Abstract
This paper presents highly robust, novel approaches to solving the forward and inverse problems of an Electrical Capacitance Tomography (ECT) system for imaging conductive materials. ECT is one of the standard tomography techniques for industrial imaging. An ECT technique is nonintrusive and rapid and requires a low burden cost. However, the ECT system still suffers from a soft-field problem which adversely affects the quality of the reconstructed images. Although many image reconstruction algorithms have been developed, still the generated images are inaccurate and poor. In this work, the Capacitance Artificial Neural Network (CANN) system is presented as a solver for the forward problem to calculate the estimated capacitance measurements. Moreover, the Metal Filled Fuzzy System (MFFS) is proposed as a solver for the inverse problem to construct the metal images. To assess the proposed approaches, we conducted extensive experiments on image metal distributions in the lost foam casting (LFC) process to light the reliability of the system and its efficiency. The experimental results showed that the system is sensible and superior.
1. Introduction
Electrical Capacitance Tomography (ECT) is a well-established technique for imaging material distribution inside closed pipes and flasks. It is noninvasive and portable, has high-speed data acquisition and inexpensive construction, and is a safe method compared to other computed tomography technologies such as X-ray [1, 2]. Recently, many process applications, especially those which need dynamic monitoring and control, apply the ECT techniques. Multiphase flow applications, where one phase in the shape of particles, droplets, or bubbles exists in an unceasing carrier phase (i.e., gas or liquid), are good instances of the dynamic applications. The first phase could be plastics, dry powder, or even metal [3].
Typically in the ECT system, capacitance measurements are collected between groups of electrodes evenly distributed around a border of the imaging area. These measurements are proportional to dielectric properties of the materials inside the imaging area, and they are used to generate tomographic images describing the distribution of the materials [4]. The ECT system has two main computational problems: forward and inverse problems [5]. (i)The Forward Problem. The distribution of the materials is known, and the capacitance measurements have to be determined by solving partial differential equations modeling the ECT system [6].(ii)The Inverse Problem. The capacitance measurements are identified, and images of the distribution of the material have to be generated using image reconstruction algorithms [7].
Solving the forward problem requires building an accurate module describing characteristics of the ECT system and calculating capacitance measurements according to certain material distribution. A finite element linear model called a sensitivity matrix, based on linearizing the relation between the physical property and the measured capacitance, is generated to update the image [8]. The sensitivity matrix is created by dividing the domain of interest into small elements; afterward, the capacitance data are obtained as a linear sum of different perturbations composing the overall distribution. Although the use of the sensitivity matrix increases the speed of solving the forward, the results are blurred and weak due to the linear approximation.
Practically, it is challenging to have an accurate model for a complex nonlinear system [4]. For some nonlinear systems that have complex physical behavior, it can be impossible to describe them by analytical equations. Therefore, parametric model identification is a well-suited method for creating experimental models of complex systems, especially when parameters of the system are not precisely known. Compared to traditional nonlinear identification methods, fuzzy systems not only do not require detailed knowledge about the model’s structure but also provide accurate models directly from measured input/output data [9, 10]. Also, the uncertainties associated with the complex system can be represented easily using the fuzzy system theory [11]. The main advantage of the designed fuzzy model is not only precisely predicting the outputs of the actual system but also adapting to changes in the behavior of the system.
The solution of the inverse problem is a tomographic image generated after applying a predefined image reconstruction algorithm [12]. Typically, these algorithms are ill-posed since the number of unknowns (image pixels) is larger than the number of knowns (capacitance measurements). The material distribution has a highly nonlinear effect on the capacitance measurements; therefore, the inverse problem solution is a challenging task [13].
The techniques that construct fuzzy models from experimental data are called fuzzy identification algorithms [14]. There are two methods, which can be combined, to integrate between the expert knowledge and data in fuzzy modeling: (1)A group of if-then rules expressing the expert knowledge written in linguistic form is built. In this case, the fuzzy model is well defined, although the parameters such as membership function parameters of this model can be optimized and tuned based on the input-output data pairs. The optimization algorithm handles the fuzzy model as a network layered structure, and the parameters of each layer can learn the characteristics of the actual system from the collected data(2)The fuzzy model is built using system data without using any prior knowledge about the system under investigation to construct the fuzzy rules. It is assumed that the fuzzy parameters, rules, and membership functions extract the system behavior embedded in the input-output data. The expert, in this case, verifies based on his knowledge whether the fuzzy model gained the required characteristics of the actual system or not. If not, he can modify the rules or provide new data with more information
The second method for building the fuzzy models, in most complex applications, is more practical than the expert knowledge-based method [9, 10]. It avoids the knowledge extracting process from the expert, which is known as the most challenging stage in the reconstruction of the fuzzy model. The fuzzy systems designed based on extracting the information from the supplied data are used for the given problem as well as changing design parameters and operating conditions.
This paper presents novel methods for solving the forward and inverse problems of the ECT system for conductive imaging materials in the lost foam casting (LFC) process. A better understanding of the characteristics of the molten metal inside the foam pattern is needed to reduce the fill-related defects and to improve the final casting [11]. One of the main problems associated with the ECT system for conductive imaging materials is shielding the sensors and makes it blind to any change. A nonlinear forward solver called Capacitance Artificial Neural Network (CANN) has been developed to solve the forward problem to overcome such limitations. It is based on the Feed-Forward Neural Network (FFNN) model. The use of FFNN combines the advantages of solving the forward problem nonlinearly with high speed and accuracy. The training data are generated based on different models developed to describe the behavior of the molten metal during the casting process [15–17]. All of them are stimulated by a finite element method to calculate the capacitance measurements related to all different distributions. To increase the accuracy of the neural network, some random metal distributions are generated and used to train the neural network.
Furthermore, a novel method for solving the inverse problem is presented. This method applies a trained fuzzy system to enhance the quality of the reconstructed images describing the metal distribution in the LFC process. The method consists of three phases, as shown in Figure 1. The first phase is a preprocessing where the capacitance measurements are filtered and normalized. These measurements are the inputs to the Metal Filled Fuzzy System (MFFS) in the second phase. The final reconstructed image is generated in the last phase. This system takes and produces the metal distribution , where is an image with an dimension. represents the presence of the metal in each pixel of an image. In our case, and are 66 and 16, respectively. This system should be able to predict the metal distribution inside the foam pattern (imaging area).
After recalling the lost foam casting process in Section 2 and the components of the ECT system in Section 3, the three stages of the framework are theoretically and practically presented and explained each in a separate section. Numerical results explained in Section 7 show good performances achieved by applying the proposed framework. The final section will conclude the proposed work.
2. Lost Foam Casting (LFC)
During the past two decades, many imaging techniques applied in industrial process applications have been developed [18, 19]. One of the primary process applications is lost foam casting (LFC). The LFC is a type of evaporative pattern casting process that has been used heavily in automotive applications, for example, cast iron, aluminum alloys, steels, and nickel. It is a simple and affordable technique at a low cost for casting very complex molds made from foam. The foam pattern is placed surrounded by compact sand inside a flask. Afterward, molten metal is poured into decomposing the foam pattern and produces a final casting [20, 21]. The steps of the LFC process are shown in Figure 2, while an example of a final cast (cylinder valve and the head) and its foam mold is illustrated in Figure 3.
The LFC process has substantial environmental and energy advantages over other traditional casting processes, such as simplicity, elimination of unwanted environmental garbage, and reduction of the processing cost. Moreover, the LFC can cast complex and delicate detailed geometries and produces castings with smooth surface finishing, which significantly reduces the machining times. It is crucial in the LFC process to monitor the behavior of the molten metal while filling the foam pattern to discover and correct process faults to enhance the final casting [23]. Therefore, capturing the metal characteristics eliminates a variation in the process parameters and leads to a consistent filling process.
Usually, it uses inferred cameras and X-ray technology for imaging a metal fill profile during the casting [24]. The X-ray tomography has many disadvantages, such as the hazard radiation, the big size, the fixed structure, and a high cost. These disadvantages of the X-ray tomography drived manufacturers to look for a novel feasible technology that can capture the flow of the molten metal during the casting process. Recently, because of the advantages of the ECT system compared with the inferred and X-ray technologies, the LFC process implements the ECT system for imaging the metal during the casting process [25].
3. ECT System
Typically, the ECT system contains three parts, an array of electrodes, a data acquisition unit, and a computing device [26]. Assume we have a group of electrodes, thus, each two formulate a capacitive sensor. There is an external shield (flask) around these electrodes to eliminate external noises and stray capacitance effects [27], as shown in Figure 4. The mutual capacitance is measured between every two electrodes where one active electrode works as a transmitter and the other grounded electrodes work as receivers. Therefore, the number of independent capacitance measurements for one distribution is computed as in the following: where is the total collected capacitance and represents electrodes’ number. The array of sensors applied in this work has electrodes evenly distributed around the foam pattern in Figure 4.
The linear forward model of the ECT is expressed as [12] where is the measurements, is the image matrix, is the number of images’ pixels which is around 256 pixels for a image, and the sensitivity matrix is calculated for each element as follows: where are capacitance when the imaging region is full by a low permittivity material and are capacitance when filled by the high permittivity one.
As shown in Equation (2), the number of image pixels is more significant than the measured data. Therefore, the problem is ill-posed, and any small change in the measurements can cause a significant error in the image. Moreover, the sensitivity matrix is not square, and the reconstructed image cannot be computed by using [28]. Hence, the reconstruction algorithms are classified into two types: noniterative and iterative algorithms. Linear Back Projection (LBP), Equation (4), is one of the noniterative algorithms which usually creates blurred images but applies low computations:
While iterative algorithms such as Iterative Linear Back Projection (ILBP), shown in Equation (5), provide more accurate images, its time complexity is high and linearly proportional with the number of iterations : where is the relaxation parameter, is the forward problem solution, and is the iteration number [28].
Also, it is essential to mention that applying the ECT technique in the LFC process is a challenging task since, typically, the ECT works for imaging the nonconductive materials [7, 29]. Existence of the grounded metal inside the imaging area affects the sensitivity matrix dramatically. Therefore, applying the traditional image reconstruction algorithms such as ILBP and Tikhonov algorithm is not feasible. Figure 5 illustrates this dramatic change between the linear sensitivity matrix and the actual sensitivity matrix for a single piece of grounded metal put almost in the center of the imaging area. The sensitivity matrix between sensor pair (1-6) appears in Figure 5(a), while the sensitivity values between pair (1-5) are shown in Figure 5(b).
As shown in Figure 5, it is impossible to depend on the linear sensitivity matrix in solving the ECT inverse problem with conductive imaging materials. Also, in computing the actual sensitivity matrix, each iteration is time-consuming. Therefore, building novel reconstruction algorithms that do not use the sensitivity matrix is crucial, and the soft-computing techniques can play a vital role in solving this problem.
The proposed LFC process has extraordinary characteristics compared to the other industrial processes. One of these characteristics is the high temperature, which can reach up to 1400°F. Therefore, a particular ECT hardware system is required for the harsh foundry environment that was built. The new capacitance measuring circuits are designed to take advantage of the high molten metal temperature [30].
4. Forward Problem Solution
In this work, a soft-computing solution to the forward problem is proposed. The solution will start by modeling the ECT problem. The process requires an in-depth knowledge of the ECT system as well as the metal fill problem. Initially, the ECT forward model is built using ANSYS Finite Element (FE) software. This model is used to generate the capacitance measurements corresponding to any metal distribution in the imaging area [4, 21]. Therefore, much data is created where the inputs are predefined metal distributions and the outputs are the resultant measurements.
4.1. Data Generation
How the molten metal replaces the foam pattern during the casting process stimulates the training data generation. The movement of the molten metal is correlated with the styles of the decomposition of the foam pattern. There are several physical mechanisms to decompose the foam pattern, which depend on different factors such as the kind of foam beads as well as the location of the gate where the metal enters the foam [31]. The foam decomposition is classified into three modes: contact mode, gap mode, and collapsed mode [15]. (1)Contact mode is shown in Figure 6(a), where the molten metal flows and presses gradually against the foam(2)Gap mode happens when a finite gap of air caused by polymer vapor between the molten metal and the decomposing foam pattern as illustrated in Figure 6(b)(3)Collapse mode happens because of the foam joining interbead pattern porosity to the coated surface, as shown in Figure 6(c). The liquid metal forms shapes like fingers inside the foam pattern and the air exhausts through the coating
![(a) Contact mode as shown in the X-ray image of a regular and smooth filling of 8 mm thick foam plate [17]](https://static.hindawi.com/articles/js/volume-2019/4208349/figures/4208349.fig.006a.jpg)
(a) Contact mode as shown in the X-ray image of a regular and smooth filling of 8 mm thick foam plate [17]
![(b) Gap mode as shown in the neutron radiograph image of a side-filling of 12 mm thick foam plate [16]](https://static.hindawi.com/articles/js/volume-2019/4208349/figures/4208349.fig.006b.jpg)
(b) Gap mode as shown in the neutron radiograph image of a side-filling of 12 mm thick foam plate [16]
![(c) Collapse mode as shown in the X-ray image of a rapid and irregular filling of 8 mm thick foam plate [16]](https://static.hindawi.com/articles/js/volume-2019/4208349/figures/4208349.fig.006c.jpg)
(c) Collapse mode as shown in the X-ray image of a rapid and irregular filling of 8 mm thick foam plate [16]
Actually, the implementation of the ECT system for conductive imaging materials is entirely different from the traditional ECT for imaging the dielectric constant of nonconductive materials. After the foam decomposition happened, the grounded metal shields the electrical field between the transmitting electrodes and the receivers, causing significant changes in the capacitance measurements. We capture these measurements and use it in generating the distribution of the metal inside the imaging area. Therefore, the dielectric constant of the decomposed foam during the three decomposition modes does not have that much effect on the measurements. However, we used it for simulating the behavior of the molten metal during the casting process to generate the required data for training the neural network in the next section.
An FE model is built to simulate the flow of the molten metal in these different foam decomposition modes. Typically, the starting point of the metal filling is the gate; then, the shape of how the metal replaces the foam depends on the foam decomposition mode. According to the metal distribution, capacitance measurements are computed and normalized from the FE model, as given in where and are capacitance measurements when the imaging area is empty and filled by the metal, respectively, and is the measurement vector equivalent to a predefined metal distribution. The gate is placed at various locations while generating the data to simulate all probable metal distributions during the casting process. For example, to simulate the collapsed mode, the molten metal is filled from different gates simultaneously to represent the random motion of the metal in this mode.
For illustration, some metal distributions simulating the contact mode of the filling are listed in Figure 7, while the normalized capacitance data computed from the FE model and corresponding to the second and last distributions are displayed in Figure 8. According to the second metal distribution, where a small piece of the metal exists between electrodes 6 and 7, the measured capacitance between electrodes 6 and 7 and the others is almost saturated and reaches maximum value one, while the rest of the measurements are small since there is no metal at their sensing region, as shown in Figure 8(a), whereas filling half of the imaging area by metal, the last metal distribution shown in Figure 7, makes many of the capacitance values equal to one, as shown in Figure 8(b).
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(b)
4.2. The CANN: Forward Problem Solver
The forward problem of the ECT system is a highly nonlinear problem, where choosing a modeling method for this problem is a big challenge. Artificial Neural Network (ANN) is an appropriate candidate for modeling this problem to overcome the high nonlinearity of the system. The ANN has proven to be successful in modeling the dynamics of nonlinear and sophisticated systems. Historically, ANN was used to model many manufacturers processed with a high level of nonlinearity with very accurate results. The ANN consists of I/O neurons, transfer function, and hidden layers. One of the crucial stages in the development of the ANN is the network topology, where selecting an optimal number of hidden neurons is crucial. Typically, this number intensely affects the complexity of the system.
In this work, there is only output neuron corresponding to the number of measurement ; hence, a Multi-Input Multi-Output (MIMO) structure had been utilized to model the system. ANN with one hidden layer structure is implemented since it arbitrarily performs the process mapping in a sufficient way [32].
A trial and error approach is adopted to identify the optimal topology of the ANN. The number of neurons increased from five to forty in the trial stage, which significantly decreased the error. The ANN model is considered as an effective representation of the process and its data. Initially, weights of the ANN are arbitrarily selected, and through multiple runs during the training phase, these weights are optimized. The final CANN model is composed of one hidden layer with twenty-five neurons for best modeling the forward problem in the ECT system.
4.3. Model Estimation
Another critical stage of the ANN deployment is applying an efficient iterative search algorithm for updating the weights of the ANN. The batch Back Propagation (BP) [33] learning algorithm is very robust, and it provides a fast convergence. Therefore, it is applied as a search algorithm for updating the weights. The mean square error (MSE) shown in Equation (7) is used as an evaluating criteria for the ANN outputs: where is the total number of the dataset, is the actual target value, and is the corresponding estimated value.
In this research work, we adopted a fixed ANN structure with a variable number of neurons in the hidden layer to choose the best number based on the normalized training error. Figure 9 shows the convergence of the CANN system with different numbers of neurons (15, 25, 50, and 66). Also, the numerical values of changing the number of neurons in the hidden layer on the normalized errors during the training and testing phases are listed in Table 1. Since the error difference between 50 and 66 neurons is too small, we adjusted the size of the hidden layer at 50 neurons. Our research experiments show that the adopted structure is adequate for modeling the problem. Figures 10 and 11 show the convergence of the CANN system with 50 neurons with single and multiple filling points, respectively.
The training and validation are crucial stages during the development of the ANN. A set of I/O data is used during the training phase to adjust the weights until the specified input yields the desired output with an acceptable error or a predefined iteration number is reached. The validity of the model is tested by measuring the MSE between the target and the network output. The modeling process is repeated if the model cannot reach precise accuracy. A simple fitness function to estimate the computed capacitance is given in where is the normalized error in and and are the actual and estimated capacitance output from the CANN system. Table 2 states the computed normalized error for the forward systems with single and multiple filling points. The outputs of the CANN system compared with the actual capacitance measurements are shown in Figure 12. Figures 12(a) and 12(b) show training and testing examples for the single filling points, while Figures 12(c) and 12(d) show some other examples of the multiple filling points.
(a) Single filling (training case)
(b) Single filling (testing case)
(c) Multiple filling (training case)
(d) Multiple filling (testing case)
5. Fuzzy Logic Modeling
5.1. Fuzzy Identification
A complex modeling problem can be decomposed into several simpler subproblems using fuzzy models. The fuzzy identification method is considered as a useful decompression tool for a nonlinear system modeling where the available data are partitioned into subsets. A simple linear model [34] approximates each subset. This way, we achieve a reasonable balance between the complicity and the accuracy of the model. A fuzzy clustering tool can smoothly divide the data into subsets without any sudden transitions and also integrate diverse types of knowledge in the same framework. In contrast to the traditional clustering methods, the fuzzy clustering allows the individuals to become useable by several groups simultaneously with different membership degrees. The fuzzy modeling process consists of two stages [9, 35]: (1)Identifying the operating regions using heuristics or data clustering techniques(2)Determining the parameters of each submodel using optimization techniques such as the least-squares technique
From this point of view, fuzzy identification is considered as a sophisticated search method that can build an accurate model from complex data. Fuzzy logic is used in this research to formulate a binary image model by aggregating a set of linearized local subsystems which identify the nonlinear dynamics of the metal filling process. Methods that can automatically generate fuzzy models from measurements, mostly from the system data, have been developed. A comprehensive methodology combining fuzzy sets and system identification tools is applied in this work. A MATLAB toolbox called Fuzzy Modeling and Identification (FMID) [35] is implemented based on this methodology. The fuzzy clustering method is applied to build linear subsets from the existing data. Afterward, the association between the offered identification algorithm and the linear regression is implemented.
5.2. Fuzzy Clustering
Many clustering algorithms have been introduced in the literature [36]. Based on the similarities among objects, a cluster analysis method is applied to classify these objects and organize them into groups. Fuzzy clustering algorithms are useful in situations where a little prior knowledge exists since they do not rely on the underlying statistical distribution of data. The clustering algorithms can reveal the underlying structures in data as well as reduce the complexity in modeling and optimization processes. These techniques are usually used with data representing observations of some physical process. Typically, a cluster can be defined as a group of individuals similar to each other more than individuals in other clusters [37]. Clusters with different geometrical shapes, sizes, and densities can be exposed to the data. In particular, an objective function to evaluate the interest of partitions is applied via the fuzzy clustering algorithm. Nonlinear optimization algorithms, such as the least-squares optimization, are utilized to search for the local extreme of the objective function. Hence, clustering with fuzzy objective function has strong relationships with the statistical regression and system identification methods [35].
One of the main concerns when using fuzzy identification via data clustering is the selection of the optimal number of clusters . In many cases, data does not have any prior information about its structure, and thus, the number of essential subsets in the data is estimated during the clustering process. There are two approaches to decide the proper number of clusters in data [36]: (1)The first approach is to cluster the available data at different values of and afterward compute validity measures to evaluate the quality of the attained clusters. There are many scalar validity measures in the literature [38](2)The second approach is to work with an appropriately large number of , then repeatedly decrease this number by combining similar clusters based on some predefined measures
The second algorithm works as follows: (1)Given the input-output sequences of measurements , to be clustered based on a chosen number which is the same as the fuzzy rules in this case(2)Divide the data into a group of local linear submodels by the Gustafson-Kessel (GK) algorithm [39, 40] (applied in this study), in order to compute the fuzzy partition matrix, , with and , the prototype matrix, , and cluster covariance matrices ( are positive definite)(3)Compute the antecedent membership functions from the cluster parameters (, , and )(4)Estimate the consequent parameters and by the least-squares method
5.3. Identification of TS Fuzzy Model
Methods for implementing Takagi-Sugeno (TS) fuzzy models from the fuzzy clusters are reviewed in this section. The structure of the TS model is described, and methods for creating the antecedent membership functions and assessing the consequent parameters are discussed.
Each cluster attained from the clustering process is considered as a local approximation of the regression hypersurface. Analytically, the antecedent membership functions are represented by calculating the distance of from the prototypes and then calculating the membership degree in inverse proportion to this distance. Therefore, if , includes all but the last column and the last row of the cluster covariance matrix , the corresponding norm inducing matrix is given by
Let denote the projection of the cluster center onto . Then, the distance between the antecedent vector and the projection of the cluster center is calculated as
This distance is transformed into the membership function by where is the fuzziness parameter in the GK algorithm. Note that the sum is one when adding up the membership degrees of all the rules; therefore, this equation calculates the correlation degree between all the rules.
On the other side, the consequent parameters and are estimated using the weighted ordinary least-squares method. Let represent the matrix and a diagonal matrix in which is the membership degree as its diagonal element, then is the least-squares solution of where columns of are linearly independent and data pair is weighted by .
This algorithm for building the TS fuzzy model is developed and implemented by Babuska [41]. He developed a MATLAB toolbox called the Fuzzy Modeling and Identification (FMID) toolbox, which is used in this work.
6. Inverse Problem Solution
The use of the fuzzy logic approach is proposed for the first time in the literature to solve the inverse problem of the ECT system design to solve the image conductive materials. Building a fuzzy system with the advantage of decomposing the domain of nonlinear to several sublinear domains based on the number of the membership function is proposed. Fuzzy logic shows a significant level of accuracy and meanwhile provides a logical relationship between the system inputs and outputs [42–46].
Figure 13 shows the steps of the fuzzy modeling process [47], and they can be listed as follows: (1)Create a set of capacitance measurements corresponding to predefined grounded metal distributions using the CANN system described in Section 4.2(2)Propose a suitable solution model for the inverse problem, as shown in Figure 1(3)Obtain a proper number of clusters by a trial and error method to attain a suitable error value in both the training and testing phases(4)Get the product space of the possible cluster sets(5)Build the membership functions and its fuzzy rules based on the input and output patterns(6)Apply the least-squares method to estimate the consequence parameters of the model and calculate its output [41]
6.1. The MFFS: Inverse Problem Solver
The traditional design of the fuzzy systems and its applications were presented in [48]. Typically, the Metal Filled Fuzzy System (MFFS) consists of if-then fuzzy rules, which have two parts: the antecedent and the consequence. The antecedent part contains information about the operation conditions of the 440 process, while a valid linear regression model represents the consequent part.
Consider a nonlinear system has inputs variables and one output variable, existing of the metal in one pixel. The function transferring these inputs to the output is represented as
The main goal is the development of a fuzzy model describing the nonlinear function between the input and the output . The fuzzy rules for a nonlinear multi-input/single-output (MISO) fuzzy system are represented as where is the rule number. For instance, in a model with three inputs and a single output system, there are two rules . The system is as follows: where the output is equivalent to the input domain variables , , and applying membership function , , and . The TS inference system generating the output using the weighted mean measure of the inputs is given as where is the degree of fulfillment of the th rule. There is a special case when the consequences of TS rules are linear, and the system parameters are as follows:
6.2. MFFS Parameter Estimation
The membership function defines the relation between the input and the output of the fuzzy model. When applying the fuzzy membership functions, the rule becomes valid over a fuzzy region for the input and output variables defined in the antecedent part of the rule. The TS fuzzy model is an appropriate choice for modeling dynamic datasets.
In the rule-based fuzzy systems, the following quantities entail to be identified: (1) the antecedents, (2) the membership functions of the consequent, and (3) the parameters of the consequent membership functions. Usually, the user selects the number of rules (clusters).
6.3. Validation Criteria
The normalized error, described in Equation (19), is a common validation measure to quantify the quality of the generated metal fill distributions and works as a normalized fitness function to calculate the reliability of the results: where is the normalized error in and and are the actual and the MFFS generated grounded metal distribution, respectively. A normalized fitness function to calculate the reliability of the results is commonly used in this case.
7. Experiments and Results
7.1. Experimental Setup
To test the tomography system and the reconstruction technique, a hardware model was used for simulating a casting process in the lab. The model consists of a cylindrical tube working as a flask. Figure 14(a) shows 12 electrodes mounted on the circumference of the flask around the foam pattern and four pieces of metal embedded in the sand in front of electrodes. Every two electrodes comprise a capacitive sensor at a time, with one electrode connected to the source signal working as a transmitter and the other works as a receiver. Each sensing set is connected to a measuring circuit to measure the mutual capacitance between the two electrodes, as shown in Figure 14(b). Another circuit (Mote) is used to send the measured data wirelessly to the base station attached to each measuring circuit. A control unit is also used to send signals to the measuring units which collect the data from the sensors. A stop command is used to stop the process.
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(b)
A LabVIEW program is implemented on the control unit to send the commands to the measuring circuits, collect the data, and make some preprocessing operation on the data to be suitable for the reconstruction algorithm. The vector of measurements consists of 66 values. These values are collected when the imaging area is full of foam and after replacing all the foam by grounded metal, as calculated in Equation (6).
The range of capacitance measurements changes depending on the distance between the grounded metal and each sensor. For example, if the metal is located very close to one sensor, the normalized capacitance values from that sensor will be very high, almost one. Meanwhile, the measurements from the neighboring sensors will be medium, and the measurements will be shallow from sensors that are far away. If the molten metal is concentrated in the middle of the imaging area, the normalized measurements will be shallow from all the sensors.
7.2. Setup of the MFFS
This section describes the structure of the proposed MFFS system for solving the inverse problem and how we generate images from the capacitance measurements . The proposed method was assessed based on data generated from the CANN system.
Our model has 256 fuzzy subsystems created that is equal to the number of pixels in the generated image. Each fuzzy subsystem represents the relationship between a particular pixel in the image and corresponding measurements ’s.
Although the number of fuzzy subsystems is high, the general interconnectivity of the entire fuzzy system is systematically low, which produces a smooth and straightforward relationship between the pixels and measurements.
The structure of the MFFS is presented in Figure 15, where inputs to the system are to capacitance measurements representing all the independent measurements between 12 electrodes mounted around the area of interest. The inputs for each of the 256 fuzzy subsystems are selected as a set of measurements. Each set of measurements for each pixel is selected as clarified in Section 7.3 by simulating the FMID toolbox [41].
7.3. Selection of MFFS Input Based on the Sensitivity Matrix
A sensitivity matrix is used in the selection process of the measurement set of each fuzzy subsystem. Every element in the sensitivity matrix represents the effect of altering the corresponding pixel in the distribution from foam to the metal on all sensor measurements. The sensitivity matrix is calculated using a finite element analysis package [29]. The size of the sensitivity matrix is where is the number of pixels of the image and is the number of measurements. A threshold value is applied to the sensitivity matrix to select the most significant measurements affected by changing the status of the corresponding pixel. This method cuts the number of inputs to each fuzzy subsystem. The graphical presentations of the measurements affecting and before and after applying the threshold are shown in Figures 16 and 17, respectively.
The output image from the MFFS consists of 256 pixels. Table 3 has samples of the fuzzy rules for pixels 101 and 255, which present the relations between each of the pixels in the image and its corresponding capacitance measurements. Thus, for pixel number 101, it is found that is mostly affected by the measurement ’s given in Table 3. Also, for the pixel 225, is affected by the measurements ’s given in the same Table 3. In this case, we found that the measurements which mostly affect the modeling process of the pixel 225 are , , , , , , , , , , , , , , , , , , , and .
7.4. Developed Fuzzy Models for Single/Multiple Metal Filling Points
The inverse problem for creating metal fill images is solved by MFFS explained in the above process. Different metal distributions with a single filling point as well as multiple points have been fed to the CANN model (forward problem solver), and the corresponding measurements have been collected. The system is trained by utilizing these data of the metal distributions and its corresponding measurements. Figure 18 demonstrates the normalized error during the testing phase of the single filling point, where each line in the figure denotes the average error of a certain pixel. Figure 19 presents the normalized error for multiple filling points.
Table 4 contains the overall mean square error for the single and multiple filling points. The MFFS is significantly able to have a high level of accuracy and meanwhile provide a logical relationship between the system inputs and outputs.
Some of these data is shown in the top row of Figures 20(a) and 20(c) for single and multiple filling points, respectively. For testing the performance of the proposed fuzzy system, a distinct set of examples has been created but not used during the training phase. How the MFFS responses to these training patterns is shown in Figures 20(b) and 20(d).
(a)
(b)
(c)
(d)
8. Conclusion
In this research, we explored the idea of using a combination of Artificial Neural Networks (ANNs) and Fuzzy Identification System (FIS) to build a rapid and adequate solver for the forward and inverse problems of the Electrical Capacitance Tomography (ECT) system for conductive imaging materials during the lost foam casting process. We implemented the ANN system to estimate capacitance measurements of an ECT system to solve the forward problem, while the MFFS has been applied to solve the inverse problem through constructing tomographic images. The proposed approach is augmented by inspecting different modes of filling the foam patterns by molten metal, which are then stimulated by exploiting the ANSYS software. The proposed fuzzy model was shed light on, analyzed, and verified by examples of single and multiple filling points. The final results show that the system is fast and generates accurate final metal distribution images.
Data Availability
The data is available on request.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgments
The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for continuous support. This work was supported financially by the Deanship of Scientific Research at Umm Al-Qura University (Grant number: 17-ENG-1-01-0001).