ROBUST DESIGN FOR STRUCTURAL OPTIMIZATION PROBLEMS DUE TO PARAMETER UNCERTAINTIES
C. Zang Department of Mechanical Engineering, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK
M.I. Friswell Department of Aerospace Engineering, University of Bristol, Queen’s Building, Bristol BS8 1TR, UK
J.E. Mottershead Department of Engineering, University of Liverpool, Brownlow Hill, Liverpool L69 3GH, UK
ABSTRACT SS The sum of squares due to the overall experimental µy
mean Product performance is now, more than ever, a critical The number of rows in the orthogonal array matrix requirement for success in manufacturing. However, n significant uncertainty exists in material and geometrical experiments n The replication number of the design parameter X parameters, such as modulus, thickness, density and X j i residual strain, and also in joints and component assembly. i The variance in the structural dynamics (noise and vibration at the j level µ The mean for each parameter X response) is critical to the system performance. Approaches Xi i to identify and remove sources of variability play an important role in the design of products. This paper presents 2 σX The Variance for each parameter Xi a technique based on robust design for structural i ∆ The expanded standard deviation optimisation problems due to parameter uncertainties. The Xi Monte Carlo method is commonly used, but is expensive due to the large number of random samples required to GTSS The grand total sum of squares accurately simulate the structural system response to the TotalSS The sum of squares due to the variation about the uncertain parameters. In this paper, orthogonal arrays are mean used to study the effect of uncertainty in many design parameters simultaneously. The analysis of variance (ANOVA) is applied for determining the principal variables and their contribution to the dynamic response. The methodology is illustrated with an example of a spot welded 1. INTRODUCTION column model. The results show the feasibility of this approach for the robust design in order to minimise the The improvement in performance is vital in the development variance in the dynamic response due to the variability of the of any system such as a vehicle. This is usually performed parameters. using tools of engineering design optimisation in order to meet design targets. However, design optimisation may not always satisfy the desired targets due to the significant NOMENCLATURE uncertainty that exists in material and geometrical parameters such as modulus, thickness, density and residual strain, as well as in joints and component assembly. Ways to X The i-th variable parameter i minimize the effect of variations of uncertainty on the yi The system response of the i-th experiment performance are of paramount concern to researchers and µy The overall mean from the orthogonal array matrix practitioners in the academic and engineering fields.
experiment Robust design, that explicitly recognises the effects of the µ y j The average of the nX responses for each level j uncertainty variation and improves the effectiveness and X i i efficiency of design optimisation, provides an approach to achieve a more consistent performance of product features. Robustness in structural dynamics aims to make the variance of the noise and vibration response of a product 2. THE TAGUCHI METHOD insensitive to the variations of the design variables. In recent years, various approaches to robust optimisation have been Taguchi’s approach to the product design process may be developed. The genetic algorithm has been applied to the divided into three stages: system design, parameter design, robustness of optimal design solutions to reduce vibration and tolerance design [14]. System design is the conceptual transmission in a lightweight 2-D structure [1-3]. The change design stage where a system is developed that functions in the performance of the optimal structures due to small under an initial set of nominal conditions. Parameter design geometric perturbations, representing manufacturing is the stage at which a targeted concept is optimised by tolerances, was also investigated by a robustness analysis. using specific set levels that makes the system less sensitive Sandgren and Cameron [4] used a hybrid combination of a to noise, thereby enhancing the system’s robustness. genetic algorithm and non-linear programming for robust Tolerance design aims to specify the allowable deviations in design optimisation of structures with variations in loading, the parameter values, loosening tolerances where possible geometry and material properties. A fuzzy finite element and tightening where necessary. The following steps briefly approach may be used to model vague uncertainties in describe the Taguchi method for finding the optimum setting structural systems [5, 6]. Monte Carlo simulation is often of the design variables to make the response insensitive to used to propagate uncertainty, due to its generality and noise factors, which may be regarded as the design and straight-forward approach to modelling the uncertainty of the manufacturing tolerances in a numerical structural analysis. input parameters [7, 8]. The major advantage is that an accurate solution can be obtained and higher order statistical (i). To determine the response characteristic to be movements and the probability distribution of the response optimised, the design parameters and their noise quantities can be computed. However, Monte Carlo factors (uncertainty of design parameters). simulation has to simulate a large number of combinations of the uncertain inputs, called testing conditions, using a (ii). To design the experiment matrix to determine the random number generator. The value of the response is effect of the design parameters and conduct the data computed for each testing condition, and the mean and analysis procedure. After selecting the appropriate variance of the response are then calculated. To obtain orthogonal array for the noise and parameters to fit a accurate estimates of the mean and variance, the Monte specific study, variations in the response Carlo method requires evaluation of the response for a large characteristic due to the noise factors are simulated. number of testing conditions. This can be very expensive. Another popular method, the Taylor Series expansion (iii). To analyse the data and determine the optimum method (or perturbation technique) [9, 10], estimates the levels. After the experiments have been conducted, variance of the response by using the derivatives of the the optimal parameter configuration within the response with respect to each uncertain parameter. The first- experiment design must be determined. The order Taylor series expansion can only give accurate experiment results are analysed using the ANOVA estimates when the variations are small. Otherwise, higher technique and the optimum levels for the parameters order Taylor series expansions must be used, which makes are determined. the formula to evaluate the response quite complicated and computationally expensive.
In contrast to the above methods, Taguchi developed an 3. ORTHOGONAL ARRAYS FOR THE DESIGN OF efficient and systematic methodology that applies statistical EXPERIEMENTS experimental design to improve product and manufacturing design [11, 12]. The Taguchi method has been used An experimental design represents a sequence of successfully in Japan and the United States to design experiments to be performed. The most basic experimental reliable, high quality products at low cost in such areas as design method is a full factorial design which investigates all the automobile industry, consumer electronics and the possible combinations, maximising the possibility of finding a aerospace industry. Recently, Lee and Park [13] favourable result. In general, if there are s structural design implemented the Taguchi method for the robust design of parameters, each at r levels, the total number of unconstrained optimisation problems. Standard structural combinations required to guarantee completeness is r s . For examples, such as the three and ten bar truss structures, example, by varying design parameters one at a time, the were treated and illustrated as well as other engineering study of 13 design parameters at 3 levels would require applications. 1,594,323 (313) possible experimental evaluations. Obviously, the size of a full factorial experiment increases In this paper, the Taguchi method will be used for structural exponentially with the number of design parameters. Many optimisation problems with parameter uncertainties. The design parameters will lead to simulations that are dynamic structural performance of a benchmark spot welded computationally intensive and impractical. column structure, inspired by the columns in automotive bodies, is investigated as an example. The effect of The orthogonal array experiment is a method of setting up uncertainty in design parameters, such as the plate experiments that need only a fraction of the experiments thickness and the diameter of the spot welds, are studied required for a full factorial combination. Orthogonal refers to simultaneously using orthogonal arrays and the analysis of the fact that the effect of each variable can be variance (ANOVA). mathematically assessed independently, without considering the effects of the other variables. The combinations are factor effects around the overall experimental mean chosen to provide sufficient information to determine the response. Figure 1 summarizes the ANOVA relationships. variable effects using statistical analysis. Conducting matrix experiments using these special matrices (orthogonal arrays) allows the effects of several parameters to be determined efficiently. In fact, the columns in an orthogonal array are SS due to mutually orthogonal. That is, for any pair of columns, all GTSS Factor Xn combinations of factor levels occur, and they occur an equal … number of times. Assuming there is no interaction among the s design parameters, the number of combinations required SS due to based on the orthogonal array for s parameters with r levels TotalSS Factor Xi is sr(1)1−+. The number of experiments is significantly SS µ y reduced. For example, using an L27 array, 13 parameters … can be studied at 3 levels by running only 27 experiments SS due to instead of 1,594,323 (313). The procedure of constructing Factor X2 orthogonal arrays is given in reference [14]. SS due to Factor X1
4. ANALYSIS OF VARIANCE (ANOVA)
Figure 1: Decomposition as the sums of squares Analysis of variance is a useful technique to process the experimental data into a sum of squares that holds the key to For factor X , the sum of the squares due to variation about the quantification of the contribution of each design i parameter. Assume there are n rows in the orthogonal array the mean is matrix experiment, X k is the value of design parameter X 2 i i k at the k level (k=2 or 3) in the combination of design SS=− n µµ (6) Xijj yy ∑ Xi X parameters, and yi is the response of i-th experiment. j =1 i Then, the overall mean from which all the variation is where µ is the average of the n responses for each calculated is given by y j Xi X i n level j (j = 1,..,k and k = 2 or 3). n is the replication 1 X j µyi= y . (1) i n ∑ i =1 number of the design parameter Xi at the j level,
The grand total sum of squares (GTSS) is determined by The percentage contribution of each factor to the measured n response is expressed as, GTSS= y 2 . (2) ∑ i Percentage contribution = (SSX / TotalSS )× 100 (7) i =1 i The GTSS can be decomposed into two parts. One is the More detail of the ANOVA technique can be found in sum of squares due to the overall experimental mean, that is Reference [15, 16].
2 n 2 1 5. CASE STUDY: THE BENCHMARK COLUMN SSµ == n()µyi y . (3) y n ∑ STRUCTURE i =1
The other part is the sum of the squares due to the variation As a simple example of an engineering application, a model about the mean, referred to as the total sum of the squares, of a column consisting of two plates welded together with 20 spot welds is selected. The upper plate is shown in Figures 2 n and 3, which show the section and plan views. The base 2 TotalSS=−∑() yiyµ . (4) plate (not shown here) is just a rectangular flat plate with i =1 dimensions 113.7×564 mm. The nominal thickness of both plates is 1.5 mm and the spot welds have a nominal Note that diameter of 6 mm. The Young’s Modulus of the material property is taken as E = 200 GPa. The Poisson’s ratio is set GTSS=+ SS TotalSS . (5) µy equal to 0.3 and the density to 7.85 x 10-9 tonnes/mm3.
The technique of summing squares can be used to define An FE model of the structure was constructed in NASTRAN the contribution of each individual parameter within the total using Pshell and Cweld elements. The dynamic response of sum of squares. The sum of squares method is based on the first three modes (503.87 Hz, 565.74 Hz and 579.90 Hz) numerically quantifying the variation that is induced by the was computed and is plotted in Figure 4. Suppose that the material properties, such as the Young’s modulus, density, and Poisson’s ratio are unlikely to change during manufacture. The variation in the dynamic response is therefore closely related to the tolerance of the components and the assembly. For the column model the four variables listed in Table 1 are considered
Table1: Variable definition of the column model
Variable Definition X1 T1 Thickness of the bottom plate X2 T2 Thickness of the upper plate near the bends X3 T3 Thickness of the upper plate away from the bends X4 D Diameter of spot weld
2 If the µ X and σ are the mean and variance, respectively, i Xi for each variable X , we will choose three levels for each i Figure 2: Transverse section of the column example variable to estimate the response variance. The deviation ∆ from the mean µ can be obtained from the definition Xi Xi of variance to make the uniformly distribution from a designed experiment emulate a normal distribution.
σ 2 = Xi 222 µµµµµµ−∆ − + − + −∆ + ∑(()XXii X i) ()() XX ii( XX ii X i) 3 (8) Condensing the expression by cancelling the mean terms gives,
2 2()∆ X σ 2 = i . (9) X i 3 The required expanded standard deviation is, Figure 3: Upper view of the column example 3 ∆=σσ ≈1.2247 (10) Xii2 XX i
3 Thus, the three levels of the variable are µσXX− , ii2 3 µ X , µσXX+ . The levels for the four various i ii2 parameters of the column model are shown in Table 2.
Table 2: Noise factors for the column model
Factors Mean Std. Dev. Levels (mm) (mm) σ 1 2 3 X i T1 1.5 0.05 1.4388 1.5 1.5612 T2 1.5 0.05 1.4388 1.5 1.5612 T3 1.5 0.05 1.4388 1.5 1.5612 D 6 0.08 5.9020 6.0 6.0980 Figure 4: The dynamic response of the first 3 modes 4 Then, the standard L9 (3 ) orthogonal array given in Table 3 can be utilized to build up the matrix experiment. In the table, columns 2 to 5 of the matrix represent the four variables and the first three modes of frequency response in the the entries in the matrix represent the levels of the variables. column model. It dominates the effect on the first and The last three columns give the corresponding natural third modes (about 77%) and also has a significant frequencies of the first three modes of the column model. effect on the second mode (17%) of the response. Tightening the variance of T3 will significantly reduce Clearly, the dynamic performance of the column model the variance of the dynamic response of the column varies considerable with the different combinations of four model, especially in the first and third modes. variables in the nine experiments. The responses of 9 orthogonal array experiments together with the response of 4. In practice, tightening the tolerance (variance) of the initial design target are overlaid in Figure 5. components or assembly will definitely increase the cost of the production. The balance between the cost and the product performance needs to be considered. However, the possibility of reducing the performance variability without or with slight tightening of the tolerance is still under investigation. One choice is to divide the variance of sensitive variables into more detailed groups and employ different combinations of components and assembly so that the variance of performance could be reduced as much as possible. This will be reported later on.
Figure 5: Overlay of responses of 9 orthogonal array experiments
The next step is to estimate the effect of variables and to decompose the main variable effects on the response from the orthogonal experiments. The effect of a variable level is defined as the deviation it causes from the overall mean, that is ()µ j − µy . Where µ j is the mean response caused Xi Xi by the parameter Xi at the level j. The effect of the three levels of the four selected variables on the dynamic response Figure 6: The effect of uncertainties on the 1st mode of the first three modes is listed in Table 4 and plotted in Figures 6, 7, and 8. The sum of squares in the analysis of variances (ANOVA) is carried out and the result of the percentage contributions of the four variables to the first three modes is shown in Table 5. The following features were noted.
1. The dynamic response of the column model is least sensitive to the diameter of the spot weld (D). The contribution to each of the three modes, compared to other three variables, is less than 0.3%. Loosening its tolerance will hardly change the dynamic performance of the structure.
2. The thicknesses of the plates (T1, T2, and T3) are the three principal variables for the dynamic performance of the column model in this example. The thickness T1 dominated the 2nd mode (81.55%) of the response while the thickness T2 had similar effects on the 1st and 3rd modes of the response (about 18%). Figure 7: The effect of uncertainties on the 2nd mode
3. The thickness of the upper plate away from the bends (T3) has an important role in controlling the variance of [3] Anthony D. K., Robustness of Optimal Design Solutions to Reduce Vibration Transmission in a Lightweight 2-D Structure, Part 3: Using Both Geometric Redesign And the Application of Active Vibration Control. Journal of Sound and Vibration, Vol. 245, No. 3, Aug. 2001, pp417- 431. [4] Sandgren E., Cameron T. M., Robust Design Optimization of Structures through Consideration of Variation, Computers and Structures, Vol. 80, 2002, pp1605-1613. [5] Rao S. S. and Sawyer J. P., A Fuzzy Finite Element Approach for the Analysis of Imprecisely-defined Systems, AIAA J., Vol. 33, 1995, pp264-370. [6] Akpan U.O., Koko T. S., Orisamolu I. R., and Gallant B. K., Fuzzy Finite Element Analysis of Smart
Figure 8: The effect of uncertainties on the 3rd mode Structures, Smart Material and Structure, Vol. 10, 2001, pp273-284.
[7] Mahadevan S. and Raghothamachar P., Simulation 7. CONCLUDING REMARKS for System Reliability Analysis of Large Structures, Computers and Structures, Vol. 77, No. 6, 2000, pp725- This paper presents a Taguchi robust design method for 734 structural optimisation problems. The objective is to minimise [8] ST-ORM, A Meta-Computing System for Stochastic the effect on the dynamic response of variability in the design Optimization and Robustness Management, EASi parameters. Using the orthogonal array and analysis of Engineering GmbH, Germany, 2000. variance (ANOVA) techniques enables the investigation of the effect of uncertainty in multiple variables simultaneously. [9] Liu W. K., Belytschko T., and Mani. A., Random Field Another advantage, compared to the Monte Carlo simulation Finite Elements, International Journal for Numerical method, is that only a much smaller number of testing Methods in Engineering, Vol. 23, 1986, pp1831-1845 conditions are needed, without sacrificing any accuracy. [10] Schuëller G.I., A State-of-the-art Report on
Computational Stochastic Mechanics, Probabilistic The example of a column model was investigated and a Engineering Mechanics, Vol. 12, No. 4, 1997, pp197- robust design is obtained. The further consideration is also 321 discussed. [11] Taguchi G., Quality Engineering Through Design Optimization, New York: Kraus International Publication, ACKNOWLEDGEMENTS 1984 [12] Taguchi G., Systems of Experimental Design, IEEE J., The authors acknowledge the support of the EPSRC (UK) Vol. 33, 1987, pp1106-1113 through grants GR/R34936 and GR/R26818. The authors also wish to thank Mr. M. Palmonella for supplying the FE [13] Lee K. H., Eom I. S, Park G. J., and Lee W. I., Robust model of the column structure. Design for Unconstrained Optimization Problems Using the Taguchi Method, AIAA J., Vol. 34, No. 5, 1996, pp1059-1063 REFERENCES [14] Phadke M. S., Quality Engineering Using Robust Design, Prentice-Hall, Englewood Cliffs, NJ, 1989 [1] Anthony D. K., Elliott S. J., and Keane A. J., [15] Fowlkes W. Y. and Creveling C. M., Engineering Robustness of Optimal Design Solutions to Reduce Methods for Robust Product Design: Using Taguchi Vibration Transmission in a Lightweight 2-D Structure, Methods in Technology and Product Development, Part 1: Geometric Design. Journal of Sound and Addison-Wesley Publishing Company, 1995 Vibration, Vol. 229, No. 3, Jan 2000, pp505-528. [16] Creveling C. M., Tolerance Design: A Handbook for [2] Anthony D. K., Elliott S. J., Robustness of Optimal Developing Optimal Specifications, Addison-Wesley Design Solutions to Reduce Vibration Transmission in a Publishing Company, 1997 Lightweight 2-D Structure, Part 2: Application of Active Vibration Control Techniques. Journal of Sound and Vibration, Vol. 229, No. 3, Jan 2000, pp529-548.
4 Table 3: Orthogonal array L9(3 )
Exp. Variables combinations Natural frequencies of the response T1 T2 T3 D F1 (Hz) F2 (Hz) F3 (Hz) 1 1 1 1 1 484.2846 547.4707 556.5075 2 1 2 2 2 500.7466 553.6725 576.0623 3 1 3 3 3 516.9555 559.4927 595.4890 4 2 1 2 3 498.1290 564.5627 573.2171 5 2 2 3 1 514.5635 570.2413 592.7891 6 2 3 1 2 496.6674 561.8227 571.0953 7 3 1 3 2 510.9483 580.3781 588.7742 8 3 2 1 3 495.2014 569.7151 571.8398 9 3 3 2 1 511.7975 577.9926 589.3703
Table 4: Effects of a variable level on the response
Variable 1st mode frequency (F1) (Hz) 2nd mode frequency (F2) (Hz) 3rd mode frequency (F3) (Hz) Level 1 Level 2 Level 3 Level 1 Level 2 Level 3 Level 1 Level 2 Level 3 T1 500.66 503.12 505.98 553.55 565.54 576.03 576.02 579.03 583.33 T2 497.79 503.50 508.47 564.14 564.54 566.44 572.83 580.23 585.32 T3 492.05 503.56 514.16 559.67 565.41 570.04 566.48 579.55 592.35 D 503.55 502.79 503.43 565.23 565.29 564.59 579.56 578.64 580.18
Table 5: Percentage contribution of the variables
Variable Percentage contribution (%)
1st mode (F1) 2nd mode (F2) 3rd mode (F3)
T1 4.485 81.550 6.109 T2 18.090 0.970 17.849 T3 77.320 17.382 75.771 D 0.105 0.098 0.271