advanced numerical prototyping methods in modern engineering applications

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• running up an optimization procedure to find out thebest levels of the essential factors due to the expectedresponse in the evaluated region.

2. The advanced DOE approachThe classical DOE experiment is based on an

orthogonal experiment scheme where mostly points areplaced at the extreme values of the design space [13].Alternative approach and more valuable, in case ofcomputer experiments, would be based on spreadingpoints over the whole design space. This can be done by anumber of methods: random, Monte Carlo, Latin-Hypercube (LH) scheme of which the LH scheme seemsto be the most attractive.

The LH scheme belongs to the space filling DOEschemes and is a kind of a random experiment where then experiments points are scattered in the domain in anorganized way, in comparison to, e.g. purely randomexperiment (e.g. Monte Carlo). LH specification describesit as an orthogonal array that randomly places samplesover the entire design space broken down into rn equal-probability regions (where r is the number of runs, and nis the number of input variables). The LH scheme can belooked upon as a stratified Monte Carlo sampling wherethe pair-wise correlations can be minimized to a smallvalue (which is essential for uncorrelated parameterestimates) or else set to a desired value. Additionally inLH design experiment in comparison to Monte Carloapproach, the test points in design region do not cluster.LH is especially useful in exploring the interior of theparameter space, and for limiting the experiment to afixed (user specified) number of runs. The LH experimentis constructed as follows:• selecting the number of tests n that are to be simulated,• dividing each factor dimension into n equidistant

levels, • sampling for each factor n random permutation of the

levels,• combining permutation of the factors’ levels into a

simulation scheme.

In case of a non-box region, more levels than test pointsare selected and then randomly generated LH on the finerlevel grid. If the LH design is infeasible, the process isrepeated while increasing the number of levels. The LHexperiment for a non-box region is referred to asconstrained LH design. In the practise different LH design

schemes exist mainly because there are number ofpossibilities to assign levels to factor dimensions. It canbe done, for instance, uniformly or randomly. Much of theattention is directed towards so called maximum distancesimulation scheme for which the minimal distancebetween test points is maximal. The minimal distance is ameasure of the space region fillingness:

Dmin=max (1)It seems like that one of the basic drawbacks of the LHdesign is a random selection of the test points. In most ofthe engineer cases it is a rule of thumb that certain pointsare more valuable than others, especially from the expertpoint of view. Therefore it was suggested to change thestandard LH scheme so as to include the prior knowledgeof an expert on the domain behavior, while keeping thecurrent „specification” of LH or to change it slightly. Inthe following table are listed the proposed changes of thestandard LH scheme in order to cope with the aboveassumptions.

Tab. 1. The proposed changes to the standard LH designscheme (Figure 3).

The proposed LH modification ColourSelecting the "key points or sub-domains" ofthe experiment

greenpoint

Condensing the experiment points in regionswhere the high non-linearity is expected

blue lines

Including the points being close to thedomain border

yellowpoint

Excluding the sub-domains that seem to beless essential in the response analysis

redrectangle

Generating n LH design schemes andselecting the one that satisfies the Equation 1

browncircle

It was decided that the main problem of the modified LHdesign refers to the problem of sub-region condensing:• in standard LH design the sub-regions are equidistant,• in the modified LH design there is an additional

function, put over the standard LH design, based onthe elaborated family of a triangle three-parameterdensity distribution (Figure 4):

Fig. 2. The Latin Hypercube experiment scheme.Fig. 3. The modified LH design scheme

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3.2. Error function definitionIn computer simulation (deterministic by its nature)

the error is totally due to modeling error and not tomeasurement error or inner or outer noise. Therefore it isjustified to treat the error εi as a continuous function of xi:

i=i x i (6)The errors could be considered as correlated by a distancefunction between the points. If the points are closetogether, then the errors should also be similar, whichmeans high correlation. Therefore it can be assumed thatthe correlation between errors would be related to thedistance between the corresponding points. A specialweighted distance formula can be used as the distancefunction:

d x i , x j =∑h=1

k

h∣xhi −xh

j∣ph (7)

where Θ>0 and ph∈[1,2]. Using this distance function, thecorrelation between the errors can be defined as follows:

corr [ x i , x j ]= 1 ed x i , x j (8)

The defined correlation function has obvious propertiesas: in case of small distance the correlation is high whilein case of large distance the correlation will approachzero. The values of the correlation function define thecorrelation matrix R of the order n×n, which has practicalmeaning in the final response model definition:

R=[r1,1 ⋯ r1, n

⋮ . .rn ,1 . rn , n

] (9)

where

r i , j=corr [ x i , x j ] (10)Thanks to the defined correlation function and thecorrelation matrix R it is possible to get a simple linearregression model and avoid a quite complicated functionalform of the response. The evaluation of the so definedstochastic model has a very important virtue, which allowsto replace the regression terms by the constant value µ

=∑h=1

k

h f h x i ; i=1, , n (11)

and the same the Equation 5 can be rewritten as:y x i = x i ; x i N 0,2 (12)

3.3. Estimation of the interpolation errorIn order to define the stochastic model of the response

it is required to estimate 2k+2 parameters: µ, δ2, θ1,..θ2,p1,…pk. This task can be achieved by maximizing thelikelihood function F of the sample, which is defined as:

F= 1 2n/22n/2∣R∣1/2

⋅ 1

e y−1 ' R−1 y−1

22(13)

where 1 denotes the n-vector of ones and y denotes the n-vector of observed function values, given by:

y= y1 , y2 , , yn' (14)

The estimators of parameters m and d2 that maximize thelikelihood function are given in a closed form by:

=1 ' R−1 y1 ' R−1 1

2= y−1 ' R−1 y−1

n

(15)

The best linear unbiased estimator of the response value yat point x^ is defined as:

y x∧= r ' R−1 y−1 (16)where the r is the n-vector matrix given as follows:

r i x∧=corr [x∧ ,x i ] (17)It is very important to assess the estimation of theprediction accuracy at point x^, which can be evaluated asthe mean squared error s2(x^) as follows:

s2 x∧=E [ y x∧−y x∧2 ]= 2 [1−r ' R r1−1 ' R−1 r '1 ' R−11 ] (18)

The values of matrix R depend directly on parameters(θh,ph). In most of the practical application the parameterph is set to 2 while the value of Θ parameter is estimated.The Θ parameter, can be treated as a measuring factor ofthe importance of the variable xh, which can be related tostatement that even small values of

d hi , j=∣xh

i −xhj∣ (19)

may lead to large differences in the function values at xi

and xj. Therefore from the statistical sense it can be stated:• if parameter Θh is large then small values of dh

i,j wouldlead to high distance d value and hence low correlation

• if parameter Θh is small then small values of dhi,j would

lead to small distance d value and hence highcorrelation

The elaborated approach to Θ parameter estimation wasbased on the calculus of variations and finding theextreme of the defined parametric functional F(Θ , p) as atotal error of the interpolation, which was defined asfollows:

F , p =∫x '

x ' '

s x , , p d x (20)

where x is a vector of the prediction points, the s(x) is thesquare root of the mean squared error of the interpolation.According to the previous considerations it can be noticedthat the both matrix R and vector r are the functions of θand p while the estimator of δ2 is a function of thecorrelation matrix R. As the parameter p is set at value 2,therefore it was suggested that the best estimator of θ canbe found as the value that minimizes the functional:

=min

F , p=2 (21)

As we deal with the sampled data the estimation of θrequires application of numerical operations including:integrals and optimisation. The Equation 21 allows toestimate the multidimensional θ parameter as well.

3.4. Global optimization algorithmThe idea of the global optimization algorithm using

the stochastic process model is to use the interpolationcurve (predictor) together with the knowledge about theaccuracy of the interpolation (predictor) in order toiteratively add points in the design space in thoselocations where the expected improvement of theobjective function is the highest, which is illustrated in theFigure 5.

The figure presents the 5 experiment points and the

interpolation curve along with the interpolation error. Thestochastic interpolation model seems to indicate that theglobal minimum for the curve is around 3, but if we takeinto account the standard error of the prediction, which isvery high around 8, it could also be possible that theglobal minimum in our region of interest is somewherearound this value. There is some probability that thefunction value around 8 will be better that the best pointthat is predicted only based on the 5 points. The expectedimprovement can be written in the following formula:

EI x = f min−y f min−ys s f min−y

s (22)

where fmin is the best experimental function value found atthis stage, and Φ and φ represent the standard normaldensity and distribution functions. The proposed iterativeapproach for the advanced numerical prototypingconsisted of the following steps:1. generate a first set of samples, which form the basis

for a first RSM based on the stochastic process model.2. create the stochastic process model.3. using a global optimization algorithm, find m points

within the design space where the expectedimprovement will be the highest.

4. if the expected improvement is lower than a specifiedtolerance, exit, otherwise:

5. compute exact responses in the m points of interest,add them to the set of data, and go to 2.

4. Verification tests

In order to verify the elaborated methodologyconcerning the advanced numerical prototyping a numberof analytical functions were selected and tested. Theadvantage of the above approach is an ability to assess theaccuracy of the applied algorithms. This can be done by

comparing the interpolated curve with the real analyticalfunction, as follows:

e=∣∫∫ f x , y dx dy−∫∫ f i x , ydx dy∣ (23)

where f(x,y) and fi(x,y) were the real and interpolatedfunction respectively.

For the verification reasons a number of tests wereperformed with selected analytical functions: monotonic,extreme, periodic. The chosen functions representedtypical outputs that could be expected in case of realproduct or process. The verification of the advancedalgorithm was performed according to the scheme: • initial number of tests N=9 generated according to the

modified LH design scheme, as an alternative to thefull factorial for two factors and three levels,

• a total number of tests during the iterative stage N wasequal 30, in order to find out the tendency of aninterpolation error for a selected analytical function.

4.1. Monotonic function.The monotonic function (Figure 6) was defined by the

following equation:f x , y=e xe y (24)

The generated initial DOE experiment based on themodified LH scheme at N=9 tests is shown in Figure 7.

Fig. 5. Prediction and standard error of the stochasticprocess model

Fig. 7. Initial DOE experiment based on modified LH.

Fig. 6. Monotonic function.

The interpolated surface and estimated distribution ofthe interpolation error after the initial experiment arepresented in Figure 8. The initial value of parameterwas estimated to be equal 1 and the final 1, as well.

The interpolation error as a function of a number ofexperiment tests N during the iterative procedure is shownin Figure 9.

It can be noticed that in case of the monotonicfunction the applied advanced numerical prototypingmethod gave very good agreement between the analyticaland interpolated curve even after the initial experiment(the estimated interpolation error e=0.07). During theiterative procedure the interpolation error was quicklydecreasing and after N=20 tests it reached the plateauregion.

4.2. Extreme functionThe extreme function (Figure 10) was defined by the

following equation:f x , y =e{−20 ⋅[ x−0.652x−0.702]} (25)

The generated initial DOE experiment based on themodified LH scheme at N=9 tests is shown in Figure 11.

The interpolated surface and estimated distribution ofan interpolation error after the initial experiment arepresented in Figure 12. The initial value of parameterwas estimated to be equal 8 and the final 11.

Fig. 10. Extreme function.

Fig. 8. The interpolated monotonic function after theinitial DOE experiment and the estimated distribution ofan interpolation error.

Fig. 9. The dependence of an interpolation error on anumber of tests N.

Fig. 12. The interpolated extreme function after theinitial DOE experiment and the estimated distribution ofan interpolation error.

Fig. 11. Initial DOE experiment based on modified LH.

The interpolation error as a function of a number ofexperiment tests N during the iterative procedure is shownin Figure 13.

It can be noticed that in case of the extreme functionthe applied advanced numerical prototyping method gave,similarly to the monotonic function case, pretty goodagreement between the analytical and interpolated curvealready after the initial experiment (the estimatedinterpolation error e=0.12). During the iterative procedurethe interpolation error was quickly decreasing but theplateau region was reached after N=30, which is more incomparison to the monotonic function case. Additionallythe estimated interpolation error e was still roughly 10times higher than for the previous case.

4.3. Periodic functionThe periodic function (Figure 14) was defined by the

following equation:f x , y =esin 10 xe y (26)

The generated initial DOE experiment based on themodified LH scheme at N=9 tests is shown in Figure 15.The interpolated surface and estimated distribution of aninterpolation error after the initial experiment are

presented in Figure 16. The initial value of parameterwas estimated to be equal 23 and the final 16.

The interpolation error as a function of a number ofexperiment tests N during the iterative procedure is shownin Figure 17.

Fig. 14. The periodic function

Fig. 15. Initial DOE experiment based on modified LH.

Fig. 17. The dependence of an interpolation error on anumber of tests N.

Fig. 16. The interpolated monotonic function after theinitial DOE experiment and the estimated distribution ofan interpolation error.

Fig. 13. The dependence of an interpolation error on anumber of tests N.

It can be noticed that in case of the periodic functionthe applied advanced numerical prototyping method gavethe worst results. The interpolated surface after the initialexperiment (Figure 16) does not reflect the real behaviourof the original analytical function (Figure 14) Theestimated interpolation after the initial experiment wasroughly 10 times higher than for the monotonic functioncase. Though during the iterative procedure theinterpolation error was decreasing it has not reached theplateau region even after N=30 tests. In fact theinterpolation error at N=11 and N=21 increased incomparison to the initial experiment. This is mainly due tothe changing value of the estimated Θ parameter aftereach run of the iterative procedure.

Finally it can be concluded that the one-dimensionalKriging interpolation method seem not to be appropriatein case of functions with different direction dependence,e.g. linear response in one direction and periodic responsein the other. The above can be improved by introducingthe multidimensional Θ parameter estimation. Theimplementation of the multidimensional Θ parameterestimation will be described in a separate paper.

5. Conclusions

One of the bottlenecks of wide exploitations ofnumerical prototyping methods is the efficiency ofdeveloping accurate response surface models. Unlike theconventional DOE method, the preliminary results of ourapproach proposed in this paper allow substantialreduction of DOE, and at the same time, the accuracycriteria of the response surface models can be satisfied.

6. Acknowledgments

The advanced numerical prototyping methoddescribed in this paper is part of the project results ofMEVIPRO, financed by the EC in the 5th EuropeanResearch Program. The authors would like to thank all theMEVIPRO team members for their valuable inputs andcontributions.

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