Approximate probabilistic optimization using exact-capacity-approximate-response-distribution (ECARD)

Transcription

1 Struct Multidisc Optim (009 38: DOI /s z RESEARCH PAPER Approximate probabilistic optimization using exact-capacity-approximate-response-distribution (ECARD Sunil Kumar Richard J. Pippy Erdem Acar Nam H. Kim Raphael T. Hatka Received: 31 March 008 / Revised: 0 July 008 / Accepted: 5 August 008 Springer-Verlag 008 / Published online: 3 September 008 Abstract Probabilistic structural design deals with uncertainties in response (e.g. stresses and capacity (e.g. ailure stresses. The calculation o the structural response is typically expensive (e.g., inite element simulations, while the capacity is usually available rom tests. Furthermore, the random variables that inluence response and capacity are oten disjoint. In previous work we have shown that this disjoint property can be used to reduce the cost o obtaining the probability o ailure via Monte Carlo simulations. In this paper we propose to use this property or an approximate probabilistic optimization based on exact capacity and approximate response distributions (ECARD. In Approximate Probabilistic Optimization Using ECARD, the change in response distribution is approximated as the structure is re-designed while the capacity distribution is kept exact, thus signiicantly reducing the number o expensive response simulations. ECARD may be viewed as an extension o SORA (Sequential Optimization and Reliability Assessment, which proceeds with deterministic optimization iterations. In contrast, ECARD has probabilistic optimization iterations, but in each iteration, the response distribution is approximated so as not to require additional response calculations. The use o inexpensive probabilistic optimization allows easy incorporation o system reliability constraints and optimal allocation o risk between S. Kumar R. J. Pippy N. H. Kim (B R. T. Hatka University o Florida, Gainesville, FL 3611, USA nkim@ul.edu E. Acar TOBB University o Economics and Technology, Sogutozu, Ankara 06560, Turkey ailure modes. The method is demonstrated using a beam problem and a ten-bar truss problem. The ormer allocates risk between two dierent ailure modes, while the latter allocates risk between members. It is shown that ECARD provides most o the improvement rom risk re-allocation that can be obtained rom ull probabilistic optimization. Keywords Reliability-based optimization Probabilistic design Uncertainty Risk allocation Approximation Nomenclature x r1 vector o those input random variables whose mean values cannot be controlled x r vector o those input random variables whose mean values can be controlled u vector o design variables u 0 vector o initial design variables u p vector o design variables at the end o the previous iteration g(x r1 ; x r ; u perormance unction c(x r1 ; x r ; u capacity (ailure stress, allowable displacement, etc r(x r1 ; x r ; u response (structural stress, maximum delection, etc r(μ xr1 ; μ xr ; u response at the mean values o random variables r ECARD (x r1 ; x r ; u approximate response used in ECARD iteration F Cj ( cumulative distribution unction o c(x r1 ; x r ; u or jth mode

2 614 Rj ( probability density unction o r(x r1 ; x r ; u or jth mode P FSU upper bound o system probability o ailure P FS system probability o ailure P det probability o ailure at the deterministic optimum design approximate probability o ailure P det FSU P det FS ε 1 Introduction upper bound o system probability o ailure at the deterministic optimum design system probability o ailure at the deterministic optimum design relative change in response convergence threshold Probabilistic structural design deals with uncertainties in responses (e.g. stresses and capacities (e.g. ailure stresses. The calculation o the structural response is typically expensive (e.g., inite element simulations, while the capacity is usually available rom tests. Furthermore, the random variables that inluence response and capacity are oten disjoint; i.e., the random variables that aect response do not aect capacity and vice versa, and they are also statistically independent. Even i the perormance unction is not in disjoint orm, i.e., we have random variables aecting both capacity and response; we can oten separate the variables to achieve disjoint capacity and response. 1 This disjoint property was used to reduce the cost o obtaining the probability o ailure via Separable Monte Carlo simulations proposed by Smarslok et al. (006. In this paper we propose to use this property or an approximate probabilistic optimization based on exact capacity and approximate response distributions (ECARD. In ECARD the change in response distribution is approximated as the structure is re-designed, while the exact distribution o capacity is used. It turns out that this approach can signiicantly reduce the number o expensive response simulations in calculating probability o ailure. ECARD may be viewed as an extension o sequential optimization and reliability assessment (SORA; 1 For instance, i perormance unction is g (r, c = r/(1 + c c, then we can multiply the equation by (1 + c and make the unction disjoint with new capacity o c(1 + c. However, we recognize that there are exceptions like when we have size eect on strength or implicit, black-box transer unctions. S. Kumar et al. Du and Chen 004; Ba-abbadetal.006 methods. These methods reduce the cost o reliability calculations by stipulating the motion o the most probable point (MPP as the structure is being re-designed so that the calculation o the reliability using FORM is trivial. ECARD may use similar assumptions or the response, but or the capacity, which is usually computationally cheaper, it does not make any assumptions. Like SORA, ECARD requires several iterations o optimization, with each iteration requiring a single accurate reliability assessment. SORA methods do not allocate risk between ailure modes in a structure where many components can ail simultaneously (Ba-abbad et al Ba-abbad et al. (006 proposed a modiied SORA using FORM to reallocate risk between multiple ailure modes, but this requires adding the individual ailure modes as additional variables. It also does not cater to the correlation between various modes o ailure. The objective o the present paper is to demonstrate that ECARD can provide near optimal risk allocation between ailure modes at a raction o the cost o ull probabilistic optimization. The remainder o the paper is organized as ollows. Section proposes an approximate method that allows probabilistic design based only on probability distribution o the capacity. The applications o the method to a beam problem and a ten-bar truss problem are presented in Sections 3 and 4, respectively. Finally, the concluding remarks are listed Section 5. Approximate probabilistic optimization using exact-capacity-approximate-response-distribution (ECARD.1 Approximation o change in response distribution In probabilistic optimization, the system constraint is oten given in terms o ailure probability o a perormance unction. We consider here a speciic orm o perormance unction, given as g (x; u = c (x c ; u r (x r ; u (1 where c(x c ; u andr(x r ; u are capacity and response, respectively. Both the capacity and response are random because they are unctions o random variables x c and x r, respectively, and both may depend on design variables u. The system is considered to be ailed when the response exceeds the capacity. We assume that the probability distribution o c(x c ; u is well known, while that o r(x r ; u requires a large number o analyses to deine. The separable orm in (1 isusedhere

3 Approximate probabilistic optimization using exact-capacity-approximate-response-distribution (ECARD 615 or simplicity. However, as explained later, the only requirements are that x c and x r are independent, and that the calculation o the limit state g(x, u isinexpensive given r and c. Furthermore, we can separate random variables, x, into those variables whose mean values cannot be controlled, x r1, and those variables whose mean values can be controlled, x r. Thereore, vector u contains only the remaining deterministic design variables. This distinction acilitates the approximation o response distribution by measuring changes in response due to changes in mean values o random variables that can be controlled, x r. Since the capacity and response are independent, the probability o ailure jth mode may be calculated as (Melchers 00a P j = Pr [ g j (x r1 ; x r ; u 0 ] = F Cj (ξ Rj (ξ dξ. ( In the above equation, F Cj (ξ isthecumulativedistribution unction (CDF o capacity or jth mode, and R (ξ is the probability density unction (PDF o response or the same mode. When F Cj (ξ is given, evaluation o the integral using Monte Carlo simulation (MCS o the response is known as the conditional MCS. That is, with N simulations o the response system probability o ailure is given by (3, where m is number o modes o ailure. This equation is based on the act that or a given response the capacities or the modes are independent. P = 1 N m { 1 1 FCj (r i } (3 N j=1 Smarslok et al. (006 showed that the conditional MCS is much more accurate than the traditional MCS where (1 is used. For a more general limit state, it is still possible to use a similar approach, called separable MCS (Smarslok et al When design variables are changed, the distributions o both capacity and response may change. We consider the case that it is inexpensive to update the distribution o the capacity but it is expensive to update the distribution o the response. There are dierent ways to update the distribution o the response. For instance, Nikolaidis et al. (008 proposed a method to update the system ailure probability when the mean values o random variables change according to redesign and there are no other changes. However, here we use the simplest possible approach, with the aim o showing that even with a very crude approximation it can provide reasonable risk allocation between ailure modes. Fig. 1 Distributions o response beore and ater redesign. Redesign only changes the mean value. Distribution o capacity is also shown In ECARD, the change in response distribution is represented by the change in the mean value μ xr o the random variable x r and changes in u. Thatis,ir(x r1 ; x r ; u 0 is the response distribution at a given design u 0, then the response distribution at a new design u,canbe approximated by r (x r1 ; x r ; u = r ( x r1 ; μ xr0 ; u 0 + ; = r ( μ xr1 ; μ xr ; u r ( μ xr1 ; μ xr0 ; u 0 (4 Figure 1 illustrates the change in response distribution, along with the distribution o capacity. The change in response distribution is approximated such that the standard deviation o the response remains constant in (4. This assumption may be acceptable when the changes in design variables are small. We start the procedure with the deterministic design. Since the probabilistic optimum design can usually be ound close to the deterministic optimum design, the above assumption will not cause large error. In addition, since ECARD recalculates r(x r1 ; x r ; u at every ECARD iteration, the error will be reduced toward the probabilistic optimum design. This simple approximation may not always work and should be replaced with more sophisticated approximations i needed. In act convergence o ECARD iterations is not guaranteed. But even this simple minded approximation worked We know that SORA developers aced convergence problems and this algorithm may also jump between easible and ineasible regions or some problems.

4 616 very well with ECARD or the two examples shown here. Another important aspect o the proposed ECARD is that it directly perturbs the distribution o output response, rather than input design variables. This contributes the signiicant eiciency o the proposed method because it does not require calculating propagation o input uncertainty to the output response distribution. The new response distribution in (4 can be obtained with a single deterministic analysis o r(x r1 ; x r ; u. We have also considered a similar approximation where the response is scaled by 1 + / r (x r1 ; x r ; u 0 so that the coeicient o variation is preserved in Acar et al. (007. The dierence in perormance was not signiicant. The change in the response distribution will change the probability o ailure. Combining (3 and (4, the probability o ailure at the new design is approximated by = 1 N N F C (r i + (5. Approximate probabilistic optimization using ECARD In this section the ECARD optimization process is explained. The goal is to urther reduce the objective unction rom the deterministic optimal design, while maintaining the same level o probability o ailure. This can be achieved by re-allocating risk between either dierent ailure modes or dierent structural members. The process takes the ollowing steps: 1. Perorm deterministic optimization with saety actors to obtain a deterministic optimum design u det and objective unction W det.. Set the initial ECARD design u 0 = u det. The probabilistic design starts rom the deterministic optimum design; u 0.Setp = 0. Perorm conditional MCS to generate distribution o response r(x r ; u 0 and its probability o ailure, P det = P 0.Itis S. Kumar et al. possible to use the First-Order Reliability Method (FORM instead o MCS (see Acar et al At the current design, u p, calculate the response at the mean value o the random variables, r μ = r ( μ X ; u p. 4. Obtain optimum design u opt and optimum objective unction W opt by solving the ollowing deterministic optimization problem with target probability P det = P 0 : Minimize u s.t. where W (μ X,u P det = r ( μ xr1 ; μ xr ; u r ( μ xr1 ; μ xr0 ; u 0 r ECARD (x r1 ; x r ; u = r ( x r1 ; μ xr0 ; u 0 + = 1 N N ( F C r ECARD i (6 (7 5. Perorm MCS to generate accurate distribution o response r(x r ; u opt and its probability o ailure P, at u opt. This step requires ull reliability analysis, which is the expensive part. 6. Check accuracy and convergence: The ECARD approximation is considered accurate when P ECARD P ε, and convergence is assumed when the changes in design variables are small. I the process is accurate and converged, stop the process. Otherwise, set p = p + 1, u p = u opt and go to Step 3. The accuracy o ECARD to locate the true optimum depends on the magnitudes o errors involved in the approximations. The simple approximation used here is accurate i changes in distribution parameters o the response due to redesign are small. In addition, the accuracy in estimating the probability o ailure aects the convergence rate o the proposed method. However, more accurate approximations such as Nikolaidis et al. (008 can be used to improve accuracy. Note that (7 is easily applicable to a single mode o ailure. For multiple modes o ailure we need to use the joint distribution o the capacities to calculate the system ailure probabilities or a vector o response quantities. In the examples used in this paper we use instead the Ditlevsen upper bound (Melchers 00b. However, the proposed method should work even L=100" Fig. Cantilever beam: geometry and loadings t w F Y F X

5 Approximate probabilistic optimization using exact-capacity-approximate-response-distribution (ECARD 617 Table 1 The mean and coeicient o variation o the random variables x Random variable Mean Coeicient o variation F X (lb 500 0% F Y (lb 1,000 10% E (psi.9e7 5% σ (psi 40,000 5% All variables ollow normal distribution with the more accurate system probabilities obtained rom MCS. 3 Application o ECARD to beam optimization problem Deterministic optimization based on saety actors typically may have sub-optimal risk allocation between ailure modes compared to probabilistic optimization. In this section, design optimization o a cantilever beam is presented in order to show the risk allocation capability o ECARD optimization. In order to gage the eectiveness o ECARD compared to ull probabilistic optimization, we compare both to the deterministic optimization. 3.1 Problem description The cantilever beam design problem (Fig. has been analyzed by many researchers including Wu et al. (001, Qu and Hatka (004, and Ba-abbad et al. (006. The beam has two ailure modes: stress ailure and excessive displacement. The minimum weight design is sought by varying the width w and thickness t o the beam; i.e., u =[w, t]. The applied loads F X and F Y along with the elastic modulus E and ailure stress σ are random variables. All random variables are assumed normally distributed with mean and coeicient o variation as listed in Table 1. The beam width w and thickness t are modeled as deterministic variables. The perormance unction corresponding to stress ailure mode can be written as ( 600 g 1 = σ wt F Y w t F X c 1 (x r 1 (x; u, (8 where c 1 and r 1 are the capacity and response o g 1. Similarly, the perormance unction corresponding to displacement ailure mode can be written as g = D 0E 4L 3 1 wt ( FY + t ( FX w c (x r (x; u, (9 where c and r are the capacity and response o g,and D 0 (here.535 is the critical displacement. We rearranged the perormance unctions so that the capacity is independent o design variables. The probabilities o ailure, P 1 and P, can be calculated using the conditional MCS in (5. 3. Deterministic optimization I the deterministic optimization is perormed with the two perormance unctions in (8 and(9, the probability o ailure at the optimum design will be close to 50%. The deterministic design takes into account the eect o random variables using a saety actor. We employ a saety actor, S F = 1.5, to the applied loads that is typical o aerospace structural design. Then the deterministic optimization problem or minimum weight can be written as Minimize w,t A = w t s.t. c 1 S F r 1 0 c S F r 0 (10 The deterministic optimization problem in (10 is solved using the Sequential Quadratic Programming algorithm in MATLAB (using the unction mincon. The results o deterministic optimization are listed in Table. The probabilities o ailure are calculated using conditional MCS. The probabilities o ailure or stress and displacement constraints are denoted by P det 1 and P det, respectively. We use one million samples or the conditional MCS. With the traditional MCS, the coeicient o variation associated with probability o ailure o stress constraint would have been 10% and % or displacement constraint. Numerical experiments showed that with the conditional MCS the corresponding coeicients o variation are 3% and 1%, Table Deterministic optimum results o the cantilever beam problem w (in t (in A (in P det 1 Conditional MCS E 05.66E 03.76E 03 The subscript 1 corresponds to stress ailure and the subscript to displacement ailure P det P det FSU

6 618 S. Kumar et al. Table 3 Comparison o system probability o ailure P det FS problem using one million traditional MCS P 1 P P det FSU and its upper bound Pdet FSU or the deterministic optimum o cantilever beam P det FS P det FSU Pdet FS Traditional MCS 8.50E 05.61E 03.69E 03.66E E 05 Small dierences with Table are due to the use o traditional MCS. The subscript 1 corresponds to stress ailure and the subscript to displacement ailure respectively. We used MCS with one million samples to calculate also the system probability o ailure P det FS and compare it in Table 3 to the Ditlevsen s irstorder upper bound P det FSU that estimates the system ailure probability as the sum o the individual ailure probabilities (Melchers 00b. The traditional MCS has poorer accuracy than the conditional MCS, so the numbers are slightly dierent. However, they still show that the ailure modes are well correlated because the Ditlevsen approximation error is close to hal o the lower probability o ailure (see Table 3. Note that the deterministic optimization allocates risk between two ailure modes such that the probability o ailure or the displacement mode is 7 times higher (see values rom Table than that o the stress mode. This is based on the use o the same saety actor or both ailure modes. Its contributions to the dierent ailure modes are dierent. We will show that the probabilistic optimization reduces the dierence in ailure probabilities signiicantly in the ollowing section. 3.3 Probabilistic optimization The probabilistic optimization is ormulated with the same objective unction. However, the constraint is imposed on the approximate (upper bound system probability o ailure. The goal is to reduce the objective unction urther by reallocating the risk between the two ailure modes, while the system probability o ailure should not be greater than that o the deterministic optimization. The probabilistic optimization problem is then ormulated as Minimize w,t s.t. A = w t P FS = ( P 1 + P P det FSU (11 The results o probabilistic optimization are listed in Table 4. The probabilistic optimization is computationally expensive, requiring 61 reliability assessments. As can be seen rom the table, the design obtained at the end o the optimization satisied the probability constraint with a small margin. Compared with the deterministic optimum (Table, the probabilistic optimization reduced the weight by 6%, while reducing the system ailure probability by %. The reduction in both the weight and the system probability o ailure is obtained by risk reallocation between two ailure modes. The deterministic design leads to smaller probability o ailure or the stress mode than that o the displacement mode, while the situation is reversed or the probabilistic design. This is based on the act that displacement is cheaper to control than stress in terms o weight expenditure. For example, i w and t are scaled uniormly, then the stress is proportional to the cube o the scale actor while the displacement is proportional to the 4th power. Similar results are also reported by Ba-abbad et al. (006 using a modiied SORA. Table 5 compares the system probability o ailure and its upper bound used in probabilistic optimization procedure. It is seen that again the error is a substantial raction o the lower o the two probabilities o ailure, indicating substantial correlation. 3.4 Approximate probabilistic optimization using ECARD The approximate probabilistic optimization problem is ormulated based on (11 by replacing the probability o ailure with approximate one, as Minimize w,t s.t. A = wt ( FSU = 1 + P det FSU (1 Table 4 Probabilistic optimum results o the cantilever beam problem w (in t (in A (in P 1 P P FS Conditional MCS E E 4.70E 3 The subscript 1 corresponds to stress ailure and the subscript to displacement ailure

7 Approximate probabilistic optimization using exact-capacity-approximate-response-distribution (ECARD 619 Table 5 Comparison o system probability o ailure P FS and its bound P FSU or the probabilistic optimum o cantilever beam problem using one million traditional MCS P 1 P P FSU P FS P FSU P FS Crude MCS.38E E 04.7E 03.61E E 04 Small dierences with Table 4 are due to the use o the traditional MCS. The subscript 1 corresponds to stress ailure and the subscript to displacement ailure where 1 and are, respectively, approximations o P 1 and P using the ECARD method described in Section.. The approximate probabilities o ailure are calculated as 1 = 1 N = 1 N N ( F C1 r i N ( F C r i + (13 where F C1 and F C are CDFs o the two capacities. The optimization in (1 is solved iteratively until the convergence criterion in Section. is satisied. We have used a gradient based method, SQP implemented by MATLAB s mincon unction. However, in general any optimization method can be used. Figure 3 shows the deterministic and probabilistic optimum designs, along with the iteration history o the ECARD design. The contour lines o P FSU = and A = in are shown or reerence. It is seen that the approximate ECARD probability o ailure at iteration 1 has an error compared to the accurate one. However, it is gradually reduced in the ollowing iterations. Table 6 lists the designs obtained during iterations o the ECARD optimization using conditional MCS. Ater the 4th iteration, the ECARD optimization was considered converged because the design variables changed by less than 0.1% and the approximate and exact probabilities o ailure were close. The ECARD design was close to the probabilistic optimum. The dierence between approximate and ull probabilistic optimizations can be explained by the convergence tolerance ε, error in probability estimation, and small change in objective unction near the optimum design. However, in general, in the limit the error may not be negligible or it may take more number o iterations to converge depending on the tightness o convergence criteria. In act we might need better approximation to deal with the issue o accuracy and eiciency. Table 7 compares the system probability and its upper bounds or the ECARD design. Comparing to the probabilistic optimum, the ECARD optimum is about 0.3% heavier, while having the same system probability o ailure. As in the probabilistic optimization, this is achieved by increasing the risk o the more expensive (in terms o weight stress mode and decreasing the risk o the cheaper displacement mode. Note that ECARD requires only our reliability assessments as opposed to 61 used by ull probabilistic optimization. Every ECARD iteration needs a reliability assessment in order to correct errors introduced by the approximations involved in the response distribution. In the irst two iterations, steps are larger and accuracy is not good. It improves in subsequent iterations. 4 Application o ECARD to ten-bar truss problem Fig. 3 ECARD design history with deterministic and probabilistic designs The second example is a ten-bar truss problem, as shown in Fig. 4. First, we present deterministic optimization o the problem. Then, probability o ailure calculation using MCS is discussed. Finally, probabilistic optimization is perormed using ECARD, and the accuracy and eiciency o the method is evaluated.

8 60 S. Kumar et al. Table 6 Iterations o approximate ECARD probabilistic optimization or the cantilever beam problem using conditional MCS Iter. w (in t (in A (in P 1 1 P P FSU PFSU ECARD E E 05.76E 03.76E 03.86E 3.70E E E E 04.30E E 3.70E E E E E 03.40E 3.70E E E E E 04.80E 3.70E E E E E 04.60E 3.70E Problem description The problem deinition or the ten-bar truss problem is taken rom Hatka and Gurdal (199 (page 37. The ten-bar truss structure in Fig. 4 is under two loads, P 1 and P. The design goal is to minimize the weight, W, by varying the cross-sectional areas, A i, o the truss members. The design should satisy stress constraints and minimum gage constraints. Input data are summarized in Table 8. In the traditional deterministic design o aircrat structures, uncertainty in material ailure stress is taken into account using a knockdown actor, with so-called A-basis or B-basis properties (Little 003. The relationship between allowable stress and the mean value o ailure stress is σ allowable = K dc σ (14 where K dc is the knockdown actor. In this example, A- basis allowable stress is used in which the knockdown actor becomes K dc = In addition, uncertainties in applied loads and errors in stress calculations are considered by multiplying the applied loads by a saety actor o Deterministic optimization The deterministic optimization protects against uncertainties using saety actor and knockdown actor. The deterministic optimization problem can be ormulated as min A i W = s.t. 10 ρl i A i N i (S F P 1, S F P, A A i = σ i (σ allowable i, i = 1,...,10 (15 where L i, N i,anda i are, respectively, the length, member orce, and cross-sectional area o element i. A is the vector o cross-sectional areas, σ i and (σ allowable i are the stress and allowable stress o an element, respectively. In this example, (σ allowable i corresponds to the capacity, while σ i to the response. The applied loads are multiplied by a saety actor in order to consider the eects o uncertain parameters. The analytical solutions or the member orces are given in Appendix. The results o deterministic optimization are listed in Table 9. It is noted that Element 5 is a zero-orce member and the cross-sectional areas o Elements, 5, and 6 reach the minimum gage. The probabilities o ailure o the elements, given in the second last column o Table 9, are calculated using the conditional MCS discussed in the next section. Last column o Table 9 presents component probability o ailure calculations using 10 million samples o traditional MCS, and it is similar to the values obtained with conditional MCS. However, the system ailure (.90E 0 in the last row is signiicantly lower than the upper bound obtained by adding elemental probabilities o ailure (4.03E 0. It is noted that three elements (, 6, and 10 contribute about 80% o the system probability o ailure (approximated with the Ditlevsen bounds. Since Elements and 6 are at minimum gage and Element 10 is close to it, it is easy to reduce the system probability o ailure by reallocating small additional weights to these members. However, the deterministic optimization did not do it because it applies uniorm saety actors and knockdown actors to all members. 4.3 Calculation o probability o ailure using conditional MCS Many uncertainties are involved in the design o ten-bar truss, such as variability rom manuacturing, loading, Table 7 Probability o ailure o system at last iteration o ECARD or cantilever beam problem using one million traditional MCS P 1 P P FSU P FS P FSU P FS Traditional MCS 1.85E E 04.65E 03.47E E 04 The subscript 1 corresponds to stress ailure and the subscript to displacement ailure

9 Approximate probabilistic optimization using exact-capacity-approximate-response-distribution (ECARD 61 areas. The perormance unction or Element i can be arranged in a orm (i.e., in a orm that allows the use o conditional MCS as g i = ( σ i (1 + e σ ( σ FEA 1 e [ (1 + e P1 P 1, (1 + e P P, (1 + e A A ] i c i r i (19 Fig. 4 Geometry and loadings o the ten-bar truss example and material properties, and errors rom modeling and numerical calculation. In the ollowing, we will detail these uncertainties. Failure o an element is predicted to occur when the stress in the element is larger than its ailure stress. That is, the perormance unction or an element can be written as g = ( σ true σ true (16 where the subscript true stands or the true value o the relevant quantity, which is dierent rom its calculated (or predicted value due to errors. Introducing errors, (16 can be re-written as g = ( 1 e σ (1 + e σ σ (17 Here, e is the error in ailure stress prediction (e.g., due to size eects, e σ is the error in stress calculation (e.g., due to errors in mathematical model. We ormulated the errors such that positive errors correspond to a conservative decision. Hence, the sign in ront o error in stress is positive, while that in ailure stress is negative. Even though our stress calculation is exact, we pretend that we have error, e σ, considering the analysis o a more complex structure where the stresses are calculated using Finite Element Analysis (FEA. The calculated stress can be written in a compact orm as σ = σ FEA [ (1 + ep1 P 1, (1 + e P P, (1 + e A A ] (18 where σ FEA [ ] stands or calculated stresses using FEA, e P1 and e P are errors in loads P 1 and P,ande A is the vector o errors corresponding to 10 cross-sectional where c i and r i are, respectively, the capacity and response. In addition to the errors, variability is also present in the perormance unction through σ, P 1, P,and A, which are modeled as random variables. These errors and variability are considered random variables. The distribution types and probabilistic parameters o errors and variability are listed in Table 10. The probabilities o ailure o the elements are calculated using conditional MCS in (5.Thesystemprobability o ailure is again approximated by Ditlevsen s irst-order upper bound 10 ( P FSU = P (0 i 4.4 Probabilistic optimization Starting rom the deterministic optimum design, the probabilistic optimization problem can be ormulated such that the weight o the structure is minimized (calculated using mean values o the areas, while maintaining the same system probability o ailure with that o the deterministic optimum design. Thus, we have Minimize A i W = 10 ρl i A i s.t. P FSU P det FSU (1 where P FSU is the upper bound on the system probability o ailure. The design variables are the mean values Table 8 Input data or ten-bar truss problem Parameters Values Dimension, b 360 in Saety actor, S F 1.5 Load, P kips Load, P kips Knockdown actor, K dc 0.87 Density, ρ 0.1 lb/in 3 Modulus o elasticity, E 10 4 ksi Allowable stress, σ allowable 5 ksi a Minimum gage 0.1 in a For Member 9, allowable stress is 75 ksi

10 6 Table 9 Results o deterministic optimization o the ten-bar truss problem The probability o ailure is calculated using conditional MCS with 10 6 samples Element A det i ( in W i (lb Stress (ksi Conditional MCS ( P det i E 03.10E E E E E E 03.30E E E E E E E E E E E E E 0 Total 1, E E 0 System probability o ailure.90e 0 S. Kumar et al Samples o traditional crude MCS ( P det i Table 10 Probabilistic distribution types and parameters or errors and variability in the ten-bar truss problem Uncertainties Distribution type Mean Scatter Errors e σ Uniorm 0.0 ±5% e P1 Uniorm 0.0 ±10% e P Uniorm 0.0 ±10% e A (10 1 vector Uniorm 0.0 ±3% e Uniorm 0.0 ±0% Variability P 1, P Extreme type I kips 10% c.o.v. A (10 1 vector Uniorm A(10 1 vector ±4% σ Lognormal 5/K dc ksi or 75/K dc ksi 8% c.o.v. Table 11 Results o probabilistic optimization o the ten-bar truss problem using conditional MCS with 10,000 samples Elements i A i Mean Stresses Conditional Conditional 10 7 Samples crude in ksi at A i MCS ( P det MCS i ( P MCS i ( P i E E E E E E E E E E 03.15E 03.10E E E E E 0.14E 03.00E E E E E E E E E E E E E 03 Total 1,497.6 lb 1, lb 4.10E E E 0 System probability o ailure using crude MCS 3.37E 0 A det

11 Approximate probabilistic optimization using exact-capacity-approximate-response-distribution (ECARD 63 Table 1 Results o approximate probabilistic optimization and progress toward the accurate optimum Element Determ. des. iter_01 iter_0 iter_03 iter_04 iter_05 Areas (in Weight (lb 1,497 1,384 1,43 1,408 1,40 1,414 Mean stresses (ksi Approximate PF E E E E E E E E 03.50E 03.70E 03.60E 03.60E E E E E E E E 03.40E 03.00E 03.0E 03.10E 03.10E E E E E E E E E 03.70E 03.90E 03.80E 03.80E E E E E E E E E E E E E E E E E E E E E E E E E 03 System 4.18E E E E E E 0 Actual PF E E E E E E E E E 03.50E 03.70E 03.60E E E 0 5.0E E E E E 03.0E E 03.30E 03.00E 03.0E E 04.40E E E E E E E E 03.70E 03.90E 03.80E E E E E E E E E E E E E E E E E E E 03 System 4.18E E E E 0 4.0E 0 4.8E 0 o the areas. Results o the probabilistic optimization are shown in Table 11. The table shows the change in area as well as the stresses corresponding to the mean values o the variables, and the change in probability o ailure o each element. It also presents evaluation o system probability o ailure using 10 million samples o traditional MCS at the probabilistic optimum design. Overall weight was reduced by 6% (90.47 lb, while maintaining the same bound on the system probability o ailure as the deterministic optimum design. We used 10,000 samples or conditional MCS, and optimization problem converged ater 59 iterations. A total o 78

12 64 reliability assessments were required during the probabilistic optimization. We used ixed random variables to eliminate the eect o random noise. In the probabilistic optimum design, the areas o three small-area elements (, 6, and 10 that have high probability o ailure in the deterministic optimum are slightly increased and the corresponding stresses are slightly reduced. However, the reduction in the probability o ailure is signiicant or these elements. Thus, the risk reallocation between dierent members occurs by moving small amount o weight to the small-weight, high-risk members. By reallocating the risk, the total weight is saved by 6%. 4.5 Approximate probabilistic optimization using ECARD The approximate probabilistic optimization problem can be written based on (1 as Minimize A i W = s.t. 10 FSU ρl i A i P det FSU ( where the approximate system probability o ailure is the sum o individual contributions. The optimization in ( is solved iteratively until convergence criterion in Section. satisies. In order to approximate the response o a design when the cross sectional areas change rom A det to A we use the ollowing equations: r ( p 1 ; p ;...; A = r ( p 1 ; p ;...; μ A + ; = r ( μ p1 ; μ p ;...; μ A r ( μ p1 ; μ p ;...; μ A det (3 Table 1 shows the results o approximate probabilistic optimization and progress toward the probabilistic optimum shown in Table 11. For this example, the ECARD method converged ater ive iterations when the changes in design variables and weight were below 0.4% and the accuracy o the system probability o ailure was about 5%. In addition, the errors in element ailure probability approximations are less than 7%. Since the probability o ailure o the Element 9 is very small, the error in its probability o ailure is not taken into account. As expected, the mean stresses S. Kumar et al. Table 13 Probabilities o ailure o each element and system probability o ailure ater last Iteration o ECARD using crude MCS Elements A i 10 7 Samples 10 4 Separable traditional MCS (P i MCS (P i E E E 03.60E E E E 03.0E E E E 03.80E E E E E E E E 03 Total 1,414.0 lbs 4.18E 0 4.8E 0 System ailure probability 3.33E 0 in the light weight elements decrease, while the mean stresses in the heavier elements increase. This relects the act that having higher saety actor to low-weight elements is more weight eicient than to high-weight elements in the overall risk allocation. Note that while the total weight is very similar to the exact probabilistic design, the individual areas are quite dierent. For example, in the probabilistic design, the stress reduction in Element is achieved by tripling its area, while in the ECARD design the stress is reduced without changing the area. Also notice that the total approximate ailure probability bound stays close to with small variations due to numerical noise in the algorithm. The true probability bound however changes as design changes. The ECARD optimization required only ive reliability assessments one or each o the ive iterations, which is a signiicant reduction rom the probabilistic optimization that consumed 78 reliability assessments. Table 13 shows system probability o ailure using 10 million samples at the design obtained in last iteration o ECARD Method. Again, the system ailure probability is substantially lower than the Ditlevsen bound. This suggests that it may be worthwhile to apply ECARD in the uture with conditional MCS so as to allow the use o a system probability constraint. 5 Concluding remarks An approximate probabilistic optimization method using exact-capacity-approximate-response-distribution

13 Approximate probabilistic optimization using exact-capacity-approximate-response-distribution (ECARD 65 (ECARD is presented. The proposed method signiicantly reduces expensive reliability calculations (typically done via MCS. ECARD was demonstrated with the simplest approximation o the response distribution one that translates the entire distribution based on the mean values o the random variables. Two examples were used to test ECARD. First, probabilistic optimization o a cantilever beam was perormed, where risk was allocated between two dierent ailure modes. Then, probabilistic optimization o a ten-bar truss problem was perormed, where risk was allocated between ten truss members. From the results obtained in these two demonstration problems, we reached the ollowing conclusions. 1. In both problems, ECARD converged to near optima that allocated risk between ailure modes much more eiciently than the deterministic optima. The dierences between the accurate and approximate optima were due to the errors in probability o ailure estimations, which led to errors in the derivatives o probabilities o ailure with respect to design variables required or risk allocation.. ECARD signiicantly reduced the expensive reliability calculations. In the cantilever beam problem, it required only our reliability assessments compared to 61 or ull probabilistic optimization. Similarly, in the ten-bar truss problem we needed only ive reliability assessments, which is almost two-orders o improvement rom Large dierences between the Ditlevsen bound and the system probability o ailure or the 10-bar truss suggest the desirability o applying ECARD with system probability constraints using the separable MCS (Smarslok et al. 006 instead o the more restrictive conditional MCS. Acknowledgements This work has been supported in part by the NASA Constellation University Institute Program (CUIP, Ms. Claudia Meyer program monitor, Mr. Vinod Nagpal o N&R engineering and NASA Langley Research Center grant number NAG , Dr. W.J. Stroud program monitor. Appendix: calculation o member orces o ten-bar truss Analytical solution to ten-bar truss problem is given in Elishako et al. (1994. The member orces satisy the ollowing equilibrium and compatibility equations. Note: Values with * are incorrect in the reerence. The correct expressions are: N 1 = P 1 N 8, N = 1 N 10, N 3 = P 1 P 1 N 8, N 4 = P 1 N 10, N 5 = P 1 N 8 1 N 10, N 6 = 1 N 10, N 7 = (P 1 + P + N 8, N 8 = b 1a a 1 b a 11 a a 1 a 1, N 9 = P + N 10, N 10 = a 11b a 1 b 1 a 11 a a 1 a 1 where a 11 = ( , A 1 A 3 A 5 A 7 a 1 = a 1 = 1, A 5 ( a = , A A 4 A 5 A 6 A 9 A 10 b 1 = ( P P 1 + P P (P1 + P, A 1 A 3 A 5 A 7 ( b = P P 4P A 4 A 5 A 9 Reerences Acar E, Kumar S, Pippy RJ, Kim NH, Hatka RT (007 Approximate probabilistic optimization using Exact-Capacity- Approximate-Response-Distribution (ECARD. In: AIAA paper , 48th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conerence, Waikiki, Hawaii, 3 6 April 007 Ba-abbad MA, Nikolaidis E, Kapania RK (006 New approach or system reliability-based design optimization. AIAA J 44(5: , May Du X, Chen W (004 Sequential optimization and reliability assessment method or eicient probabilistic design. ASME J Mech Des 16(:5 33 Elishako I, Hatka RT, Fang J (1994 Structural design under bounded uncertainty optimization with anti-optimization. Comput Struct 53(6: Hatka RT, Gurdal Z (199 Elements o structural optimization, 3rd edn. Kluwer, New York Little RE (003 Mechanical reliability improvement. CRC, Boca Raton, pp A 8

14 66 S. Kumar et al. Melchers RE (00a Structural reliability analysis and prediction. Wiley, New York, p 17, Eq. (1.18 Melchers RE (00b Structural reliability analysis and prediction. Wiley, New York, pp Nikolaidis E, Saleem S, Farizal, Zhang, Mourelatos Z (008 Probabilistic reanalysis using Monte Carlo simulation. SAE Special Publication 170, paper No , pp 1 14 Qu X, Hatka RT (004 Reliability-based design optimization using probabilistic suiciency actor. J Struct Multidisc Opt 7(5: Smarslok BP, Hatka RT, Kim NH (006 Taking advantage o conditional limit states in sampling procedures. AIAA Paper , 47th AIAA/ASME/ASCE/ AHS/ASC Structures, Structural Dynamics, and Materials Conerence, Newport, RI, May 006 Wu YT, Shin Y, Sues R, Cesare M (001 Saety Factor Based Approach or Probability-based Design Optimization. In: Proceedings o 4nd AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conerence, Seattle, WA, 001, AIAA Paper