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An ANOVA approach for accounting for life-cycle uncertainty reduction measures in RBDO: the FSAE brake pedal case study Jos Romero1 & Nestor V. Queipo2 Received: 4 September 2017 / Revised: 26 March 2018 /Accepted: 27 March 2018 /Published online: 12 April 2018 # Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract Accounting for uncertainty reduction measures (URMs) is critical to maximize the potential benefits of probabilistic design methods such as reliability-based design optimization (RBDO) and tackle the challenges in the design and construction of lightweight, high quality and reliable products. This work formulates and solves the RBDO of a Formula SAE (FSAE) brake pedal model with two failure modes (stress-Smax and buckling-fbuck) accounting for uncertainty reduction measures (URMs) throughout the product lifecycle while establishing the URMs global relative contributions to weight savings (expected value and variability) and computational expense. Given a set of URMs such as number of coupon tests, mesh refinement and manufacturing control, the solution approach includes: i) modeling structural analysis errors, ii) construction of surrogate models for the functions of interest, e.g., mass-M, Smax, fbuck and the corresponding error functions, iii) modeling pre-design and post-design URMs, such as material property density functions from coupon tests, and manufacturing tolerances (quality control), iv) solving the RBDO problems associated with each of the entries in a DOE with replication, and v) using ANOVA to compute main effects of most significant URMs on selected performance measures, i.e., mean and standard deviation of brake pedal mass, and computational expense. Results show that in the context of the brake pedal case study: the adoption of URMs led to reductions of up to 15 and 85% of mass mean and standard deviation, respectively, design and post-design URMs were responsible for 77 and 19% of the maximum mass reduction, respectively, and it was possible to set preliminary guidelines for URMs allocation and meet a particular performance objective under alternative URMs. Keywords Brake pedal . FSAE . Design under uncertainty . Reliability-based design optimization . Uncertainty reduction measures . Surrogate models . ANOVA Nomenclature ci Design parameter i Cov(.) Covariance function CT Computational time d Vector of random design variables di Random design variable i E Young's modulus f(.) Regression function in Kriging model fbuck(.) Buckling load factor function Gj(.) Limit state function for failure mode j G1(.) Limit state function of stress condition G2(.) Limit state function of buckling condition I(.) Indicator function p Vector of random design parameters M(.) Brake pedal mass function M^ : Random brake pedal mass function n Number of sample points nrv Number of random variables P(.) Probability of the statement within the braces to be true Pfsys Probability of system failure PfT Target probability of system failure R(.) Correlation function Sy Material yield strength Responsible Editor: W. H. Zhang * Nestor V. Queipo n..o@ica.luz.edu.ve Jos Romero j..o@ica.luz.edu.ve 1 Department of Mechanical Engineering, University of Zulia, Maracaibo, Venezuela 2 Applied Computing Institute, University of Zulia, Maracaibo, Venezuela Structural and Multidisciplinary Optimization (2018) 57:2109-2125 https://doi.org/10.1007/s00158-018-1983-6 Smax(.) Maximum von Mises stress function toli Manufacturing tolerance probability distribution of di x Vector of random variables, x = [d, p] or input variables vector xi Random variables or vector of the input variable at the ith sample point xA Lower [1] or higher [1] level correponding to number of coupon tests xB Lower [1] or higher [1] level correponding to the mesh refinement xC Lower [1] or higher [1] level correponding to the DOE size for the construction of surrogate model xD Lower [1] or higher [1] level correponding to the degree of accuracy of the selected surrogate model xE Lower [1] or higher [1] level correponding to the manufacturing quality control xURM Set of uncertainty reduction measures level combination xURM = {xA, xB, xC, xD, xE} yCT Response surface model of computational expense - total run time of RBDO yM Response surface model of the mean of the brake pedal mass yM Response surface model of the standard deviation of the brake pedal mass ^y : Surrogate prediction function Z(.) Random function with mean zero, and nonzero covariance 0, i , ii,ij Polynomial regression coefficients mb Buckling load factor surrogate model error mM Mass surrogate model error mS Maximum von Mises stress surrogate model error rb Buckling prediction error due to the mesh refinement level rS Maximum von Mises stress prediction error due to the mesh refinement level d Vector of mean values of random design variables M Mass mean value p Vector of mean values of random design parameters Poisson's ratio 2 Process variance in Kriging model M Mass standard deviation value 1 Introduction Accounting for uncertainty reduction measures (URMs) throughout products life cycle is critical to maximize the benefits (lighters and more reliable products) of adopting probabilistic methods such as RBDO in the automotive (Acar and Solanki 2009), aeronautic (Venter and Scotti 2012) and aerospace industries (Villanueva et al. 2014). Common URMs are, for example, manufacturing quality control-material, material selection quality control, coupon tests (predesign), improved structural analysis and failure modeling (design), manufacturing quality control-elements, element tests, inspection, health monitoring and maintanance (post-design). On the other hand, probabilistic designs often show significant gains (weight reduction) for a given probability of failure with deterministic designs offering excessive protection (deficient risk allocation) with respect to alternative failure modes (Acar and Haftka 2007; Villanueva et al. 2009; Venter and Scotti 2012; Suzuki and Haftka 2014; Romero and Queipo 2017). Reaping the benefits of accounting for URMs throughout product life-cycles can provide an additional and definitive incentive to the widespread adoption of probabilistic approaches. Significant progress for adopting URMs in probabilistic design has been made, in particular, providing quantitative evidence of the benefits of adopting URMs to weight reductions/costs throughout product life-cycles. For example, Qu et al. (2003) in the RBDO of composite laminates for cryogenic environments (analytical model) achieved a reduction of 25% in the laminate thickness through the quality control to 2 of the transverse failure strain instead of reducing the variability of tensile strain (predesign). Using the above-referenced case study, Acar et al. (2007) demonstrated that incorporating quality control to 2 of the transverse failure strain and reducing errors in structural analysis from 20 to 10% (design) isolately and jointly may lead to weight reductions of 12, 20, and 36%, respectively. Acar et al. (2006) in the Reliability based design (RBD) of a sandwich structure (analytical model) reported that an improved model for fracture toughness (design) might increase by 26.5% the allowable flight loads in aircrafts. In addition, Kale et al. (2008) in the RBDO of a fuselage panel with an analytical model for fatigue crack growth show that inspection-type sequence and time (post-design) can lead to about 25% reduction in its lifetime cost. Furthermore, Acar and Solanki (2009) in the RBDO of a 1996 Dodge Neon structure (numerical model) considering full frontal and side impacts demonstrated that improving material selection quality control (predesign) and reducing empirically model errors in structural analysis (design) may lead to 3.3 and 4.2% weight reductions, respectively. Acar et al. (2010) in the RBDO of a single aircraft component (analytical model) showed that increasing the number of coupon tests from 50 to 80 (predesign) or element tests from 3 to 5 (postdesign) depending on empirically estimated structural analysis error (5% vs. 10%) may result, respectively, in weigh reductions of 0.13-0.33% (5%) or 0.24-0.45% (10%); using a 2110 J. Romero, N. Queipo Bayesian formulation, element tests allow for a redesign with an updated failure stress. Using RBDO, Acar (2011) minimizes lifetime cost of wing and tail structures (analytical model) and found the optimal values for the number of coupon tests (predesign), knockdown factor (design), and number of element tests (postdesign) to be equal to (72, 0.9711, 4)-wing, and (76, 0.8963, 3)-tail, respectively. Venter and Scotti (2012) in the RBDO of a stepped cantilevered beam with three segments (analytical model) show that including the results of proof/acceptance tests (postdesign) during the design process can result in an 8% weight reduction. Villanueva et al. (2014) in the RBDO of an integrated thermal protection system (numerical model) found that empirically reducing errors in structural analysis from 12 to 6% (design) may lead to a weight reduction of 1.9%; when the probability of redesign is set to 0.5 (postdesign) the weight reduction was substantially lower (0.2%). Previous works, however, have been limited to: analytical models (except for Acar and Solanki 2009 and Villanueva et al. 2014), empirically estimated structural analysis errors, and, specially, one-at-a-time local sensitivity analysis/2D response surface analysis that make it difficult to establish URMs trade-off and allocation strategies (shortcomings). This work formulates and solves the RBDO of a Formula SAE (FSAE) brake pedal model with two failure modes (stress and buckling) accounting for uncertainty reduction measures (URMs) throughout the product lifecycle such as number of coupon tests, mesh refinement and manufacturing control, among others, and, through an ANOVA approach, establishes response surface models with the URMs global relative contributions to weight savings (expected value and variability) and computational expense that can help the above-referenced shortcomings. The RBDO of a brake pedal of FSAE vehicles represents: i) an easy to understand structural problem among engineering students and practitioners around the world yet relevant to aerospace and automotive industrial environments, and ii) an opportunity to illustrate an approach to quantitatively assess the weight savings potential of the adoption of URMs throughout a product life-cycle. The Formula SAE project (SAE International 2016) with an annual participation of 2470 students from 120 university teams, promotes careers and excellence in engineering in the automotive industry; in particular, each component in a single seater FSAE vehicle must be safe and lightweight in order to decrease the vehicle mass, and take the most advantage of the engine power. The following sections are entitled as: 2) RBDO accounting for URMs - problem formulation, 3) 3D numerical modeling under uncertainty for stress and buckling analysis, 4) product lifecycle URMs and design of experiments with replication for analysis of variance, 5) accounting for URMs throughout the product life-cycle in RBDO - solution methodology, and 6) results and discussion. 2 RBDO accounting for URMs - problem formulation Given a FSAE brake pedal model with a set of design variables and geometric parameters (e.g., Fig. 1), the problem of interest can be formulated as follows: Find d d1; d2; ...; d8 that Min M d; p; xURM 1 Subject to: Pfsys P G 1 d; p; xURM 0 G2 d; p; xURM 0 PfT dL;i di dU;i; i 1; 2; ...; 8 2 where superscripts L and U denote lower and upper bounds, respectively, and: G1 d; p; xURM Sy xA Smax d; p; xB; xC; xD; xE 1 rS xB ... 1 mS xB; xC; xD 3 G2 d; p; xURM f buck d; p; xURM 1 rb xB 1 mb xB; xC; xD 1 4 M d; p; xURM M d; p; xC; xD; xE 5 Note that URMs are incorporated in the RBDO formulation through the variable xURM. For example, an instance of xURM is (xA, xB, xC, xD, xE) =(1,1,1,1,1) representing an scenario with the highest levels of uncertainty reduction for the adopted measures. 3 Brake-pedal 3D numerical model under uncertainty The parametrization of the FSAE brake pedal geometrical model includes a set of 8 random design variables and 1 random parameter (Fig. 1 and Table 1) and corresponds to a non-optimized reduced configuration as reported by Romero and Queipo (2017). The design variable bounds allowing a significant number of alternative shapes are shown in Table 1, while deterministic geometric parameters and probability distributions of material property values are reported in Tables 2 and 3, respectively. The material used in the brake pedal design was Aluminum 6063-T5 with probability distributions for selected mechanical properties identified (Fig. 2) by Barroso (2015) from experimental data sets (ASTM 2007) and their suitableness confirmed using the Anderson-Darling goodness-of-fit test (D'Agostino 1986). An ANOVA approach for accounting for life-cycle uncertainty reduction measures in RBDO: the FSAE brake... 2111 3.1 Assumptions, load and boundary conditions The assumptions are: material is homogeneous and isotropic, maximum load uniformly distributed and gradually applied (no impact), finite element (FE) stress analysis was static and linear, and FE buckling analysis was nonlinear with enabled large deflections. The brake pedal shall withstand a force of 2000 N without any failure of the brake system or pedal box (SAE International 2016), with pin and planar boundary conditions as shown in Fig. 3. Note that the domain of the brake pedal numerical model must be complete (Fig. 3) to perform a correct buckling analysis, that is, a reduction of the computational time by the application of the symmetry principle is not an option. 4 Brake-pedal lifecycle URMs and design of experiments with replication for analysis of variance The URMs under consideration are: number of coupon tests to determine mechanical properties (predesign), finite element mesh refinement, design of experiments (DOE) size for the construction of surrogate models and degree of accuracy of the selected surrogate model (design), and manufacturing quality control (post-design)); note that since no real element tests (post-design) were available the uncertainty in the model to predict failure was no accounting for. All URMs under consideration have associated two potential values (levels) as presented in Table 4. & Number of coupon tests (A): The low and high levels were defined as below and above the minimum sample size values [25-30] frequently use to reasonable represent univariate normal distributions. Fig. 1 FSAE brake pedal geometrical model with design variables and parameters Table 1 FSAE brake pedal probabilistic design variables and parameter Design variable Bounds (mm) Manufacturing tolerance (mm) lower upper coarse fine d1 27 33 U(0.8,0.8) U(0.15,0.15) d2 70 80 U(0.8,0.8) U(0.15,0.15) d3 70 87 U(0.8,0.8) U(0.15,0.15) d4 2 4 U(0.3,0.3) U(0.05,0.05) d5 8 18 U(0.5,0.5) U(0.1,0.1) d6 80 115 U(0.8,0.8) U(0.15,0.15) d7 2 8 U(0.5,0.5) U(0.1,0.1) d8 2 6 U(0.3,0.3) U(0.05,0.05) c23 d8 U(0.3,0.3) U(0.05,0.05) 2112 J. Romero, N. Queipo & Mesh refinement (B): The low and high levels correspond to the finite element maximum length specified as default by the structural analysis package (ANSYS Workbench) and a compromise solution between accuracy and computational cost, respectively. & DOE size for the construction of surrogate models (C): The sample size for the low level was established as 20% higher than the minimum recommended by Jones et al. (1998), while the sample size for the high level was set to five times the low one. & Degree of accuracy of the selected surrogate model (D): Polinomial regression-PRG (approximating surrogate) and Kriging-KRG (interpolating surrogate) have associated the low and high level for this factor, respectively. & Manufacturing quality control (E): The lower and higher levels were represented as coarse and fine manufacturing tolerances in the geometrical brake pedal model (Table 1), respectively, as defined in the ISO 2768-1 standard (ISO 1989). Given the fact that all URMs have been specified with two levels (low and high) and the stochastic nature of the dependent variables of interest, a factorial 2k design of experiments-DOE with replication is considered, with k being the number of URMs. Yates' method (Montgomery 2013) was used for defining the arrangement of URMs level combinations in a factorial 25 DOE (Table 6 - Appendix). Each experiment was replicated giving a total of 64 RBDO runs. 5 Accounting for URMs throughout the product life-cycle in RBDO - solution approach In the above-referenced Yates' arrangement (DOE), each experiment represents a RBDO problem accounting for URMs with the performance measures, namely, mass mean value, mass standard deviation value and computational expense - total run time of RBDO calculated as described in Sections (steps) 5.1-5.5. In Section 5.6, an analysis of variance is conducted to quantify the effects of the URM factors and their interactions and their statistical significance in linear models of the performance measures. Please note that since the ANOVA approach requires a DOE replication, steps 5.4 and 5.5 are executed twice. Figure 4 depicts the proposed approach. Table 2 FSAE brake pedal deterministic design parameters Design parameter Value c1 12.7 mm c2 15.875 mm c3 10 mm c4 37.32 mm c5 10 mm c6 48 mm c7 10 mm c8 10 mm c9 75 c10 75 c11 50 mm c12 200 mm c13 3.175 mm c14 3.175 mm c15 67 mm c16 30 mm c17 2 mm c18 60 mm c19 36 mm c20 6 mm c21 4 mm c22 24 mm c24 5 mm c25 12 mm c26 5 mm c27 7.5 mm c28 13 mm c29 4 mm c30 67 mm c31 3 mm c32 67 mm c33 3 mm c34 2 mm Table 3 Al 6063-T5 - Material properties probability distributions Material properties Unit Probability distribution Mean value Characteristic value (5%) Density () Kg/m3 - 2700 - Young's modulus (E) MPa LN(11.059, 0.0709) 63,657.5 55,805.866 Yield strength (Sy) MPa N(209.69,9.002) 209.69 194.883 Poisson's ratio () - - 0.33 - An ANOVA approach for accounting for life-cycle uncertainty reduction measures in RBDO: the FSAE brake... 2113 5.1 Model the errors due to the mesh refinement level (B) in the finite element calculations of von Mises stress and buckling load factor It requires the following steps: i) Generate a sample for the random variables (design and parameters) in the brake pedal case study using a latin hypercube experimental design (LHS; McKay et al. 1979). The number of random variables is ten (10) and the DOE includes 100 samples; the samples can be obtained using the lhsdesign command in MATLAB. ii) Conduct numerical simulations (output) via the ANSYS Workbench using each sample (input) from the previous step and obtain the corresponding von Mises stress (Smax) and buckling load factor (fbuck) considering meshes with finite element maximum length equal to 1, 2 and 7 mm as specified in Section 4. iii) Using the input/output pairs in the previous step calculate error values due to the mesh refinement level (r) using the following expression: r;i yi;1mmyi;selec yi;selec 6 where y is the output value of interest (Smax o fbuck) and the subindices i, 1 mm, and selec identify the sample under consideration, through its number i, and the mesh used which can be of 1 mm or either 2 mm or 7 mm (select) finite element maximum length. iv) The error distribution for each of the mesh refinement levels (7 mm-low and 2 mm-high) was identified (maximum likelihood estimation) using the fitdist (data, distname) command in MATLAB with distname being the probability distribution (normal, lognormal, Weibull, Gamma) that best approximated the available data according to the Kolmogorov-Smirnov goodness of fit test (kstest command in MATLAB). 5.2 Model surrogate prediction errors It involves the following steps: i) Generate a latin hypercube design (LHS) for each of the two levels (1,+1) of DOE size (120, 600) under consideration for the construction of the surrogate models; the LHS can be implemented by, for example, the lhsdesign command of MATLAB. ii) Compute the mass M and, Smax and fbuck values (output) for each sample in the DOE identified in the previous step, considering the two alternative levels of mesh refinement; the outputs of interest were Fig. 2 Histogram of mechanical properties, a Young's modulus (E), b yield strength (Sy) Fig. 3 Structural analysis domain with applied load and boundary conditions of the FSAE brake pedal numerical model 2114 J. Romero, N. Queipo Yes Yes Solve the RBDO problem , , x Subject to: d d Model the mechanical properties error and variability , Model manufacturing tolerance Provide DOE for 2 level URMs (25 ) x x x x x x Model FE analysis error due to the mesh refinement level Model surrogate prediction error End Store the solution for RBDO problem corresponding to ith row of Z and j replication *Value of optimal solution 5.1 5.2 5.3 5.4 5.5 Conduct analysis of variance (ANOVA) on the performance measures No No 5.6 Fig. 4 FSAE Brake pedal RBDO accounting for URMs - solution methodology Table 4 Values for low and high levels of the URMs (factors) Stage of product life URM Level Low (1) High (+1) Predesign Number of coupon tests (A) 10 50 Design Improved structural analysis Mesh refinement-finite element maximum length (B) 7 mm 2 mm DOE size for the construction of surrogate models (C) 120 600 Degree of accuracy of the selected surrogate model (D) PRG KRG Post-design Manufacturing quality control (E) Coarse Fine An ANOVA approach for accounting for life-cycle uncertainty reduction measures in RBDO: the FSAE brake... 2115 determined using the mass property and FE analyses in the ANSYS Workbench, respectively. iii) For each level of mesh refinement and DOE size, using the input/output pairs identified in the previous step, construct surrogate models for M, Smax and fbuck based on Polynomial Regression and Kriging. A brief description of each of the surrogate models under consideration is next. A. Polynomial regression (PRG) In this work the regression models (Draper and Smith 1969) considered are of linear (L), ^y x 0 nrv i1ixi 7 and quadratic forms (Q): ^y x 0 nrv i1ixi nrv i1iixi 2 nrv1 i1 nrv ji1ijxixj 8 where nrv is the total number of random variables, xi, and the parameters 0, i, ii and ij are estimated by the least-squares method implemented in MATLAB (regress command). B. Kriging model (KRG) A Kriging model (Sacks et al. 1989) postulates a combination of a polynomial model f(x) and an error model of the form, Z(x) as: ^y x f x Z x 9 where ^y x is the unknown function of interest, f(x) is a regression function of x and Z(x) is a random function (stochastic process) with mean zero, and nonzero covariance given by the following expression: Cov Z xi; xj 2 R xi; xj 10 Where, 2 is the process variance and R is the correlation function. In this work, consideration is given to constant (0) and linear (1) regression functions, and Gaussian (G), exponential (E), linear (L) and spherical (S) correlation functions. The parameters of correlation and regression functions are identified using a maximum likelihood estimator. The Kriging models were implemented using the Design and Analysis Computational Experiments (DACE) toolbox developed by Lophaven et al. (2002) and written in MATLAB. iv) Using leave one-out cross-validation, prediction errors, such as percentage maximum absolute error (MAXE %), percentage root mean square error (RMSE%), the coefficient of determination (R2 ) and errors m (11) were estimated for each of the surrogate-models under consideration. m x B;xC;xD ysimysurr ysurr 11 where m represents a vector of surrogate model prediction errors, ysurr is the surrogate model predictions of M, Smax or fbuck and ysim is the vector of corresponding values obtained through finite element analysis (output). v) The error distribution for each combination of surrogate model and mesh refinement level was modeled as a non parametric probability distribution with a Gaussian kernel using the fitdist command in MATLAB. 5.3 Model manufacturing tolerance The manufacturing tolerances (Table 1) are modeled as uniform distributions following the principle of maximum entropy. Note that the error distributions of finite element mesh refinement (5.1), surrogate prediction (5.2) and manufacturing tolerance (5.3) are estimated only once (outside the ANOVA loop). With respect to URMs /statistical analysis, the ANOVA loop only includes an inexpensive operation (predesign) - taking samples of statistical distributions of mechanical properties (yield strength and Young's modulus). 5.4 Model mechanical properties error and variability The estimated distribution of mechanical properties from coupon tests is called the posible true distribution (PTD) of the parameter, this essentially becomes a distribution of distributions and it is modeled as a Normal distribution ^c;true (Park et al. 2014): ^c;trueN ^c;true; ^c;true 12 where ^c;true and ^c;true are, respectively, the mean and standard deviation of ^c;true and can be estimated as: ^c;trueN c;test; c;test ffiffiffiffi nc p 13 ^c;true c;test ffiffiffiffiffiffiffiffiffi nc1 p nc1 14 2116 J. Romero, N. Queipo where c, test and c, test are, respectively, the mean and standard deviation of mechanical properties, i.e., yield strength or Young's modulus, in nc coupons, (nc 1) is a chi-distribution of order nc 1 (Evans et al. 1993). Note that ^c;true and ^c;true depend on the numbers of coupon tests, hence with an infinite number of coupons, ^c;true will become a deterministic value, i.e., no sampling error. The PTD of the material properties can be obtained using a double-loop Monte Carlo simulation (MCS). The outer loop generates N samples of the two distribution parameters, from which N pairs of Normal distributions N(i, i ), can be defined. In the inner loop, M samples of material property values are generated from each N(i, i ). Then, all NxM samples are used to obtain the PTD of material property, which includes both material variability and sampling errors. The values of N and M were select such as 100,000 samples are obtained. 5.5 Solve the RBDO problem The brake pedal RBDO problem (1-2) is solved using sequential quadratic programming (SQP; Powell 1978) as implemented in the MATLAB fmincon command with randomly selected starting points (50). The probability of failure for failure mode j was estimated using Monte Carlo simulations (MCS; Rubinstein 1981) as: P Fj 1 n n i1I Gj xi < 0 15 where, xi is a vector of the input variable at the ith sample point, n is the total number of samples, Gj(xi) is the limit state function for failure mode j, and I is the indicator function, which equals 1 if the condition is true and 0 if the condition is false. The target probability of failure was assumed as PfT = 103 considering an acceptable design and a medium severity level (Goh et al. 2009), and the system probability of failure was obtained with 105 MC simulation cycles. Since two failure modes (stress and buckling condition) are accounted for, a system reliability approach is necessary for estimating the probability of failure for the brake pedal design under consideration. In this case, system failure occurs when any of the failure modes is active, hence it is referred to as a series system (or a weak link system) and defined as the probability of the union of the individual failure modes (Haldar and Mahadevan 2000) as shown below: P F P G 1 d; p 0 G2 d; p 0 P G 1 d; p 0 P G 2 d; p 0 P G 1 d; p 0 G2 d; p 0 16 Once obtained the RBDO solutions, the probabilities for the failure modes and the system failure are evaluated with 106 MC simulation cycles and the higher level of number of coupon tests (xA = 1) to ensure the accuracy of the results. Note that the number of Monte Carlo simulations was in the order of 105 -106 and found to be enough to obtain accurate results with an affordable computational expense. 140 142 144 146 148 150 152 154 156 158 160 120 600 M (g) DOE size (C) =9.323 g 140 142 144 146 148 150 152 154 156 158 160 7 mm 2 mm M (g) Finite element maximum length (B) =5.546 g 140 142 144 146 148 150 152 154 156 158 160 PRG KRG M (g) Surrogate model (D) =5.390 g 140 142 144 146 148 150 152 154 156 158 160 coarse fine M (g) Manufacturing quality control (E) =4.842 g Fig. 5 URMs main effects in the mean of brake pedal mass of (C), (B), (D), (E); these effects represent a reduction of 15.52% (25.101 g) with respect to the scenario where none of the URMs are adopted (level 1) An ANOVA approach for accounting for life-cycle uncertainty reduction measures in RBDO: the FSAE brake... 2117 In this work, unbiased estimators of the mean value, and standard deviation of the random brake pedal mass function M^ x were obtained using the MCS method with 105 simulation cycles (Papadrakakis and Lagaros 2002): M 1 n n i1M^ xi 17 M ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 n1 n i1 M^ xi M 2 r 18 with xi being vector of the input variable at the ith sample point, n is the total number of samples, M^ xi M xi 1 mMi , and M(x) and mM denoting the mass surrogate model, and surrogate error probability distribution, respectively. 5.6 Conduct analysis of variance (ANOVA) on the performance measures Assuming a linear model for the performance measures (M, M, CT) as a function of the URM factors and their interactions, the ANOVA allows to: i) identify linear models for the performance measures with statistically significant coefficients, ii) establish the relative contribution of the URM factors and interactions on the performance measures total variability, and iii) compute main effects, i.e., the average changes in the performance measures when each URM (control factor) is changed from the low setting (1) to the high setting (+1). The ANOVA is performed through the following steps, for each of the URM factors and their interactions: i) compute the effects and sum of squares using the Yates method (Montgomery 2013), ii) calculate the mean squares as the sum of squares divided by the Fig. 7 Critical and alternative buckling load factors interference models; the alternative buckling load factors correspond to the brake pedal RBDO with URMs at the lowest (1) and highest (+1) levels Fig. 6 Stress-strength interference models for the brake pedal RBDO with URMs at the lowest (1) and highest (+1) levels 2118 J. Romero, N. Queipo corresponding number of degrees of freedom, iii) determine the F-statistic dividing each of the mean of squares of factors and interactions by the so called mean of squares error, and iv) compute the p-value for the calculated F-statistic; a factor or interaction is considered statistically significant if its p-value is lower than a given threshold, usually set as = 0.05. The sum of squares error represents the sum of squares that is not accounted for the linear model with degrees of freedom equal to the difference between the total degrees of freedom (2.25 -1) and the sum of the degrees of freedoms of factors and interactions. On the other hand, the p-value is defined as the probability of observing a value greater than the calculated F-statistic assuming it follows a F probability distribution with primary and secondary parameters equal to the factor and error degrees of freedom, respectively. As an example, Table 7 (Appendix) presents the ANOVA corresponding to the mean of the brake pedal mass in a replicated 25 Yates's full factorial DOE. 6 Results and discussion This section shows the results of adopting the proposed ANOVA approach on the RBDO of the FSAE brake pedal case study. Specifically, it includes: URMs effects in the mean and standard deviation of brake pedal mass, computational expense - RBDO total run time and guidelines for allocating URMs and conducting trade-off analysis. 6.1 URMs effects in the mean of brake pedal mass The response surface model yM (g) identified for this performance measure is: yM 2:773xB4:662xC2:695xD2:421xE0:617xAxC 0:478xBxC0:873xBxD 1:227xCxD 0:620xBxCxE0:565xCxDxE 0:52xAxBxCxDxE 150:476 19 The terms in yM represent statistically significant (pvalue < 0.05) URMs and their interactions; the model exhibited an R2 = 0.979, adjusted R2 = 0.921, MAXE% = 1.65 and RMSE% = 0.664. The URMs DOE size for the construction of surrogate models (C), mesh refinement-finite element maximum length (B), degree of accuracy of the selected surrogate models (D) and manufacturing quality control (E) and (C) (D) interaction account for 92% of the total variability; more precisely, the contributions are 44.375, 15.700, 14.830, 11.969 and 3.071%, respectively. On the other hand, the main effects of the cited URMs-(C), (B), (D) and (E), i.e., the average changes in the mean of brake pedal mass when each URM (control factor) is changed from the low setting (1) to the high setting (+1) are 9.323, 5.546, 5.390 and 4.842 g, respectively, which account for 95.71% of the total mass mean reduction when all URMs are adopted (Fig. 5). The adoption of all URMs led to a reduction of 26.226 g representing approximately 15% of the mean of the mass obtained in the scenario where none of the URMs were adopted. Note that C, B, D and E URMs at their highest levels all are related although in different ways with the construction of more accurate surrogate models (predictions) for the maximum von Mises stress and buckling load factor. This translates into density functions for the former (Fig. 6) and the latter (Fig. 7) closer to their critical values and hence less conservative designs. 0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000 -1 +1 M (g) (C)(D)(E) =0.214 g Fig. 9 Second most significant term (URMs), i.e., interaction (C)(D)(E), in the reduction of standard deviation of brake pedal mass 0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000 coarse fine M (g) Manufacturing quality control =3.568 g Fig. 8 URMs main effect in standard deviation of brake pedal mass An ANOVA approach for accounting for life-cycle uncertainty reduction measures in RBDO: the FSAE brake... 2119 6.2 URMs effects in the standard deviation of brake pedal mass The response surface model yM (g) identified for this performance measure is: yM 0:044xB0:093xC1:784xE0:045xCxE 0:071xDxE0:107xCxDxE 2:732 20 As previously stated, the terms in yM represent statistically significant (pvalue < 0.05) URMs and their interactions; the model exhibited an R2 = 0.998, adjusted R2 = 0.997, MAXE % = 8.333 and RMSE% = 3.187. The URM manufacturing quality control (E) account for 98.586% of the total variability of the standard deviation of brake pedal mass. On the other hand, the main effect for the cited URM, i.e., the average change in the standard deviation of brake pedal mass when (E) is changed from the low setting (1) to the high setting (+1) is 3.568 g, which represents 87.97% of the reduction obtained when all URMs are adopted (Fig. 8); the second most significant term correspons to the interaction (C)(D)(E) which accounts for 5.28% of the total possible mass uncertainty reduction (Fig. 9). The adoption of all URMs led to a reduction of approximately 85% of the standard deviation of the mass obtained in the scenario where none of the URMs were adopted. Note that E (level +1) is associated with better manufacturing quality control and consequently a less uncertain mass density function (Fig. 10). 6.3 URMs effects in computational expense - RBDO total run time The response surface model yCT (s) identified for this performance measure is: yCT 40816:879xC 63829:55xD 40694:094xCxD 66482:251 21 The terms in yCT represent statistically significant (pvalue < 0.05) URMs and their interactions; the model exhibited an R2 = 0.995, adjusted R2 = 0.994, MAXE% = 11.701 and RMSE % = 4.376. The URMs, degree of accuracy of the selected surrogate models (D), DOE size for the construction of surrogate models (C) and (C)(D) interaction account for 99% of the total variability of computational expense; more precisely, the contributions are 54.678, 22.359, and 22.225%, respectively. On the other hand, the main effects for the cited URMs (D) and (C) are 127,659 and 81,634 s, respectively, which account for 100% of the difference between the computational expense when all (level +1) and none (level 1) of the URMs are adopted (Fig. 11). The computational expense of alternative URMs can exhibit differences of order of magnitude. Note that D and C at the highest levels all are related with the construction of more accurate surrogate models (predictions) for the contraints and objective function with the corresponding high computational cost of RBDO runs. Fig. 10 Mass probability density functions for low (1) and high (+1) levels of the manufacturing quality control (E) with all other URMs at their highest levels 0 20000 40000 60000 80000 100000 120000 140000 120 600 CT (s) Finite element maximum length (C) =81633 s 0 20000 40000 60000 80000 100000 120000 140000 PRG KRG CT (s) Surrogate model (D) =127659 s Fig. 11 URMs main effects in computational expense - total run time of RBDO 2120 J. Romero, N. Queipo 6.4 Guidelines for URMs trade-off and allocation strategies Table 8 (Appendix) shows a summary of all posible combinations of URMs and the corresponding values for the performance measures, namely, mean and standard deviation of brake pedal mass and computational expense-RBDO total run time. Hence, for example, the maximum (163.985 g) and minimum (137.759 g) for the mean of brake pedal mass are obtained using the levels xURM = [+1, 1, 1, 1, 1] and xURM = [+1, +1, +1, +1, +1], respectively. Note that it is possible to obtain similar values for the performance measures using alternative URMs; for instance, the set of URMs xURM = [+1, +1, +1, 1, +1], xURM = [+1, +1, +1, +1, 1] and xURM = [+1, 1, +1, +1 + 1] all lead to similar values for the mean of the brake pedal mass but with significant differences in the standard deviation of the brake pedal mass and computational expense. On the other hand, the number of coupon tests (A) under consideration did not influence any of the performance measures. Hence, the URMs should be selected depending on the particular design objectives, that is weight savings, improve manufacturing quality, reduce computational or manufacturing costs (Table 5). If, for instance, reducing the mean of the mass of the structural element is of interest, the maximum finite element length and DOE size (surrogate modeling) should be the minimum and greatest possible, respectively. In practice, tough, the URMs adopted will typically represent a compromise solution that fit the cost and time constraints of industrial environments. 7 Conclusions This works presents an ANOVA approach for quantitatively assess the impact of uncertainty reduction measures (URMs) during the pre-design, design- and post-design stages (life-cycle) of a structural element that can lead to allocation and trade-off strategies; its effectiveness was demonstrated using a the RBDO design of a FSAE brake pedal with two failure modes (stress and buckling). The URMs included the number of coupon tests to determine mechanical properties (predesign), finite element mesh refinement, design of experiments (DOE) size for the construction of surrogate models and degree of accuracy of the selected surrogate model (design), and manufacturing quality control (postdesign). While the proposed procedure can be computationally expensive its application is critical for understanding the effects of URMs and in budget/resource allocations and worthwhile considering the significant weight reductions that can be achieved. The URMs with the highest impact on the reduction of the mean of the brake pedal mass were the DOE size for the construction of surrogate models, mesh refinement-finite element maximum length, degree of accuracy of the selected surrogate models (design) and manufacturing quality control (post-design); these URMs are responsible for 77% (design) and 19% (post-design) of the maximum mass reduction. On the other hand, reductions of the standard deviation of brake pedal mass can be almost uniquely attributed to the manufacturing quality control (post-design) URM, and changes in the computational expense are related to selected design URMs, i.e., accuracy of the selected surrogate models, DOE size for the construction of surrogate models.. The results of the quantitative assessment of the adoption of URMs in RBDO also show that given a particular design objective, e.g., weight savings i) there are alternative URMs that could fit the purpose but these alternative will surely differ in other criteria such as computational expense and/or manufacturing cost (trade-off strategies), and ii) it was possible to set preliminary guidelines for URMs allocation througout a product life-cycle stages. Future research should evaluate the proposed under a broader scope of URMs and structural elements so that, more general guidelines for URMs allocation are made available to researchers and practitioners and the potential benefits of adopting probabilistic design methods can be maximized (hence promoting their widespread use). Table 5 Preliminary URMs allocation guidelines for RBDO surrogate-based probabilistic designs Design objectives URMs allocation Weight savings Low Weight variability Low computational expense Low manufacturing cost Mesh refinement DOE size Model accuracy Manufacturing quality control + ++ + + ++ + + + - + + + - - + + - + + + - - + ++ - An ANOVA approach for accounting for life-cycle uncertainty reduction measures in RBDO: the FSAE brake... 2121 Appendix Table 6 Yates' arrangement for the factorial (25 ) design of the URMs in the FSAE brake pedal case study Exp. Yates A B C D E 1 (1) 1 1 1 1 1 2 a +1 1 1 1 1 3 b 1 +1 1 1 1 4 ab +1 +1 1 1 1 5 c 1 1 +1 1 1 6 ac +1 1 +1 1 1 7 bc 1 +1 +1 1 1 8 abc +1 +1 +1 1 1 9 d 1 1 1 +1 1 10 ad +1 1 1 +1 1 11 bd 1 +1 1 +1 1 12 abd +1 +1 1 +1 1 13 cd 1 1 +1 +1 1 14 acd +1 1 +1 +1 1 15 bcd 1 +1 +1 +1 1 16 abcd +1 +1 +1 +1 1 17 e 1 1 1 1 +1 18 ae +1 1 1 1 +1 19 be 1 +1 1 1 +1 20 abe +1 +1 1 1 +1 21 ce 1 1 +1 1 +1 22 ace +1 1 +1 1 +1 23 bce 1 +1 +1 1 +1 24 abce +1 +1 +1 1 +1 25 de 1 1 1 +1 +1 26 ade +1 1 1 +1 +1 27 bde 1 +1 1 +1 +1 28 abde +1 +1 1 +1 +1 29 cde 1 1 +1 +1 +1 30 acde +1 1 +1 +1 +1 31 bcde 1 +1 +1 +1 +1 32 abcde +1 +1 +1 +1 +1 2122 J. Romero, N. Queipo Table 7 Analysis of variance for the mean of the brake pedal mass; pvalue < 0.05 in bold Source of variation Sum of squares Degrees of freedom Mean squares F-statistic pvalue A 2.942 1 2.942 0.918 3.452E-01 B 492.063 1 492.063 153.514 9.394E-14 C 1390.774 1 1390.774 433.893 3.551E-20 D 464.812 1 464.812 145.012 1.999E-13 E 375.138 1 375.138 117.036 3.197E-12 AB 3.742 1 3.742 1.168 2.880E-01 AC 24.369 1 24.369 7.603 9.548E-03 AD 4.589 1 4.589 1.432 2.403E-01 AE 1.171 1 1.171 0.365 5.499E-01 BC 14.633 1 14.633 4.565 4.037E-02 BD 48.698 1 48.698 15.193 4.663E-04 BE 5.127 1 5.127 1.599 2.151E-01 CD 96.260 1 96.260 30.031 4.914E-06 CE 0.097 1 0.097 0.030 8.632E-01 DE 4.340 1 4.340 1.354 2.532E-01 ABC 0.305 1 0.305 0.095 7.598E-01 ABD 0.050 1 0.050 0.016 9.013E-01 ABE 2.494 1 2.494 0.778 3.843E-01 ACD 2.881 1 2.881 0.899 3.502E-01 ACE 4.854 1 4.854 1.514 2.275E-01 ADE 0.016 1 0.016 0.005 9.437E-01 BCD 2.563 1 2.563 0.799 3.779E-01 BCE 24.648 1 24.648 7.690 9.186E-03 BDE 9.760 1 9.760 3.045 9.058E-02 CDE 20.409 1 20.409 6.367 1.679E-02 ABCD 3.734 1 3.734 1.165 2.885E-01 ABCE 4.085 1 4.085 1.274 2.673E-01 ABDE 4.562 1 4.562 1.423 2.416E-01 ACDE 4.665 1 4.665 1.455 2.365E-01 BCDE 0.499 1 0.499 0.156 6.957E-01 ABCDE 17.320 1 17.320 5.404 2.660E-02 Error 102.571 32 3.205 Total 3134.170 63 An ANOVA approach for accounting for life-cycle uncertainty reduction measures in RBDO: the FSAE brake... 2123 References Acar E (2011) Reliability-based structural design of a representative wing and tail system together with structural tests. 52ndAIAA/ASME/ ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, USA, AIAA 2011-1761 Acar E, Haftka RT (2007) Reliability-based aircraft structural design pays, even with limited statistical data. 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Biom J 11(6): 427-427 Table 8 For all posible combinations of URMs, expected values for the performance measures and percentage reduction with respect to the adoption of URMs worst scenario; italic data are associated with high levels (+1) for the URMs URM levels Values for performance measures (percentage reduction with respect to the URMs worst scenario) xA xB xC xD xE M g (%) M g (%) CT s (%) +1 +1 +1 +1 +1 137.759 (15.99) 0.73 (84.75) 211,823 (0.00) 1 +1 +1 +1 +1 137.953 (15.87) 0.73 (84.75) 211,823 (0.00) +1 +1 +1 +1 1 141.451 (13.74) 4.46 (6.81) 211,823 (0.00) +1 +1 +1 1 +1 142.531 (13.082) 0.802 (83.24) 2775 (98.69) 1 +1 +1 +1 1 143.725 (12.35) 4.46 (6.81) 211,823 (0.00) +1 1 +1 +1 +1 143.727 (12.35) 0.818 (82.91) 211,823 (0.00) 1 +1 +1 1 +1 144.805 (11.70) 0.802 (83.24) 2775 (98.69) 1 +1 1 +1 +1 145.475 (11.29) 1.22 (74.51) 48,801 (76.96) +1 +1 1 +1 +1 145.669 (11.17) 1.22 (74.51) 48,801 (76.96) 1 1 +1 +1 +1 146.001 (10.97) 0.818 (82.91) 211,823 (0.00) +1 +1 +1 1 1 146.043 (10.94) 4.388 (8.31) 2775 (98.69) 1 +1 +1 1 1 146.237 (10.82) 4.388 (8.31) 2775 (98.69) +1 1 +1 1 +1 147.087 (10.30) 0.89 (81.40) 2775 (98.69) 1 1 +1 1 +1 147.281 (10.19) 0.89 (81.40) 2775 (98.69) 1 +1 1 +1 1 149.387 (8.90) 4.342 (9.28) 48,801 (76.96) +1 1 +1 1 1 150.999 (7.92) 4.476 (6.477) 2530 (98.81) +1 +1 1 +1 1 151.661 (7.52) 4.342 (9.28) 48,801 (76.96) +1 1 +1 +1 1 151.979 (7.32) 4.548 (4.97) 211,823 (0.00) 1 1 1 +1 +1 152.011 (7.30) 1.308 (72.67) 48,801 (76.96) 1 1 +1 +1 1 152.173 (7.20) 4.548 (4.97) 211,823 (0.00) 1 +1 1 1 +1 152.895 (6.76) 0.864 (81.95) 2530 (98.81) 1 1 +1 1 1 153.273 (6.53) 4.476 (6.477) 2775 (98.69) +1 1 1 +1 +1 154.285 (5.92) 1.308 (72.67) 48,801 (76.96) +1 +1 1 1 +1 155.169 (5.38) 0.864 (81.95) 2530 (98.81) 1 1 1 +1 1 155.523 (5.16) 4.43 (7.44) 48,801 (76.96) +1 1 1 +1 1 155.717 (5.04) 4.43 (7.44) 48,801 (76.96) 1 1 1 1 +1 158.019 (3.64) 0.952 (80.11) 2530 (98.81) +1 1 1 1 +1 158.213 (3.52) 0.952 (80.11) 2530 (98.81) 1 +1 1 1 1 161.147 (1.73) 4.698 (1.84) 2530 (98.81) +1 +1 1 1 1 161.341 (1.61) 4.698 (1.84) 2530 (98.81) 1 1 1 1 1 161.711 (1.39) 4.786 (0.00) 2530 (98.81) +1 1 -1 -1 -1 163.985 (0.00) 4.786 (0.00) 2530 (98.81) 2124 J. 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