Calibration of a constitutive model using genetic algorithms

ELSEVIER

Computers and Geotechnics. Vol. 19, No. 4, 325-348, 1996 Copyright 0 1996 Elsevier Science Ltd

Printed in Great Britain. All rights reserved PII:SO266-352X(96)00006-7 0266-352X/96/$15.00+0.00

Calibration of a Constitutive Model Using Genetic Algorithms

Surajit Pal,” G. Wije Wathugala” & Sukhamay Kundub

‘Louisiana Transportation Research Center, Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803, U.S.A.

‘Department of Computer Science, Louisiana State University, Baton Rouge, LA 70803, U.S.A.

(Received 12 July 1995; revised version received 26 October 1995; accepted 9 April 1996)

ABSTRACT

Before any constitutive model can be used in a numerical procedure, the model needs to be calibrated using laboratory test results. The traditional calibration techniques use stress and strain levels at certain states that a material undergoes during certain types of laboratory tests. Sometimes this method of calibrating a constitutive model fails to capture the overall behavior of a material, i.e. behavior at every point in stress/strain path. In this paper, we have shown how a random search technique, genetic algorithm (GA), can be used to calibrate constitutive models. The advantages of using GA are that it considers the overall behavior of a material, not the behavior at some specific states as the traditional method does, and it can work with many types of laboratory tests. The concept is applied to calibrate the hierarchical single surface (Hiss) 6, model for geologic materials. Three cases have been studied where the dtflerence is in the type of test data used to calibrate the model. These three dtrerent test data are: (1) simulated conventional test data, (2) simulated cyclic test data, and (3) real test data. A comparison of two dtxerent cross-over schemes, one-point and six-point, has been made. Copyright 0 1996 Elsevier Science Ltd.

INTRODUCTION

Any statically indeterminate load-deformation problem requires constitutive models for the stress-strain behavior of all the materials involved. Stresses are related to strains or vice versa of an engineering material via a constitutive model. Simple constitutive models are based on the theory of linear elasticity, and the calibration of these models is quite simple. Unfortunately, most engineering materials, especially those which are geologic, do not obey the theory of elasticity. The constitutive models for those materials have been

325

326 S. Pal et al

developed based on the theory of elasticity and plasticity. The theory of plasticity requires definitions of a yield surface, a plastic potential surface (which is the same as yield surface when the flow rule is associative), and a hardening law. In most cases, definitions of a yield surface, a plastic potential surface, and a hardening law require quite a number of material parameters which makes the calibration of the model quite complicated. Here calibration of a constitutive model is defined as determination of appropriate values for the material parameters so that the observed stress-strain behavior of the material matches with the stress-strain behavior predicted by the constitutive model.

When constitutive models are developed for a material or a class of materials, we need to introduce several material parameters. The selection of these parameters at this stage depends mainly on the improvements these parameters would add to the prediction capabilities of the model. However, before we can use these models, they need to be calibrated based on laboratory tests. The traditional approach is to find certain well-defined states in certain tests where the behavior of a material is controlled by only a couple of material parameters. Then the stress and strain tensors and other history parameters at these states can be used to find those material parameters. Sometimes it is found that adding certain material parameters to a model improves the prediction capability of the model substantially, however, it is not possible to find an easy way to find these parameters from laboratory tests. The effects of those parameters may be for the whole stress-strain response and in that case we may not be able to isolate certain states where these parameters fully control the response. Since genetic algorithm (GA) can use the whole stress-strain response of all the tests simultaneously, we can use any material parameter in the model without worrying about how it can be determined from laboratory test data. Therefore, we have more flexibility in developing constitutive models when we can use GA to calibrate them.

In the present work, we show how a GA can be used to calibrate constitutive models. The method has been applied to the 6, model based on the hierarchical single surface concept (HISS) developed by Desai and co-workers [14]. The basic structure of GA is based on the basic mechanism of natural evolution. It is a selective, efficient random global search algorithm and it is widely used in optimization and machine learning problems in Artificial Intelligence, commerce and engineering. GA has been used to solve various problems in different fields of Civil Engineering, such as geotechnical engineering [5], structural engineering [6-lo], transportation engineering [I l-131, etc.

Genetic algorithm employs the Darwinian survival-of-the-fittest theory to promote the best characteristics among a population of partial (semi-optimal) solutions [14-193. It performs a random information exchange (cross-over and mutation) to create new potential solution points or offspring. There are

Calibration using genetic algorithms 327

three fundamental steps associated with a genetic algorithm. The first step is coding of the parameters by binary strings, called chromosomes. One can also use direct coding with real numbers [20,21]. The second step is defining an evaluation function which gives the fitness value for each set of parameters (chromosomes). The third step is the search algorithm for finding the opti- mum solution (chromosomes) in which a population of chromosomes is considered at a time, and new populations are created by “cross-over” (and other operations) on the chromosomes in the previous population. The details of each of these steps are given later.

HIERARCHICAL SINGLE SURFACE (HiSS) S, MODEL

The main ingredient of a plasticity model is the yield function (F). Together with flow rule and consistency condition, yield function describes the stress- strain behavior of a material. The yield function for the HiSS models is defined as:

where

(2)

F, = (1 - pS,)“t (3)

(4)

In the above expressions, J, is the first invariant of stress tensor, J2D is the second invariant of deviatoric stress tensor and J3D is the third invariant of deviatoric stress tensor; m, n, p, y are material parameters and (Y is hardening function defined here as:

where al and 7, are two material parameters associated with hardening and 5 is the trajectory of plastic strain.

For non-associative plasticity, the yield function and the plastic potential function are not the same and in that case the plastic potential function is defined as:

328 S. Pal et al.

where CQ is defined as:

fXe = (Y + K(CQ - Cf) (1 - YI,) (7)

K is a non-associative material parameter, CQ is the value of (Y at the beginning of shear loading and Y, = &J&, where ,.$ = trajectory of total plastic strain and 5, = trajectory of volumetric plastic strain. Therefore, there are seven material parameters, in addition to two elastic parameters, associated with HiSS 6, constitutive model: y, p, m, n, a,, ql and K. A brief description of the traditional method [3] for determining these material parameters is given below. Then the proposed GA is described.

TRADITIONAL METHOD FOR DETERMINATION OF MATERIAL PARAMETERS

Ultimate parameters (7, 0 and m)

The value of m for many geologic materials is found to be equal to -0.5. parameters p y evaluated as

(Y = 0 from following be

stress for least two with different (i.e. compression extension) are Eqn can solved for y p. If ultimate stresses more two stress paths available, an optimization scheme such as least squares [22] may be used to evaluate y and p. If all the test data have same S,. (i.e. triaxial compression only), then additional assumptions, such as friction angle in compression, 4,. and friction angle in extension, c#+ are same, are used [3,23].

Phase change parameter (n)

The value of n is determined at the state of stress at which the plastic volume change (d&P,,) is zero. Stress points where this phase change occurs for associative materials lie on a straight line passing through the origin. This


straight line is called “phase change line” and slope of this line is found as [23]:

[yF,(l-:)1’_ tan 8 (9)

Using known values of 7, /3 and stresses at which plastic volume change (d&‘,) is zero, n can be determined from the above expression (Eqn 9). Assumption of associative flow rule in Eqn (9) could lead to some error for material with a high value of K.

Hardening parameters (a, and 77,)

Taking natural logarithm of both sides of Eqn (5) yields

In (Y = In a, - vi In [ (10)

Using the consistency condition in the theory of plasticity, (Y for any point in a stress control laboratory test can be found from

(11)

5 = I& = S(dEp, d&fy2 can also be computed from the laboratory test data. Substituting the values of cx and 5 in Eqn (10) leads to the number of simul- taneous equations equal to the number of points in the observed data. These equations can be solved for In a, and q, using the least square procedure.

Nonassociative parameter (K)

From eqn (7) K can be expressed as

K= (a0 - a:(1 - Y,) (“Q - a)

(12)

ap which is a function of (a, E, /3, y, n) (Eqn 7) can be calculated at each point of the observed curve. Then K can be determined at each point from Eqn (12). Generally, the portion of E” - E, or I,’ - 41zDp curve near the ultimate condition is used for determining K because the deviation (coo - (Y) is the greatest in the ultimate zones.

CODING AND DECODING OF PARAMETERS

We use here a simple binary coding of the parameter values. The length of binary string to be used to code a parameter depends on the required accuracy.

330 S. Pal et al.

A real variable X whose range is a < X I b can be discretized and coded using binary string of length L as:

(13)

where C = value of the parameter the string represents and B = decimal integer value of the binary string. As an example, for L = 6 the string 000000 corresponds to the lower bound a and the string 111111 corresponds to the upper bound 6. In this case C can be one of the 26 = 64 values equally spaced between the lower bound, a, and the upper bound, b. If a = 0, b = 10 and the binary string is 010110, the decimal value of the string, B = 22 and C = 3.492. Note that a more general scheme would be to use a nonlinear discretization where a table of size 2L can be used to translate each binary string to a real number in the range a I X I b. Such methods are useful when different points of the range a 5 X I b need to be searched or examined at different resolution.

In general, a constitutive model always has more than one parameter. To code the multiple parameters, binary strings representing the parameters are simply concatenated in a certain order and a single string is generated which represents all the parameters. For example, if we have six parameters A,, AZ, A,, A,, A, and A, and each one is coded with six bits (L = 6), the final string will contain 36 bits. Suppose, A, = 110001; A, = 010101; A, = 101010; A4 = 111000; A5 = 000111; A, = 100110, then the string becomes

110001 010101 101010 111000 000111 100110

Al A2 A3 ‘4 A5 A6

One can, of course, use different string length (L) for different parameters depending on the required precision. To decode such a string, we decom- pose it into components of appropriate length and then decode each part by Eqn (13).

There are seven material parameters associated with the HiSS S, model in addition to the two elastic parameters E (Young’s modulus) and v (Pois- son’s ratio). The determination of elastic parameters is very simple for the HiSS model because it uses linear elastic relationship. Therefore, we did not use them in our GA based search method. Out of these seven parameters, the parameter m is taken as a constant (= -0.5) for every material. This leaves six parameters: ultimate parameters y and p; phase change parameter n; hardening parameters al and vl, and non-associative parameter K.

Since the range of hardening parameter a, is very large, we use binary string to code ln(a,) instead of al. In the present study, each material parameter has been represented by 10 bits; therefore, each string contains


total 60 bits, giving a search space of 2@’ possible combinations of the values of six parameters. In this study, the maximum number of generation used is 500 and the size of the population used is 50. Therefore, 25,000 sets of parameters are examined out of 260 parameter sets in the search space. Computing the fitness value for a given parameter set takes approximately 0.3 s CPU time on an IBM RS/6000-35 computer.

FITNESS FUNCTION

In our problem, a binary string or chromosome represents a set of material parameters. The fitness function evaluates a scalar value called fitness value for each set of parameters or chromosome. The fitness value represents the quality of the binary string as a solution and the goal of genetic algorithm is to minimize or maximize this fitness value depending upon the problem. In the calibration problem, our goal is to find a set of material parameters for the constitutive model which predicts the measured stress-strain behavior of laboratory tests as closely as possible. So, the fitness value could be obtained by comparing the observed stress-strain behavior of laboratory tests with the one predicted by the constitutive model using the set of material parameters represented by a string.

In the present study, a strain path is predicted for the stress path used in the laboratory test using the HiSS model. We use three types of stress-strain plots. They are Z, vs Ji, dZzD vs dJ2,, and dZzD vs -I,, where Z, = first invariant of strain tensor, ZzD = second invariant of deviatoric strain tensor, J, = first invariant of stress tensor, JzD = second invariant of deviatoric stress tensor. These three curves are used because they represent the stress-strain behavior of a material in full stress and strain spaces. For each stress-strain curve, we define a fitness value as the ratio of the area between the predicted curve and the laboratory test curve to the area of the rectangle generated by the maximum and the minimum values of stresses and strains of the laboratory test, (area of rectangle = (maximum stress - minimum stress) X (maximum strain - minimum strain)) as shown in Fig. 1. Note that the ratio is inde- pendent of the scales used for stress and strain. We denote this fitness value as Fi, where i = 1, 2, 3 representing the three curves. For cyclic tests, this definition is valid for each cycle and the rectangle which contains the cycle is used in the denominator. It should be noted here that since we are inter- ested in the minimization of the area-ratio, the higher the value of fitness function, the lower the fitness of the parameter set. In short the fitness value represents the degree of “unfitness” in this problem. Now, we check the ultimate strain values of the predicted curves and add a penalty depending on the deviation of the predicted ultimate strain values from the

332 S. Pd et al.

curve obtained from laboratory stress-strain measurements

Invariant

of Stress

curve predicted by GA

Area ODE (shaded)

Fitness Value =

Area OABC (rectangle)

.

0 Invariant of Strain A

Fig. 1. Definition of$tness value.

observed ultimate strain values. A quadratic penalty function is used and defined as:

Penalty = y, 1 - ( piJ++$ (14)

where dIpzD.u,, = Predicted ultimate value of dZ,,, dPZD,u,, = Observed ultimate value of dIzD, ZP,,u,, = Predicted ultimate value of I,, IOl,U,, = Observed

ultimate value of I,, and yI and y2 are two scalar multipliers. A similar penalty function is used by Adeli and Cheng [24,25] to optimize space structures. The penalty is related to the horizontal distance between the points B and D in Fig. 1, and forces these points to be close to each other (in addition to minimizing the shaded area). The deviation of the ultimate strain value of the predicted test result from the real test result indicates that the set of parameters corresponds to a stronger or weaker material than the real one, and by adding a penalty depending on that eliminates such bad parameter sets, i.e. strings from the population. The authors have found that adding the penalty improves the accuracy of the solution considerably. Therefore, the final fitness value, f for each test becomes:

f=CF;+ y, l- ( ST+ y2 (I-%r wherei= 1,2,3. (15)

For multiple test data, we average the fitness values of all the tests and take that average value as the final fitness value. Finally, the fitness value is mul- tiplied by 100 to represent it as a percentage. Our goal is to find the optimal


combination of the six parameter values that minimizes this final fitness value using genetic algorithm.

From many laboratory test results, the stress tensors associated with the stress path and the corresponding strain tensors can be evaluated. Thus genetic algorithm can be used to evaluate material parameters from many conventional laboratory tests. Therefore, the restriction of using certain types of tests to find material parameters by traditional methods can be eliminated resulting in flexibility to the development of constitutive models. However, it should be noted here that if a parameter has little effect on the behavior of a material for a certain stress path, then the accuracy of that parameter found from that stress path will be much less.

GENETIC ALGORITHM (GA)

We are using here a simple form of genetic algorithm. There are more sophisticated variations available [15,18]. The three basic operations for creating a new population from an old population are: (1) reproduction, (2) cross- over and (3) mutation. Reproduction is a process in which decision is made on the number of copies of each string (chromosome) that will go into the mating pool. This decision is based on the fitness value of the string. In a maximization problem, strings with higher fifitness value have higher probability to survive and, consequently, proportionately more copies are made in the mating pool to create the next generation. Similarly, in a minimization problem, such as the one considered here, a string with less fitness value has higher probability to survive and more copies are produced in the mating pool. For that reason we redefine the fitness value of each chromosome as

f;. =fmm -A (16) where f,,, is the maximum fitness value in the population. For a population of size n, the probability of ith string to survive is:

f;.

pi=cfi (17)

wheref; is the fitness value of ith string. The number of copies for ith string to be made for the mating pool is given by (approximately, differing at most by l),

n, = n X pi (18)

This process is analogous to “natural selection” in the real world. In more complex form of GA certain percentage of the population may be copied directly to the next generation in order to make sure that some of the best chromosomes survive and are not destroyed by mutation or cross-over.

334 S. Pal et al.

The fitness values are resealed so that the best chromosome will have a probability to survive by a factor, called MAX MIN factor, times the survival probability of the worst chromosome. The MAX MIN factor induces a relative measure; how good the best chromosome is compared to the worst chromosome. In the present study a MAX MIN factor of 2 has been used for all the runs.

The next generation is created by cross-over from the mating pool and followed by mutation. This is the same as mutation first and then cross- over. Cross-over is a random and relatively simple process in which no prior knowledge about the fitness value of individual string is necessary. In this process two mating strings swap certain number of bits and produce two off-spring which have new characteristics derived from their parent strings. There are many ways a cross-over operation can be implemented in a GA. In the present work, cross-over is performed as follows. First, two partici- pating parent strings for cross-over are chosen at random from the mating population. Then a position (the same position for both the strings) in the strings is chosen at random about which bits are exchanged after the crossover point between the two parent strings. The process is depicted in Fig. 2. Much of the power of genetic algorithm relies on reproduction together with cross-over.

In the present problem, each chromosome has six components representing six material parameters. The cross-over in the present study has been implemented by choosing a random position for each component and swapping bits of that component of the two parent chromosomes; i.e. cross- over takes place at six points in a chromosome. Figure 2 can be considered as the representative of one component.

The mutation operation is used mainly to prevent loss of some characteristics of the population and to reduce duplicate members in the population.

Parents: 011000

Chosen randomly from the mating population

0101 10

I- Random position, about which

bits are exchanged

Off-spring: 011010

010100

Fig. 2. Schematic diagram of cross-over.


The rate of mutation is generally low which conforms to the fact in the real world where mutation takes place only occasionally [14]. After reproduction, if some bits have the same value (0 or 1) for all the strings in the mating population then the same would be true for all offspring and for all succes- sive generations if we do not use mutation. Mutation, which is a process in which some bits depending on the mutation rate are changed in all the strings, acts as a safeguard for that situation. It could also make two strings different which were similar before the mutation operation. Though mutation is a secondary operation in GA, sometimes it plays a vital role in the search operation, especially in the situations mentioned above. A mutation rate of 2% is used in the present study. The cross-over rate used is lOO%, i.e. each and every string (chromosome) in a population participates in cross-over operation, The initial population is created randomly. We tried with different initial population, and as one might expect, the results do vary from one run to the other. In our case, except for K all other parameters showed very small variations (<5%), but K varied between 10% and 40%. A flow chart of genetic algorithms is shown in Fig. 3.

I I INITIAL

POPULATION

EVALUATION

MATING

POOL

NEW GENERATION

Fig. 3. Flow chart of genetic algorithm.

336 S. Pal et al.

EXAMPLES

In this section, three numerical examples of the calibration of the HiSS 6, model using GA are presented. In the first example three simulated conventional laboratory test data have been used to calibrate the model. The reason for using the simulated test data is that the answer is known in this case. Thus. it makes it possible to compare the parameters obtained by GA with the input parameters and to test the robustness and accuracy of the algorithm. In the second example, simulated cyclic test data have been used. In the traditional method, the material parameters cannot be evaluated from a cyclic test result. Here the cyclic test data have been chosen to test the usefulness of the present approach in a non-traditional situation. The reason for choosing the simulated data is the same as before. In the last example, three conventional real test data have been used to show the applicability of the algorithm to a real problem.

Simulated test data

Three triaxial tests were simulated: (1) triaxial compression (TC, J, = constant, u, increasing, a, = a, decreasing) test at a confining pressure of 100 kPa, (2) conventional triaxial compression (CTC, q increasing, a, = a, constant) test at a confining pressure of 200 kPa, and (3) triaxial extension (TE, J, = constant, (T, decreasing, a, = a, increasing) test at a confining pressure of 200 kPa. Later, the data from these simulated tests were used to evaluate material parameters using GA. The purpose of this study is to evaluate the performance of the algorithm, i.e. to see whether GA is able to give back the original parameters used to simulate the tests. The multipliers y, and y2 for the penalty function were taken as 10. Material parameters by conventional approach were also evaluated from these simulated test data. Table 1 shows the material parameters used to create the simulated test data, those obtained from GA and the material parameters determined by the traditional approach [3]. Note that the nearest values the parameters can take on in GA due to discretization match with the real values up to the third decimal place. Therefore, they are not shown in this table. It can be observed from this table that the material parameters obtained using GA are closer to the actual parameters used to simulate the triaxial compression tests. Figures 4-6 show the predicted curves by GA and the traditional method together with the simulated test data. It can be observed from Figs 4-6 that GA in general predicted strains near failure better than the traditional method. For example, in each of Figs 4-6, the termination points of the curves for simulated test data and that obtained by GA are very close to each other. However, for some of the predictions, the traditional method had predicted better at low strains. In GA, by using a suitable fitness function it is possible to give weightage to any property of the stress-strain

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curve. In the present study, we have given more weightage for the ultimate strains through the penalty function in Eqn (15). Overall, both the methods gave acceptable results in this example. However, it was found that for materials with higher K values, GA gives better results than the traditional method. These results are not presented here for the sake of brevity.

500-

Li 300-

CkPa)

200 1

100 -1

0 -e-T_, I , r , I-r, 0.0 0.1 ii2 0.3 0.4

Fig. 4(a). dZlo - dJzD curves for simulated CTC test (confining pressure 150 kPa).

4

-0.6

-0.7'

Fig. 4(b). dIzD - I, curves for simulated CTC test (confining pressure 150 kPa).


Simulated cyclic test

In the traditional calibration method, we cannot use a single cyclic test to find material parameters. However, GA could be used even for this case. A cyclic triaxial test (a, cyclic, u2 = a3 = constant) was simulated with the material

300

(kpa)

200

-t- Traditional Method

0 0.0 0.1 0.3 0.4

Fig. S(a). dLD - dJIo curves for simulated TC test (confining pressure 100 kPa).

0.00 0

-0.02

-0.04

A -0.06

-0.06

-0.10

-0.12

-0.14

-0.16-l

Fig. 5(b). d12, - I, curves for simulated TC test (confining pressure 100 kPa).

340 S. Pal et al.

parameters given in Table 2. The test was not run up to failure. Table 2 also shows the material parameters predicted by GA. The multipliers yI and y2 were taken as 20 for this run. These values were taken because we want to match the ultimate strain values closer since the test was not run up to failure. The plots of a a,-0, and E,-E~ were used instead of dJ,, and ‘II,, to

180

160

100

Wa) 80

60

40

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22

Fig. 6(a). dIzD - dJzD curves for simulated TE test (confining pressure 200 kPa).

0.1

1

0.0 ITr-_T--T--’ -/- -,

0 6 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22

Z 1 -0.1

-0.41

Fig. 6(b). dZ2D - I, curves for simulated TE test (confining pressure 200 kPa).


TABLE 2 Material parameters for simulated cyclic test

Material parameters and their ranges used in GA

(0.04-0.11) 2; (0.2-0.7) n (2.14.0) In a, (-11.513 to -6.908) 7, (0.4-l .O) K (O-0.5)

Parameter values used

for simulated test

Parameter values obtained by using GA and percentage d@erence

0.089 0.092 (+3.37%) 0.442 0.413 (-6.56%) 3.0 2.751 (-8.3%)

-8.634 -8.202 (-5.0%) 0.850 0.970 (+14.12%) 0.25 1 0.327 (+30.28%)

capture the sign (positive or negative) of these quantities. Figure 7 shows the simulated test data and the predictions by GA. Material parameters obtained from GA are close to the material parameters used to simulate the test except for ql and K, especially the deviation for K is very large. The reason for this large deviation for K could be that the cyclic test was not run up to failure and the effect of K in the behavior of the material is maximum near failure. The deviation of q, could be a shortcoming of the fitness function. Despite the fact that only one simulated non-traditional test result was used and that was also not run up to failure, GA was able to estimate the parameters satisfactorily.

Fig. 7(a). E,-E~ - a,-a2 curves for simulated cyclic test.

342 S. Pal et al

Fig. 7(b). E, - .E? - E,. curves for simulated cyclic test.

Real test data

Three laboratory triaxial tests on Leighton Buzzard Sand [4] were used in this example. These three tests were two CTC tests at confining pressures of 90 kPa (13 psi) and 34.5 kPa (5 psi), and one triaxial extension (TE) test at a confining pressure of 138 kPa (20 psi). The multipliers y, and y2 for the penalty function were taken as 10 in this case. Table 3 shows the material parameters obtained using GA and those obtained by traditional approach. To compare how well GA evaluates material parameters compared to the traditional approach, predictions have been made for all the tests using the material parameters obtained by the traditional method and those obtained by GA. Figures 8 to 10 show plots of predicted curves together

TABLE 3 Material parameters obtained from real tests

Material parameters and their ranges used in GA

Parameter values obtained Parameter values obtained by using GA by traditional method

y (0.04-O. 11) 0.098 0.089 p (0.2-0.7) 0.45 0.442 n (2.14.0) 2.56 3.0 a, (1.0 X 10 5-1.O X 10 ‘) 3.98 x IO 4 1.78 x IO4 7, (0.4-l .O) 0.918 0.852 K (O-0.5) 0.356 0.25 1


with laboratory test data. It is observed from these figures that the curves predicted by GA are closer to the laboratory test data than the curves predicted by the traditional method, except for the lower portion of the curves in Figs 8a. 9a and 10a. Note that in these figures, the deviation of the upper portion of the curves predicted by the traditional method is larger than the

go-

0

0.00 0.01 0.02 0.03 0.04 0.05

Fig. g(a). dIzo ~ dJID curves for real CTC test (confining pressure 34.5 kPa).

1 r , 1 ( 1, ” r / 1 ” 1

0.02 0.03 0.04 0.05

-0.006 1

-0.008

-0.010 I

-0.012

-0.014-

Fig. 8(b). dIzo - I, curves for real CTC test (confining pressure 34.5 kPa).

344 S. Pal et al.

deviation in the lower portion showed by GA. Also in Figs 8b, 9b and lob, the deviation of the curves predicted by the traditional method is significant, whereas the curves predicted by GA are very close to the actual material behavior. Considering the overall behavior, the material parameters obtained by GA are more acceptable than those obtained by the traditional method.

JJa (kf’a)

200

180

180

140

120

100

80

80

40

20

I 3 I 1 1 r / 1 I -I

0.00 0.01 0.02 0.03 0.04 0.05 0.08 0.07

i-z

Fig. 9(a). dIzn - dJ2D curves for real CTC test (confining pressure 89.6 kPa).

Fig. 9(b). dI,, - I, curves for real CTC test (confining pressure 89.6 kPa).


PERFORMANCE OF TWO DIFFERENT CROSS-OVER SCHEMES

In conventional GA, cross-over is done with respect to one point. To compare the performance of one-point cross-over with six-point cross-over, we ran one problem with the simulated test data using one-point cross-over.

o- I ’ ’ I ’ ’ I I

0.00 0.01 0.02 0.03 0.04

JL,

Fig. 10(a). d12, - ‘JJz, curves for real TE test (confining pressure 138 kPa).

Fig. 10(b). dI,, - I, curves for real TE test (confining pressure 138 kPa).

346 .S. Pal et al.

1

0.0

Six point cross-over One point cross-over

0 100 200 300 400 500

Generation

Fig. 11. Performance of one-point cross-over and six-point cross-over schemes.

Figure 11 shows the plot of the best fitness value in a generation vs that generation number for both cross-over methods. It can be observed that the six-point cross-over improves the convergence rate. It also achieves a lower fitness value which results in more accurate material parameters. It should be noted that the set of parameters corresponding to the lowest fitness value in the whole generations is reported as the best set; not the best parameter set in the last generation. For the one-point cross-over scheme the best parameter set was achieved in the vicinity of the generation 400 (Fig. 1 I).

DISCUSSION AND CONCLUSION

In the present study, a random search technique, genetic algorithm (GA) is used to calibrate the Hierarchical Single Surface (HISS) 6, model. The results of this study demonstrate how a modern and relatively simple random search algorithm, GA can be used to determine material parameters for a constitutive model. The main advantage of using GA over the traditional method is that it takes care of the global characteristics of the test results, i.e. behavior at each and every point in stress or strain path, not the characteristics of the test result at some specific points or states, such as ultimate, phase change, etc. Apart from


that, GA can be used to find material parameters when only non-traditional test results are available in which case a traditional method cannot be used.

The traditional method of calibration is sequential; i.e. it finds one or more parameters first and later uses these values to find the other parameters. If any error induces in the parameters found initially, the parameters found later are affected. Sometimes, two or more parameters are interdependent and there are no explicit relations available to solve for all the parameters. Some assumptions are generally made to handle such situations. For example, we must know the value of the phase change parameter, it in order to find the value of the nonassociative parameter, K. However, the assumption that the volumetric plastic strain changes sign at the phase change line is not true for the nonassociative model. The higher nonassociativeness may give an erroneous n value which eventually triggers an erroneous K value. In such a situation GA works better than the traditional method.

The way fitness value of a string is evaluated in the present study is capable of capturing the overall trend of the laboratory test results. The penalty depending on the ultimate strain values of the predicted result and the real test result improves the accuracy of the solution considerably. For the present problem, this deviation indicates that the set of parameters corresponds to a stronger or weaker material than the real one and by adding a penalty depending on that eliminates such bad parameter sets, i.e. strings from the population.

The penalty function used in the present study may not be the best one. It should be noted that one may wish to consider the behavior of a material at specific states (e.g. phase change, ultimate, etc.). In that case, an appropriate penalty function can be added to the definition of fitness function to capture that behavior. The authors are currently working on defining the best fitness function for the calibration of the HiSS S, model.

REFERENCES

1. Desai, C. S., Somasundaram, S. & Frantziskonis, G. N., A hierarchical approach for constitutive modelling of geologic materials. International Journal of Numerical Analytical Methods in Geomechanics, 10 (1986) 225-257.

2. Frantziskonis, G. N., Desai, C. S. & Somasundaram, S., Constitutive modelling for nonassociative behavior. Journal of Engineering Mechanics, ASCE, 9 (1986) 932-946.

3. Desai, C. S. & Wathugala, G. W., Hierarchical and unified models for solids and discontinuities (joint/interfaces). Short Course Notes, 2nd Int. Conf. on Constitutive Laws for Engineering Materials: Theory and Applications, Tucson, AZ, U.S.A., 1987.

4. Hashmi, Q. S. E., Nonassociative plasticity model for cohesionless materials and its implementation in soil-structure interaction, Ph.D. Thesis, University of Arizona, Tucson, AZ, U.S.A., 1986.

348 S. Pal et al.

5. Simpson, A. R. & Priest, S. D., The application of genetic algorithms to optimization problems in geotechnics. Computers and Geotechnics, 15 (1993) l-19.

6. Wu, S. & Chow, P., Steady-state genetic algorithms for discrete optimization of trusses. Computers and Structures, 56 (1995) 979-991.

7. Hajela, P. & Lee, E., Genetic algorithm in truss topological optimization. International Journal of Solids and Structures, 32 (1995) 3341-3357.

8. Tesar, A. & Drzik, M., Genetic algorithm for dynamic tuning of structures. Computers and Structures, 57 (1995) 287-295.

9. Koumousis, V. K. & Georgiou, P. G., Genetic algorithms in discrete optimization of steel truss roofs. Journal of Computing in Civil Engineering, 8 (1994) 309-325.

10. Adeli, H. & Cheng, N. T., Concurrent genetic algorithm for optimization of large structures. Journal of Aerospace Engineering, 7 (1994) 276296.

11. Chakroborty, P., Deb, K. & Subrahmanyam, P. S., Optimal scheduling of urban transit system using genetic algorithm. Journal of Transportation Engi- neering, 121 (1995) 544-553.

12. Chan, W. T., Fwa, T. & F. Tan, C. Y., Road-maintenance planning using genetic algorithms: I. Journal of Transportation Engineering, 120 (1994) 693- 709.

13. Fwa, T. F., Tan, C. Y. & Chart, W. T., Road-maintenance planning using genetic algorithms: II. Journal of Transportation Engineering, 120 (1994) 710-722.

14. De Jong, K. A., An analysis of the behavior of a class of genetic adaptive system. Ph.D. Thesis, University of Michigan, Ann Arbor, MI, U.S.A., 1975.

15. Goldberg, D. E., Genetic Algorithm in Search, Optimization and Machine Learn- ing. Addison-Wesley Publishing Company, Inc., Reading, MA, 1989.

16. Goldberg, D. E. & Samtani, M. P., Engineering optimization via genetic algorithm. Proceedings of the Ninth Conference on Electronic Computation, 1988, pp. 471482.

17. Syswerda, G., Uniform crossover in genetic algorithms. Proceedings of the Third International Conference on Genetic Algorithms, George Mason University, Morgan Kaufmann Publishers, Inc., 4-7 June 1989, pp. 2-9.

18. Davis, L., Handbook of Genetic Algorithms. Van Nostrand Reinhold, New York, 1991.

19. Hajela, P., Genetic search- an approach to the nonconvex optimization problem, AZAA J., 28 (1990) 1205-1210.

20. Eshelman, L. J. & Schaffer, J. D., Real-coded genetic algorithms and interval- schemata. In Foundations of Genetic Algorithms II (Edited by D. Whitley). Morgan Kaufman, San Mateo, CA, 1995, pp. 187-202.

21. Wright, A. H., Genetic algorithms for real parameter optimization. Foundations of Genetic Algorithms (Edited by G. J. E. Rawlins). Morgan Kaufman, San Mateo, California, 199 1, pp. 205-2 18.

22. Desai, C. S. & Siriwardane, H. J., Constitutive Laws for Engineering Materials. Prentice-Hall Inc., Englewood Cliffs, NJ, 1984.

23. Wathugala, G. W., Finite element dynamic analysis of nonlinear porous media with applications to piles in saturated clays. Ph.D. Thesis, The University of Arizona, Tucson, AZ, 1990.

24. Adeli, H. & Cheng, N. T., Integrated genetic algorithm for optimization of space structures. Journal of Aerospace Engineering, ASCE, 6 (1993) 315-328.

25. Adeli, H. & Cheng, N. T., Augmented lagrangian genetic algorithm for structural optimization. Journal of Aerospace Engineering, ASCE, 7 (1994) 104-l 18.

Calibration of a constitutive model using genetic algorithms

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