Abstract:

In accordance with the extreme value condition for optimization, the backcalculation of pavement layer moduli is converted into the solution of a zero point problem for nonlinear mapping. A backcalculation model based on homotopy method is established by using a numerical differential method to solve the first and the second partial derivatives of deflections with respect to moduli. Then Li-Yorke algorithm is applied in solving initial value problems for differential equations to trace homotopy curves. A relevant program HMBPLM is developed for the backcalculation of moduli. The precision and the reliability of backcalculation results with the homotopy method are verified by backcalculating seven kinds of pavement layer moduli according to the deflection basins measured with Falling Weight Deflectometer (FWD) and comparing to other common backcalculation programs. In the meantime, the property of a wide range of convergence and the stability of backcalculation results with the homotopy method are checked through comparing to the results of different initial values. This shows that the homotopy method for backcalculation of pavement layer moduli has effectively solved the problems of initial value and local convergence in traditional optimization algorithms, so the method is high in precision, fast in speed, great in efficiency, stable in result and wide in convergent range.

Keywords:
Homotopy method; Backcalculation of moduli; Pavement; Deflection; Falling weight deflectometer (FWD)
*[Project 50208004 supported by National Natural Science Foundation of China]

1. Preface

Since the application of Falling Weight Deflectometer (FWD) in the 1970s for pavement evaluations, FWD has been regarded as a non-destructive testing device in a wider range of application for its accuracy, fast speed, safety, labor saving, convenience, non-destruction to pavement layers and other advantages. It was introduced into China in the late of the 1980s. At present there are about 30 FWDs in various kinds in China. In order to determine the rigidity and the strength of pavements, the resilient moduli of pavement layers must be backcalculated and analyzed from the deflection basins on pavement surface measured with FWD. Therefore, the backcalculation of pavement layer moduli has become a research hotspot in pavement engineering.

The backcalculation of pavement layer moduli is a very complex problem. At present the backcalculation methods have mainly mathematic programming (Van Cauwelaert et al., 1989; Harichandran et al., 1993; Sivaneswaran et al., 1993; Ni et al., 1994), database searching (Anderson, 1989; Scullion et al., 1990), system identifications (SIDs) (Wang F. et al., 1993), artificial neural networks (ANNs) (Meier et al., 1997; Zha et al., 2002) and genetic algorithms (GAs) (Fwa et al., 1997; Zha et al., 1998). The first three methods are the optimization ones based on heuristic searching techniques, and there exist such problems as initial value and local minimization. If the initial value is chosen not well, the backcalculation result will easily occur divergent or irrational. The ANNs are a kind of nice method for real-time treatment, but their precision and scope of application are greatly influenced on network architectures, training sets, learning algorithms and noisy treatments. The GAs are a kind of optimization method for simulating the natural selection law, and in fact they are the heuristic probabilistic searching algorithms of iterative self-adaptation. The GAs have advantage of good global convergence but a shortcoming of slow calculation speed, so it is very difficult to meet the requirement for a large number of backcalculation.

In order to solve the above problems such as the initial value and the local convergence for searching methods, the precision for ANNs and the speed for GAs, the paper researches homotopy method for the backcalculation of pavement layer moduli. The homotopy method developed in the 1970s is a kind of numerical method for solving nonlinear problems (Wang Zeke et al., 1990). Its characteristic is normally a wide range of convergence and easy to carry out parallel computations. It has been widely used in solving multiple zero points of mapping, fix points of mapping, complementation problems, optimization problems, boundary value problems and characteristic value of matrix.

2. Basic fundamentals

2.1 Mathematic model for backcalculation of moduli
The backcalculation of pavement layer moduli from measured deflection basins with FWD is generally described as an optimization problem (Zha et al., 1998):

(1)

In which, W_i is computed in accordance with the elastic layered system theory:

(2)

Where Y is the relative square error between measured deflections and theoretic ones. W_i, L_i, r_i and q_i are respectively the theoretic deflection (0.01mm), the measured deflection (0.01mm), the distance to load center (cm) and the weight in each testing point. h_j, E_j and m_j are respectively the thickness (cm), the resilient moduli (MPa) and the Poisson's ratio for each pavement layer. p and a are respectively the load (MPa) and the load radius (cm). i and m are respectively the serial number and numbers of testing points. j and n are respectively the serial number and numbers of pavement layers.

For the practical backcalculation, E_j is the variable, and other parameters the constant values. From Equation (1), it can be seen that the backcalculation of pavement layer moduli is a nonlinear optimization problem.

2.2 Fundamentals of homotopy method
Inverse image theorem
(Wang Zeke et al., 1990): Supposing X is a smooth manifold with boundary in dimension k, and Y is a smooth manifold with boundary in dimension l. k > l. f : X (r) Y is a smooth mapping. If y Î Y is simultaneously a canonical value of the mapping f: X (r) Y as well as that of the boundary ¶ f : ¶X (r) Y, the inverse image f ^-1(y) will either be a empty set or a manifold with boundary in dimension k - l.

Sard theorem:
If X is a smooth manifold with boundary, Y is a smooth manifold and f: X (r) Y is a smooth mapping, the measure in Y for the critical value set of f and that of ¶ f will be all zero.

The homotopy method is a kind of nonlinear numerical method for solving zero points or fix points of mapping. The basic thinking is that constructing a new mapping makes the dimensions of definition domain for the new mapping be the same as that of definition domain for the original mapping, and the dimensions of range reduce one dimension. Multiple non-intersecting smooth curves are obtained with the application of Sard theorem and inverse image theorem of canonical value. Some end points of the curves are zero points or fix points of the original mapping. From the given points of curves and through solving initial value problems for differential equations, the curves will be traced to zero points or fix points of the original mapping. Therefore, the homotopy method does not require strict to the initial value and is a kind of algorithm with a wide range of convergence.

Definition:
Supposing X and Y are nonempty subsets of Rⁿ. f, g: X (r) Y are smooth mappings. With regard to arbitrary (t, x) Î [0, 1] x X (r) Y , if the following will be tenable,

(3)

the smooth mapping H: [0, 1] x X (r) Y is called linear homotopy between f and g.The parameter t Î [0,1] is named as homotopy parameter.

A smooth auxiliary mapping g: Rⁿ (r) Rⁿ is chosen to make the zero point of g distinct. For example, the following is selected:

(4)

Where a is a constant vector in Rⁿ, then there will be only one zero point of x = a in g. g is named as ordinary mapping. We wish the work starts from the ordinary mapping g(x) = 0 and traces gradually to the object problem f(x) = 0. In other words, the tracing is gradually from the ordinary mapping g to the object mapping f so as to start from the zero point of the ordinary mapping g to reach the zero point of the target mapping f. From Equation (3), the following is obtained,

(5)

which is to say, the zero point of homotopy mapping H at t = 0 is the same as that of the mapping f; while the zero point at t = 1 the same as that of the mapping g . So the basic thinking of homotopy can be realized from the mapping in Equation (3).

From the above, the zero set H^-1(0) of homotopy H is the smooth simple curves. Tracing a curve in H^-1(0) from (1, a), the zero point of the object mapping f will be solved by starting from the zero point of the ordinary mapping g . Introducing a arc length parameter s, the tracing curve l(s) will be as follows:

(6)

Because l(s) is a curve in H^-1(0), it meets the following equation:

(7)

The derivatives with respect to s are solved from both sides of the equation, so the solution of the curve l(s) becomes a initial value problem for differential equations:

(8)

Where is the derivative of x(s) with respect to s. is the derivative of t(s) with respect to s.

Because 0 is a canonical value of homotopy H at [0, 1] x Rⁿ, the matrix in Equation (8) is nonsingular. There is only one solution at [0, 1] x Rⁿ for the initial value problem in Equation (8). Therefore, the method of a initial value problem for differential equations is applied in solving Equation (8) to obtain one zero point of the mapping f, which is to say, the homotopy method is come true.The detail computation can be referred to the Li-Yorke algorithm in relevant references (Wang Zeke et al., 1990).

2.3 Realization of homotopy method for backcalculation of moduli
In accordance with the extreme value condition for optimization, the optimum solution of Equation (1) must meet the following requirement:

(9)

Thus Equation (1) can be converted into solving the zero point of nonlinear equations in Equation (9). Introducing a homotopy parameter t and an ordinary mapping,

(10)

a linear homotopy mapping will be constructed as the following:

(11)

According to initial value problems for differential equations, the tracing of curve H^-1(0) is carried out from (1, E⁰) to t = 0 of (0, E*) with the Li-Yorke algorithm. Thus the zero point in Equation (9) will be obtained, and the relevant E* will be the backcalculation result.

In order to realize the homotopy method, the first and the second partial derivatives of the relative square error Y with respect to E must be computed. From Equation (1), the following will be reached:

(12)

Due to many special functions and infinite integral in the theoretical deflection W for the elastic layered theory, it is very difficult to derive and compute the first and the second partial derivatives of W with respect to E. In order to solve the computation of partial derivatives, the numerical differential method (Zha et al., 1996) is applied. According to the definition of derivatives and the given step h_j, the middle difference formulas will be reached:

(13)

When j =j', the following will be reached:

(14)

The numerical testing shows that the partial derivatives with satisfactory precision can be obtained when the step h_j is selected from the range of 0.1~0.001E_j. In the meantime, in order to reduce the computing quantities of the initial value problem, it can be estimated with Euler method. Then the backcalculation of pavement layer moduli will be realized according to the homotopy method. In line with the Li-Yorke algorithm, the relevant backcalculation program HMBPLM has been developed with Visual FORTRAN 5.0.

The value of ¶ Y /¶ E is usually smaller than that of g during tracing the curve. In order to make the difference between the two become smaller, the homotopy method will be modified through selecting a suitable ordinary mapping g . Because there is no possibility of negative moduli, g can be selected as a natural logarithm function:

(15)

The practical computation shows that the ordinary mapping in Equation (15) not only has effectively improved the tracing capability of the curve, but also reduced the influence on the precision of the numerical difference.

3. Comparison of examples

In order to test the capability of backcalculation of moduli with the homotopy method, the deflection basins measured with FWD must be selected to backcalculate and compare with other backcalculation programs. Therefore, 7 measured deflection basins are selected to analyze from 7 kinds of pavement structures, such as 2 layered, 3 layered and 4 layered system. The data of the measured deflection basins are shown in Table 1.

The data of deflection basins in Table 1 are quoted from a example database in LEEPWIN software developed by U. S. Army Engineer Waterways Experiment Station (WES), and the backcalculation program in the software is named as WESDEF (Van Cauwelaert et al., 1989). There are 7 deflections in each deflection basin, and the testing points are located at an interval of 12in. within the range of 0~72in. apart from the load center, including 0in. (0cm), 12in. (30.48cm), 24in. (60.96cm), 36in. (91.44cm), 48in. (121.92cm), 60in. (152.40cm) and 72in. (182.88cm) in proper order. The load is a circle, vertical and uniform one, and the load radius 5.9in. (15cm). Due to the widely used British unit system in backcalculation programs, the system is directly accepted for the practical backcalculation to compare easily. In the meantime, supposing there is a rigid under-layer beyond a certain depth of pavement surface, 24ft (731.52cm) will be selected in the practical analysis. When the elastic layered system theory is used for the calculation, the foundation will usually be considered as a semi-space layer with very hard rigidity, and its moduli generally is regarded as a constant of 1000ksi (6894.76 MPa) and the Poisson's ratio 0.50. Therefore, the above explanations will be followed during the backcalculation comparisons.

In addition, the commonly used programs for backcalculation of moduli are chosen for comparisons, including EVERCALC of WSDOT (Sivaneswaran et al., 1993), WESDEF of WES (Van Cauwelaert et al., 1989), MODULUS of TXDOT (Scullion et al., 1990), and MGABPLM (Zha et al., 1998) developed with the modified genetic algorithm by ourselves. In addition to HMBPLM and MGABPLM compute with our program for the elastic layered system, other backcalculation programs use the WESLEA program. As EVERCALC, WESDEF and MODULUS backcalculate with traditional searching methods, the convergence of results will depend on the selection of initial value. Therefore, the backcalculation moduli of minimum errors respectively will be chosen as the final results. The compared results of the backcalculation moduli are shown in Table 2. The moduli in the table are converted from the British unit system into the metric unit system, and the weight q_i for each testing point is 1. The definition of the relative average error d is as follows:

(16)

As far as the backcalculation results in Table 2 is concerned, there is no much difference among the results of soil subgrade moduli in each program. It also shows that the backcalculation results of soil subgrade moduli are of good stability, and the backcalculation moduli of pavement layers are slightly less stable than that of the subgrade. Through comparisons, the backcalculation results with the homotopy method and that with the genetic algorithm have kept high consistent, showing that the results with the homotopy method are of high precision. However, they have also kept consistent with that of EVERCALC in a good way. Compared with WESDEF and MODULUS, others are basically the same except that there is greater difference among the backcalculation results of the sixth deflection basins. It shows that the backcalculation moduli with the homotopy method are reliable.

As for the same deflection basin, the moduli are backcalculated with Pentium II/233 CPU computers. It will take about 20~30s with the homotopy method to backcalculate the moduli for one 2 layered system, and as to 3 and 4 layered system, it will usually take about 1min to complete. With the adoptable precision control, the computing speed of the homotopy method will be accelerated, and the speed completely meets the requirement of a large amount of backcalculation. The speed of the homotopy method is at least 10 times as fast as that of the genetic algorithm, while it is compared with the traditional optimization methods, the speed is rather slow. Although it is so, a reliable and stable result of moduli can be obtained with the homotopy method within 1min, whereas there will be the needs to repeatedly adjust the initial value within the given moduli range with the traditional optimization methods, and only in this way can the stable result be obtained. In fact, from the reliability of results, the efficiency of the homotopy method will obviously higher than that of the traditional optimization algorithms. Therefore, the backcalculation of moduli with the homotopy method is a stable and reliable way, and its speed can completely meet the requirement of engineering.

4. Comparison of convergence

In order to check a wider range of convergence of the homotopy method for backcalculation of moduli, the third deflection basin in Table 1 has been chosen for the backcalculation comparison. From the results in Table 2, it can be seen that moduli E are about 500/100/20 ksi (3447.38/689.48/137.90 MPa), and five groups of moduli produced randomly within the range of 0.001~1000E. The groups and E together serve as the initial values for the backcalculation comparison, and the compared results are shown in Table 3. The tracing curves with the homotopy method are illustrated in Figure 1 to Figure 6, where the dashed, the solid and the dotted curve are respectively presented as the tracing curve of E_1, E₂ and E₀ among the homotopy parameter t. The moduli in Table 3 and in Figure 1 to Figure 6 have been converted into the metric units.

Because the genetic algorithm has nothing to do with the selection of initial values, the results of MGABPLM has not been listed in Table 3. In the meantime, as MODULUS is backcalculated, the pavement layer moduli will be selected within a certain range of 10~10000ksi (68.95~68947.57MPa) in general, while as to the selection of the initial value for soil subgrade moduli, the range will be 1~100ksi (6.89~689.48MPa). Therefore, when MODULUS is used for the backcalculation, the range of pavement layer moduli will chose within 10~10000ksi, regardless of the initial values. The soil subgrade moduli will only be taken into consideration, and the backcalculation can be carried out when the moduli are within 1~100ksi. WESDEF needs not only the initial value but also the range, and the moduli range will select from 0.001~1000E in practical backcalculation.

From Table 3, it can be seen that EVERCALC is greatly influenced on initial values. This is mainly because the program uses the modified Gauss-Newton algorithm, and only can the reliable result be achieved when it starts at a good initial value. The convergence of WESDEF is better, but the reliability of the results worse; and as to the fourth initial value, it can not be backcalculated. The convergence of MODULUS is relatively much better. It is mainly because of the more strict range of moduli and the database searching method for the program. This way has actually reduced the capability and the range of backcalculation. As to the backcalculation results with the homotopy method, it can be seen with Figure 1~Figure 6 that it can all converge by selecting different initial values within the rang of 0.001~1000 times object moduli, and the backcalculation results is completely the same.

Fig 1: Tracing curves (Initial value 1).

Fig 2: Tracing curves (Initial value 2).

Fig 3: Tracing curves (Initial value 3).

Fig 4: Tracing curves (Initial value 4).

Fig 5: Tracing curves (Initial value 5).

Fig 6: Tracing curves (Initial value 6).

This shows that it is of very good stability and convergence. Therefore, the wider range of convergence and the reliability and the stability of results for the homotopy method have effectively solved the problems of initial values and the moduli range of each layer needed to select for the traditional optimization algorithms. During the practical backcalculation, there will be no need to make any fixed initial value as the homotopy method is applied, and it will be chosen by computers at random so as to greatly improve the efficiency of the backcalculation of moduli.

5. Conclusions

The following conclusions can be achieved through the analysis on the backcalculation of pavement layer moduli with the homotopy method:

1. According to the mathematical model for optimization, the backcalculation of moduli is converted into the solution of nonlinear equations problem. Due to the homotopy mapping and the tracing of curve H^-1(0), it has involved the first and the second partial derivatives for the object function with respect to the moduli. Because there are complex infinite integral and special functions in the theoretical solution for the surface deflection of the elastic layered system, it is very difficult to obtain the analytic solution of the partial derivatives. Then the numerical differential method has been used to compute the partial derivatives in an approximate way, and the backcalculation model of pavement layer moduli with the homotopy method has been established.
2. In accordance with the Li-Yorke algorithm, the program HMBPLM based on the homotopy method has been developed with Visual FORTRAN 5.0 to backcalculate pavement layer moduli. The numerical testing shows that it is proper that the numerical differential step is selected within 0.1~0.001E_j in order to successfully trace the curve H^-1(0). The capability of tracing curves will be effectively improved by changing the ordinary mapping.
3. The results consistence for the homotopy method with the genetic algorithm has been kept well through a lot of backcalculation comparisons for the measured deflection basins, and this shows that the results of the homotopy method are of high precision. In the meantime, as compared with those of EVERCALC, WESDEF and MODULUS, the backcalculation results are basically the same, and it illustrates that the backcalculation moduli with the homotopy method are reliable. Although the computing speed of the homotopy method is slower than that of the traditional one, it is much quicker than that of the genetic algorithm, and it can completely meet the requirement of engineering. The computing efficiency is much greater than that of the traditional optimization methods.
4. It shows that the backcalculation results from different initial values with the homotopy method are completely the same when the initial values are selected from a range of 0.001~1000 times object moduli. So the homotopy method is of a good wide range of convergence and the stability and reliability of the results. Such problems as initial value, local convergence and the needs of the moduli range of each layer for the traditional optimization algorithms have been effectively solved.

Therefore, the homotopy method is a kind of algorithm for backcalculation of pavement layer moduli with high precision, fast speed, great efficiency, stable result and a wide range of convergence. It can be applied in the backcalculation and analysis on a large number of measured deflection basins with FWD to evaluate the strength and rigidity of pavement structures.

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