shown below:
i n x1 p 1 X i;2 pi pi i
ðxpi i Þ þ e ðx2pi þ x2p i;2 2xi xi;2 Þ pi 2 i¼1 i
ð7Þ
Now the xpi i terms can be combined as follows: g~ðX Þ ¼ gðX2 Þ
þ
n X i¼1
1 p n n X ogðX2 Þ xi;2 i pi 1 X ðxi;2 Þ þ e ðx2pi Þ pi oxi 2 i¼1 i;2 i¼1 1 p ogðX2 Þ xi;2 i pi 1 i ðxi Þ þ eðx2p
2xpi i xpi;2i Þ oxi 2 i pi
!
ð8Þ From the above equation the required constants can be obtained as: 2 ! p og xi;2 1 pi 2 og xi;2i
2xpi;2i þ exi;2
; oxi pi 2 e oxi pi i¼1 p 2 og xi;2i and ci ¼
2xpi;2i e oxi pi
a ¼ gðX2 Þ þ e b¼ ; 2
n X
Therefore, there is a slight overestimation of the bound when the interval arithmetic is applied on each approximation. In order to reduce the overestimation, the weighting function Wk ðX Þ in Eq. (2) is estimated at the central value of the bound and this would tend to underestimate the bound. A combination of the weighting function evaluated at the central value and the local approximations evaluated using the bounds, resulted in a conservative estimation of the response bound. Fig. 1 pictorially demonstrates the proposed algorithm for estimating the failure probability interval.
6. Numerical examples This method can produce a bound on the probability of failure by considering variables with distributions and intervals simultaneously. The interval of failure probability obtained by using MPA of the failure surface is compared with the actual failure probability interval. The error is estimated for both the upper and lower limits individually. While performing the reliability analysis, the values of the interval variables are set to central values. 6.1. Cantilever beam A cantilever beam shown in Fig. 2 subjected to a tip load P is considered in this example. The limit state is defined as tip displacement greater than 0.15 in. (failure). G1 ðX Þ ¼
1 p n n X X ogðX2 Þ xi;2 i pi ogðX2 Þ ðxi;2 Þ þ g~ðX Þ ¼ gðX2 Þ
p ox oxi i i i¼1 i¼1
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4PL3
0:15 6 0:0 Ebh3
where L, b, and h are the length, width and height of the beam with mean values of 30, 0.8359 and 2.5093 in., respectively and the Young’s Modulus, E, is 107 psi. The length, width and height of the beam are considered as the random variables, and the standard deviations are, rL ¼ 3:0 in., rb ¼ 0:08 in. and rh ¼ 0:25 in., respectively. Both L and h are considered as log-normally distributed and b is considered as a normal distribution. The load P is available as an interval instead of a constant value. The interval considered for P is [60, 100] lb. Therefore, for every possible value of this load, there would be a corresponding failure probability and using the proposed method the bound of this failure probability is estimated. Initially, a reliability analysis was performed with a load of 80 lb., which resulted in an MPP and six intermediate design points during the MPP search. After obtaining the MPP, DOEs was used to obtain seven additional points distributed around the MPP in the additional dimensions introduced by the interval variables. The local TANA2 were constructed using these
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Fig. 1. Proposed algorithm details.
Fig. 2. Cantilever beam.
design points and they are blended into one MPA. This MPA was used to determine the bounds on the failure probability. The failure probabilities for the designs vary from 0.004963 to 0.08116 when the load is considered to vary as an interval. The design had higher failure probability for a load of 100 lb. and it was the lowest for a load of 60 lb. at the tip. The obtained results were obvious from the equation of the limit-state function. However, when dealing with implicit nonlinear limit-state functions and multiple interval variables, the minimum values of the interval variables do not always result in a minimum failure probability. The variation of failure probability with the variation in loading is a nonlinear function as shown in Fig. 3. Therefore, a linear approximation at
Fig. 3. Failure probability vs. tip load.
the MPP for the random variables and central values for the interval variables would result in an inaccurate estimation of the failure probability. As shown in Table 1 the failure probability bound was [0.005134, 0.08192] using MPA and it was [0.004963, 0.08116] using the actual function. The results show that there is a great variation in the failure probability and the proposed method is capable of capturing the variation accurately and efficiently. The obtained results are after a convergence in the failure probability and also
R.C. Penmetsa, R.V. Grandhi / Computers and Structures 80 (2002) 1103–1112 Table 1 Cantilever beam results
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is set to the central value and the random variables are used to obtain the failure probability. The MPA values for the current example were conservative and they followed the true trend for the failure probability.
Limit-state function
Failure probability bound
% Difference Lower limit
Upper limit
Monte Carlo MPA
[0.0049, 0.0811] 200,000 simulations [0.0051, 0.0819] 13 simulations
–
–
6.2. Ten-bar truss structure
3.44
0.92
The failure probability bounds of the 10-bar truss, shown in Fig. 5, were calculated in this example. The cross-sectional areas of all the ten truss members are lognormally distributed random variables with 2.5 in. mean value and 0.5 standard deviation. The Young’s modulus is 107 psi and the forces applied are P1 ¼ P2 ¼ 105 lb, as shown in Fig. 4. The limit-state considered in this example is the displacement limit on the maximum displacement of the tip of the truss structure which should be less than 1.8 in. The lengths L1, L2, L3, L4 and L5 (as shown in Fig. 5) are considered uncertain in the interval [350, 370].
after adding new design points from the unselected domains in the Design of Experiment technique. Seven additional design points were considered along with the six intermediate design points to obtain the results, which were obtained using 100,000 Monte Carlo simulations on the approximate model. The proposed method required 13 actual function evaluations and at each function evaluation the gradients are available analytically. However, using the combinatorial approach for the interval analysis to explore the entire domain results in 200,000 exact function evaluations provided 100,000 are required for each reliability analysis of lower and upper bounds. Moreover, this does not consider the possibility of the upper or lower limit of the response lying within the interval of the uncertain variables. Fig. 4 shows the variation of the probability of failure with the uncertainty introduced by the interval loading. As the uncertainty in loading increases, the failure probability interval varies as a nonlinear function of uncertainty. The value of failure probability for the load of 80 lb. is the crisp value when the interval variable
Fig. 5. Ten-bar truss.
Fig. 4. Failure probability variation with uncertainty in loading.
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Table 2 Ten-bar truss results Limit-state function
Failure probability bound
% Difference
MPA For Li ¼ 35000 Monte Carlo
[0.00015, 0.0162]
–
MPA For Li ¼ 36000 Monte Carlo MPA For Li ¼ 37000 Monte Carlo MPA
G1 ðX Þ ¼
0.00039 100,000 simulations 0.00043 204 simulations 0.00229 100,000 simulations 0.00242 204 simulations 0.0133 100,000 simulations 0.0142 204 simulations
)9.30
)5.67
)6.76
Dtip
1:0 6 0:0 1:8
Table 2 compares the results obtained using the Monte Carlo simulation on the actual limit-state and MPA. Since this is a problem with an implicit limit-state function, TANA2 are constructed at the six data points obtained in the process of searching for the MPP and the additional 28 points obtained from the DOEs technique. These local approximations are blended together into one MPA. The construction of TANA2 required the evaluation of gradients at the design points. The gradients of the random variables (cross-sectional areas) were available analytically from the finite element software; however, the gradients of the interval variables were numerically evaluated. The five interval variables would lead to 32 (25 ) different settings for a combinatorial estimation of the bounds of the failure probability. Therefore, in this problem the accuracy of the bound estimated using the MPA is demonstrated by comparing the results of failure probability for different interval settings. Three different settings are explored; one where the interval variables are assigned the central values and the other two where the interval variables are assigned the lower and upper limits respectively. For the case where the values of Li s are considered to be 360 in., which are the central values, the MPA produced an estimate that had 5.6% error. For the interval variable setting of 350 in., the error in the estimation was 9.3%. Moreover, for the case where the interval variables were assigned 370 in. the error was 6.76%. This error is introduced by the approximate function; however, the final result of the interval bound for failure probability
would be a conservative estimate because of the slight overestimation due to the use of interval arithmetic. In our experience, the difference in the prediction of failure probability using this method was found to be less than 10%. In this example, a total of 34 function evaluations are required at the data points. Moreover, since all the gradients are not available analytically, the gradients of the function due to the lengths were obtained numerically. Therefore, 170 additional simulations were required for obtaining the gradients. However, the number of exact simulations is considerably less compared to the Monte Carlo simulation, which would require one reliability analysis (around 100,000) for each of the 32 combinations of the uncertain interval variables. Therefore, by using the proposed method the bounds on failure probability can be obtained efficiently. From this example, it can be seen that the lower/ upper limit of the interval variables would not result in the lower/higher limit of the failure probability. This is because the tip displacement for the truss would be maximum for a particular combination of the interval variables, which is different from the lower or upper limit settings for all the variables. Due to this reason it can be seen that all the possible combinations of the interval variables have to be explored in order to obtain the bounds on the failure probability. However, the use of interval analysis and high quality approximations would result in a conservative estimate of the bounds of failure probability with considerably less computational effort. 6.3. Wing structure example In this example a wing with three spars and five ribs is considered. Fig. 6 shows the wing with the loading at each of its nodes. The ribs and spars are modeled using shear elements, the posts are modeled as rod elements
Fig. 6. Box wing example.
R.C. Penmetsa, R.V. Grandhi / Computers and Structures 80 (2002) 1103–1112
1111
and the skins are modeled as membrane elements. The thicknesses of chord wise skins are physically linked to have the same thickness, this is assumed to be a random variable. The skin thickness is normally distributed with a mean of 1.5 in. and a standard deviation of 0.15 in. Each spar is linked to have the same thickness and the same is done with the ribs. This linking resulted in three random variables for spars and five random variables for the ribs. Each of these eight random variables are normally distributed with a mean of 0.5 in. and a standard deviation of 0.05 in. The cross-sectional area of the posts is assumed to be deterministic. The linking of the variables resulted in 12 random variables and the loading is considered to be available as an interval. As shown in Fig. 6, the load is applied on a set of nodes. The load on the leading edge has one interval value, the load on the trailing edge has one interval, and the intermediate nodes have a certain interval loading. The limit-state considered in this example is the tip displacement >1.0 in. represents a failure.
along with the interval variables to obtain the expanded domain of interest. M þ N þ P (¼ 19) data points are used in the initial analysis. Then one data point is added in every iteration until there is a convergence in the failure probability values. Eighteen iterations were required for the convergence of the failure probability, therefore a total of 37 exact simulations were required to obtain the bounds using the proposed method. However, additional 111 (37 data points 3 variables) simulations were required to determine the gradients of the tip displacement with respect to the load. Considering the low error (<6%) in the results obtained and the number of exact simulations required for obtaining them, it can be seen that the proposed method is capable of producing bounds for the failure probability efficiently and accurately.
Dtip 1:0 6 0:0 ðsafeÞ
A methodology was developed for a general uncertain analysis problem. Both random distributions and intervals of input variables are handled simultaneously with considerable accuracy and efficiency. The use of MPA has enabled modeling of the failure region in multiple dimensions in order to produce accurate results efficiently. The local TANA2 capture the nonlinearity of the failure surface accurately near the expansion point and the comparison point. The proposed method is capable of handling a wide range of random variable distributions. It can also be used for nonlinear problems with more than a dozen random variables. The interval uncertainty can also be handled by using the Fuzzy definition for the uncertain variables that are given as intervals. When dealing with fuzzy numbers, the current estimation would be estimation for a zero degree of possibility. For different degrees of possibility, the bounds of failure probability can be obtained.
Table 3 shows the comparison between the bounds of failure probability estimated by first using Monte Carlo Simulation on exact limit-state function and later on MPA of the limit-state. This example is a special case where the effect of the intervals of loading is known in advance and two Monte Carlo simulations are sufficient to obtain the bounds on the failure probability. One simulation with the loads L1 and L2 set equal to maximum and L3 set to minimum, this would give the maximum deflection, and the other is when the loads L1 and L2 are set to minimum and L3 are set to maximum, this would result in minimum displacement. Each Monte Carlo simulation bound was obtained by running 100,000 actual function evaluations for each of the interval variable settings (one is the lower limit and the other is upper limit). A total of 200,000 exact simulations were performed. The MPP search technique has converged in eight iterations, which gives eight data points. Then perturbing the MPP in each dimension by two standard deviations expands the domain around MPP. This is used
Table 3 Wing tip displacement results Limit-state function
Failure probability bound
% Difference Lower Limit
Upper Limit
Monte Carlo MPA
[0.00119, 0.0422] 200,000 simulations [0.00126, 0.0434] 156 simulations
–
–
)5.80
)1.98
7. Summary
Acknowledgements This research work has been sponsored by the Air Force Office of Scientific Research under contract F4962-00-1-0377. The support for the Graduate Research Assistant has been provided by the Dayton Area Graduate Studies Institute (DAGSI).
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