Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 124042, 15 pages

http://dx.doi.org/10.1155/2015/124042

## Back Analysis of the Permeability Coefficient of a High Core Rockfill Dam Based on a RBF Neural Network Optimized Using the PSO Algorithm

^{1}Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian 116024, China^{2}College of Civil and Architecture Engineering, Heilongjiang Institute of Technology, Harbin 150050, China

Received 15 June 2015; Revised 13 October 2015; Accepted 15 October 2015

Academic Editor: Antonino Laudani

Copyright © 2015 Shichun Chi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

It is important to determine the seepage field parameters of a high core rockfill dam using the seepage data obtained during operation. For the Nuozhadu high core rockfill dam, a back analysis model is proposed using the radial basis function neural network optimized using a particle swarm optimization algorithm (PSO-RBFNN) and the technology of finite element analysis for solving the saturated-unsaturated seepage field. The recorded osmotic pressure curves of osmometers, which are distributed in the maximum cross section, are applied to this back analysis. The permeability coefficients of the dam materials are retrieved using the measured seepage pressure values while the steady state seepage condition exists; that is, the water lever remains unchanged. Meanwhile, the parameters are tested using the unstable saturated-unsaturated seepage field while the water level rises. The results show that the permeability coefficients are reasonable and can be used to study the real behavior of a seepage field of a high core rockfill dam during its operation period.

#### 1. Introduction

The earth-rock dam design is widely adopted locally and abroad because of low investment, locally produced raw materials, and simple construction. As a result of complex situations, many influencing factors, and poor management, a number of accidents have occurred over the past several years. Many accidents have made it clear that the disaster caused by a flood once a dam breaks is unimaginable. According to the International Commission on Large Dams [1, 2], the number of earth-rock dam breaks accounted for 70 percent of the total number of dam breaks; 25 percent of the earth-rock dam breaks were caused by infiltration or deformation and the percentage was up to 40 percent in China [3]. Seepage failure is one of the main causes of collapse of an earth-rock dam; therefore, it is very important to use mature theories and methods to analyze and evaluate leak and deformation conditions.

Generally speaking, there are three types of methods used in permeability coefficient identification: analysis, testing, and back analysis methods. An analysis method, which is based on numerous hypotheses, is difficult to adapt to a dam with a complex structure. Some permeability coefficients can be measured in the field; however, such measurements are usually scarce and are expensive to obtain. Therefore, a back analysis method to fill in the missing parameters is typically conducted before an analysis of the seepage flow. During the past three decades, extensive research has been conducted to study the problem of permeability coefficient estimation for seepage flow models. This research resulted in the development of numerous techniques that employ search methods to find parameter values that minimize the difference between the observed and calculated hydraulic head values. A brief review of the approaches employed in these techniques is provided herein.

In 1986, research about numerical methods for permeability coefficient identification was conducted [4]. Tsurumi et al. [5] used a finite element method to identify parameters in groundwater hydrology. Zijlstra and Dane [6] used a quasi-Newton method to identify hydraulic parameters in layered soils. El Harrouni et al. [7] presented a method of estimating the groundwater parameter using optimization. Prasad and Rastogi [8] studied an inverse method of estimating the hydraulic conductivity values based on a genetic algorithm. Zio [9] studied the inverse problem of parameter estimation in groundwater models using artificial neural networks. At home, He and Zheng [10] deduced a series of formulas about numerical inversion of Laplace transform solutions for the dynamics of porous flow. Liu and Wang [11] studied an improved genetic algorithm for back analysis of seepage parameters. Liu et al. [12, 13] studied the back analysis of seepage with an ANN based on alternative and iterative algorithm ANN based on the simulated annealing Gauss-Newton algorithm. Li et al. [14] proposed ant colony optimization to estimate aquifer parameters.

In recent years, neural networks (NNs) have been successfully used in the field of civil engineering to directly map nonlinear complex relations. As is stated in [15], the mapping capabilities of a NN are strictly related to the nonlinear component found in the activation function of the neurons. A feed forward neural network (FFNN) is a NN where the inner architecture is organized in subsequent layers of neurons, and the connections are made according to the following rules: every neuron of a layer is connected to all (and only) the neurons of the subsequent layer. The commonly used backpropagation algorithm for FFNN training suffers from slow learning speed and being liable to be trapped in a local minimum. The radial basis function (RBF) neural networks are normally considered as universal approximators [16–18], which can approximate any function to an arbitrary accuracy, provided that the size of network is not constrained. Applications of RBF neural network are widespread and can be found in prediction of time series [19], function approximation problems [20], hydrological modeling [21, 22], and system modeling [23]. General description of the RBF neural network can be found in many of the standard ANN text books [24]. The present work proposes the use of RBF neural network (choosing Gaussian function as its activation function) for mapping the complex nonlinear relations between water heads and permeability coefficients. The RBF neural network has a simple structure, succinct training, fast convergence speed, and the ability to represent any complicated nonlinear function relations. It creates a radial basis network one neuron at a time. Neurons are added to the network until the mean square error falls beneath an error goal or a maximum number of neurons are reached. As you know, the more the input vectors are, the more the neurons are needed to achieve a better fitting effect and forecast precision. It easily leads to dimension disaster [25]. To avoid this phenomenon, this paper selects an optimizing algorithm to determine the RBF neural network structural parameters, which can achieve a better fitting effect and forecast precision with fewer hidden layer neurons.

Optimum parameters of RBF neural network can be determined by many methods of varying complexity, some of which are noniterative clustering [26], -means clustering [27], and orthogonal least squares (OLS) [28, 29]. Evolutionary algorithms also have been very popular in determining the parameters of RBF networks. A genetic algorithm was used in [30] to evolve the network for function approximation. The genetic algorithm population consisted of individual kernels instead of entire networks. A similar strategy has been used in [31] which makes use of an OLS procedure and singular value decomposition to evaluate the contribution of individual kernels. In [32] the number of kernels is also determined dynamically using a genetic algorithm. Other methods (e.g., [33]) evolve an entire population of RBF neural networks. PSO is similar to the genetic algorithm in the sense that they are both population-based search approaches and that they both depend on information sharing among their population members to enhance their search processes using a combination of deterministic and probabilistic rules. Although PSO and the GA on average yield the same effectiveness (solution quality), PSO is more computationally efficient (uses less number of function evaluations) than the GA [34]. Its small computational requirement makes it a good candidate for solving optimization problems. Solutions computed by PSO are derived from both local and global searches. This allows PSO to consider solutions near the vicinity (local) of the starting point, in addition to a possible solution located within a global region. The search process for new point(s) includes consideration of the previous best solutions. Thus, PSO is used to train the neural network to obtain the optimal values of the structure parameters for the RBF neural network in this paper.

In this paper, a back analysis model is proposed with the radial basis function neural network optimized using the particle swarm optimization algorithm (PSO-RBFNN) and the technology of finite elements for solving the saturated-unsaturated seepage field. The recorded osmotic pressure curves of osmometers, which are distributed in the maximum cross section, are applied to this back analysis. The permeability coefficients of the dam materials are retrieved from the prototype data measured while the dam was in the steady state seepage condition; that is, the water lever remained unchanged. Meanwhile, the parameters are tested using the unstable saturated-unsaturated seepage field while the water level rises. The results show that the permeability coefficients are reasonable and can be used to study the real behavior of a seepage field of a high core rockfill dam while operating.

#### 2. Mathematical Model

The objective of the inverse model or operator is to obtain optimal estimates of unknown permeability coefficients that minimize the difference between observed and calculated hydraulic head values. In this case, the optimization mathematical model of the back analysis of permeability coefficients is as follows [35]:In the above model, (1) is the objective function which achieves minimum of the mean squared error of water heads; and are the th measured and computed water head, respectively; is the th weight (equal weights were used in this study); is the number of water head measuring points. Equation (2) gives the range of permeability coefficients, ; and are the maximum and minimum of , respectively.

#### 3. Seepage Flow Model

##### 3.1. Partial Differential Water Flow Equations [36]

The flow of water through both saturated and unsaturated soil follows Darcy’s law which states thatwhere is the specific discharge; is the hydraulic conductivity; and is the gradient of total hydraulic head.

According to the water continuity and Darcy’s law, a two-dimensional unsteady flow through an earthfill dam can be described by the Richards equation aswhere is the total head; and are the hydraulic conductivity in the - and -directions, respectively; is the volumetric water content; is the applied boundary flux; and is the time.

Equation (4) can be employed to simulate two-dimensional unsteady state water flow through nonhomogeneous, anisotropic, saturated-unsaturated porous media receiving lateral flow. It states that the difference between the flow (flux) entering and leaving an elemental volume at a point in time is equal to the change in storage of the soil systems. More fundamentally, it states that the sum of the rates of change of flows in the - and -directions plus the external applied flux is equal to the rate of change of the volumetric water content with respect to time.

Under steady state condition, the flux entering and leaving an elemental volume is the same at all times. The right side of (4) consequently vanishes and the equation reduces to

Changes in volumetric water content are dependent on changes in the stress state and the properties of the soil. The stress state for both saturated and unsaturated conditions can be described by two state variables. These stress state variables are and where is the total stress, is the pore-air pressure, and is the pore-water pressure. It is usually assumed that the total stress and the pore-air pressure are both constants during transient processes. This means that remains constant and has no effect on the change in volumetric water content. Changes in volumetric water content are consequently dependent only on changes in the stress state variable, and, with remaining constant, the change in volumetric water content is a function only of pore-water pressure changes. As a result, the change in volumetric water content can be related to a change in pore-water pressure by the following equation:where is the slope of the storage curve.

The total hydraulic head, , is defined aswhere is the unit weight of water; is the pore-water pressure; and is the elevation.

Equation (7) can be rearranged as .

Substituting (7) into (6) gives the following equation:which now can be substituted into (4), leading to the following expression:

##### 3.2. Finite Element Water Flow Equations [36]

Applying the Galerkin method of weighted residual to the governing differential equation, the finite element method for two-dimensional seepage equation can be derived aswhere is the gradient matrix; is the element hydraulic conductivity matrix; is the vector of nodal heads; is the vector of interpolating function; is the unit flux across the edge of an element; is the thickness of an element; is the time; is the storage term for a transient seepage equal to ; is a designation for summation over the area of an element; and is a designation for summation over the edge of an element.

In an abbreviated form, the finite element seepage equation can be expressed aswhere is the element characteristic matrix; is the element mass matrix; is the element applied flux vector.

#### 4. Prediction Model of RBFNN Optimized by PSO

##### 4.1. Radial Basis Function Neural Network (RBFNN)

The radial basis function neural network (RBFNN) used in this study is a feed forward neural network with one radial basis layer. It can uniformly approximate any continuous function with a specified accuracy [16–18]. General description of the RBFNN can be found in many of the standard ANN text books [23]. Figure 1 is a radial basis network with inputs [37].