The Open Civil Engineering Journal, 2009, 3, 83-9283Open AccessBackcalculation of Non-Linear Pavement Moduli Using Finite-Element Based Neuro-Genetic Hybrid Optimization Kasthurirangan Gopalakrishnan. Department of Civil, Construction and Environmental Engineering, Iowa State University, Ames, Iowa, USA Abstract: The determination of pavement layer stiffness is an essential step in evaluating the performance of existing road pavements and in conducting pavement design and analysis using mechanistic approaches.

Over the years, several methodologies involving static, dynamic, and adaptive processes have been developed and proposed for obtaining in-situ pavement layer moduli from Falling Weight Deflectometer (FWD) test deflection data through inverse analysis and parameter identification routines. In this paper, a novel pavement analysis toolbox combining the strengths of Finite Element (FE) modeling, Neural Networks (NNs), and Genetic Algorithms (GAs) is described. The developed user-friendly automated pavement evaluation toolbox, referred to as Neuro-Genetic Optimization Toolbox (NGOT) can be used on a real-time basis for accurate and rapid transportation infrastructure evaluation. It is shown that the developed toolbox backcalculates non-linear pavement layer moduli from actual field data with better accuracy compared to regression and conventional backcalculation approaches.INTRODUCTION According to recently published data 1, billions of U.S. Dollars are needed annually to improve transportation infrastructure conditions nationally.

Thus, consistent, costeffective, and accurate monitoring of pavement is necessary for improving the performance and serviceability of pavements and to schedule proactive pavement repair and maintenance activities. About 93% of the paved roads in the US are reported to be composed of flexible pavement 2.

Flexible pavements are multi-layered, heterogeneous structures that are designed to “flex” under repeated traffic loading. A typical flexible pavement structure consists of the surface course (typically Hot-Mix Asphalt) at the top, underlying base and subbase (optional) courses (typically unbound granular material), and a subgrade (typically soil) at the bottom. In the field, Non-Destructive Testing (NDT) of in-service pavements using a Falling Weight Deflectometer (FWD) equipment is carried out to measure the deflection response of the pavement structure to applied dynamic load that simulates a moving wheel.

The deflected shape of the basin is predominantly a function of the thickness of the pavement layers, the moduli of individual layers, and the magnitude of the load. The surface deflections are typically measured at radial offsets of 0 mm (D0), 300 mm (D1 or D300), 600 mm (D2 or D600), 900 mm (D3 or D900), 1200 mm (D4 or D1200), and 1500 mm (D5 or D1500) from the center of FWD load plate. “Backcalculation” is the accepted term used to identify a process whereby the elastic (Young’s) moduli of individual.Address correspondence to this author at the Civil Engineering, 354 Town Engineering Bldg. Iowa State University, Ames, IA; USA; Tel: (515) 294-3044; Fax: (515) 294-8216; E-mail:1874-1495/09pavement layers are estimated based upon measured FWD surface deflections. As there are no closed-form solutions to accomplish this task, a mathematical model of the pavement system (called a forward model) is constructed and used to compute theoretical surface deflections with assumed initial layer moduli values at the appropriate FWD loads (referred to as forward calculation).

Through a series of iterations, the layer moduli are changed, and the calculated deflections are then compared to the measured deflections until a match is obtained within tolerance limits. Most of the commercial backcalculation programs currently in use utilize an Elastic Layer Program (ELP) as the forward model to compute the surface deflections. Many studies have addressed the interpretation of FWD pavement deflection measurements as a tool to characterize pavement-subgrade systems 3-5. The backcalculated in-situ pavement layer moduli from measured deflections (inverse analysis) are by themselves indicators of pavement layer condition as well as necessary inputs for conducting mechanistic pavement structural analysis and remaining life calculations 6, 7. Over the years, several techniques have been proposed for back-calculation of pavement layer moduli such as the least-squares (parameter identification), database search, Neural Networks (NNs), neuro-fuzzy systems, and recently Genetic Algorithms (GAs) 8-21.

The hybrid approach presented in this paper represents the latest development in backcalculating the mechanical properties of flexible pavement systems. This innovative approach takes advantage of the combined efficiency and accuracy achieved by integrating advanced pavement numerical modeling schemes, computational intelligence based surrogate mapping techniques, and heuristics based global optimization strategies, and yet provides a userfriendly pavement evaluation toolbox for the pavement engineer to use on a real-time basis for accurate infrastructure evaluation. 2009 Bentham Open84 The Open Civil Engineering Journal, 2009, Volume 3PAVEMENT TIONLAYERMODULIBACKCALCULA-Considering the complex nature of the backcalculation problem, numerous approaches have been proposed and efforts have also been made to develop a standardized approach 22, 23. The discrepancies among the numerous backcalculation techniques developed for the backcalculation of pavement layer moduli arise from the type of the pavement (forward) response model and the optimization procedure utilized for the determination of appropriate layer modulus values 21. Existing methods of backcalculation rely on some assumptions and simplifications that should be made to facilitate the backcalculation process. Some of the major difficulties faced by researchers in using the different commercial backcalculation programs include: (1) ability to handle only limited and idealistic solutions due to closedform nature of solutions, (2) longer computational time in using conventional optimization techniques for backcalculation, (3) convergence of backcalculation solutions to local optima or non-unique nature of solutions depending on seed moduli (initial values), etc.

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Most of the conventional commercial backcalculation programs involve Multi-Layer Elastic Theory (MLET) in their forward calculation routines and assume that pavement materials are linear-elastic, homogenous, and isotropic resulting in unrealistic backcalculated pavement layer moduli. Several research studies have shown that pavement geomaterials used in the underlying pavement layers exhibit non-linear, stress-sensitive behavior under repeated traffic loading. Unbound aggregates used in pavement base/subbase course exhibit stress-hardening and fine-grained soils show stress-softening-type behavior 24-26. Research studies have shown that non-linear analysis using FE based approach increases the precision of the forward model 21. Majority of the commercial backcalculation programs employ an iterative approach (see Fig. 3) which is known to suffer from limitations such as dependency on the initial seed moduli and the possibility of local minimum solutions 27.

Other limitations include computational ineffectiveness or lack of robustness leading to divergence in some cases where no solution is obtained, requiring a high level of userinvolvement during the backcalculation process making it less amenable to automation, number and thickness of layers used in the analyzed pavement system, etc. To overcome the limitations associated with existing commercial backcalculation programs, a new toolkit referred to as Neuro-Genetic Optimization Toolbox (NGOT) was developed through FE-based neuro-genetic integrated systems approach. Such a hybrid approach towards backcalculation has many advantages which include realistic prediction of non-linear pavement layer moduli, rapid prediction ability and the provision to model uncertainties in the deflection data including noise, errors, etc. And derivation of global optimum solutions.

Backcalculation

In addition, such an intelligent, yet user-friendly toolbox can provide pavement engineers and designers with the ability to rapidly evaluate the infrastructure condition in real-time without requiring them to have in-depth knowledge of the actual modeling of the problem.Kasthurirangan GopalakrishnanNEURO-GENETIC OPTIMIZATION The framework of NGOT as discussed in this paper is shown in Fig. As mentioned previously, depending on the problem under consideration, the inputs and outputs will vary. The problem under consideration is to backcalculate pavement layer moduli from non-destructive test deflection data acquired using the FWD device. The major modules integrated into NGOT are the GA module, NN module and FE module. Each of these modules with a brief introduction and background and the steps involved in developing the NGOT toolbox are discussed in the following sub-sections. Genetic Algorithms Genetic algorithms are a part of evolutionary computing, a rapidly growing area of artificial intelligence.

Categorized as global search heuristics, GAs use techniques inspired by evolutionary biology such as inheritance, mutation, selection, and crossover (also called recombination). Being robust search and optimization techniques, GAs are finding applications in a number of practical problems where calculus-based search methods are inefficient in searching for the optimal solution in a complex multi-modal search space 28-31. In recent years, researchers started exploring the feasibility of using GAs for pavement layer moduli backcalculation 14-18, 32, 33.

First, the fundamentals of GA theory are briefly discussed. In the Simple Genetic Algorithms (SGA) evolutionary search process (see the ‘GA module’ in Fig. 4), the population size is selected and all the individuals in the population are randomly initialized.

During each generation, the solution represented by each individual is evaluated, and solutions are selected for reproduction based on their fitness. Selection embodies the principle of ‘survival of the fittest’, wherein good solutions are selected for reproduction while bad solutions are eliminated.

The fitness value determines the ‘goodness’ of a solution. The selected solutions then undergo recombination under the action of the crossover and mutation operators. The iterative process of evaluation and genetic manipulation is continued until convergence is reached 30. In his original work, Holland 31 established the basic theoretical foundation of the genetic algorithms by the use of the schema theory. The Schema Theorem first divides the search space into subspaces and then quantifies the subspaces and explains the movement mechanism of individuals between subspaces. The overall framework for NGOT is constructed in MATLAB® and the GA module in NGOT is based on the Genetic Algorithm Optimization Toolbox (GAOT) developed at North Carolina State University and implemented in MATLAB® 34. Backcalculation of pavement layer moduli from FWD deflections can be treated as a global optimization problem where the objective is to determine the unknown pavement layer moduli that minimize the difference between measured and computed deflections.

In this paper, the implementation of NGOT is discussed for a three-layered flexible pavement structure (see Fig. 1) although it can be used for other pavement types with varying number of layers owing to its flexible and integrated modular systems approach. AlthoughBackcalculation of Non-Linear Pavement Moduli Using Finite-Elementthe deflection-based objective function can be defined in a number of ways, a simple objective function representing sum of the squared differences between measured and computed deflections as shown in Eq. 11 was selected for this study (where n = 6):The Open Civil Engineering Journal, 2009, Volume 3850.80 0.80nf = 0001 ( Di 0002 d i ) 2(1)Fitness = 1 /(1 + f )(2)i =10.80Fig. NN surrogate forward model training progress.Fig.

Trailer-mounted Falling Weight Deflectometer (FWD) equipment.The input variables to NGOT (see Fig. 4) include six FWD measured deflections (see Fig. 2), AC surface and base layer thicknesses and the corresponding ranges of pavement layer moduli.

The use of ranges of values for the optimized unknowns rather than seed values used in the conventional backcalculation approach makes the search for an optimal solution more powerful in the GA approach since a global solution can be obtained and divergence or local optima can be prohibited when compared to conventional methods. Neuro‐Genetic Optimization Toolbox0.80 (1) Deflection Data (2) Pavement Layer PropertiesGenetic Algorithms Optimization(3) GA ParametersFinite Element Based Neural Networks Pavement Response Forward ModelPredicted Pavement Layer ModuliFig. Neuro-genetic hybrid optimization framework.The GA module implemented in this study is capable of using either a floating point representation or a binary representation. First, the starting population is randomly generated. Each individual in the population, representing a set of pavement layer moduli, is passed on to the ANN module (which is described in the next section) for computing deflections which are then passed back to the GA module for fitness evaluation. Using the fitness function (Eq. 12), the GA module performs simulated evolution to determine the fitness of the solution strings.Fig.

ANN surrogate forward model prediction accuracy.In the current GA implementation, it is possible to use a variety of crossover and mutation functions which include arithmetic crossover, heuristic crossover, simple crossover for crossover operator and boundary mutation, multi-nonuniform mutation, non-uniform mutation, and uniform mutation for mutation operator. Similarly, the implemented selection schemes include: roulette wheel, normalized geometric select, and tournament 34.86 The Open Civil Engineering Journal, 2009, Volume 3Kasthurirangan GopalakrishnanReferences 17, 18, and 35 discuss the selection of optimal GA parameters for the pavement moduli backcalculation problem. These studies provided some guidelines in optimizing the GA parameters and operators for the current study.

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Apart from that, a preliminary sensitivity study was conducted to determine the optimal settings for the GA module. The current study described in this paper was conducted using the normalized geometric selection scheme with a probability of 0.08, arithmetic crossover and nonuniform mutation operators with variable probabilities.

The size of the population and generation size were set to 80 and 100, respectively. Since, the ANN and FE modules are interlinked, they are presented together in the next section. Finite Element Modeling and Neural Networks The NN module is an important component of the NGOT which significantly reduces the computational time required for forward calculation of deflections for each of the individuals in the generation.

Reliability

A brief discussion on the ANN background is first presented followed by details related to the development of NN based surrogate forward models incorporated into NGOT. NNs are parallel connectionist structures constructed to simulate the working network of neurons in human brain.

They attempt to achieve superior performance via dense interconnection of non-linear computational elements operating in parallel and arranged in a pattern reminiscent of a biological neural network. The perceptrons or processing elements and interconnections are the two primary elements which make up a neural network.

A single perceptron is mathematically represented as follows 36:0004 n 0006 y k = 0003 (v j ) = 0003 0001 xi wij 0002 b j 0005 i =1 0007(3)where xi is input signal, wij is synaptic weight, bj is bias value, vj is activation potential, 0001 is activation function, yk output signal, n is the number of neurons for previous layer, and k is the index of processing neuron. Multilayer perceptrons (MLPs), frequently referred to as multi-layer feedforward neural networks, consist of an input layer, one or more hidden layer, and an output layer. Learning in a MLP is an unconstrained optimization problem, which is subject to the minimization of a global error function depending on the synaptic weights of the network 21. For a given training data consisting of input-output vectors, values of synaptic weights in a MLP are iteratively updated by a learning algorithm to approximate the target behavior. This update process is usually performed by backpropagating the error signal layer by layer and adapting synaptic weights with respect to the magnitude of error signal 21. Reference 37 presented the first backpropagation (BP) learning algorithm for use with MLP structures.

A general schematic of a multi-layer feedforward neural networks with one output neuron trained using error BP learning algorithm is displayed in Fig. The backpropagation training algorithm for a simple three-layer MLP structure (one input layer, one hidden layer, and one output layer) is described as follows. The networkis initially presented with an input vector (x1, x2, x3, xN) augmented by a bias x0 = 1. The net activations of the hidden neurons and the outputs from the hidden layer are calculated as follows:0003 N 0005 I j = 0002 (neth j )= 0002 0007 0001 v ji xi 0004 i =0 0006(4)where i varies from 0 to N and j varies from 1 to L hidden neurons. The synaptic weights of the interconnections between the inputs and the hidden neurons are represented by vji.

Among the nonlinear activation functions, the sigmoid (logistic) function is the most usually employed in ANN application. The presence of a nonlinear activation function, 0001, is important because, otherwise, the input-output relation of the network could be reduced to that of a single-layer perceptron. The computation of the local gradient for each neuron of the multilayer perceptron requires that the function 0001 be continuous. In other words, differentiability is the only requirement that an activation function would have to satisfy.

The sigmoid function is a bound, monotonic, nondecreasing function that provides graded, nonlinear response within a specified range, 0 to 1. The sigmoid nonlinear activation function is given by:0003 (neth j )=1 1 + exp(0001 0002 neth j )(5)where 0001 is a parameter defining the slope of the function. The net activations for the neurons in the output layer and the outputs are calculated as follows:0003 L y k = 0002 (netok )= 007 0001 wkj I j 0004 j =00005 0006(6)where k varies from 1 to M output neurons. The synaptic weights of the interconnections between the hidden neurons and the output neurons are represented by wkj. The system error is then computed by comparing the actual outputs (yk) with the desired outputs (dk). The error terms for the output hneurons (0001 ko ) and the hidden neurons ( 0001 j ) are given by:0002 ko = ( d k 0001 y k )0003 ' ( netok )(7)M0002 hj = 0003 ' (neth j )0001 0002 ko wkj(8)k =1where the sigmoid activation function is differentiated as follows:0002 ' ( neto k ) = 0002 (neto k )(1 0001 0002 (netok )) = y k (1 0001 y k )(9)0002 ' (neth j ) = 0002 (neth j )(1 0001 0002 (neth j )) = I j (1 0001 I j )(10)Then, the synaptic weights are updated for each neuron in the hidden layer and the output layer.

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Efforts have been made in this paper to backcalculate the in situ elastic moduli of asphalt pavement from synthetically derived falling weight deflectometer (FWD) deflections at seven equidistant points. An artificial neural network (ANN) is used as a tool for backcalculation in this work. The ANN is observed to backcalculate layer moduli, both from normal as well as noisy deflection basins, with better accuracy compared with other software, namely, EVERCALC and ExPaS. EVERCALC is a backcalculation software downloaded from the Internet and ExPaS is a backcalculation algorithm developed in-house, based on a “search and expand” approach. Work have been extended further to develop ANN models that can predict a possible rigid layer at the bottom of the pavement and can directly predict the remaining life of the pavement without backcalculating the layer moduli. Finally, a reliability analysis is performed to quantify the performance of backcalculation using an ANN.Keywords:,.

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