Prediction of stress intensity factors in pavement cracking with neural networks based on semi-analytical FEA

Expert Systems with Applications xxx (2013) xxx–xxx

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

Prediction of stress intensity factors in pavement cracking with neural networks based on semi-analytical FEA Zhenhua Wu a,⇑, Sheng Hu b, Fujie Zhou b a b

Department of Engineering, Virginia State University, P.O. Box 9032, Petersburg, VA 23806, United States Texas Transportation Institution, The Texas A&M University System, College Station, TX 77843, United States

a r t i c l e

i n f o

Keywords: Prediction Stress intensity factor Pavement cracking Semi-analytical FEA Neural networks

a b s t r a c t Computation of the stress intensity factors (SIFs) at the crack tip is the basis for pavement crack propagation analysis. Due to the three-dimensional (3-D) nature of cracked pavements and traffic loading, twodimensional (2-D) finite element analysis (FEA) may be too simple to precisely predict SIFs, and the best choice for calculating the SIFs seems to be 3-D FEA programs. However, the 3-D FEA solutions are often computationally heavy. We had previously developed a semi-analytical FEA with multi-variable regression approach to fill this gap, but its accuracy still needs to be improved. To address this problem, a neural network approach based on semi-analytical FEA is presented in this paper: firstly, a SIFs database was generated through analyzing varieties of pavement structures using elastic semi-analytical FEA program; secondly, from the results in the database, neural network (NN) based SIF equations were developed for practical applications. The determination coefficients (R2) of all the developed NN models were greater than 0.99 and mean square error (MSE) values were less than 1e4. The comparisons between the prediction results from NN models and multivariable regression models also showed the advantage of NN over multivariable regression on the prediction accuracy. This proposed NN SA-FEA SIF prediction approach has been developed as a pavement crack propagation analysis tool. Published by Elsevier Ltd.

1. Introduction Cracking is one of the primary forms of distress in hot-mixed asphalt (HMA) pavements. It will lower the ride quality; reduce service life with the penetration of water and foreign debris into these cracks accelerating the deterioration of the HMA layer and the underlying pavement. Pavement research engineers are making significant efforts to better design and reduce the road distresses. In recent years, Paris’ law, which is based on linear elastic fracture mechanics, has been adopted to analyze pavement cracking problems. However, the difficulty of using the well-known Paris’ law is to accurately and quickly determine the cracking tip’s stress intensity factor (SIF) values, which is a parameter that amplifies the magnitude of the applied stress. Three-dimensional (3-D) finite element analysis (FEA) is necessary to accurately determine the SIF values of a cracked pavement due to the nature of 3-D pavement structure, but its heavy computation requirement prohibits its practical application. Meanwhile, two-dimensional (2-D) FEA approach is too simple to fit the pavement’s 3-D nature. Semi-analytical FEA (SA-FEA) is an approach between 2-D and 3-D FEA. It may be a promising approach to remedy 2-D FEA’s low

⇑ Corresponding author. Tel.: +1 804 524 8989x1125; fax: +1 804 524 6732. E-mail address: [email protected] (Z. Wu).

accuracy and 3-D FEA’s heavy computation requirement on pavement analysis. Under a research project sponsored by the Texas Department of Transportation, millions of SIF values for varieties of pavement structures have been generated using the SA-FEA approach (Hu, Hu, Zhou, & Walubita, 2008; Zhou, Hu, Hu, & Scullion, 2009). Past experiences (Zhou et al., 2009) also indicated that although the multivariable regression equations can quickly determine the SIF values based on the FEA results, the multivariable regression model’s accuracy needs improvement. How to bridge the gap of precisely estimating the SIF values for pavement applications becomes the new question for this research to address. Literature review (Adeli, 2001; Flintsch, 2003) suggests that neural network has recently been applied together with fracture mechanics to predict the cracking growth, or the stress intensity factors, or the stress concentration factors (SCFs) etc. The NNs have been shown to be capable of building a class of flexible models which can be used for a variety of different applications, such as nonlinear regression and discrimination analysis (Li, Wu, & Zhang, 2008; Sarkar, Ben Ghlia, Wu, & Bose, 2009). With its ability to learn from the sample training set in a supervised or unsupervised manner, the NN might provide a promising alternative for the conventional regression equation approach on estimating SIFs. Das and Parhi (2009) had applied neural network technique for diagnosing a cracked cantilever beam. In his NN model, there were six input parameters, which were the relative deviations of the first

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Please cite this article in press as: Wu, Z., et al. Prediction of stress intensity factors in pavement cracking with neural networks based on semi-analytical FEA. Expert Systems with Applications (2013), http://dx.doi.org/10.1016/j.eswa.2013.07.063

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three natural frequencies and the first three mode shapes; and two output parameters, which were relative crack depth and relative crack location. Theoretical expressions were developed to calculate the effect of crack depths and locations on natural frequencies and mode shapes. Strain energy release rate at the crack section of the beam was used for calculating the local stiffness of the beam. He summarized different boundary conditions on the crack location, derived several training patterns, and designed the neural network models accordingly. Finally he developed experiments to verify the robustness of the developed NN models. The accuracy of NN outputs to the real results illustrated the developed system’s feasibility on predicting the location and depth of the crack. Ince (2010) had developed a NN model to analyze effective crack model (ECM) for determining the fracture parameters of cement-based materials. To characterize the fracture of concrete structures, the ECM needs two fracture parameters: the effective crack length and the critical stress intensity factor, which require a closed-loop testing system and heavy computational loads. For this reason, ECM was simulated with an artificial neural network (ANN) in his study. The authors collected 464 noisy test data from the literature, which were obtained via different test methods in different laboratories. The ANN was built directly on these data by using the ANN’s self-organizing capabilities. ANN had been applied to recognize the stress intensity factor in the interval from micro-crack to fracture on compact tension specimens using acoustic emission (AE) measurements (Kim, Yoon, Jeong, & Lee, 2004). The specimens were made from structural steel SWS490B. The ANN had a 5-14-1 structure: five neurons in the input layer, which were the AE parameters including ringdown counts, rise time, energy, event duration and peak amplitude, 14 neurons in the hidden layer, and 1 neuron in the output layer which represents the SIF. The performance of the ANN was tested using a specific set of the AE data. Kutuk, Atmaca, and Guzelbey (2007) presented the application of NNs to express the explicit formulation of stress intensity factor (SIF) for the opening mode (KI) of fracture mechanics. Explicit formulations for KI values were obtained using the parameters from the trained NNs. Some numerical applications were performed to show the generalization capability of the trained NNs. He derived an SIF formulation for three commonly used geometries in fracture mechanics. It showed that the results of the explicit formulation were in good agreement with the ANSYS FEA. Chiew, Gupta, and Wu (2001) had developed an approach using NN to estimate the SCFs of multi-planar tubular XT-joints. To train and test the network, SCF data, which covers a wide range of geometrical parameters for the XT-joints with three axial loads, was generated using the FEA. In the NN model, the geometrical properties of the tubular joints were used as the training input parameters, and the FEA SCFs were used as the training output parameters. Different network configurations were also tested for the best possible results. The results illustrated that a trained NN

can predict the SCFs for the various load cases with a high level of accuracy. Ceylan, Gopalakrishnan, and Lytton (2011) had applied the NN methodology to model the SIF as cracks grow upward through a HMA overlay as a result of both load and thermal effects with and without reinforcing interlayers. Nearly 100,000 runs of a finite-element program were conducted to calculate the SIFs at the tip of the reflection crack. The coefficients of determination (R2) of all the developed NN models were above 0.99 except one model. Comparing to his work, the research presented in this paper has a larger dataset with 2,000,000 finite element analysis runs. Our study’s input/output variables’ data ranges were much wider than his, so the modeling process was also more complicate. Besides this, they only analyzed the reflective cracking in the pavement, but we covered both the fatigue and reflective cracking. On the NN modeling, we provided more detailed work on (1) setting neural network’s topology including input variables and output variables, number of neurons in each layer, types of neuron functions in each layer, and (2) deciding weights and biases with different learning and training functions. Fathi and Aghakouchak (2007) had developed four multiple layer perceptron (MLP) networks to predict weld magnification factor for weld toe cracks in T-butt joints under membrane and bending loading. He obtained the training data for these networks from the FEA modeling. Two types of neural networks including MLP and radial basis function (RBF) were developed to predict stress intensity modification factors for deepest point of fatigue cracks in tubular T-joint, under axial loading. The results of these networks were used to predict fatigue lives of tubular T-joints. The comparison between network outcomes and fatigue results in experiments shows that NN is a successful technique in predicting weld magnification factor. Most of the aforementioned applications follow the similar procedures: they first get the data through real experiments or FEA simulation, then use the NN to identify the pattern of the fracture mechanics. Initiated by these, we were enlightened to develop neural network based models to estimate of SIFs for analyzing two common pavement crack types: fatigue cracking and reflective cracking, see Fig. 1 (Courtesy: ‘‘http://www.pavementinteractive.org’’). The study was performed as part of a larger endeavor of using fracture mechanistic models to characterize defects on pavements. The NN models use SA-FEA profiling to predict SIFs in pavement cracking. For the comparison purposes, both NN and multivariate nonlinear regression models were developed and compared on the estimation performance. The NN prediction approach remedied the precision deficiency of multivariable regression equations on estimating SIF values, also has the advantage of less computation burden. The rest paper is organized as follows. Section 2 discusses the proposed approach including: (1) how to determine SIFs from SA-FEA, and (2) how to build neural network and select proper

Fig. 1. (a) Fatigue cracking, (b) reflective cracking.

Please cite this article in press as: Wu, Z., et al. Prediction of stress intensity factors in pavement cracking with neural networks based on semi-analytical FEA. Expert Systems with Applications (2013), http://dx.doi.org/10.1016/j.eswa.2013.07.063

Z. Wu et al. / Expert Systems with Applications xxx (2013) xxx–xxx

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equation, can account for the complex nature of HMA pavement cracking analysis.

dc n ¼ AðDkÞ dN

(a)

(b)

ð1Þ

where c is the crack length, N is the number of loading cycles, Dk is SIF amplitude, A and n are material constants of asphalt mixture determined by lab testing. From Eq. (1), it can be seen that the key issue of applying Paris law in pavement cracking analysis is to identify a simple way of determining the stress intensity factors under various combinations of traffic loads and pavement structures. In order to achieve the computation efficiency under the accuracy constraint, neural network based semi-analytical finite element analysis (SA-FEA) approach was applied in this study to determine bending mode stress intensity factor KI and shearing mode stress intensity factor KII. This proposed approach will be specified in Sections 2.2 and 2.3. 2.2. Determine SIFs from SA-FEA

(c)

(d) Fig. 2. (a) Bending mode in fracture mechanics, (b) shearing mode in fracture mechanics, (c) pavement loading type in bending mode, and (d) pavement loading type in shearing mode.

parameters from the computer simulation. In Section 3, the SIF prediction results for fatigue cracking and reflective cracking on different types of pavements are presented. Comparisons between NN models and regression models are illustrated on different performance metrics including determination coefficient (R2), absolute average error (AAE), and standard error of predicted values divided by the standard deviations of measured values (Se/Sy) etc. Section 4 summarizes the findings of this study and outlines some future work.

2. Using NN approach to predict SIFs based on the SA-FEA 2.1. Paris’ law’s application on pavement cracking estimation In the pavement research, we are interested in two types of cracking modes: bending and shearing. In fracture mechanics, these two load types are categorized as Mode I, or II. Mode I, shown in Fig. 2(a), is an opening (tensile) mode where the crack surfaces move directly apart. Mode II, shown in Fig. 2(b), is a sliding (inplane shear) mode where the crack surfaces slide over one another in a direction perpendicular to the leading edge of the crack. Fig. 2(c) and (d) show the corresponding pavement loading types (in bending mode and in shearing mode) respectively. Note that in Fig. 2(c), the crack front is in the middle of the axle load area, while in Fig. 2(d), the crack front is at the edge of the axle load area. The combined mechanism of crack propagation can be modeled by fracture Modes I (bending) and II (shearing) makes the fracture mechanics approach very attractive for modeling HMA pavement cracking. The application of Paris’ law, expressed in the following

The purpose of applying FEA is to calculate stress intensity factor, which is one of the key parameters in applying fracture mechanics to analyze and design pavement. Currently, two categories of FEA based SIF computation tools are available. The first category includes commercial FEA packages (such as ABAQUS and ANSYS), which are general or, rather, multipurpose. These universal FEA packages are powerful and accurate, but also complex and user-unfriendly. The complexity requires intensive training which is time consuming and often not ideal for most practicing pavement engineers and researchers. Furthermore, these commercial FEA packages are relatively costly and require licenses; only large engineering firms and institutions own these commercial FEA packages. Thus, in real practice, these commercial FEA packages are not readily used for routine crack propagation analysis and pavement design. The second category includes those specialized FEA tools for pavement SIF computation. Currently, two computation programs have already been developed for pavement analysis. The first one is CRACKTIP (Chang, Lytton, & Carpenter, 1976), which is a twodimensional (2-D) FEA program, and it modeled a single vertical crack in the HMA layer via a crack tip element. This program has been successfully used to develop the thermal SIF model for low-temperature cracking prediction under SHRP A-005 research (Lytton et al., 1993). However, the difference between 2-D plane strain conditions and the 3-D nature of a cracked pavement and traffic loading often leads to a significant over-estimation of the displacements and SIF values under the same load. The other pavement SIF program, CAPA (computer aided pavement analysis), (Scarpas, Blaauwendraad, De Bondt, & Molenaar, 1993; Scarpas, De Bondt, & Gaarkeuken, 1996) is a 3-D program with some special functions to address the cracking issue. These functions make the CAPA 3-D program a good option for crack propagation analysis. Unfortunately, due to its 3-D characteristics, its high hardware and computation-time demands render it suitable primarily for research purposes. There are currently few 3-D FEA tools that are relatively inexpensive and would allow the practitioners to interpret routinely these complex but frequently encountered situations of pavement crack analysis. Thus, there is great need to find a means both to improve the calculation speed and to reduce the resource requirement without the loss of accuracy. One of the methods that seem most promising to achieve the aforementioned objective is the method known as semi-analytical (SA) FEA method. This method can effectively transform a 3-D pavement analysis problem to an equivalent 2-D model pavement at a significant savings in relation to the computational effort (Zienkiewicz, 1977). We developed a

Please cite this article in press as: Wu, Z., et al. Prediction of stress intensity factors in pavement cracking with neural networks based on semi-analytical FEA. Expert Systems with Applications (2013), http://dx.doi.org/10.1016/j.eswa.2013.07.063

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Table 1 Pavement database for SIF prediction models.

Table 2 Range of input parameters in fatigue cracking models.

SIF types

Pavement properties

Parameters

Fatigue cracking Bending SIF Shearing SIF

E1-H1-Ebase-Hbase-Esub-c-c/H E1-H1-Ebase-Hbase-Esub-c-c/H

Reflective cracking AC over AC bending SIF AC over AC shearing SIF AC over PCC bending SIF AC over PCC shearing SIF

E1-H1-E2-H2–Ebase-Hbase-c-C-C/H E1-H1-E2-H2–Ebase-Hbase-c-C-C/H E1-H1-E2-H2–Ebase-Hbase-c-C-C/H E1-H1-E2-H2–Ebase-Hbase-c-C-C/H

E1 (MPa) H1 (mm) Ebase (MPa) HBase (mm) Esub (MPa) c (mm) c/H

specialized pavement crack propagation analysis tool SA-CrackPro based on the SA-FEA approach. Comparing to the general purpose FEA program, this new SA-CrackPro analysis tool (1) reduced analysis dimension thus improving computational efficiency, (2) applied isoparametric quadratic quarter-point element (which is infrequently used in conventional FEA) to represent the singular crack-tip stress and strain in pavement fracture, (3) used thin-layer elements to simulate pavement layer contact condition and load transfer efficiency at joints and cracks, and (4) successfully automatically meshed and re-meshed both quarter elements and standard elements surrounding/along the crack tip (Hu et al., 2008). Regarding the concern of accuracy of replacing SA-FEA with 3-D FEA, we also verified accuracy of the SA-CrackPro with the commercial ANSYS FEA. The results were shown as in Table 1 of Hu et al. (2008). It shows that SA-CrackPro has comparable accuracy to ANSYS as the reference benchmark, with the maximum error rate of 8.8%, and most error rates of around ±5%. More detailed information about the SA-CrackPro program, pavement structure and boundary conditions, and the comparison with the ANASYS FEA program can be found in reference (Hu et al., 2008). 2.3. Using neural network model to predict the SIFs Although SA method has shortened the computing time of the 3-D FEA, the computation time of SA-FEA is still considerable due to the FEA nature. When directly integrating SA-FEA into the pavement design software to predict crack propagation, if the SA-FEA program needs to be run many times during crack propagation analysis, the total analysis time will often take hours, which is still undesirable. To be more practical, we highly recommended an approach of using SA-FEA to predict SIF: firstly, use SA-CrackPro to perform large-scale SIF calculations; secondly, develop SIF regression equations based on these calculations. By directly integrating these SIF equations into pavement design and analysis tools, the time of analyzing crack propagation can often be reduced to a few seconds. With the verified SA-CrackPro program, we first generated the SIFs prediction database determining the cracking SIFs at different

Fatigue bending SIF

Fatigue shearing SIF

Min

Max

Min

Max

1000 25 50 100 20 3.15 0.051

30,000 1600 20,000 1600 180 1520.1 0.954

1000 25 50 100 20 3.15 0.051

30,000 1600 20,000 1600 180 1520.1 0.954

situations when cracks grow up. Networked personal computers were used in making numerous runs to calculate pavement’s SIFs (close to 2 million SIFs) on various pavement structures under different traffic loads. For fatigue cracking analysis, the pavement is a three-layer structure: (1) HMA layer, (2) base layer, and (3) subgrade. For reflective cracking analysis, the pavement is a fourlayer structure: (1) HMA overlay, (2) existing asphalt concrete (AC) layer or Portland cement concrete (PCC) layer, (3) base layer, and (4) subgrade. Table 2 lists the models in the database that have been developed for predicting the SIFs as a crack grows up through the HMA layer. In Table 1, E1 and H1 are the Young’s modulus and the thickness of the HMA layer, respectively; E2 and H2 are the Young’s modulus and thickness of the existing AC or PCC layer, respectively; Ebase and Hbase are the Young’s modulus and thickness of base layer, respectively; Esub is the Young’s modulus of subgrade; c is crack length in the HMA layer; C equals c plus H2; and H equals H1(for fatigue cracking analysis) or equals H1 plus H2 (for reflective cracking analysis). These parameters are illustrated as Fig. 3, and their data ranges for the fatigue cracking and reflective cracking models are tabulated as Tables 2 and 3 respectively. In the tables, the rows are the input parameters and their ranges correspondingly. The columns are pavement types. Note that in Fig. 3(b) the Esub is not included because the sensitivity analysis results in Zhou et al. (2009) show that Esub does not have significant influence on reflective cracking SIFs. With this database, we started predicting SIFs with neural network. The goal of neural network modeling was to predict the SIFs of pavements using a function of the inputs, which were measurable variables that would act upon the output. Desired network architectures were built containing a few hidden layers and hidden nodes for a good prediction of the stress intensity factors. Fig. 4 shows the structure of the multilayer perceptron neural network. In this structure, input layer corresponds to the information from SA-FEA profile such as pavements’ mechanical or physical properties. Output is the SIF value at different loading and cracking length etc. The input was propagated forward through the network to compute the output value. The error is calculated based on the

Fig. 3. Illustration of pavement structures for fatigue cracking (a) and reflective cracking (b) analysis.

Please cite this article in press as: Wu, Z., et al. Prediction of stress intensity factors in pavement cracking with neural networks based on semi-analytical FEA. Expert Systems with Applications (2013), http://dx.doi.org/10.1016/j.eswa.2013.07.063

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Z. Wu et al. / Expert Systems with Applications xxx (2013) xxx–xxx Table 3 Range of input parameters in reflective cracking models. Parameters

E1 (MPa) H1 (mm) E2 (MPa) H2 (mm) Ebase(MPa) HBase(mm) c (mm) C (mm) C/H

Reflective-AC over AC bending SIF

Reflective-AC over AC shearing SIF

Reflective-AC over PCC bending SIF

Reflective-AC over PCC shearing SIF

Min

Max

Min

Max

Min

Max

Min

Max

1000 25 700 50 50 100 1.25 51.25 0.088

30,000 1200 20,000 600 3500 1500 1140 1740 0.998

1000 25 700 50 50 100 1.25 51.25 0.088

30,000 1200 20,000 600 3500 1500 1140 1740 0.998

1000 25 20,000 200 100 150 1.25 201.25 0.24

30,000 800 40,000 400 15,000 1500 760 1160 0.997

1000 25 20,000 200 100 150 1.25 201.25 0.24

30,000 800 40,000 400 15,000 1500 760 1160 0.997

difference between the calculated output value and the desired value. In order to get the mean square error (MSE) between the actual and desired output values as close as possible to zero, back propagation algorithm (White, Wooldridge, Gallant, Hornik, & Stinchcombe, 1992) was applied by adjusting the weights and biases associated with each link of the network. The back propagation training process is illustrated as Fig. 5. During the backward pass, the error terms were computed when the hidden units and the weights and biases were updated. The output was then compared to the desired output and the MSE was computed. If the error was zero or close to preset values, the network training process stopped. Otherwise desired weights and biases would be searched with different learning and training functions such as gradient descent algorithm etc. In the neural network modeling process, two aspects of information need to be decided: (1) the neural network’s topology including input variables and output variables, number of neurons in each layer, types of neuron functions in each layer; and (2) weights and biases with learning and training functions used in the neural network model. The following sections detailed how to decide these two kinds of information.

variables may be removed with sensitivity analysis in NN models; we will investigate that in the future research. There is no hard-and-fast rule for determining the number of hidden layers and the number of nodes within each layer. Only rules of thumb are converged MSE values and required computation time for different structures. A few topologies (7-50-1, 7-50-50-1, and 7-60-60-1) for fatigue cracking, and (9-50-1, 9-50-50-1, and 9-60-60-1) for reflective cracking were investigated. Mean error rate was used to determine which topology would work best. The training results show that only structures of 7-60-60-1 for fatigue cracking and 9-60-60-1 for reflective cracking will converge to our desired MSE error (1e4), thus we decided NN model’s topological structure as 7-60-60-1 or 9-60-60-1, and the neuron functions in the hidden layers and output layer as ‘‘log-sigmoid’’, ‘‘log-sigmoid’’, and ‘‘pure-linear’’ respectively. The input variables used for predicting SIFs on different types of pavements are listed in Table 4.

(1) Set up the network topology: The input nodes were determined according to the importance of parameter’s contribution to SIF prediction. The selection of the input parameters to NN model is based on the sensitivity analysis on regression equations in reference (Zhou et al., 2009). We identified that those variables are necessary to guarantee the prediction accuracy in regression equations. Some of dependent

Fig. 4. A multilayer perceptron neural network with error back propagation.

Fig. 5. Flow chart of the back propagation neural network.

Please cite this article in press as: Wu, Z., et al. Prediction of stress intensity factors in pavement cracking with neural networks based on semi-analytical FEA. Expert Systems with Applications (2013), http://dx.doi.org/10.1016/j.eswa.2013.07.063

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(2) Determine parameter settings: For every link between layers, there are two parameters to be estimated namely biases and weights need to be updated with NN parameters. Parameters settings such as training functions and learning functions were used to determine the optimum biases and weights for the NN. Training and learning functions are mathematical procedures used to automatically adjust the network’s weights and biases. The training function dictates a global algorithm that affects all the weights and biases of a given network. The learning function can be applied to individual weights and biases within a network. While training the neural network, each training function is developed based on a specifically designed learning algorithm. The training functions and learning functions evaluated were those available in MATLAB’s Neural Network Toolbox (Demuth, Beale, & Hagan, 2007).This toolbox supports a variety of training algorithms, including gradient descent methods, conjugate gradient methods, the Levenberg–Marquardt algorithm (LM), and the resilient back propagation algorithm (Rprop) etc. Matlab simulation experiments were used to determine the best training function and learning function. Table 5 lists the available training functions and their performance on SIFs prediction. The simulation results show that MSE error will converge to the desired error only with the ‘‘Trainlm’’ function.

Table 5 List of the available training functions and their performance on SIF prediction. Algorithm

Performance

Traingd

Gradient descent back propagation

Traincgf Trainscg

Conjugate gradient back propagation with Fletcher–Reeves updates Scaled conjugate gradient back propagation

Trainoss

One step secant back propagation

TrainBFG

BFGS quasi-Newton back propagation

TrainCGB

Conjugate gradient back propagation with Powell–Beale restarts Gradient descent with momentum back propagation Levenberg–Marquardt back propagation RPROP back propagation

Not converge Not converge Not converge Not converge Not converge Not converge Not converge Converged Not converge Not converge

Traingdm Trainlm Trainrp Traingdx

Gradient descent with momentum & adaptive learning back propagation

Table 6 List of the available learning functions and their performance on SIF prediction.

After we decided ‘‘Trainlm’’ as the training function in our NN model, we started to select the proper learning function to embed in the training function. Table 6 lists the available learning functions in Matlab and their error performances and computation time for desired error on the pavement data. The results illustrated that the learning function won’t affect the precision that much. It only affected the calculation speed of NN model. Comprehensively considering error performance and computation time, we selected ‘‘Learngd’’ as our learning function. (1) Decide the output function: After deciding the NN topology and training the NN weights and biases, the network output, SIFp, as the predicted stress intensity factor value, can be expressed by a function f (X, w) of the input data X = [x1, . . ., xm]T and the network parameters w commonly called weights. The neural network output is given by network h i weights vector w ¼ k1 ; . . . ; kH2 ; b11 ; . . . ; bH1 H2 ; c11 ; . . . ; cH1m ,   and the biases vector B ¼ b11 ; . . . ; b1H1 ; b21 ; . . . ; b2H2 ; b31 . The scalars H1 and H2 denote the number of nodes in hidden layers 1 and 2 of the network, respectively. g1(x), g2(x) and g3(x), which are selected to be the ‘‘log-sigmoid’’, ’’log-sigmoid’’ and ’’pure-linear’’ functions, are neuron functions attached to nodes in hidden layer 1, layer 2 and output layer. The neural network can be interpreted as a parametric nonlinear regression of SIF on X. The network model is completely determined once the number of nodes in hidden layers 1 and 2 has been determined and the network weights have been calculated. Finally, the predicted SIFp can be calculated as below Eq. (2).

Training function

Learning function

Algorithm

Performance

Time (s)

Learncon Learngd

Conscience bias learning function Gradient descent weight/bias learning function Gradient descent with momentum weight/bias learning function Hebb weight learning function Hebb with decay weight learning rule learn is instar weight learning function Kohonen weight learning function LVQ1 weight learning function LVQ2 weight learning function Outstar weight learning function Perceptron weight and bias learning function Normalized perceptron weight and bias learning function Self-organizing map weight learning function Batch self-organizing map weight learning function Widrow–Hoff weight/bias learning function

0.000997 0.000992

673 197

0.000911

373

0.000997 0.000877

132 468

0.000837 0.000959 0.000982 0.000986 0.000829

468 163 269 140 526

0.000991

442

0.00107

326

0.000911

153

0.001

247

Learngdm Learnh Learnhd Learnk Learnlv1 Learnlv2 Learnos Learnp Learnpn Learnsom Learnsomb Learnwh

SIF p ¼ f ðX;wÞ ¼ g 3

H2 X

kh 2 g 2

h2 ¼1

H1 X

bh2 h1 g 1

h1 ¼1

m X

!

!

ch1 i xi þ b1h1 þ b2h2 þ b3

! ð2 - 1Þ

i¼1

1 1 þ expðxÞ 1 g 2 ðxÞ ¼ 1 þ expðxÞ g 3 ðxÞ ¼ x

ð2 - 2Þ

g 1 ðxÞ ¼

ð2 - 3Þ ð2 - 4Þ

of input variables that have high influence.

Table 4 List of critical input variables on the SIF predication. Pavements

NN structure

Input variables

Fatigue-bending SIF Fatigue-shearing SIF Reflective-AC over AC bending SIF Reflective-AC over AC shearing SIF Reflective-AC over PCC bending SIF Reflective-AC over PCC shearing SIF

7-60-60-1 7-60-60-1 9-60-60-1 9-60-60-1 9-60-60-1 9-60-60-1

E1-H1-Ebase-Hbase-Esub-c-c/H E1-H1-Ebase-Hbase-Esub-c-c/H E1-H1-E2-H2–Ebase-HBase-c-C-C/H E1-H1-E2-H2–Ebase-HBase-c-C-C/H E1-H1-E2-H2–Ebase-HBase-c-C-C/H E1-H1-E2-H2–Ebase-HBase-c-C-C/H

Please cite this article in press as: Wu, Z., et al. Prediction of stress intensity factors in pavement cracking with neural networks based on semi-analytical FEA. Expert Systems with Applications (2013), http://dx.doi.org/10.1016/j.eswa.2013.07.063

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When we were performing the NN based SIF prediction, simulation results show that once the weights and biases were trained in the neural network structure, the trained NN models computed their results based on Eq. (2) very quickly (more than 30,000 cases in less than a second). The NN approach would be very computationally efficient for SIF prediction.

3. Results and comparisons 3.1. Results

completely different from the training data, but can represent all features contained in the trained NN model. These 40 NN SIF prediction models were successfully developed and summarized in Table 7. The goodness-of-fit statistics for the NN model predictions in arithmetic scale were performed by using statistical parameters such as the determination coefficient R2 (overall R2 for both training and validating) and MSE. The R2 is a value between 0 and 1, and it is a measure of correlation between the predicted and the measured values, thus determining accuracy of the fitting model (the higher R2, the higher accuracy).

"

N X 2 R ¼1 ð^xi  xÞ 2

Due to data size consideration, the 6 models in Table 1 were expanded to 40 models based on different traffic load levels (expressed by different tire lengths) and different existing pavement condition (expressed by different joint load transfer efficiency values). The minimum data size for these 40 sets is 36,750, and the maximum data size is 57,600. For each dataset, we randomly divided it into training and validating parts as the ratio of 80% over 20%. The training data set is used to build/calibrate NN models and validating data set is for validating against overfitting. To achieve this, we used the Matlab function of ‘‘dividevec’’. It supports random partition of original data to avoid the overfitting on the final results (Demuth et al., 2007).The validation data is

i¼1

,

N X ðxi  xÞ2

# ð3Þ

i¼1

A sample of R2 prediction performance of NN based SIFs prediction models for fatigue cracking bending and shearing SIFs is shown in Fig. 6. Fig. 7 is a sample prediction performance for reflective cracking bending and shearing SIFs on the AC over AC pavement; and Fig. 8 is for reflective bending and shearing SIF on the AC over PCC pavement. It shows that NN based models all have R2 values higher than 99% on the training, validating and overall data, and the MSE values are all smaller than 1e4. It illustrated the true generalization capabilities of the NN models within the domain of the available data.

Table 7 Summarization of the NN modeling results on all SIF types. SIF Types

R2

MSE

Data size

Fatigue cracking Bending-tirelength40 mm Shearing-tirelength40 mm Bending-tirelength112 mm Shearing-tirelength112 mm Bending-tirelength184 mm Shearing-tirelength184 mm Bending-tirelength256 mm Shearing-tirelength256 mm

0.991797 0.997721 0.990841 0.996842 0.990443 0.996443 0.990901 0.996403

9.47e5 8.95e5 9.57e5 9.88e5 9.72e5 9.95e5 9.52e5 9.50e5

36750 36750 36750 36750 36750 36750 36750 36750

Reflective cracking AC over AC bending-tirelength40 mm AC over AC bending-tirelength112 mm AC over AC bending-tirelength184 mm AC over AC bending-tirelength256 mm AC over AC shearing LTE = 0.1 tirelength40 mm AC over AC shearing LTE = 0.1 tirelength112 mm AC over AC shearing LTE = 0.1 tirelength184 mm AC over AC shearing LTE = 0.1 tirelength256 mm AC over AC shearing LTE = 0.5 tirelength40 mm AC over AC shearing LTE = 0.5 tirelength112 mm AC over AC shearing LTE = 0.5 tirelength184 mm AC over AC shearing LTE = 0.5 tirelength256 mm AC over AC shearing LTE = 0.9 tirelength40 mm AC over AC shearing LTE = 0.9 tirelength112 mm AC over AC shearing LTE = 0.9 tirelength184 mm AC over AC shearing LTE = 0.9 tirelength256 mm AC over PCC bending-tirelength40 mm AC over PCC bending-tirelength112 mm AC over PCC bending-tirelength184 mm AC over PCC bending-tirelength256 mm AC over PCC shearing LTE = 0.1 tirelength40 mm AC over PCC shearing LTE = 0.1 tirelength112 mm AC over PCC shearing LTE = 0.1 tirelength184 mm AC over PCC shearing LTE = 0.1 tirelength256 mm AC over PCC shearing LTE = 0.5 tirelength40 mm AC over PCC shearing LTE = 0.5 tirelength112 mm AC over PCC shearing LTE = 0.5 tirelength184 mm AC over PCC shearing LTE = 0.5 tirelength256 mm AC over PCC shearing LTE = 0.9 tirelength40 mm AC over PCC shearing LTE = 0.9 tirelength112 mm AC over PCC shearing LTE = 0.9 tirelength184 mm AC over PCC shearing LTE = 0.9 tirelength256 mm

0.991299 0.991996 0.992474 0.993929 0.998361 0.998081 0.997901 0.997601 0.997981 0.997581 0.997422 0.996703 0.998481 0.997681 0.997741 0.997681 0.996723 0.996683 0.996603 0.996383 0.998381 0.997701 0.997222 0.996942 0.9988 0.99862 0.998101 0.997681 0.99952 0.99948 0.99944 0.99938

9.80e5 9.39e5 9.98e5 9.43e5 9.22e5 9.64e5 9.97e5 9.91e5 9.46e5 9.14e5 9.09e5 9.69e5 8.23e5 9.36e5 9.35e5 9.90e5 9.53e5 9.45e5 9.33e5 9.90e5 9.88e5 9.98e5 9.89e5 9.62e5 9.82e5 9.58e5 9.99e5 9.95e5 9.33e5 9.92e5 9.82e5 9.73e5

57600 57600 57600 57600 57600 57600 57600 57600 57600 57600 57600 57600 57600 57600 57600 57600 43200 43200 43200 43200 43200 43200 43200 43200 43200 43200 43200 43200 43200 43200 43200 43200

Note: LTE is the load transfer efficiency of joint/crack of the existing AC or PCC.

Please cite this article in press as: Wu, Z., et al. Prediction of stress intensity factors in pavement cracking with neural networks based on semi-analytical FEA. Expert Systems with Applications (2013), http://dx.doi.org/10.1016/j.eswa.2013.07.063

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Z. Wu et al. / Expert Systems with Applications xxx (2013) xxx–xxx

Fig. 6. Fatigue cracking bending (a) and shearing (b) NN SIF prediction performance.

3.2. Comparison between NN with statistical nonlinear regression As stated in the introduction part, we had previously developed a set of nonlinear regression equations with least square error approach to predict SIF, the detail on these regression equations can be found in reference [1]. Now we can compare the NN approach with the nonlinear regression method on SIF prediction. The comparison metrics include determination coefficient (R2), average absolute error (AAE) and Se/Sy. Their mathematic definitions are as Eqs. (3)–(5). Suppose^ xi is predicted values and xi is the true value, N is the sample size for the available data, then we have:

AAE ¼

N X j^xi  xi j

, N

ð4Þ

i¼1

Se=Sy ¼ ðstandard error of predicted valuesÞ=ðstandard deviation of true valuesÞ

ð5Þ

The comparison results are summarized in Table 8. We can see that the best nonlinear regression model of stress intensity factors in overlays that we could achieve had an R2 value of 0.985 and the lowest R2 is 0.958, while NN models constantly have R2 values higher than 0.99. NN models also constantly have lower AAE values than

Fig. 7. Reflective cracking (AC over AC) bending (a) and shearing (b) NN SIF prediction performance.

Please cite this article in press as: Wu, Z., et al. Prediction of stress intensity factors in pavement cracking with neural networks based on semi-analytical FEA. Expert Systems with Applications (2013), http://dx.doi.org/10.1016/j.eswa.2013.07.063

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Z. Wu et al. / Expert Systems with Applications xxx (2013) xxx–xxx

Fig. 8. Reflective cracking (AC over PCC) bending (a) and shearing (b) NN SIF prediction performance.

Table 8 Comparison between NN and nonlinear regression’s performance on SIF prediction. Pavements

Fatigue-bending Fatigue-shearing Reflective-AC over Reflective-AC over Reflective-AC over Reflective-AC over

NN model

AC bending AC shearing PCC bending PCC shearing

Nonlinear regression

R2

AAE

Se/Sy

R2

AAE

Se/Sy

0.993 0.997 0.991 0.998 0.997 0.998

0.192 0.037 0.119 0.04 0.046 0.027

0.0052 0.0052 0.0041 0.0041 0.0048 0.0048

0.97 0.985 0.958 0.984 0.961 0.983

0.352 0.1183 0.273 0.152 0.17 0.114

0.0052 0.0052 0.0042 0.0042 0.0048 0.0048

corresponding nonlinear regression models. On Se/Sy, the two approaches have the same performance. The comparison results achieved with all the cases clearly demonstrated the success of NN-based modeling approach over nonlinear regression. 4. Conclusions Preliminary models for HMA pavement cracking analysis have been developed with 2-D/3-D FEA approaches. However, those studies have the limitation of either over-estimation or heavy computation requirements. To remedy this, this study developed NN-based models which can be used in mechanistic-empirical procedures to predict and analyze SIFs of HMA cracking. Systematical modeling approaches including: (1) calculating SIF with SA-FEA program, (2) deciding NN structure topology, (3) selecting input variables, and (4) deciding training functions and learning functions, were explicitly described in this paper. Forty types of SIFs on various pavement structures under different traffic loading conditions were predicted with the NN models. Results show that NN models have significantly high accuracy in predicting SIFs with R2 values of 0.99. In order to justify the proposed NN approach’s advantage on SIFs prediction, comparisons were also made between NN and nonlinear regression approach on performance metrics on R2, AAE and Se/Sy. The comparison results achieved with all the cases clearly demonstrated the success of NN-based

modeling approach over nonlinear regression. Specific achievements for this study were to: (1) successfully apply the SA-FEA on the computation of SIFs, and validate the accuracy; (2) solidly model and predict stress intensity factors using NN and statistical multivariate approaches; and (3) systematically assess the value of these two approaches and identify areas for further investigation. For the future research, the authors are planning to further explore NN’s application on SIFs from two aspects: (1) Use sensitivity analysis or principal component analysis (PCA) to reduce the input variables in NN models. In the experiment, it was found that the NN training computation complexity and load were cursed with the number of input variables. When input variable increases, the computation load also increases exponentially. How to reduce the input variable while retaining the most useful information will be carried out in the future work. (2) Extend the NN models to predict thermal SIFs due to daily temperature variations. Further research can be performed on identifying parameters that will affect thermal SIFs and design proper NN algorithms for precisely predicting it. Acknowledgments The authors would like to sincerely thank the anonymous reviewers for their many invaluable comments and suggestions, which have greatly improved the quality of this article.

Please cite this article in press as: Wu, Z., et al. Prediction of stress intensity factors in pavement cracking with neural networks based on semi-analytical FEA. Expert Systems with Applications (2013), http://dx.doi.org/10.1016/j.eswa.2013.07.063

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Please cite this article in press as: Wu, Z., et al. Prediction of stress intensity factors in pavement cracking with neural networks based on semi-analytical FEA. Expert Systems with Applications (2013), http://dx.doi.org/10.1016/j.eswa.2013.07.063