ABSTRACT - WIT Press · number of iterations required to achieve convergence. In these numerically...

The effectiveness of optimisation techniques

for material property characterisation

E.J. O'Brien, J.J. O'Donnell

Department of Civil, Structural and Environmental

Engineering, Trinity College, Dublin, Ireland

ABSTRACT

The paper describes the application of a technique whereby the parameters

which characterise a physical process can be found. The technique is based on

the use of optimisation to find those values for the characteristic parameters

which give a best fit of the measured results to a chosen numerical model.

Typically these parameters represent the physical properties of the materials

involved in the process. Factors which influence the effectiveness of the

technique are described and methods are developed for the assessment of its

suitability for various applications.

INTRODUCTION

There are many situations where the parameters which characterise a physical

process are difficult to determine directly. When modelling such a process

numerically, it is sometimes convenient to determine characteristic parameters

indirectly from measured results. This can be achieved by finding those values

of the characteristic parameters which give a best fit of the measured results to

the numerical model.

In any numerical model of a physical process, three groups of parameters can

be identified. The first group consists of the characteristic parameters, x. ; i =

1, 2, ..., n, the parameters for which values are sought. The second, consists

Transactions on Modelling and Simulation vol 5, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

46 Computational Methods and Experimental Measurements

of those parameters used in the 'best fit' process, i.e., those for which

differences between measured values, ff ; i = 1, 2, ..., p, and values foundfrom the numerical model, f. ; i = 1, 2, ..., p, are minimised. All other

parameters belong to the third group, y. ; i = 1, 2, ..., m, and values for these

are either measured, known or assumed. Parameters of the second group, as

calculated using the numerical model, can be expressed as:

for i=l,2, ..., p

The best fit process consists of minimising the sum of squares of differences

between these calculated values and the corresponding measured values, i.e.,

Find x. ; i = 1, 2, ..., n

to minimiseP 22^ ̂ " *i '

The minimisation problem can be solved using standard optimisation

algorithms.

A MEASURE OF ILL-CONDITIONING

Two problems, each involving two characteristic parameters, x^, x^, two

parameters of the second group, f^, f^, and one other parameter, y^, are

illustrated in Figures la and Ib. As only two parameters are used in the best fit

process, a unique solution can be found at P. In many cases, there are errors in

the numerical model and/or the measurements (f^ and f^ in these examples).

As the curves in Figure la are almost parallel at P, the solution (namely, the

coordinates at P) is sensitive to such errors and the problem can be termed ill-

conditioned. In the general case, where the solution is being found by

optimisation, it can be quite difficult to obtain an accurate solution to ill-

conditioned problems such as this. The problem of Figure Ibis less sensitive

to such errors and is, in contrast, well conditioned.


Computational Methods and Experimental Measurements 47

a) Ill-Conditioned b) Well Conditioned

Figure 1. Conditioning of Two-Dimensional Problems

A measure of the difference in slope of the two curves at P is the area of the

parallelogram constructed about the unit normal vectors as illustrated in the

figure. Clearly, a small area implies an ill-conditioned problem while an area

close to unity implies that the problem is well conditioned. This area can be

calculated numerically as the determinant of the matrix, [G], where,

[G] =

i, i

|Vfi

h, 2

VfJ

(1)

In this equation, f. . denotes the partial derivative off with respect to x., i.e.,



and JV fi| is the length of the gradient vector for f., i.e.,

N = V#l(3)

For a problem in which n parameters are used in the best fit process to

determine n characteristic parameters, the corresponding matrix is, [H], where,

fl, 2 fl, n- • • • -

[H] =

|v f i| |v f i| |v fi|

jVfJ

fn, 1 fn, 2

IvTIlVfl.r N |v in|

(4)

and where,

v ̂ - v&r 9x2,(5)

The determinant of [H] has been used by the authors as a comparative measure

of the conditioning of problems. As for two-dimensional problems, a

determinant close to zero implies ill-conditioning while a value close to unity

implies a well conditioned problem.

REDUNDANCY OF PROBLEMS

In many cases it is possible to use many more parameters in the best fit process

than there are characteristic parameters, i.e.,

p > n

In such cases, it is often not possible to obtain exact agreement between all

measured values and those calculated using the numerical model.



A two-dimensional example with a redundancy of one is illustrated in Figure 2.

The best fit solution, P, clearly does not satisfy any of the equations exactly. A

simple measure of how closely the numerical model fits the measured results is

the sum of squares of differences between measured and calculated values.

This is, of course, the value of the objective function at the optimum solution.

Figure 2. Redundant Problem in Two Dimensions

As before, a measure of ill-conditioning is the area of a parallelogram or the

multi-dimensional equivalent of this. However, for redundant problems in

n-dimensions, there are more than n normal vectors. It can be seen in Figure 2

that the curves defined by,

and

fi(xi, x:, yi) =

, X2, yi) =

are almost parallel. However, as the third curve is approximately perpendicular

to them at P the problem is well conditioned. It can be seen that the

parallelogram defined by the vectors, nj and ng is an appropriate measure of



conditioning for this example. In general for two-dimensional problems, the

parallelogram of greatest area defined by any pair of vectors would seem to be

most appropriate.

The unit normal vectors are illustrated in Figure 3 for a problem where n = 3

and p = 5. As each of these vectors is of unit length, each end point is located

on a sphere of unit radius as illustrated in Figure 3a and 3b. The points A, B, C

and D in this figure are defined by the intersection with the sphere of planesparallel to the x^ and x^ axes (two parallel to each). The volume of a solid

defined by the vectors PA, PB, PC (and PD) would give some measure of the

conditioning of this problem as all unit normal vectors would be 'enclosed' by

this. However, there is a skew in the orientation of the unit normal vectors

a) Unit Normal Vectors

b) Sphere of Unit Radius

and Planes Parallel to

Original Axes

c) Sphere of Unit Radius

and Planes Parallel to

Transformed Axes

Figure 3. Unit Normal Vectors in 3 - Dimensional Problem



which this volume would not reflect. Therefore, linear regression has been

applied to identify the plane passing through P, which gives a best fit to the end

points of the unit vectors. The axes are then transformed so that the x^ and x%

axes lie in this plane and the, x^ axis is perpendicular to it. Then, planes

parallel to the transformed axes are found which just enclose the end points of

all unit normal vectors. The result is illustrated for the three-dimensional

problem in Figure 3c where it can be seen that the appropriate solid is defined

by the vectors PE, PF, PG (and PH).

THE INFLUENCE OF SCALING

The authors have used the conjugate directions method [1] to solve the

optimisation problem. While this technique is relatively inefficient, it has been

found to be extremely robust for problems involving discontinuities of first

derivative and/or sudden changes in second derivative. For the subroutine

used, the rate of convergence was found to be sensitive to the relative scaling ofthe variables. For example, convergence was found to be poor when x^ ranged

in value from -3000 to +3000 while x% ranged from -2 to +6. Speed of

convergence was restored when the initial step length used in the search

procedure for each variable was calculated as a fixed percentage of the expected

range.

The scaling of the variables clearly also affects the angle between unit normal

vectors such as those illustrated in Figure 2. To remove this effect in a manner

consistent with that used for the optimisation, each partial derivative in matrices

[G] and [H] is multiplied by the range of the appropriate variable. Thus, if xj isthe expected range of the variable, x., then the matrix [H] becomes:



f 1,2X2

gl

f2, 1

|gd

n, 1 Xi fn, 2 X2 In, n :• • • • —i—r

Ignl Ignl

(5)

where,

(6)

THERMAL CONDUCTION

The temperature increase at an internal node, i, on the axis of a hydrating

concrete prism can be expressed as,

where

fi = xi exp [x2 (l/y2 - 1/yi)] + xs ys (7)

y^, y^, y^ = known parameters (over which there is control)

Xj, x^, x^ = characteristic parameters

In a real thermal conduction problem, Equation 7 cannot be applied directly as it

does not allow for heat loss through the ends and sides of the prism. However,

it is used here to illustrate the principles described above. Three numerical

examples are considered first. In each case, temperature increases were

calculated at three different nodes along the prism. In each example,

'measured' temperatures were generated numerically using values for thecharacteristic parameters of, x^ = 1.630 x 10" , x^ = -5683 and x^ = 1.520 x

10" . This exact solution to the optimisation problem was subsequently used toassess the accuracy of the optimisation procedure. The variables, y , y^, and

yy can be controlled by the experimenter. For the three examples considered,

different combinations of values were chosen as presented in Table 1. The



Table 1. Details of Thermal Conduction Examples

Example

1

2

3

EquationNumber, i(Node No.)

1

2

3

1

2

3

1

2

3

Group 3 variables

?i

303

303

303

303

303

303

333

333

333

?2

303

302

301

303

298

293

333

303

273

?3

99

100

101

90

100

110

100

200

300

# ff

3.135

3.052

2.974

2.998

2.710

2.531

3.150

3.341

4.598

determinant of matrix [FT], as given by Equation 5, was calculated for each

example. The results are given in Table 2. It can be seen from the table that,

while none of the examples are very well conditioned, the third is significantly

better than the other two. The ill-conditioning of the former two is evident from

the relatively large optimum value for the objective functions and the large

number of iterations required to achieve convergence. In these numerically

generated examples the exact solution has been used to calculate the % errors in

the optimal solutions. It can be seen that the errors in the values inferred for the

characteristic parameters vary from up to 38% in Example 1 to 0.1% in

Example 3. Clearly, there is a strong correlation between accuracy and the

determinant of [FT].

In a fourth example, experimental data was used to determine values for the

same characteristic parameters as considered above. Details of the experiment

in which measured temperature changes at nine locations were used to infer

values for six characteristic parameters are given in [2]. For this example,

temperature changes over a one hour period were considered. In order to



Table 2. Results of Thermal Conduction Examples

Eg.

1

2

3

Det [H']

xlO^

0.84

104

986

No.

Iters.

6798

3500

392

Objective

Fn. x 10^

588

122

0.82

*i(Error)

1.034x 10-4

(37%)

1.480x 10-4

(9.2%)

1.628x 10'4

(0.1%)

*2

(Error)

-7254

(2896)

-6293

(10.7%)

-5682

(0.196)

*3

(Error)

2.105 x lor*

(38%)

1.691 x 10-G

(11.3%)

1.519 x 10~*

(0.196)

provide a comparison with the examples considered above only three of the

unknown parameters were treated as characteristic parameters. For the other

three, the values inferred from a previous run were taken to be known. Thus,

the problem became one of using nine equations (nodes) to infer values for

three parameters. The determinant of [H'J was found for this fourth example to

be 338 x 10't Comparison with the results presented in Table 2 would suggest

an error of not greater than 10% for each parameter. This inference is of course

only valid if measured temperature changes and the values assumed for other

parameters are exact.

CREEP DEFLECTION MODEL

In this example the measured quantities are deflections at various points and for

successive construction stages of a post-tensioned balanced cantilever bridge

deck. Full details are presented elsewhere in this publication [3]. There are

again three characteristic parameters representing, in this case, elastic stiffness

and creep properties of the concrete. A computer program was used to calculate

theoretical deflections for given values of the characteristic parameters. The

problem considered is to find those values for the characteristic parameters

which result in a best fit to measured deflection data. The bridge was

constructed in stages with one segment being added at each stage. The

deflections at the end of each segment at each stage of construction were the

parameters used in the best fit process. Thus, one parameter was available after



Stage 1 (Segment 1, Stage 1), three parameters were available after Stage 2

(Segment 1, Stages 1 and 2 and Segment 2, Stage 2) and so on. For the

purposes of this study, values for the three characteristic parameters were

determined four times using all data available after each of Stages 4, 5, 6 and 7

(10, 15, 21 and 28 parameter values respectively).

The results are presented in Table 3. It was expected that the conditioning of

the problem would improve as an increasing amount of data became available.

This however, would not appear to be the case as the number of iterations

required increases with the quantity of data used. In the third column of Table

3, the determinants of [H'], as evaluated at the solution in each case, are

presented. The general trend apparent in these values is consistent with the

increasing number of iterations required to determine the solutions, ie.,

Table 3. Results of Creep Deflection Model

Problem No. j

(No. Params.)

1(10)

2(15)

3(21)

4(28)

No.

Iters.

147

171

197

317

At Solution

to No. j

0.904

0.861

0.293

0.233

Determinants

At Solution

to No. 1

0.904

0.896

0.928

0.951

At Solution

to No.4

0.047

0.233

0.233

0.233

successive problems are less well conditioned. The fact that the determinants

reduce as data is added has been found to be due largely to the fact that the

optimal solutions for each problem are quite different from one another. In the

fourth column of Table 3, the determinants are all evaluated at the point, (43.29

x 10̂ , 1.341, 0.841), which is the optimal solution for Problem No. 1. The

final column of the table contains the determinants as evaluated at the point,

(31.92 x 10f, 0.385, -4.04), which is the optimal solution for Problem No. 4.



Clearly, for these problems the optimal solution changes as more data is

considered. It can be seen that this has a significant effect on the values of the

determinants. The general trend evident in each of columns 4 and 5 is one of

increasingly well conditioned problems. However, a comparison of

corresponding values in different columns indicates that the conditioning of the

problems are quite variable in the vicinity of the solutions. For example,

Problem No. 1 is well conditioned near its own solution but is quite poorly

conditioned at the optimal solution to Problem No. 4. Clearly such a possibility

must be considered when the determinant of [H'] is used as a measure of the

conditioning of problems.

CONCLUSIONS

A technique is described for the determination of the parameters which

characterise a physical process. The determinant of a matrix of partial derivative

terms is proposed as a measure of the conditioning of such problems. A

method of catering for highly redundant problems is described. Examples

illustrate the application of the method to two different physical processes.

REFERENCES

1. Rao S.S. Optimization Theory and Applications, 2nd Edition, Halsted Press

New York, 1984.

2. O'Brien, E.J., O'Donnell, J.J., Waldron P., Lahlouh, El-H., "Non-linear

Conduction Modelling of Concrete Walls under the Influence of Heat of

Hydration of Cement" in Advanced Computational Methods in Heat Transfer II,

Vol 1, (Ed. Wrobel, R.C., Brebbia, C.A. & Nowak A.J.) pp. 121-130.

Computational Mechanics Publications & Elsevier Applied Science, 1992.

3. Flanagan J.W., O'Brien E.J. "Numerical Modelling of Time Dependent

Deflections during Construction of Prestressed Concrete Bridges", in CMEM

93 - Computational Methods and Experimental Measurements, Siena, 1993

(Ed. Brebbia, C.A. and Carlomagno, G.M.). Computational Mechanics

Publications, 1993.


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