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Stochastic model updating: Part 2—application to a set of physical structures
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Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 20 (2006) 2171–2185 www.elsevier.com/locate/jnlabr/ymssp
Stochastic model updating: Part 2—application to a set of physical structures J.E. Mottersheada,, C. Maresb, S. Jamesa, M.I. Friswellc a
Department of Engineering, University of Liverpool, Brownlow Hill, Liverpool L69 3GH, UK b Brunel University, School of Engineering and Design, Uxbridge, Middlesex UB8 3PH, UK c Department of Aerospace Engineering, University of Bristol, Queen’s Building, University Walk, Bristol BS8 1TR, UK Received 22 October 2004; received in revised form 22 June 2005; accepted 26 June 2005 Available online 16 September 2005
Abstract The application of a stochastic model updating technique using Monte-Carlo inverse propagation and multivariate multiple regression to converge a set of analytical models with randomised updating parameters upon a set of nominally identical physical structures is considered. The structure in question is a short beam manufactured from two components, one of folded steel and the other flat. The two are connected by two rows of spot-welds. The main uncertainty in the model is concerned with the spot-weld but there is also considerable manufacturing variability, principally in the radii of the folds. r 2005 Elsevier Ltd. All rights reserved. Keywords: Uncertainty; Variability; Model updating
1. Introduction For complicated physical structures differences in dynamic behaviour are often related to manufacturing and functioning as well as to variations in the material properties of the components. Other important errors occur in mathematical modelling by making simplifications in the representation of actual physical phenomena. These ‘model-structure errors’ are present especially when trying to model different types of joints and interfaces: bolted joints, welds, Corresponding author. Tel.: +44 0151 794 4827; fax: +44 0151 794 4848.
E-mail address: [email protected] (J.E. Mottershead). 0888-3270/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2005.06.007
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adhesively connected surfaces, etc. A stochastic model updating method was described in the companion paper [1] which allows for manufacturing variability and modeling uncertainty so that a set of analytical models having randomised parameters are ‘corrected’ by converging them upon a set of experimental results from a collection of nominally identical test pieces. In this paper the inverse error propagation and multivariate multiple regression is carried out to correct the finite element meta-model of a benchmark structure with spot-welds. Two main types of spot-weld models exist [2,3], namely those that require the stress within the weld spot to be calculated and those that do not. The former require detailed modelling so that a smooth stress field is computed, whereas in the case of the latter the estimation of stiffness and mass distributions is sufficient. When modelling spot-welds, it is difficult to take into account the many local effects such as geometrical irregularities, residual stresses, material in-homogeneities and welding defects. Furthermore, a detailed and complex local model of the joint characteristics is not possible in industrial applications where typically thousands of spot-welds are used to manufacture automotive parts. These conceptual modelling uncertainties are incorporated in an interface finite element [4] which may be used to connect panels at spot-weld locations. The material properties of the element determine the local stiffness in the tangential and normal directions at the surface. The interface element stiffnesses are used together with other parameters to update a structural and statistical model of a short beam manufactured from two components, one of folded steel and the other flat. The other parameters are two elastic moduli of curved regions in the folded plate. Taken all together they become the randomised parameters used in the Monte-Carlo simulation. In the following sections the stochastic model updating of a collection of nominally identical spot-welded structures, representative of spot-welded beam-like parts in car bodies, is carried out. Details of the finite element model, choice of randomised parameters, manufacturing details and application of the method are described.
2. The set of benchmark structures A set of ten nominally identical benchmark structures were built, each one consisting of two parts, a folded steel plate with a ‘hat’-shaped cross-section and a flat plate, connected by two rows of spot-welds along the flanges as shown in Fig. 1. It was intended to represent the structural dynamics and the manufacturing variability of beam-like structures, such as door pillars and roof supports that occur commonly in car-body shells. A manufacturing specification was developed with the intention of reducing manufacturing variability, which included the following instructions: (i) standard rolled sheet steel of nominal thickness 1.5 mm to be used; (ii) all plates to be cut from the same sheet; (iii) no material to be used within 30 mm of the edge of the sheet; (iv) all plates to be aligned lengthways parallel to the sheet rolling direction; (v) all hats to be folded using the same former, and by the same person. After cutting and bending, all ten plates and hats were measured, numbered and returned for spot-welding as numbered pairs. In order to isolate the spot-weld modelling uncertainties from any manufacturing variability present in the set of the benchmark structures, the plate and hat belonging to one specimen were tested separately before being welded together. The finite element models of the hat and flat plate were then updated deterministically using measured data from the
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Fig. 1. Benchmark structure.
chosen specimens. These updated models provided the starting point for updating the complete model stochastically using the full set of benchmark structures. The purpose was to isolate the spot-weld modelling uncertainty, since errors in plate thickness or fold radius were supposed to have already been corrected at the first step of updating the separate components. This is the usual approach when model updating is carried out on multi-component structures using a buildingblock approach.
3. Updating the separate components A modal test with free–free boundary conditions, 30 roving hammer points and two measurement points was carried out on the chosen flat plate specimen and the identified modes are presented in Fig. 2. The thickness of the plate was measured at different places and a mean value of 1.448 mm was used in the finite element model. With a mass and Young’s modulus correction the model was found to be very accurate for the bending modes but less accurate in torsion. The frequency errors and MAC values are presented in Table 1. The standard deviation of the plate thickness measurements was 0.0075 mm. The free–free modal test on the chosen hat component was also of the roving hammer type, this time using 40 hammer point and 4 accelerometers. The identified modes are shown in Fig. 3. Measurement of the thickness at different points on the hat showed a variation both along the span of the model and on the cross-section, the main differences appearing at the fold radii. The specified radii of 5 mm were in fact found to be approximately 4 mm and when this change was incorporated in the finite element model the correlation was improved considerably. The hat was divided in 5 regions, as indicated in Fig. 4. The thicknesses of the plate in those regions were used to update the model by minimising the predicted natural frequency errors. A floating-point genetic algorithm [5] with 30 individuals and 50 generations was used for the optimisation. The frequency errors and MAC values for the best solution are presented in Table 2. The genetic algorithm optimisation loop led to a number of sub-optimal solutions from which the one reducing the error in the first mode (with the greatest initial error) was chosen. In the case
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Fig. 2. Plate modes for the chosen specimen.
Table 1 Natural frequencies for the flat plate Mode
Test
FE
Error (%)
MAC
1 2 3 4 5 6
23.74 65.95 76.78 130.23 156.14 216.33
23.92 66.28 74.48 130.73 151.94 217.02
0.76 0.50 3.00 0.38 2.69 0.32
100 100 100 100 99 99
of updating the two components alone, there is an aleatory uncertainty related to the dimensions and material properties of the shells especially in the curved areas of the hat and in the updating process one faces an epistemic uncertainty (ignorance or lack of knowledge about the system due to limited experimental data). It is also possible that extension–shear-bending coupling effects reduce the performance of the shell element used and this might represent a conceptual modelling
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Fig. 3. Hat modes for the chosen specimen.
Fig. 4. Different regions of the specimen hat component.
uncertainty. The shell element that was implemented in MATLAB was known to be less accurate than the MSC-NASTRAN CQUAD element, which had been used at a preliminary stage. Choosing the best solution based on the frequency distance for a single sample and using these component models for all 10 samples means that in the next step, the modelling errors related to
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Table 2 Measured natural frequencies and finite element predictions in Hz for the specimen hat Mode
1 2 3 4 5 6 7 8 9 10 11
Exp
71.631 256.83 275.23 338.37 403.65 631.01 644.23 757.81 783.77 791.23 982.23
Initial
Updated
FE
Error (%)
MAC
FE
Error (%)
MAC
67.994 257.80 274.00 345.80 407.87 632.06 642.96 783.26 794.68 815.51 995.68
5.08 0.38 0.45 2.19 1.04 0.17 0.20 3.36 1.39 3.07 1.37
95 93 94 95 94 86 85 95 97 92 80
70.025 255.13 272.92 343.57 409.53 629.59 641.56 781.70 797.99 820.31 997.10
2.24 0.66 0.84 1.53 1.46 0.23 0.41 3.15 1.81 3.68 1.51
95 93 94 96 94 86 86 96 97 92 80
Table 3 Thickness in mm of the different hat regions after updating Upper curved zone
Lower curved zone
Vertical walls
Upper plate
Contact zone
1.400
1.400
1.593
1.512
1.600
this sample and not an average error, are propagated through the updating loop. The modification of the initial measured average thickness of 1.468 mm for all the zones is presented in Table 3.
4. Modal tests on the complete set of structures A free–free roving-hammer modal test with 70 hammer points and 4 measurement points was carried out on the specimen assembled from the chosen tested parts. For the other nine specimens a reduced test was performed with only 2 hammer points and 4 measurement points. The natural frequencies were matched based on a reduced wire-frame mode shape with the objective that any mode swapping or other type of experimental uncertainty would be investigated by further tests if needed. The mode shapes of the test for the chosen specimen are presented in Fig. 5 and were used later for MAC checks during the updating loop of the whole model. The natural frequencies for all the samples are shown in Table 4 where one can observe a standard deviation of about 1% for all the modes of the sample.
5. Stochastic model updating The spot-weld material constants ks1 ; ks2 ; kn represent tangential shear and normal direct stiffnesses for a special interface element described in detail in Appendix A. These three
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Fig. 5. Welded structure modes for the chosen specimen.
interface-element stiffnesses defined in Eq. (A.8), also in Appendix A, were selected as updating parameters. The three alone proved insufficient to give an acceptable updated structural and statistical model and it was decided therefore to include a further two parameters; the elastic moduli of the plate material at the upper and lower folds. The five chosen parameters are all equivalent ones, which it was hoped would enable the spot-weld uncertainty, its diameter, precise location, variations in material properties at and around the nugget, and variability in the fold radii and thickness of the steel plate to be represented in the updated model. In order to quantify the input parameters that account for a high percentage of the output variability in natural frequencies, a cloud of models was created by varying the 5 material constants according to a normal distribution. The assumption of normality for the input variables is dictated by the lack of a probabilistic model validated by experimental data from the spotwelds. Also the interface element is an equivalent model of the spot-welds as are the elastic moduli that represent variability in fold radii. Each spot-weld in the finite element model was represented by four interface elements with stiffness parameters having initial mean values shown in Table 5, based on an elastic modulus of 25% of the usual value for steel and allowed to vary between 5% and 150%. The elastic moduli of the folds were allowed to vary by 50% about the initial mean,
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Table 4 Natural frequencies in Hz for the complete sample Spec no.
1 2 3 4 5 6 7 8 9 10 Mean Std
Mode 1
2
3
4
5
523.39 521.88 526.8 525.99 512.19 523.74 517.09 522.06 526.97 528.90 522.90 5.04 (0.96%)
589.06 588.45 594.17 592.72 580.24 590.74 586.05 589.71 592.12 596.16 589.94 4.49 (0.76%)
600.24 588.45 602.73 595.92 580.24 599.64 586.05 590.56 606.56 604.48 595.48 8.76 (1.28%)
682.14 689.14 688.81 679.4 680.71 678.83 675.64 677.63 689.82 688.52 683.06 5.45 (0.80%)
713.45 708.87 715.16 706.3 702.19 706.68 702.84 706.06 720.77 718.52 710.08 6.50 (0.91%)
Table 5 Initial and updated mean parameter values
Initial Final
E1
E2
Ks1
Ks2
Kn
2.10E+11 1.89E+11
2.10E+11 2.31E+11
5.00E+10 5.63E+10
5.00E+10 2.50E+10
5.00E+10 4.69E+09
being the usual nominal value for steel plate. The initial variance matrix contained the same data given in Table 4. Updating was carried out on the basis of the first five measured modes from the sample, mean values being given in Table 6. The number of observations for the experimental and analytical models influences the size of the confidence ellipse and a convergent shape is obtained from the T 2 statistic with degrees of freedom p; n1 þ n2 2 as described in Appendix B. It is shown in Fig. 6 that the stabilisation of the statistic T 2 is obtained closely when n2 ¼ 200 analytical models and n1 ¼ 10 experimental tests. n2 ¼ 200 becomes the dimension of the meta-model, to be displaced to the location in the output space given by the experimental data by variation of the randomised updating parameters. The reduction of the Euclidian distance between the models in the output space was obtained by using the gradient and regression method described in the companion paper [1]. In the simulation, ten iterations were completed and at each step the covariance was updated a posteriori, the minimum being obtained after 5 iterations. The scatter plots of the initial and final cloud are presented in Fig. 7 and in separate planes in Figs. 8–10 and the mean natural frequencies of the model, the mean parameter values and standard deviations of the initial and final iterations are presented in Table 5–7. The strength of the relationship between input and output variables in the model has been measured by the Spearman correlation matrix presented in Fig. 11. The analysis of the material
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Table 6 Mean natural frequencies Mode
1 2 3 4 5
Initial
Final
Test
Frequency (Hz)
Error (%)
Frequency (Hz)
Error (%)
Frequency (Hz)
533.28 627.10 685.89 768.82 775.88
1.98 6.29 15.18 12.55 9.26
512.48 593.99 604.77 687.28 692.59
1.99 0.68 1.55 0.61 2.46
522.90 589.94 595.49 683.06 710.08
Fig. 6. T2-statistic for determining the optimal sample number.
coefficients shows that the most important parameters at the beginning of the updating loop are the material constants of the curved areas while closer to the test data the significance of the input variable related to the normal displacement between the welded shells becomes more important. The material constant related to the sliding between the two surfaces are less important, because the modes concerned involve almost no shear behaviour. The distance between the means of the parameters of the experimental and analytical samples can be determined from the confidence ellipsoid. Then, following the analysis given in Appendix B, the norm of the observed difference between the mean outputs from the final analytical
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Iteration #1
Iteration #5
720 640
700
630
680
620 610
640
600
f3
f3
660
620
590
600
580
580
570 560 620
660
560
540
460
480
500
520
590
(b)
580 570
490
500
520
510
530
540
f
580
600 f2
f2
(a)
610
560
1
600
540
f
620
1
640
Fig. 7. Analytical cloud and target cloud in the output space together with their projection on each plane (a) initial; (b) after 5 iterations.
660 600
640
595 f2
f2
620 600
590
580
585
560 480
(a)
500
520 f1
540
580
560
(b)
505
510
515 f1
520
525
530
Fig. 8. Scatter plot of the analytical cloud and the target cloud in the (f1–f2) plane: (a) initial; (b) after 5 iterations.
population and the experimental data is d ¼ j¯ytest y¯ analysis j ¼ 23:40 and the eigenvalues of the covariance matrix SD ¼ ð1=n1 þ 1=n2 ÞSpl are 38:390; 0:831; 0:140; 0:015; 0:010 . The semi-axes of the 95% confidence ellipsoid are found to be ci ¼ f 21:00; 3:09; 1:27; 0:42; 0:34g. Therefore for the hypothesis to be accepted the origin should lie between jdj 21:00 ¼ 2:40 and jdj þ 21:00 ¼ 44:40, which it does not. This analysis shows that da0 and the means of the two populations in the final iteration are therefore not equal at the 95% confidence level. However, it can be observed from the scatter plots, shown in Figs. 8–10 that considerable convergence of the cloud of analytical results upon the test data has occurred and that the sizes and orientations of the ellipses shown in Figs. 8–10 are similar, though not exactly superimposed one on the other of course. The final set of analytical models has natural frequencies with means and standard deviations comparable, from an engineering point of view, with the experimental data from the ten specimens. The frequency errors are in fact much reduced with respect to the test mean and the mode shapes are in very good agreement. The parameter with the most
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700 680
620
660
610 f3
f3
2181
640
600
620 590
600 580
580 500
480
520 f1
(a)
540
560
505
510
515
520
525
530
f1
(b)
Fig. 9. Scatter plot of the analytical cloud and the target cloud in the (f1–f3) plane: (a) initial; (b) after 5 iterations.
630 700
620
680
610 f3
f3
660 640
600
620
590 600
580 580
580 560
580
600
620
640
660
f2
(a)
585
590
595
600
f2
(b)
Fig. 10. Scatter plot of the analytical cloud and the target cloud in the (f2–f3) plane: (a) initial; (b) after 5 iterations. Table 7 Standard deviation
Initial Final
E1
E2
Ks1
Ks2
Kn
4.20E+10 4.85E+09
4.20E+10 1.15E+10
1.00E+10 7.90E+09
1.00E+10 9.06E+09
1.00E+10 7.36E+08
important variation is the material constant kn and this with all the other parameters shows reductions in the standard deviation as a result of convergence towards the test data. The changes to the spot-weld stiffnesses incorporate most of the variation in the tested specimens. Therefore the likely reason that updated population is different from the test population is that the updating parameters are not capable of truly representing all the differences between the test structures and the meta-model. The same parameters describe all the spot-welds although in reality each spot-weld is different. If the spot-welds were parameterised separately each one would have a
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1.2
1.2 1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2 1
1 Up 2 da tin 3 gv 4 ar iab les
5 4 3 5
2 1
es
Mod
Up 2 da tin 3 gv ar iab
5 4 les
3
4 s ode
5
2
M
1
Fig. 11. Spearman correlation matrix between the input and output variables for the initial and final cloud.
similar effect, therefore giving rise to ill-conditioning. There may of course be other variations with random distributions sufficiently dissimilar to the spot-weld and fold errors so that the two statistical populations will be different and cannot be made the same by adjusting these equivalent parameters. This appears to be an important result, which confirms what we have already understood about model updating, that engineering judgement is of paramount importance in its application.
6. Conclusions The stochastic model updating method has been applied to a set of benchmark structures with spot-welds. The spot-welds are modelled by using an interface zero-thickness element with the properties of a spring membrane at the contact between shell elements. A hierarchical approach is used in the study by analysing different sources of modelling uncertainty and manufacturing variability and their effect on model updating and inverse error propagation through a finite element model, which is converged upon multiple sets of experimental data. Results are obtained which show that the means of the meta-model fail to converge upon set of test structures within a 95% confidence limit. Never-the-less good convergence is obtained from the engineering viewpoint. The results confirm what we already know about model updating; that engineering judgement is extremely important. This observation is perhaps especially relevant when the purpose is not only to converge a structural model but also a probabilistic one.
Acknowledgements The research reported in this paper is supported by Engineering and Physical Sciences Research Council (EPSRC) grants GR/R26818 and GR/R34936.
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Appendix A. Equivalent spot-weld finite element model A generally applicable and simple joint/interface element for 3D analysis is described by Beer [4]. The element can model interfaces between solid and shell finite elements. The formulation is quite general, the element has zero thickness and the derived element is isoparametric. On the contact surfaces of the two shell elements, the displacements at any point are given by 8 9 8 9 ( ) > < ui > = = X > X di ai v ¼ Ni v i Ni ½ v2i v1i TOP , (A.1) bi > > 2 :w > ; : > ; TOP w TOP i TOP 8 9 8 9 > = < ui > = X > X di v ¼ Ni vi þ Ni ½ v2i > > 2 : > ; :w > ; w BOT i BOT
( v1i BOT
ai bi
) ,
(A.2)
BOT
where the Ni are the isoparametric shape functions, the v1i ; v2i are the unit vectors in the direction of the local tangent axes, ai ; bi are the rotation angles for the same axes and d i is the thickness of the shell at each node. These equations may be rewritten in the form 8 9 > = v ¼ NTOP;BOT aTOP;BOT , (A.3) > : > ; w TOP;BOT
aTOP;BOT ¼ ½a1 ; a2 ; . . . ; an TOP;BOT ;
8 9 ui > > > > > > > > > vi > > < > = a i ¼ wi , > > > > ai > > > > > > > :b > ;
(A.4)
i
NTOP;BOT ¼ ½N1 ; N2 ; . . . ; Nn TOP;BOT , di di Ni ¼ N i I; v2i ; v1i . 2 2
ðA:5Þ
The displacements in the directions normal and tangential to the contact surface are obtained by 8 9 8 9 > > = = v ¼ ½ s1 s2 n v , (A.6) > > ; : > : > ; w w where s1 ; s2 ; n are the tangent vectors and the normal vector to the surface area. The relative displacements at the surface are slip in the two tangent directions and the normal convergence or
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separation: 8 9 8 9 8 9 > > = = < ds1 > = > ds2 ¼ v v . > > : > ; : > ; :d > ; > w TOP w BOT n
(A.7)
The following relationships are defined between the tractions acting on the contact and relative displacements: 8 9 2 38 9 ks1 0 0 > > < ts1 > = < ds1 > = 6 7 ts2 ¼ 4 0 ks2 0 5 ds2 ; t ¼ Dd, (A.8) > > > :t > ; : ; 0 0 k d n
n
n
where ts1 ; ts2 ; tn are the shear tractions and contact pressure and ks1 ; ks2 ; kn are the shear and normal stiffnesses.
Appendix B. Aspects of statistical inference Hypothesis testing may be used to assess the quality of a model and its parameterisation. In the case of multivariate normal populations these tests are sufficient and numerical studies may be used to confirm robustness to deviations. When a large number of variables are used for simulation, data reduction becomes essential for numerical stability, Principal Component Analysis [6] being the main tool for this type of analysis. Measures of association such as the Pearson and Spearman coefficients [7,8] can identify those inputs that account for a high percentage of the output variability. In the present gradient-and-regression approach a test on the means (without reference to the covariances) may be carried out using a T 2 distribution due to Hotteling in 1931 [9]. The hypothesis, H 0 , that the means of the sampled outputs from two data sets originate from the same parent distribution is tested by the statistic: 1 1 1 n1 n2 T 2 þ ð¯y1 y¯ 2 Þ ¼ D2 , (B.1) Spl T ¼ ð¯y1 y¯ 2 Þ n1 n2 n1 þ n2 M Spl ¼
1 ½ðn1 1ÞS1 þ ðn2 1ÞS2 ; n1 þ n2 2
(B.2)
where the y¯ 1 ; y¯ 2 ; S1 ; S2 are the mean vectors and the covariance matrices for each sample, Spl is the pooled sample covariance matrix, DM is the Malhalanobis distance [7,8] and the T 2 is expressed in ‘characteristic’ form. When H 0 is true the quantity T 2 is distributed as T 2p;n1þn22 where the distribution is indexed by two parameters, the dimension p and the degrees-of-freedom n 1 for one sample or n1 þ n2 2 for two samples. It is necessary that n1 þ n2 240 for Spl to be nonsingular. Since an observed output is generally not equal to the true value, inference upon the equality of the means y¯ 1 ; y¯ 2 (or covariance matrices) of the two populations requires an estimate of uncertainty. The concept of the confidence region applied to the distance between population means (described by the Mahalanobis distance) shows that the confidence ellipse converges upon the stabilisation of the T 2p;n1þn22 statistic and this result should be used for choosing an optimum
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number of samples for the analytical population. The difference between the true means of two populations l1 ; l2 is bounded more accurately by using the confidence ellipsoid [10]: 1 1 1 T 2 T ¼ ½ð¯y1 l1 Þ ð¯y2 l2 Þ þ ½ð¯y1 l1 Þ ð¯y2 l2 Þ (B.3) Spl n1 n2 or T 1 T 2 ¼ ðd dÞT S1 D ðd dÞ ¼ D SD D,
(B.4)
d ¼ y¯ 1 y¯ 2 ; d ¼ l1 l2 ; 1 1 SD ¼ Spl . þ n1 n2
(B.5)
D ¼ d d,
(B.6)
For the T 2 statistic a 100ð1 aÞ% confidence region is defined by, 2 ðd dÞT S1 D ðd dÞpT a;p;n1þn22
(B.7)
By determining the eigenvalues and eigenvectors ðK; AÞ of SD , the confidence region is bounded by a p-dimensional hyperellipsoid centred at d with semi-axes given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (B.8) ci ¼ li T 2a;p;n1þn22 ; i ¼ 1; :::; p. The hypothesis on equal means d ¼ l1 l2 within a 100(1a)% confidence interval is then accepted if the origin is enclosed by the hyperellipsoid such that jdj ci p0pjdj þ ci
(B.9)
for any one of the semi-axes. The hypothesis is rejected if the origin lies outside the hyperellipsoid.
References [1] C. Mares, J.E. Mottershead, M.I. Friswell, Stochastic model updating: Part 1—theory and simulated examples, Mechanical Systems and Signal Processing, submitted for publication doi:10.1016/j.ymssp.2005.06.006. [2] M. Palmonella, M.I. Friswell, J.E. Mottershead, A.W. Lees, Finite element models of spot welds in structural dynamics: review and updating, Computers and Structures 83 (8–9) (2005) 648–661. [3] M. Palmonella, M.I. Friswell, J.E. Mottershead, A.W. Lees, Guidelines for the implementation of CWELD and ACM2 spot weld models in structural dynamics, Finite Elements in Analysis and Design 41 (2) (2004) 193–210. [4] G. Beer, An isoparametric joint/interface element for finite element analysis, International Journal for Numerical Methods in Engineering 21 (1985) 585–600. [5] C. Houck, J. Joines, M. Kay, A genetic algorithm for function optimization: a MATLAB implementation, NCSUIE TR 95-09, 1995. [6] I.T. Jolliffe, Principal Components Analysis, Springer, Berlin, 1986. [7] A. Rencher, Methods of Multivariate Analysis, Wiley, New York, 1995. [8] A. Rencher, Methods of Statistical Inference and Applications, Wiley, New York, 1998. [9] H. Hotteling, The generalization of student’s ratio, Annals of Mathematical Statistics 2 (1931) 360–378. [10] I. Doltsinis, F. Rau, On the ordinary distance between multivariate random systems in engineering, Computer Methods in Applied Mechanics and Engineering 191 (1–4) (2001) 133–156.
Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 20 (2006) 2171–2185 www.elsevier.com/locate/jnlabr/ymssp
Stochastic model updating: Part 2—application to a set of physical structures J.E. Mottersheada,, C. Maresb, S. Jamesa, M.I. Friswellc a
Department of Engineering, University of Liverpool, Brownlow Hill, Liverpool L69 3GH, UK b Brunel University, School of Engineering and Design, Uxbridge, Middlesex UB8 3PH, UK c Department of Aerospace Engineering, University of Bristol, Queen’s Building, University Walk, Bristol BS8 1TR, UK Received 22 October 2004; received in revised form 22 June 2005; accepted 26 June 2005 Available online 16 September 2005
Abstract The application of a stochastic model updating technique using Monte-Carlo inverse propagation and multivariate multiple regression to converge a set of analytical models with randomised updating parameters upon a set of nominally identical physical structures is considered. The structure in question is a short beam manufactured from two components, one of folded steel and the other flat. The two are connected by two rows of spot-welds. The main uncertainty in the model is concerned with the spot-weld but there is also considerable manufacturing variability, principally in the radii of the folds. r 2005 Elsevier Ltd. All rights reserved. Keywords: Uncertainty; Variability; Model updating
1. Introduction For complicated physical structures differences in dynamic behaviour are often related to manufacturing and functioning as well as to variations in the material properties of the components. Other important errors occur in mathematical modelling by making simplifications in the representation of actual physical phenomena. These ‘model-structure errors’ are present especially when trying to model different types of joints and interfaces: bolted joints, welds, Corresponding author. Tel.: +44 0151 794 4827; fax: +44 0151 794 4848.
E-mail address: [email protected] (J.E. Mottershead). 0888-3270/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2005.06.007
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adhesively connected surfaces, etc. A stochastic model updating method was described in the companion paper [1] which allows for manufacturing variability and modeling uncertainty so that a set of analytical models having randomised parameters are ‘corrected’ by converging them upon a set of experimental results from a collection of nominally identical test pieces. In this paper the inverse error propagation and multivariate multiple regression is carried out to correct the finite element meta-model of a benchmark structure with spot-welds. Two main types of spot-weld models exist [2,3], namely those that require the stress within the weld spot to be calculated and those that do not. The former require detailed modelling so that a smooth stress field is computed, whereas in the case of the latter the estimation of stiffness and mass distributions is sufficient. When modelling spot-welds, it is difficult to take into account the many local effects such as geometrical irregularities, residual stresses, material in-homogeneities and welding defects. Furthermore, a detailed and complex local model of the joint characteristics is not possible in industrial applications where typically thousands of spot-welds are used to manufacture automotive parts. These conceptual modelling uncertainties are incorporated in an interface finite element [4] which may be used to connect panels at spot-weld locations. The material properties of the element determine the local stiffness in the tangential and normal directions at the surface. The interface element stiffnesses are used together with other parameters to update a structural and statistical model of a short beam manufactured from two components, one of folded steel and the other flat. The other parameters are two elastic moduli of curved regions in the folded plate. Taken all together they become the randomised parameters used in the Monte-Carlo simulation. In the following sections the stochastic model updating of a collection of nominally identical spot-welded structures, representative of spot-welded beam-like parts in car bodies, is carried out. Details of the finite element model, choice of randomised parameters, manufacturing details and application of the method are described.
2. The set of benchmark structures A set of ten nominally identical benchmark structures were built, each one consisting of two parts, a folded steel plate with a ‘hat’-shaped cross-section and a flat plate, connected by two rows of spot-welds along the flanges as shown in Fig. 1. It was intended to represent the structural dynamics and the manufacturing variability of beam-like structures, such as door pillars and roof supports that occur commonly in car-body shells. A manufacturing specification was developed with the intention of reducing manufacturing variability, which included the following instructions: (i) standard rolled sheet steel of nominal thickness 1.5 mm to be used; (ii) all plates to be cut from the same sheet; (iii) no material to be used within 30 mm of the edge of the sheet; (iv) all plates to be aligned lengthways parallel to the sheet rolling direction; (v) all hats to be folded using the same former, and by the same person. After cutting and bending, all ten plates and hats were measured, numbered and returned for spot-welding as numbered pairs. In order to isolate the spot-weld modelling uncertainties from any manufacturing variability present in the set of the benchmark structures, the plate and hat belonging to one specimen were tested separately before being welded together. The finite element models of the hat and flat plate were then updated deterministically using measured data from the
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Fig. 1. Benchmark structure.
chosen specimens. These updated models provided the starting point for updating the complete model stochastically using the full set of benchmark structures. The purpose was to isolate the spot-weld modelling uncertainty, since errors in plate thickness or fold radius were supposed to have already been corrected at the first step of updating the separate components. This is the usual approach when model updating is carried out on multi-component structures using a buildingblock approach.
3. Updating the separate components A modal test with free–free boundary conditions, 30 roving hammer points and two measurement points was carried out on the chosen flat plate specimen and the identified modes are presented in Fig. 2. The thickness of the plate was measured at different places and a mean value of 1.448 mm was used in the finite element model. With a mass and Young’s modulus correction the model was found to be very accurate for the bending modes but less accurate in torsion. The frequency errors and MAC values are presented in Table 1. The standard deviation of the plate thickness measurements was 0.0075 mm. The free–free modal test on the chosen hat component was also of the roving hammer type, this time using 40 hammer point and 4 accelerometers. The identified modes are shown in Fig. 3. Measurement of the thickness at different points on the hat showed a variation both along the span of the model and on the cross-section, the main differences appearing at the fold radii. The specified radii of 5 mm were in fact found to be approximately 4 mm and when this change was incorporated in the finite element model the correlation was improved considerably. The hat was divided in 5 regions, as indicated in Fig. 4. The thicknesses of the plate in those regions were used to update the model by minimising the predicted natural frequency errors. A floating-point genetic algorithm [5] with 30 individuals and 50 generations was used for the optimisation. The frequency errors and MAC values for the best solution are presented in Table 2. The genetic algorithm optimisation loop led to a number of sub-optimal solutions from which the one reducing the error in the first mode (with the greatest initial error) was chosen. In the case
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Fig. 2. Plate modes for the chosen specimen.
Table 1 Natural frequencies for the flat plate Mode
Test
FE
Error (%)
MAC
1 2 3 4 5 6
23.74 65.95 76.78 130.23 156.14 216.33
23.92 66.28 74.48 130.73 151.94 217.02
0.76 0.50 3.00 0.38 2.69 0.32
100 100 100 100 99 99
of updating the two components alone, there is an aleatory uncertainty related to the dimensions and material properties of the shells especially in the curved areas of the hat and in the updating process one faces an epistemic uncertainty (ignorance or lack of knowledge about the system due to limited experimental data). It is also possible that extension–shear-bending coupling effects reduce the performance of the shell element used and this might represent a conceptual modelling
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Fig. 3. Hat modes for the chosen specimen.
Fig. 4. Different regions of the specimen hat component.
uncertainty. The shell element that was implemented in MATLAB was known to be less accurate than the MSC-NASTRAN CQUAD element, which had been used at a preliminary stage. Choosing the best solution based on the frequency distance for a single sample and using these component models for all 10 samples means that in the next step, the modelling errors related to
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Table 2 Measured natural frequencies and finite element predictions in Hz for the specimen hat Mode
1 2 3 4 5 6 7 8 9 10 11
Exp
71.631 256.83 275.23 338.37 403.65 631.01 644.23 757.81 783.77 791.23 982.23
Initial
Updated
FE
Error (%)
MAC
FE
Error (%)
MAC
67.994 257.80 274.00 345.80 407.87 632.06 642.96 783.26 794.68 815.51 995.68
5.08 0.38 0.45 2.19 1.04 0.17 0.20 3.36 1.39 3.07 1.37
95 93 94 95 94 86 85 95 97 92 80
70.025 255.13 272.92 343.57 409.53 629.59 641.56 781.70 797.99 820.31 997.10
2.24 0.66 0.84 1.53 1.46 0.23 0.41 3.15 1.81 3.68 1.51
95 93 94 96 94 86 86 96 97 92 80
Table 3 Thickness in mm of the different hat regions after updating Upper curved zone
Lower curved zone
Vertical walls
Upper plate
Contact zone
1.400
1.400
1.593
1.512
1.600
this sample and not an average error, are propagated through the updating loop. The modification of the initial measured average thickness of 1.468 mm for all the zones is presented in Table 3.
4. Modal tests on the complete set of structures A free–free roving-hammer modal test with 70 hammer points and 4 measurement points was carried out on the specimen assembled from the chosen tested parts. For the other nine specimens a reduced test was performed with only 2 hammer points and 4 measurement points. The natural frequencies were matched based on a reduced wire-frame mode shape with the objective that any mode swapping or other type of experimental uncertainty would be investigated by further tests if needed. The mode shapes of the test for the chosen specimen are presented in Fig. 5 and were used later for MAC checks during the updating loop of the whole model. The natural frequencies for all the samples are shown in Table 4 where one can observe a standard deviation of about 1% for all the modes of the sample.
5. Stochastic model updating The spot-weld material constants ks1 ; ks2 ; kn represent tangential shear and normal direct stiffnesses for a special interface element described in detail in Appendix A. These three
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Fig. 5. Welded structure modes for the chosen specimen.
interface-element stiffnesses defined in Eq. (A.8), also in Appendix A, were selected as updating parameters. The three alone proved insufficient to give an acceptable updated structural and statistical model and it was decided therefore to include a further two parameters; the elastic moduli of the plate material at the upper and lower folds. The five chosen parameters are all equivalent ones, which it was hoped would enable the spot-weld uncertainty, its diameter, precise location, variations in material properties at and around the nugget, and variability in the fold radii and thickness of the steel plate to be represented in the updated model. In order to quantify the input parameters that account for a high percentage of the output variability in natural frequencies, a cloud of models was created by varying the 5 material constants according to a normal distribution. The assumption of normality for the input variables is dictated by the lack of a probabilistic model validated by experimental data from the spotwelds. Also the interface element is an equivalent model of the spot-welds as are the elastic moduli that represent variability in fold radii. Each spot-weld in the finite element model was represented by four interface elements with stiffness parameters having initial mean values shown in Table 5, based on an elastic modulus of 25% of the usual value for steel and allowed to vary between 5% and 150%. The elastic moduli of the folds were allowed to vary by 50% about the initial mean,
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Table 4 Natural frequencies in Hz for the complete sample Spec no.
1 2 3 4 5 6 7 8 9 10 Mean Std
Mode 1
2
3
4
5
523.39 521.88 526.8 525.99 512.19 523.74 517.09 522.06 526.97 528.90 522.90 5.04 (0.96%)
589.06 588.45 594.17 592.72 580.24 590.74 586.05 589.71 592.12 596.16 589.94 4.49 (0.76%)
600.24 588.45 602.73 595.92 580.24 599.64 586.05 590.56 606.56 604.48 595.48 8.76 (1.28%)
682.14 689.14 688.81 679.4 680.71 678.83 675.64 677.63 689.82 688.52 683.06 5.45 (0.80%)
713.45 708.87 715.16 706.3 702.19 706.68 702.84 706.06 720.77 718.52 710.08 6.50 (0.91%)
Table 5 Initial and updated mean parameter values
Initial Final
E1
E2
Ks1
Ks2
Kn
2.10E+11 1.89E+11
2.10E+11 2.31E+11
5.00E+10 5.63E+10
5.00E+10 2.50E+10
5.00E+10 4.69E+09
being the usual nominal value for steel plate. The initial variance matrix contained the same data given in Table 4. Updating was carried out on the basis of the first five measured modes from the sample, mean values being given in Table 6. The number of observations for the experimental and analytical models influences the size of the confidence ellipse and a convergent shape is obtained from the T 2 statistic with degrees of freedom p; n1 þ n2 2 as described in Appendix B. It is shown in Fig. 6 that the stabilisation of the statistic T 2 is obtained closely when n2 ¼ 200 analytical models and n1 ¼ 10 experimental tests. n2 ¼ 200 becomes the dimension of the meta-model, to be displaced to the location in the output space given by the experimental data by variation of the randomised updating parameters. The reduction of the Euclidian distance between the models in the output space was obtained by using the gradient and regression method described in the companion paper [1]. In the simulation, ten iterations were completed and at each step the covariance was updated a posteriori, the minimum being obtained after 5 iterations. The scatter plots of the initial and final cloud are presented in Fig. 7 and in separate planes in Figs. 8–10 and the mean natural frequencies of the model, the mean parameter values and standard deviations of the initial and final iterations are presented in Table 5–7. The strength of the relationship between input and output variables in the model has been measured by the Spearman correlation matrix presented in Fig. 11. The analysis of the material
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Table 6 Mean natural frequencies Mode
1 2 3 4 5
Initial
Final
Test
Frequency (Hz)
Error (%)
Frequency (Hz)
Error (%)
Frequency (Hz)
533.28 627.10 685.89 768.82 775.88
1.98 6.29 15.18 12.55 9.26
512.48 593.99 604.77 687.28 692.59
1.99 0.68 1.55 0.61 2.46
522.90 589.94 595.49 683.06 710.08
Fig. 6. T2-statistic for determining the optimal sample number.
coefficients shows that the most important parameters at the beginning of the updating loop are the material constants of the curved areas while closer to the test data the significance of the input variable related to the normal displacement between the welded shells becomes more important. The material constant related to the sliding between the two surfaces are less important, because the modes concerned involve almost no shear behaviour. The distance between the means of the parameters of the experimental and analytical samples can be determined from the confidence ellipsoid. Then, following the analysis given in Appendix B, the norm of the observed difference between the mean outputs from the final analytical
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Iteration #1
Iteration #5
720 640
700
630
680
620 610
640
600
f3
f3
660
620
590
600
580
580
570 560 620
660
560
540
460
480
500
520
590
(b)
580 570
490
500
520
510
530
540
f
580
600 f2
f2
(a)
610
560
1
600
540
f
620
1
640
Fig. 7. Analytical cloud and target cloud in the output space together with their projection on each plane (a) initial; (b) after 5 iterations.
660 600
640
595 f2
f2
620 600
590
580
585
560 480
(a)
500
520 f1
540
580
560
(b)
505
510
515 f1
520
525
530
Fig. 8. Scatter plot of the analytical cloud and the target cloud in the (f1–f2) plane: (a) initial; (b) after 5 iterations.
population and the experimental data is d ¼ j¯ytest y¯ analysis j ¼ 23:40 and the eigenvalues of the covariance matrix SD ¼ ð1=n1 þ 1=n2 ÞSpl are 38:390; 0:831; 0:140; 0:015; 0:010 . The semi-axes of the 95% confidence ellipsoid are found to be ci ¼ f 21:00; 3:09; 1:27; 0:42; 0:34g. Therefore for the hypothesis to be accepted the origin should lie between jdj 21:00 ¼ 2:40 and jdj þ 21:00 ¼ 44:40, which it does not. This analysis shows that da0 and the means of the two populations in the final iteration are therefore not equal at the 95% confidence level. However, it can be observed from the scatter plots, shown in Figs. 8–10 that considerable convergence of the cloud of analytical results upon the test data has occurred and that the sizes and orientations of the ellipses shown in Figs. 8–10 are similar, though not exactly superimposed one on the other of course. The final set of analytical models has natural frequencies with means and standard deviations comparable, from an engineering point of view, with the experimental data from the ten specimens. The frequency errors are in fact much reduced with respect to the test mean and the mode shapes are in very good agreement. The parameter with the most
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700 680
620
660
610 f3
f3
2181
640
600
620 590
600 580
580 500
480
520 f1
(a)
540
560
505
510
515
520
525
530
f1
(b)
Fig. 9. Scatter plot of the analytical cloud and the target cloud in the (f1–f3) plane: (a) initial; (b) after 5 iterations.
630 700
620
680
610 f3
f3
660 640
600
620
590 600
580 580
580 560
580
600
620
640
660
f2
(a)
585
590
595
600
f2
(b)
Fig. 10. Scatter plot of the analytical cloud and the target cloud in the (f2–f3) plane: (a) initial; (b) after 5 iterations. Table 7 Standard deviation
Initial Final
E1
E2
Ks1
Ks2
Kn
4.20E+10 4.85E+09
4.20E+10 1.15E+10
1.00E+10 7.90E+09
1.00E+10 9.06E+09
1.00E+10 7.36E+08
important variation is the material constant kn and this with all the other parameters shows reductions in the standard deviation as a result of convergence towards the test data. The changes to the spot-weld stiffnesses incorporate most of the variation in the tested specimens. Therefore the likely reason that updated population is different from the test population is that the updating parameters are not capable of truly representing all the differences between the test structures and the meta-model. The same parameters describe all the spot-welds although in reality each spot-weld is different. If the spot-welds were parameterised separately each one would have a
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1.2
1.2 1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2 1
1 Up 2 da tin 3 gv 4 ar iab les
5 4 3 5
2 1
es
Mod
Up 2 da tin 3 gv ar iab
5 4 les
3
4 s ode
5
2
M
1
Fig. 11. Spearman correlation matrix between the input and output variables for the initial and final cloud.
similar effect, therefore giving rise to ill-conditioning. There may of course be other variations with random distributions sufficiently dissimilar to the spot-weld and fold errors so that the two statistical populations will be different and cannot be made the same by adjusting these equivalent parameters. This appears to be an important result, which confirms what we have already understood about model updating, that engineering judgement is of paramount importance in its application.
6. Conclusions The stochastic model updating method has been applied to a set of benchmark structures with spot-welds. The spot-welds are modelled by using an interface zero-thickness element with the properties of a spring membrane at the contact between shell elements. A hierarchical approach is used in the study by analysing different sources of modelling uncertainty and manufacturing variability and their effect on model updating and inverse error propagation through a finite element model, which is converged upon multiple sets of experimental data. Results are obtained which show that the means of the meta-model fail to converge upon set of test structures within a 95% confidence limit. Never-the-less good convergence is obtained from the engineering viewpoint. The results confirm what we already know about model updating; that engineering judgement is extremely important. This observation is perhaps especially relevant when the purpose is not only to converge a structural model but also a probabilistic one.
Acknowledgements The research reported in this paper is supported by Engineering and Physical Sciences Research Council (EPSRC) grants GR/R26818 and GR/R34936.
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Appendix A. Equivalent spot-weld finite element model A generally applicable and simple joint/interface element for 3D analysis is described by Beer [4]. The element can model interfaces between solid and shell finite elements. The formulation is quite general, the element has zero thickness and the derived element is isoparametric. On the contact surfaces of the two shell elements, the displacements at any point are given by 8 9 8 9 ( ) > < ui > = = X > X di ai v ¼ Ni v i Ni ½ v2i v1i TOP , (A.1) bi > > 2 :w > ; : > ; TOP w TOP i TOP 8 9 8 9 > = < ui > = X > X di v ¼ Ni vi þ Ni ½ v2i > > 2 : > ; :w > ; w BOT i BOT
( v1i BOT
ai bi
) ,
(A.2)
BOT
where the Ni are the isoparametric shape functions, the v1i ; v2i are the unit vectors in the direction of the local tangent axes, ai ; bi are the rotation angles for the same axes and d i is the thickness of the shell at each node. These equations may be rewritten in the form 8 9 > = v ¼ NTOP;BOT aTOP;BOT , (A.3) > : > ; w TOP;BOT
aTOP;BOT ¼ ½a1 ; a2 ; . . . ; an TOP;BOT ;
8 9 ui > > > > > > > > > vi > > < > = a i ¼ wi , > > > > ai > > > > > > > :b > ;
(A.4)
i
NTOP;BOT ¼ ½N1 ; N2 ; . . . ; Nn TOP;BOT , di di Ni ¼ N i I; v2i ; v1i . 2 2
ðA:5Þ
The displacements in the directions normal and tangential to the contact surface are obtained by 8 9 8 9 > > = = v ¼ ½ s1 s2 n v , (A.6) > > ; : > : > ; w w where s1 ; s2 ; n are the tangent vectors and the normal vector to the surface area. The relative displacements at the surface are slip in the two tangent directions and the normal convergence or
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separation: 8 9 8 9 8 9 > > = = < ds1 > = > ds2 ¼ v v . > > : > ; : > ; :d > ; > w TOP w BOT n
(A.7)
The following relationships are defined between the tractions acting on the contact and relative displacements: 8 9 2 38 9 ks1 0 0 > > < ts1 > = < ds1 > = 6 7 ts2 ¼ 4 0 ks2 0 5 ds2 ; t ¼ Dd, (A.8) > > > :t > ; : ; 0 0 k d n
n
n
where ts1 ; ts2 ; tn are the shear tractions and contact pressure and ks1 ; ks2 ; kn are the shear and normal stiffnesses.
Appendix B. Aspects of statistical inference Hypothesis testing may be used to assess the quality of a model and its parameterisation. In the case of multivariate normal populations these tests are sufficient and numerical studies may be used to confirm robustness to deviations. When a large number of variables are used for simulation, data reduction becomes essential for numerical stability, Principal Component Analysis [6] being the main tool for this type of analysis. Measures of association such as the Pearson and Spearman coefficients [7,8] can identify those inputs that account for a high percentage of the output variability. In the present gradient-and-regression approach a test on the means (without reference to the covariances) may be carried out using a T 2 distribution due to Hotteling in 1931 [9]. The hypothesis, H 0 , that the means of the sampled outputs from two data sets originate from the same parent distribution is tested by the statistic: 1 1 1 n1 n2 T 2 þ ð¯y1 y¯ 2 Þ ¼ D2 , (B.1) Spl T ¼ ð¯y1 y¯ 2 Þ n1 n2 n1 þ n2 M Spl ¼
1 ½ðn1 1ÞS1 þ ðn2 1ÞS2 ; n1 þ n2 2
(B.2)
where the y¯ 1 ; y¯ 2 ; S1 ; S2 are the mean vectors and the covariance matrices for each sample, Spl is the pooled sample covariance matrix, DM is the Malhalanobis distance [7,8] and the T 2 is expressed in ‘characteristic’ form. When H 0 is true the quantity T 2 is distributed as T 2p;n1þn22 where the distribution is indexed by two parameters, the dimension p and the degrees-of-freedom n 1 for one sample or n1 þ n2 2 for two samples. It is necessary that n1 þ n2 240 for Spl to be nonsingular. Since an observed output is generally not equal to the true value, inference upon the equality of the means y¯ 1 ; y¯ 2 (or covariance matrices) of the two populations requires an estimate of uncertainty. The concept of the confidence region applied to the distance between population means (described by the Mahalanobis distance) shows that the confidence ellipse converges upon the stabilisation of the T 2p;n1þn22 statistic and this result should be used for choosing an optimum
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number of samples for the analytical population. The difference between the true means of two populations l1 ; l2 is bounded more accurately by using the confidence ellipsoid [10]: 1 1 1 T 2 T ¼ ½ð¯y1 l1 Þ ð¯y2 l2 Þ þ ½ð¯y1 l1 Þ ð¯y2 l2 Þ (B.3) Spl n1 n2 or T 1 T 2 ¼ ðd dÞT S1 D ðd dÞ ¼ D SD D,
(B.4)
d ¼ y¯ 1 y¯ 2 ; d ¼ l1 l2 ; 1 1 SD ¼ Spl . þ n1 n2
(B.5)
D ¼ d d,
(B.6)
For the T 2 statistic a 100ð1 aÞ% confidence region is defined by, 2 ðd dÞT S1 D ðd dÞpT a;p;n1þn22
(B.7)
By determining the eigenvalues and eigenvectors ðK; AÞ of SD , the confidence region is bounded by a p-dimensional hyperellipsoid centred at d with semi-axes given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (B.8) ci ¼ li T 2a;p;n1þn22 ; i ¼ 1; :::; p. The hypothesis on equal means d ¼ l1 l2 within a 100(1a)% confidence interval is then accepted if the origin is enclosed by the hyperellipsoid such that jdj ci p0pjdj þ ci
(B.9)
for any one of the semi-axes. The hypothesis is rejected if the origin lies outside the hyperellipsoid.
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