0th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conerence 0 August - September 004, Albany, New Yor AIAA 004-4470 Techniques or Estimating Uncertainty Propagation in Probabilistic Design o Multilevel Systems Byeng D. Youn and Kyung K. Choi Department o Mechanical & Industrial Engineering, The University o Iowa, Iowa City, IA 54 Michael Koolaras and Panos Y. Papalambros Department o Mechanical Engineering, University o Michigan, Ann Arbor, MI 4809 Zissimos P. Mourelatos Department o Mechanical Engineering, Oaland University, ochester, MI 4809 David Gorsich Department o The Army, United States Army Tan-Automotive and Armaments Command, Warren, MI 4809 In probabilistic design o multilevel systems, the challenge is to estimate uncertainty propagation since outputs o subsystems at lower levels constitute inputs o subsystems at higher levels. Three uncertainty propagation estimation techniques are compared in this paper in terms o numerical eiciency and accuracy: root sum square (linearization), distribution-based moment approximation, and Taguchi-based integration. When applied to simulation-based, multilevel system design optimization under uncertainty, it is investigated which type o applications each method is best suitable or. The probabilistic ormulation o the analytical target cascading methodology is used to solve the multilevel problem. A hierarchical bi-level engine design problem is employed to investigate unique eatures o the presented techniques or uncertainty propagation. This study aims at helping potential users to identiy appropriate techniques or their applications.. Introduction Multilevel system design reers to the optimization process o large, complex engineering systems that are decomposed into a hierarchy o subsystems. Since the subsystems are coupled, their interactions need to be taen into consideration to achieve consistent designs. Analytical target cascading (ATC) is a methodology that taes these interactions into account during the early stages o the design optimization process []. In recent years, design guidelines and standards are being adjusted to incorporate the concept o uncertainty into the early design and product development stage []. In response to these new requirements, the ATC ormulation has been extended to solve probabilistic design optimization problems []. Probabilistic design o multilevel systems does not only entail the diiculty o ormulating and solving nondeterministic optimization problems; it is also quite challenging to model the mechanism o uncertainty propagation throughout the multilevel hierarchy. Outputs o subsystems at lower levels constitute inputs o subsystems at higher levels. It is thus necessary to estimate the statistical inormation o these outputs (that are inputs o subsystems at higher levels) with adequate accuracy without requiring a huge amount o raw data. In previous wor [4], we have taen advantage o the ATC ormulation that enables the use o irst-order Taylor series [5] or approximating nonlinear responses. The ATC consistency constraints do not allow large deviations rom the incumbent expansion point (which are the mean values o the design variables) during the optimization process. In this manner, not only can we linearize nonlinear responses, but we can also consider them as normally distributed i all the random variables they depend on were also normally distributed. Although large approximation errors o expected values or the nonlinear responses are avoided, the convergence rate o the ATC process can be low since many iterations involving small steps may be necessary. In addition, this estimation technique may exhibit relatively large errors when approximating higher-order statistical moments [,4,6]. This paper considers two alternative methods or estimating statistical moments o nonlinear responses o random variables. The irst method generates approximate probability density unctions to be numerically integrated []. The second method uses numerical quadrature rules motivated by Taguchi-type experimental designs [7]. The scope o this paper is to investigate the stability, accuracy, and eiciency o these two methods when applied on Copyright 004 by the, Inc. All rights reserved.
simulation-based, multilevel system design optimization problems, and to determine which type o applications each method is best suitable or.. Probabilistic Design o Multilevel Systems The major characteristic o hierarchically decomposed multilevel systems is that outputs (responses) o lowerlevel subsystems are inputs o higher-level subsystems. We tae advantage o this unctional dependency among levels and use analytical target cascading to solve probabilistic design problems o multilevel systems. The ATC process ormulates and solves a design optimization problem or each o the elements in a multilevel hierarchy, e.g., the one shown in Figure. system subsystem subsystem subsystem n component component component m Figure : Example o multilevel system hierarchy In the ollowing general mathematical ormulation, subscripts i and j are used to denote level and element, respectively. For each element j at level i, the set C ij includes the elements that are children o this element. esponses r ij are unctions o children responses r ( i+ ),..., r ( i+ ), local design variables x c ij, and shared design ij variables y ij, i.e., where ij { c ij } rij = ij ( r ( i+ ),...,r( ), x, y ) ( z ) i+ = ij ij ij ij, () cij C =,,,. Shared design variables are restricted to exist only among elements at the same level having the same parent. In the presence o uncertainty, random design variables and parameters are symbolized by the use o upper case letters and represented by their mean values, responses are represented by the expectations o nonlinear responses, and constraints are ormulated probabilistically. Mathematically, the probabilistic design optimization problem associated with each problem in the multilevel hierarchy is ormulated as u u r y minimize E - µ + µ µ + ε + ε r y ij ij Yij Yij ij ij µ ~, εij, εij j subject to Cij Cij µ E ( ) ( ) ε + + i i l Y ( i+ ) Y( i+ ) ( ) P > 0 gij Zij P µ µ ε l y ij r ij ()
( ) ( ) cij with ij = ij ( i+ ),..., ( i+ ), j, Yij = ij Z ij, where the vector o design variables µ ~ consists o vectors µ, µ,..., µ Z Y Y and ij ( ij ) ij ( i+ ) ( i+ ) cij g Z denote local design inequality constraints. random variable, P [ ] represents the probability o an event, and i.e., probabilities o violating constraints. Tolerance optimization variables ε r and X ij E denotes expected value o a P is a vector o assigned probabilities o ailure, y ε are introduced or coordinating responses and shared variables, respectively. Superscripts (subscripts) u() l are used to denote response and shared variable values that have been obtained at the parent (children) problem(s), and have been cascaded down (passed up) as design targets (consistency parameters). Figure illustrates the inormation low o the ATC process at element j in level i. Assuming that all the parameters have been updated using the solutions obtained at the parent- and children- problems, Problem () is solved to update the parameters o the parent- and children- problems. This process is repeated until the tolerance optimization variables in all problems cannot be reduced any urther. The top-level element o the hierarchy is a special case; the responses cascaded rom above are the given system design targets, and since this is the only element o the level, there exist no shared variables. Figure : ATC inormation low at element j, level i We optimize with respect to the means o the random variables, and utilize an analytical irst-order reliability method (FOM) to evaluate the reliability o satisying the probabilistic constraints. Speciically, we adopt the hybrid mean value (HMV) algorithm to compute the most probable point (MPP) or each constraint at each iteration o the optimization process []. It is emphasized that irst-order methods yield exact results only i the limit-state (i.e., constraint) responses are linear, and random variables are normally distributed and uncorrelated. I these assumptions are violated, the obtained results are only approximate. Nevertheless, these methods are used widely in literature due to their simplicity and eiciency despite their relative inaccuracy.. Uncertainty Propagation Techniques The solution o a probabilistic design problem requires inormation on the distribution and moments o the random design variables and parameters. Typically, this inormation is given or postulated at the bottom level o a probabilistic multilevel system design problem. However, since the outputs o lower-level problems constitute inputs to higher-level problems, we must propagate the uncertainty inormation as accurate as possible to solve the higher-level problems and the overall multilevel design problem. In this section we present two alternative techniques or estimating uncertainty propagation.
. Advanced Mean Value Based Distribution Generation and Moment Estimation The main idea o this technique is to perorm a reliability analysis on the output response (i.e., the nonlinear response o the random variables) using FOM or a suiciently large range o reliability targets, e.g., rom β = 4 (with corresponding probability o ailure P =Φ( β) = 0.0000 ) to β = 4 (with P =Φ( β ) = 0.99997 ). Once the most probable point is ound (we use the HMV method []), the output response is evaluated at this point to provide the corrected unction value or the corresponding probability o ailure [8]. With the cumulative density unction (CDF) available, one can then dierentiate numerically to obtain the probability density unction (PDF) [9]. We use central dierences to obtain second-order accurate approximations. Finally, we integrate numerically, using spline interpolation to estimate response values that lie between the available discrete points o the PDF, to compute moments. As will be shown later by means o preliminary numerical results, this method is quite accurate. However, it can be ineicient depending on how the β-range is discretized.. Taguchi-based Integration and Moment Estimation.. Output Statistical Moment Modeling: Numerical Integration on Input Domain One purpose o statistical moment estimation stems rom the robust design optimization, which attempts to minimize the quality loss [0,], which is a unction o the statistical mean and standard deviation. Several methods are proposed to estimate the irst two statistical moments o the output response. Analytically, the statistical moments are expressed in an integration orm as = E ( x) ( x) dx E ( ( x) µ ) = ( ( x) µ ) ( x) dx X X () Using numerical integration, the statistical moments o output response are approximated through numerical integration on the input domain as E [ x ] µ = ( x) Π ( x ) dx dx X i i n i= m m = w w ( µ + α,, µ + α ) j jn j n jn j = jn = n x X i i n i= m m E ( ) µ ( ( ) µ ) Π ( x ) dx dx n = w w ( µ + α,, µ + α ) µ j jn j n jn j = jn = (4) For application, Taguchi [,] proposed an experimental design approach or statistical tolerance design with a three-level (m=) actorial experiment, which are composed o low, center, and high levels as { w }, w, w, α, α, α = {,,,,0, } Three-level actorial experiment is modiied by D Errico and Zaino [] by employing distinctive weights at dierent levels as { w, w, w, α, α, α } = {, 4,,,0, 6 6 6 } Thus, the modiied three-level actorial experiment improved numerical accuracy in estimating the statistical moments o output response. In numerical integration, three weights or X i are used to approximate the probability density o X i at three dierent probability levels. From the statistical point o view, the modiied three-level actorial 4
experiment is meaningul, since many random input variables ollows the rule o high density near the mean and low density at the tail o statistical distribution, as shown in Fig.. ( x ) i ( x ) i 4/6 / /6 µ σ µ X i X + i µ σ X i µ σ µ µ + σ X ( a ) Taguchi Method [] ( b ) D Errico and Zaino Method [] Figure : Three-level numerical integration on the input domain In the experimental method, the computation o the moment could be very expensive or a large number o design and/or random parameters, since the number o unction evaluations or experiments required is N = n where n is a number o design and random parameters. Thus, this method is not used in this paper... Output Statistical Moment Modeling: Numerical Integration on Output Domain In Section.., statistical moments o output response are estimated through numerical integration on the input domain, maing it very expensive or reliability-based robust design optimization. In this paper, the proposed method directly identiies uncertainty propagation using numerical integration on the output domain. Unlie Eq. (), the statistical moment calculation is carried out by E = r () r dr = µ E ( µ ) = ( r µ ) ( r) dr (5) where () r is a probability density unction o. To approximate the statistical moments o accurately, N-point numerical quadrature technique can be used as E = µ N i= wr i i and N i i i= E µ w ( r µ ) or 5 (6) At minimum, the three-point integration (N=) is required to maintain a good accuracy in estimating irst two statistical moments. By solving Eq. (6), three levels and weights on the output domain are obtained as{ r, r, r } = { r, r ( ), r β= X β=+ } µ and {,, } {, 4, } 6 6 6 symmetrically located, as shown in Fig. 4. w w w =, as shown in Fig. 4. In general, upper and lower levels are not 5
(r) 4/6 /6 r β= µ r X β=+ r( ) Figure 4: Three-level numerical integration on the output domain Using the three-level numerical integration on the output domain, the irst two statistical moments in Eq. (5), the mean and standard variation o the output response are approximated to be 4 = µ + ( µ ) 6 β= 6 X + 6 β=+ E r r r [ µ ] = σ = ( µ ) () ( r µ ) ( ) 6 + r µ β= 6 β=+ E r r dr (7) Since the statistical moments o output response are estimated through a numerical integration on the output (or perormance) domain, this method is called perormance moment integration (PMI) method. In the PMI method, r and r are obtained through reliability analyses [,8,9] at β =± conidence levels. In this paper, the β= β=+ hybrid mean value (HMV) method is used or reliability analysis []. 4. Numerical Examples 4. Single-level Propagation Examples Two nonlinear analytical examples and one vehicle crashworthiness or side-impact simulation example are used to demonstrate the aorementioned techniques. For abbreviation purposes, the method presented in Section. is called distribution-based method (DBM) in this paper. Monte Carlo simulation (MCS) with one million samples and root sum square (SS) method are used or numerical comparison. Experimental methods [,] are not used or comparison because it would be too expensive even though it would be as accurate as MCS. Statistical nonnormality o the response unctions is represented by sewness and urtosis. Sewness is a measure o symmetry o probability density unction (a normal distribution has a sewness value o 0). Kurtosis is a measure o relative peaness/latness o probability density unction to normal distribution, which has a urtosis value o. The irst analytical example, the response is = X X ( X ) /0 (8) For this example, the input random parameters are modeled as N(5.0, 0.) or i=,. As shown in Table and Fig. 5, the probabilistic distribution o the irst response is close to a normal distribution with a moderate rate o sewness and urtosis. Thus, SS, DBM, and PMI show overall a good accuracy in estimating the irst two statistical moments o responses. The second analytical example response is X 7 ( X ) = e X + 0. (9) 6
The input random parameters are modeled as N(6.0,0.8) or i=,. As shown in Table, the SS method yields a large approximation error o 07% or the second moment, whereas the DBM and PMI methods are much more accurate or both the mean and standard deviation. The last single-level example is the pubic orce rom a side impact simulation [4], which is modeled with input uncertainties o Gumbel distribution and 0% coeicient o variation. Even though the stochastic response is highly sewed with large urtosis, the PMI method seems to predict the irst two statistical moments accurately, whereas the SS could yield larger errors. DBM results are not available or this example. Table : Single-level examples Mean Standard Deviation Sew. Kurt. SS DBM PMI MCS SS DBM PMI MCS -5.500-5.86-5.86-5.79 0.885 0.84 0.84 0.8405-0.6. Error, % 0.45-0.59-0.59 -- -0.8 0.7 0.07 -- -- --.6.609.608.497.986 0.90 0.8800 0.949-0.57 7. Error, %.96.5.77 -- 07.4 -.59-5.87 -- -.400 N/A -.45 -.49 0.06 N/A 0.0685 0.0708-0.99 4.9 Error, %.7 N/A.09 -- -0.7 N/A -.48 -- -- -- Figure 5: PDF o (let), (middle), and (right) 4. Bi-level Propagation Example In this section we demonstrate how the aorementioned techniques are used to estimate uncertainty propagation in multilevel systems. The probabilistic ormulation o the ATC process is used to solve a simple yet illustrative simulation example. An engine is considered at the system (top) level, which is decomposed into subsystems (bottom level) representing the piston-ring/cyliner-liner subassembly o the cylinders. Although an engine has multiple cylinders, they are all designed to be identical. For this reason, only one subsystem is considered. The system simulation predicts engine perormance in terms o brae-speciic uel consumption. The ring/liner subassembly simulation taes as inputs the surace roughness o the ring and the liner and the Young s modulus and hardness and computes power loss due to riction. The root mean square (MS) o asperity height is used to represent asperity roughness, which is assumed to be normally distributed with a coeicient o variation o 5-5%. The engine simulation taes then as input the power loss and computes brae-speciic uel consumption o the engine. Commercial sotware pacages were used to perorm the simulations. Detailed descriptions o the problem can be ound in [,4]. Since we have some inormation about the uncertainty associated with surace roughness o the ring and the liner, we begin at the bottom level. The bottom-level ATC problem is ormulated as 7
min µ X, µ, X x, x4 u ( E[ ] µ ) ( - ) -5 ( blow-by > 4.5 0 / ) - ( oil consumption > 5. 0 / ) s.t. P liner wear rate >.4 0 m / s P P g s P ( < µ m) ( > 0µ m) ( < µ m) ( > 0µ m) P g hr P P X P X P X P X 4 P P P P 40GPa x 80GPa 40BHV x 50BHV where = ( X, X, x, x ) 4 (0) and is solved irst assuming that u µ = 0. When the optimal solution is ound, uncertainty propagation associated with the random response must be estimated in order to solve the top-level problem, which is ormulated as min µ, ε s.t. ( E[ 0 ] T) l ( E[ ] ) where = ( ) 0 0 + ε µ ε () Once this problem is solved, we have an update or µ u, and we solve the bottom-level problem again. This process is repeated until the optimization variables in both problems don t change signiicantly anymore. It was ound that the response o the bottom-level simulation (power loss) is highly nonlinear, while the response o the top-level simulation (uel consumption) was almost linear. Thereore, we ocus on the results o the bottomlevel simulation, which are the more interesting anyway since it is the power loss that is propagated in the bi-level system (as the output o the bottom level simulation that constitutes the input o the top level simulation). The DBM method enables the generation o the highly nonlinear, multi-modal probability density unction (shown in Fig. 6). The histogram obtained using Monte Carlo simulation with one million samples conirms the accuracy o the DBMpredicted PDF. Note that the presented plots and numerical results are based on the obtained optimal ring/liner design. 0.05 0.045 0.04 0.05 0.0 0.05 0.0 0.05 0.0 0.005 0 0. 0.5 0. 0.5 0.4 0.45 0.5 Figure 6: DBM-generated PDF o power loss (let) and MCS histogram (right) 8
As shown in Table, although all methods predict the mean o power loss accurately, SS is highly inaccurate when estimating the standard deviation. PMI exhibits a smaller error, while DBM yields excellent prediction, due to the act that it generates distribution inormation. The PMI error is mostly due to the statistical nonlineariy o bimodal distribution. PMI requires two reliability analyses or each o the three measures, whereas DBM requires about reliability analyses to discretize the β-range in order to generate the probability density unction. Table : Numerical accuracy o moments or simulation-based example Mean Standard Deviation SS PMI DBM MCS SS PMI DBM MCS 4 0.950 0.90 0.9 0.9 0.048 0.097 0.00 0.0 Error, % 0.5-0. -0. - 55.0-4.6-0.45 - Table compares the optimal design o the ring/liner obtained using the three methods. It is clear that solving the probabilistic multilevel design problem using the error-prone SS method yields the wrong results, which highlights the importance o accurate estimation o uncertainty propagation. The results obtained using DBM or PMI are very similar or this example. Table : Comparison o ring/liner optimal design Design variable Initial SS optimum DBM optimum PMI optimum X : ring surace roughness, [µm] 5.000 4.64 4.000 4.000 X : liner surace roughness, [µm] 5.000 6.00 6.50 6. x : liner Young s modulus, [GPa] 00 80 80 80 x 4 : liner hardness, [BHV] 00 0.84 40 9.8 5. Discussion and Conclusions Two alternative techniques or estimating uncertainty propagation in probabilistic design o multilevel systems were presented in the paper. The irst method generates approximate probability density unctions, which are then integrated numerically to obtain statistical moments (distribution-based method, DBM). The second method uses numerical quadrature rules to estimate statistical moments o output response (perormance moment integration, PMI). The methods were successully applied to model the uncertainty propagation mechanism by estimating statistical moments eiciently and accurately. The scope o this paper was to investigate the stability, accuracy, and eiciency o these two methods when applied on simulation-based, multilevel system design optimization problems, and to determine which type o applications each method is best suited or. It was ound that both DBM and PMI estimate statistical moments accurately or nonlinear responses with high sewness and urtosis. Thus, PMI and DBM successully carried out the probabilistic design optimization o multilevel hierarchical system. The methods were compared to the oot Sum Square (SS) method and Monte Carlo simulation was perormed to compare the estimation o statistical moments. PMI and DBM are more accurate to assess statistical moments than SS. PMI can be useul or many nonlinear engineering systems, since it is computationally inexpensive yet accurate. On the other hand, DBM can be more accurate but is computationally more expensive. Finally, DBM should be used when the probabilistic design o multilevel systems requires generating distributions o nonlinear responses, i.e., when moments are not adequate to model propagation o uncertainties. Acnowledgments This research was partially supported by the Automotive esearch Center, a U.S. Army Center o Excellence in Modeling and Simulation o Ground Vehicles. This support is grateully acnowledged. eerences Kim, H. M., Michelena, N.F., Papalambros, P.Y., and Jiang, T., Target Cascading in Optimal System Design, ASME Journal o Mechanical Design, 5(): 474-480, 00. 9
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