Department of Aerospace Engineering and Mechanics, Mail Stop 9549, ... composites from obliquely incident ultrasonic bulk wave data, employing an inverse ...
Composites Part B 29B (1998) 171-180
PII: S1359-8368(97)00007-3
ELSEVIER
© 1998 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/98/$19.00
Inversion of composite material elastic constants from ultrasonic bulk wave phase velocity data using genetic algorithms
Krishnan Balasubramaniam* and Navin S. Rao Department of Aerospace Engineering and Mechanics, Mail Stop 9549, Mississippi State University, MS 39762, USA In this paper, efforts on the reconstruction of material stiffness properties of unidirectional fiber-reinforced composites from obliquely incident ultrasonic bulk wave data, employing an inverse technique based on genetic algorithms, is described. Computer-generated ultrasonic phase velocity data, as a function of the angle of refraction in both symmetry and nonsymmetry through the thickness planes of unidirectional composites were used as the input to the genetic algorithm. A simple genetic algorithm, with optimal parameters chosen from the literature and numerical analysis (reported here), was implemented for the reconstruction. The inversion using this novel technique was found to be extremely promising in the characterization of the material stiffness properties. Stability to noise in the input phase velocity data set was also investigated. Advantages and disadvantages of the genetic reconstruction technique over conventional methods are also discussed. © 1998 Elsevier Science Limited. All rights reserved. (Keywords: B. elasticity; D. ultrasonics; inverse techniques)
INTRODUCTION Fiber-reinforced composite materials have been used in many structural applications varying from swimming pool diving boards to advanced aerospace components. The primary advantage of composites includes a high stiffness to weight ratio which in the past has come at an increased cost. Continued improvements in the development of costeffective manufacturing methods and development of lowcost fibers and resin materials have increased the use of composites in infrastructural applications such as buildings and bridges i. Recently, the application of specially oriented fiber-reinforced composite wave guides for acoustical isolation has been reported 2. As fiber-reinforced composites become more widely used, the need for a reliable method to nondestructively measure the material stiffness properties and characterize material defects is becoming critical for ensuring a reliable level of performance in both structural uses as well as for noise control. The methods of ultrasonic NDE for isotropic materials such as metals have long been established. Due to the anisotropic nature of fiber-reinforced composites, the interpretation of the ultrasonic data becomes more involved. This is especially true when the thickness of the composites increase. Behavior of the elastic acoustical waves within an anisotropic structure can be predicted if the stiffness constants of the material is known 3-7. * To w h o m correspondence should be addressed
Reconstruction of elastic constants is an essential part of nondestructive ultrasonic material characterization. The idea is that ultrasonic data (usually phase or group velocities) are related to the material properties through a known mathematical model. Normally, the mathematical model defines the forward problem in that it relates known material properties to ultrasonic data. Thus, if experimentally measured ultrasonic data are available, computing the required stiffness properties is just a matter of solving the inverse problem, i.e. relating known ultrasonic data to material properties using the inverse of the same model. However, even though the forward approach might be relatively easy, the inverse step is often more difficult. Generally, the inverse problems are highly nonlinear and hence, analytically intractable. Furthermore, practical difficulties and the constraint of limited data sets (due to the NDE method of obtaining data) also increases the effort involved in reconstruction. Consequently, numerical rather than closed-form solutions appear to be the practical answer to solving such nonlinear, not well understood inverse problems. Posing the identification (inverse) problem in an optimization form is one of the most popular numerical methods. Considerable work has been done in the area of elastic constant reconstruction of composite materials from various types of ultrasonic data sets. Inversion of leaky Lamb wave 8'9 and bulk wave data 1° 14 from immersion techniques are by far the most popular and reported. In most cases, either the Simplex algorithm or a classical gradient
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Inversion of composite material elastic constants: K. Balasubramaniam and N. S. Rao technique (for example, the Newton-Raphson method) has been used for the inversion of an overdetermined data set. Two important points are worth noting here. Firstly, the use of an overdetermined set stems from a belief that such an approach helps to cancel out random noise (experimental errors) in the measured data. Secondly, the gradient technique operates on the optimization formulation of the inverse problem and hence, is often referred to as the leastsquares optimization method. For a general anisotropic material, the generalized Hooke's law in contracted indical notation is (7i = C o c j
(1)
where: ai are the stress components; ci are the strain components; and Cij is the stiffness or elastic constant matrix. Usually, composite materials systems possess at least orthotropic symmetry. Hence, nine (or less) elastic constants have to be determined to complete the stiffness matrix. In the past, despite a reasonable amount of success, certain problems have been encountered in reconstructing the elastic constants. A problem worth noting is that of elastic constant determination from ultrasonic bulk wave velocity data obtained using the immersion technique. Chu and Rokhlin 11 have had immense success in reconstruction from the two symmetry planes along and transverse to the fiber direction. In these planes of orthotropic symmetry, the displacement relations of the Christoffel equation become simpler as the particle motions decouple into a shear horizontally polarized (SH) wave and quasi-longitudinal and -transverse waves polarized in the sagittal plane. However, only seven of the nine elastic constants of an orthotropic composite material can be obtained rising this method. This is because the SH wave cannot be detected from immersion measurements. The remaining two constants C12 and C66 c a n only be found from nonsymmetry plane data where all three particle motions are coupled. Furthermore, it is also possible to find all nine of the constants from pure nonsymmetry plane measurements. Chu et al. 14 have attempted to follow exactly this approach, but with limited success. They found that the 'nonlinear least-squares optimization procedure' was highly dependent on the initial guesses and attributed this to the increased nonlinearity of the problem as compared to reconstruction from symmetry planes. To circumvent this problem, they proposed a two-step inversion procedure. In the first step, seven elastic constants are determined from symmetry planes and then with these seven stiffness properties kept constant, a second inversion of a two parameter problem in a nonsymmetry plane is performed to obtain C 12and C66. This method obviously increases the complexity and effort involved in reconstruction. B alasubramaniam and Whitney 1° also avoided nonsymmetry planes by making measurements in all the three symmetry planes, but here again, practical difficulties might be encountered with measurements in the third symmetry plane oriented normal to the thickness direction of a thin composite plate. Thus, the primary reason behind the lack of success in reconstruction from nonsymmetry planes appears to be the fact that the inversion is highly nonlinear with many local
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optima. As mentioned earlier, the 'least-squares optimization method' is a gradient-based technique and consequently, has a high probability of entrapment at a local minimum for initial guesses that are not close to the global minimum. Genetic algorithms (GAs) hold a lot of promise in searching such complex, multimodal spaces for unique or nonunique global maxima or minima. GAs are not gradientbased search techniques and no initial guesses are required. Furthermore, genetic searches begin from a set of points in the search space rather than a single point. Even more significant is the fact that their search mechanisms possess an implicit parallelism which enables a rapid sampling of the space and thus, an improved recognition of the whereabouts of the global optima. All these features tend to render GAs robust and global without the pitfall of entrapment at local optima. GAs have been applied in a wide variety of fields, notably structural optimization 15 and control system optimization 16. Within the context of inversion, Stoffa and Sen 17 have used GAs to invert plane-wave seismograms and obtain information regarding the earth's sub-surface layers. Despite the above advantages, GAs are sometimes computationally extensive for small-scale problems and their performance near to the global solution appears to be relatively imprecise when compared with conventional local gradient-based techniques. Hajela 15 states that although GAs do not reach the global solution, they get very close to it without the drawbacks of dependence on initial guesses. In case, a more accurate global solution is needed, the GA solution can always be used as the initial guess for a classical local/gradient technique. An even more significant problem with GAs is the fact that with multiparameter identification problems like the one described in this paper, the relative sensitivity of the parameters becomes an important issue. If the sensitivities of the unknown parameters (elastic constants in this case) are markedly different from each other, then the performance of GAs is not reliable from an identification point of view, even though it is quite satisfactory from an optimization perspective. Nevertheless, going by the limitations of the conventional methods to search multi-dimensional, multimodal spaces, GAs offer an improved optimization method for solving difficult nonlinear (identification) problems, especially in situations where it is difficult to estimate good initial guesses for the solutions, provided care is taken to pose a problem involving a minimum difference in the sensitivities of the unknown parameters. In this paper, reconstruction using computer-generated (hence, noise-free) bulk wave velocity data as the input is initially described and examined for four different material systems with known stiffness properties from literature. The materials examined are listed in Table 1. These materials possessed different degrees of anisotropy and belonged to either the transversely isotropic or the orthotropic class of symmetry. No experimental efforts have been reported in this paper; only the data capable of being measured from immersion experiments were used in the reconstruction. Stability to noise was also investigated by adding different
Inversion of composite material elastic constants: K. Balasubramaniam and N. S. Rao Table 1
Material systems used for genetic reconstruction
Material number
Material type
Symmetry class
Referenced from
1 2 3 4
Graphite/epoxy Glass/epoxy Graphite/epoxy Ceramic-matrix composite
Orthotropic Orthotropic Transversely isotropic Transversely isotropic
Rokhlin and Wang 13 Balasubramaniam and Whitney l° Chu and Rokhlin ll'j2 Chu and Rokhlin 1kl2
distributions and amounts of synthetic random noise to the noise-free velocity data.
GENETIC ALGORITHMS Genetic algorithms (GAs) are randomized search methods that are based on stochastically generated population models. As opposed to other random search techniques, rather than being a computationally expensive brute-force approach, GAs are an intelligent way of rapidly sampling the search space and arriving very close to optimal global solutions: both unique and nonunique ones. Furthermore, in contrast to conventional methods where the search space is treated as a real vector ~U (N being the number of parameters to be identified), GAs operate on finite length alphabet strings (chromosomes) which are coded to represent points in the equivalent vector space. Binary coding is the simplest and most frequently used, although other types of coding (real-valued and Gray coding) are not uncommon. A simple genetic algorithm (SGA) begins with a randomly generated population of points in a prescribed search space. These can be regarded as probable solutions to the given problem. Each point is coded as a chromosome composed of concatenated sub-strings that represent the real-valued parameters of the problem. Each chromosome is then divided into its sub-strings and these are decoded into their corresponding parameters. Next, the chromosome's fitness value (a measure of the distance of this probable solution from the global optima) is evaluated from the fitness function, which is nothing but the function that must be optimized. Note that, in the context of GAs, fitness is defined according to whether the problem is a minimization or maximization one. Pertaining to the reconstruction problem, the fitness function is the sum of the least-squares errors between experimentally measured and calculated data. As the fitness function has to be minimized, population members with lower errors have higher fitness. Subsequent to the random creation and fitness evaluation of the first and initial generation of probables, three basic stochastic operations--selection, crossover and mutation-are performed in that order on this first choice of candidate solutions. Selection is a procedure that decides which member of the current population should survive and he allowed to reproduce and form a mating pool, so as to create the next generation. This operator is biased towards members with above-average fitness. Hence, such individuals reproduce more often than poorly fit ones (similar to the Darwinian principle of survival of the fittest). Crossover consists of randomly choosing two individuals from the newly formed mating pool and exchanging random portions
of their strings (chromosomes), in order to form two new chromosomes (children) which concurrently replace the parents. The crossover operation is, however, performed with a specified probability Pc which is normally kept at a very high value. Mutation involves intentionally changing the value (allele) of an alphabet (gene) in a particular position (gene locus) of the chromosome string, with a specified probability P u that is usually kept very low to prevent unwanted disruption of the population. For the case of binary coding, the mutation operator just flips every one to a zero and vice versa. The purpose of mutation is to create unexceptionably fit members compared to current ones and thus allow the search to gain momentum. This innovative operation rescues the GA from pitfalls of stagnation and incorporates increasing robustness and globalism in the search. Finally, after applying the above operators on the current population, a new population is created and the next generation begins by going back to the process of dividing the chromosomes and evaluating the fitness, followed by the three stochastic operations again to create yet another generation. This process is continued iteratively until a stopping criterion on either the maximum fitness of a generation or on the number of generations themselves is attained. For a much detailed account of GAs, we suggest the texts by Goldberg TMand Mitchell 19.
THE INVERSE PROBLEM STATEMENT In the minimization form apt for inverse problems, the optimization problem can be stated as follows:
Minimize i=3j=M ERR(C)= Z ~ - [V~)-(q~, c m (~,, 0) 2 ] 2 0) 2 - V~j
(2)
i=1 j = l
Subject to CMI N < C • CMA X
where: C is the elastic constants vector [CH~ C22i C33! Cl2i C l 3 i C23~ C44i C55i C66]T;
i is an index specifying the mode of bulk wave propagation, i.e. longitudinal, fast shear or slow shear; j is an index specifying the direction of bulk wave propagation; I/~n(4, 0) is the input phase velocity (usually obtained from experimental measurements) of a particular bulk wave mode propagating in a plane at an azimuthal angle 4~ with respect to a material symmetry direction
173
Inversion of composite material elastic constants: K. Balasubramaniam and N. S. Rao 1(FIBER
I//
! Figure 1
,TRANSVERSE
0IRECTION,
41/i!1J
Schematic representation of the material symmetry coordinate system, depicting bulk wave propagation in the nonsymmetry plane
(in this case the fiber direction) and at a refraction angle 0 with respect to the normal to the surface as shown in Figure 1; V~(th, 0) is the forward calculated phase velocity of the same bulk wave mode in the same direction as V/~(th, 0), but based on the candidate solution generated by the search technique; M is the number of different directions of velocity measurement; CMIN is the lower bound on elastic constants; CMAX is the upper bound on the elastic constants; ERR(C) is the fitness function. The calculated phase velocities are obtained by solving the Christoffel equation. The solution to this equation can be easily determined using the approach outlined by Chu et al. 14. It is worthwhile to point out here that Chu et al. 14 have minimized the sum of the least-squares errors in phase velocities; while we have in effect, minimized the sum of the least-squares errors in the squares of the velocities. The motivation behind this slight change is based on the fact that it is the square of velocity which is obtained as the solution to the eigenvalue problem inherent in the Christoffel equation 5, rather than the velocity itself. It is also important to note that bounds on the elastic constants have to be imposed as the SGA cannot search in an infinite space. Herein lies a slight drawback in that physically acceptable bounds have to be chosen, but the bounds can often be close to the physically acceptable values which the elastic constants may assume for a given material system. It is not unusual for the bounds to be _+ 50% (or more) of the expected solution values.
THE GENETIC ALGORITHM USED FOR RECONSTRUCTION Binary parameter coding as described by Goldberg 18, stochastic remainder selection without replacement ts and
174
DIRECTION)
uniform crossover 19 were implemented for chromosome coding, selection and crossover, respectively. Sigma scaling t9, to keep up the variation within a population and prevent premature convergence, was also incorporated. To conserve the history of previous generations a protection scheme outlined by Stoffa and S e n 17 w a s used. In this technique, every individual in the current generation is compared to one randomly picked from the previous generation without replacement. The older member replaces the current one with a specified update probability (PuPDATE) only if it is fitter; otherwise the current one survives. A stopping criterion of a maximum of 200 generations was imposed, and the bounds on the parameters (elastic constants) were set at -+50% of the expected solution values, defining a sufficiently large search space. The numerous genetic parameters (population size, chromosome string length and probabilities) mentioned earlier have a marked influence on both the off- and on-line performance of the GA. As there exist no set rules or analytical procedures, these parameters have to be chosen from literature and numerical experiments. A large population size is normally recommended to encourage greater variation within a population and hence, a more global search. Small population sizes could lead to premature or slow convergence. However, very large populations increase the number of function evaluations resulting in increased computational costs. Bearing this in mind, a compromise was made and a moderate but satisfactory population size of 50 was chosen. The chromosome string length depends on the individual lengths of its constituent substrings. As mentioned earlier, each substring is a coded representation of a particular parameter (in this case, an elastic constant) to be identified. Thus, assuming substrings of identical length, the chromosome string length defines how coarsely or finely the independent coordinates (nine for the current problem) of the reconstruction search space are discretized. Longer strings describe a finely divided space and vice versa. Generally, very long lengths are not recommended as they
Inversion of composite material elastic constants: K. Balasubramaniam and N. S. Rao
1 0 4.0
1 0 3.0
Pc=0.80 Pupdate=0.90
\,, ~._..---"~
1 0 z.°
i~_j,-............ .... _/-.._. ,o..............................
C 101"°
Pm=0.01 ......... Pm=0.10 10 °-°
10 -l-0
10 -2.0
6 ....
Figure 2
2'5 ....
5'0 ....
7'5 . . . . 160 .... 1½5 . . . . NUMBER OF GENERATIONS
150 ....
175 ....
200
Influence of mutation rate on G A p e r f o r m a n c e
increase computational time with no significant improvements in performance, and small lengths compromise accuracy as well as the search for the global solution. Based on these facts, the chromosome was chosen to consist of equal substrings of length 10. Hence, for the current nineparameter inversion, the chromosome was 90 bits long. From numerical experiments on material no. 4 (using bulk wave phase velocity data from (~b = 0, 90 and 45 ° planes) coupled with recommended values from literature, the optimal probabilities of mutation (PM), crossover (Pc) and update (PuPDATE) w e r e found to be as follows: PM = 0.01; P c = 0.80; PUPDATE = 0.90. The influence of mutation, crossover and protection rates on the GA performance as described by the evolution of the best-fit member of each generation, is depicted in Figures 2-4, respectively. The reference convergence plot of the GA with the above optimal set of probabilities is also given in these figures. Figure 2 shows the effect of a 10-fold increase in the mutation rate, while maintaining the other two probabilities at optimal values. As expected, such high mutation rates tend to be disruptive rendering a more or less random search, and result in erratic fitness patterns with no convergence at all. The effect of perturbing the crossover rate solely from the optimal set of probabilities is shown in Figure 3. It appears that there is no significant impact of the crossover rate on the overall performance except that the number of generations to convergence tends to increase for nonoptimal values. From Figure 4, it can be seen that conserving the history
of previous generations is very essential to achieve convergence. Interestingly, an update probability of zero implies no protection (conservation) at all, and as can be observed, the search becomes directionless. Further, ideally a PUPDATE = 1.00 appears to be the best. However, a slightly lower value of PUPDATE= 0.90 was chosen according to the recommendations of Stoffa and Sen iv. This is perhaps a precautionary measure to prevent stagnation in the evolution of the population by providing some currently misfit individuals having potentially fit genes, a slight chance of survival. Figure 5 illustrates the progress of the genetic search for two parameters (C11 and C12 of material no. 1) of the nineparameter invesion. As can be seen from the figure, the genetic search begins from a set of points (in this case, 50 points corresponding to a population size of 50) scattered in the search space, and finally converges to the global solution (marked by the cross) within 200 generations.
RESULTS AND DISCUSSION Noise-free GA reconstruction on an overdetermined data set, using various combinations of planes oriented at different azimuthal angles ~b (Figure 1) was first examined on material no. 1. This particular material (graphite/epoxy) was chosen because of its high anisotropy and orthotropic symmetry. Phase velocity data of all the three bulk wave modes propagating at refraction angles between 0 = 0 and 90 ° (at 5 ° intervals) in each plane at a particular 4~, were used as the noise-free data input to the GA. To test the repeatability of the reconstruction, I0 consecutive runs for each combination of planes were made. Reconstruction
175
104'° Pm=0.01 Pupdate=0.90
103.0
10 2.0
~L......I"~, lOL°
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10 °.°
Pc=0.80 .......... Pc=0.50 - - - - - Pc=0.95
10 -L° ..
\ ...........................
10 -~.o 0
25
50
75
100
125
150
200
175
NUMBER OF GENERATIONS Figure 3
Influence of crossover rate on GA performance
104.0 Pm=0.01
Pc=0.80 103.0
/ 102.0 I
[k . \ ,
101.0
i
Pupdate=0.90
~
.......... Pupdate=0.50
-~./. ~ ~
-----
Pupdate=0.00
10 °.° :|;
t*
l
""
'
"
i
°
'
7
10 -I.o
10-2.0 0
25
50
75
100
125
150
175
200
NUMBER OF GENERATIONS
Figure 4 Influence of protection rate on GA performance
from planes at ~b = 0, 90 and 45 ° and ~b = 15, 75 and 45 ° gave excellent results in terms of accuracy and repeatability. Table 2 summarizes the noise-free (0% noise) reconstruction statistics based on 10 runs for these planes. The high repeatability can be inferred from the comparatively low standard deviations of the reconstructed stiffness constants.
176
The identification results from other combinations of planes (incidentally in purely nonsymmetry directions) are given in Table 3. As indicated by the means and the standard deviations, it can be seen that the repeatability and accuracy improves considerably if planes closer to the fiber and transverse directions are exploited for reconstruction. In
Inversion of composite material elastic constants: K. Balasubramaniam and N. S. Rao Table 2 Elastic constant
Identification results for material no. 1 from ~ = 0, 90 and 45 ° and ~b = 15, 75 and 45 ° planes Original data
0% Noise Mean
Standard deviation
(% error)
2% Noise Mean
Standard deviation
(% error)
5% Noise Mean (% error)
Reconstruction resuits for material no. 1 from C II 144.00 144.00 (0.00) C22 13.60 13.60 (0.00) C33 12.00 12.00 (0.00) Ci2 5.47 5.49 (0.37) Ct3 5.00 4.99 (0.20) C23 7.00 7.00 (0.00) C44 3.70 3.70 (0.00) C55 6.00 6.00 (0.00) C66 6.50 6.50 (0.00)
q~ = 0, 90 and 45 ° (units in GPa) 0.07 143.80 1.30 (0.14) 0.01 13.57 0.06 (0.22) 0.01 12.00 0.11 (0.00) 0.14 5.66 2.32 (3.47) 0.04 5.17 1.24 (3.40) 0.01 6.97 0.09 (0.43) 0.00 3.69 0.08 (0.27) 0.01 5.96 0.13 (0.67) 0.03 6.97 1.06 (7.23)
142.90 (0.76) 13.71 (0.81) 12.10 (0.83) - 5.61 (2.56) 5.67 (13.40) 7.24 (3.43) 3.75 (1.35) 6.48 (8.00) 6.68 (2.77)
Reconstruction resuits for material no. 1 from CI1 144.00 144.00 (0.00) C22 13.60 13.60 (0.00) C33 12.00 12.00 (0.00) C12 5.47 5.47 (0.00) C 13 5.00 5.01 (0.20) C23 7.00 7,00 (0.00) C44 3.70 3.70 (0.00) C55 6.00 6.00 (0.00) C66 6.50 6,50 (0.00)
4, = 15, 75 and 45 ° (units in GPa) 0.07 144.04 1.49 (0.03) 0.01 13.69 0.44 (0.66) 0.01 12.02 0.06 (0.17) 0.03 5.38 1.01 (1.65) 0.02 5.03 0.64 (0.60) 0.00 6.92 0,19 (1.14) 0.00 3.67 0,09 (0.81) 0.01 6.06 0.23 (1.00) 0.01 6.48 0.49 (0.31)
146.31 (1.60) 14.29 (5.07) 12.10 (0.83) 4.57 (16.45) 4.02 (19.60) 6.82 (2.57) 3.67 (0.81) 5.83 (2.83) 5.96 (8.31)
Table 3
Standard deviation
10% Noise Mean
Standard deviation
(% error) 3.22 0.60 0.30 2.72 2.04 0.34 0.10 0.41 2.70
4.81 1.64 0.19 1.50 1.74 0.22 0.09 0.23 1.47
145.39 (0.97) 13.71 (0.81) 12.52 (4.33) 3.28 (40.04) 5.62 (12.40) 6.77 (3.29) 3.78 (2.16) 5.89 (1.83) 4.42 (32.00)
5.18
148.24 (2.94) 15.23 (11.99) 12.21 (1.75) 4.79 (12.43) 4.54 (9.20) 7.10 (1.4.3) 3.88 (4.86) 6.08 (1.33) 5.69 (12.46)
8.16
1.02 0.65 1.73 1.80 0.90 0.47 1.21 2.07
3.10 0.73 2.23 2.34 0.67 0.20 1.09 2.66
Noise-free identification results for material no. 1 from other nonsymmetry planes (units in GPa)
Elastic constant
Original data
Cll
144.00
C22
13.60
C33
12.00
CI2
5.47
C13
5.00
C23
7.00
C44
3.70
C55
6.00
C66
6.50
~ = 45 ° Mean (% error) 143.58 (0.29) 13.56 (0.29) 12.00 (o.oo) 6.08 (11.15) 4.95 ( 1.00) 7.06 (0.86) 3.70 (o.oo) 6.00 (0.00) 6.74 (3.69)
Standard deviation 2.43 1.69 0.01 1.06 0.13 0.13
4) = 30, 60 ° Mean (% error) 143.97 (0.02) 13.65 (0.37) 12.00 (o.oo) 5.59 (2.19) 4.98 (0.40) 7.01
Standard deviation 0.07 0.35 0.01 1.10 0.11 0.08
(0.14) 0.01 0.01 2.03
3.70 (o.oo) 6.00 (0.00) 6.48 (0.31)
0.00 0.01 0.17
~b = 15, 75 ° Mean (% error) 143.96 (0.03) 13.60 (0.00) 12.00 (o.oo) 5.47 (0.00) 5.01 (0.20) 7.00 (0.00) 3.70 (o.oo) 6.01 (0.17) 6.50 (0.00)
Standard deviation 0.06 0.01 0.00 0.01 0.03 0.00 0.00 0.01 0,01
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Inversion of composite material elastic constants: K. Balasubramaniam and N. S. Rao 48.0 O
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Progress of the genetic search
20000 15000 10000 ERR(C) 5000 0
7
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7 100
C12 (GPa) 150 CI I (GPa)
Figure 6
200
4
3
Search space for a two-parameter inverse problem
fact, using data from a single nonsymmetry plane (in this case, ~b = 450) does not appear to yield the expected results. The above observations suggest that the relative sensitivities of the elastic constants to the fitness function (hence, the phase velocities) are instrumental in determining the accuracy and reliability of the G A identification. The surface plot in Figure 6 of the fitness function for a simple two-parameter (Cll and C12) problem space (~b = 30 and 60 °, material no. 1), illustrates the marked difference in sensitivities of the elastic constants. It can be seen that the poor sensitivity of C12 as compared to C12 is consistent with
178
7
its greater scatter in reconstruction as revealed by the standard deviations of Table 3. Hence, the primary reason for the improved reconstruction from combinations possessing planes in or close to the material symmetry directions, appears to be the exploitation of the maximum possible sensitivities (not the decoupling) of most, if not all, of the stiffness constants. Moreover, going by the success in reconstruction from purely nonsymmetry planes as well as the case with symmetry planes, reinforces the fact that it is the sensitivity to the elastic constants rather than the nonlinearity of the problem, which is the determining factor.
Inversion o f composite material elastic constants: K. Balasubramaniam and N. S. Rao 1 0 4.0
10 3.0
. ~
10% NOISE
"~/~
5% NOISE
10 z-o
C ~
10x.o
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~
10 -I.o
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Pupdate=0.90
10-2.o . . . .
z's . . . .
s'0 . . . .
7's . . . .
160 . . . .
. . . .
130 . . . .
l÷s . . . .
200
NUMBER OF GENERATIONS Figure 7
Convergence plots with and without noise
Sensitivity and stability to noise are closely linked. Usually, poor sensitivities tend to bring about reduced stability to noise. Stability to different levels (2, 5 and 10%) of noise was also investigated by adding 10 different distributions of synthetic random noise for each level. The motivation behind examining the GA performance to different random distributions for each noise level was to observe the effectiveness of an overdetermined inversion in reducing the effects of noise on the reconstruction. The reconstruction statistics for noise, from 4~ = 0, 90 and 45 °) and 0 = 15, 75 amd 45 °, are given in Table 2 for material no. 1, and the convergence plots for material no. 1 are shown in Figure 7. From these tables, it is conclusive that elastic constant identification is stable to 2 - 5 % noise of any distribution (ideal white noise). However, a noise level of 10% is also acceptable. Similar trials on other material systems of Table 1 showed that ~b = 0, 90 and 45 ° and = 15, 75 and 45 ° were the best combinations of planes for elastic constant identification. Thus, it can be confidently concluded that genetic algorithms can be used to invert experimentally measured ultrasonic bulk wave phase velocities to closely accurately evaluate the stiffness properties of unidirectional composites, provided the experimental errors in the measured input data are within 10% of actual values.
SUMMARY In this paper, an inversion method using genetic algorithms (GAs) was successfully employed to obtain the stiffness matrix of unidirectional composites from ultrasonic bulk wave phase velocity profile data sets. The algorithm was
tested on four different materials with known stiffness properties. It was demonstrated that the GA will provide a simultaneous solution set of all nine elastic constants using noisy (up to 10%) ultrasonic data sets of phase velocity profiles of the ultrasonic bulk wave modes. Genetic algorithms, unlike the gradient-based methods, do not require initial guesses, but instead work within a valid set of bounds which can often be quite broad. Furthermore, GAs also avoid entrapment at local minima. However, like all inverse methods, the sensitivity of the evaluated parameters (elastic constants) to the input data set (phase velocity profiles at specific orientation to material axes) is important in reliably providing an acceptable inverse solution for all evaluated parameters. Consequently, it was found that reliable reconstruction is possible employing data sets which have sufficient sensitivity to the elastic constants. Nevertheless, despite the success of bulk wave reconstruction, further work needs to be done in examining the material identification potential of GAs from other ultrasonic data sets such as surface and Lamb waves, which offer a much more convenient single-sided approach to material characterization.
ACKNOWLEDGEMENTS The research upon which this material is based was supported by the National Science Foundation through grant no. OSR-9108767, the State of Mississippi, and the Mississippi State University. Mention of trade names or commercial products used in research does not necessarily constitute their endorsement by the funding agencies.
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