Conjugate Cellular Automata and Neural Network Approach: Failure Load Prediction of Masonry Panels
The intricate interplay between the microscopic constituents and their macroscopic properties for masonry structures complicates their failure analysis modelling. A composite strategy incorporating neural network (NN) and cellular automata (CA) is developed to predict the failure load for masonry panels with and without openings subjected to lateral loadings. The discretized panels are modelled by the CA methodology using nine neighbour cells, which derive their state values from geometric parameters and opening location placement for the panels. An identification coefficient dictated by these geometric parameters and experimental data is fed together as the input training data for the NN. The NN uses a backpropagation algorithm and two hidden layers with sigmoid activation functions to predict failure loads. This method achieves greater accuracy in prediction when compared with the yield line and finite elemental analysis (FEA) methods. The results attained elucidate the feasibility of the current methodology to complement conventional approaches such as FEA to provide additional insight into the failure mechanism of masonry panels under varied loading conditions.
Masonry structures have a long history spanning centuries of usage due to simplicity in their construction and durability . They are continuously subjected to dynamic loadings and intermittent abusive loadings such as earthquakes and floods. Generally, structural responses, e.g., displacements, velocities, and bending, are recorded and inversely solved for dynamic loads due to technological and economic constraints in direct measurement of external dynamic loads . Various models have been proposed and can broadly be grouped under frequency-domain [3, 4] or time-domain [5, 6]. Innovative techniques such as the application of stochastic simulation of Griffith flaws  or a combination of topology optimization method and phase-field design  have been proposed to improve the modelling of the load and resistance of building materials. More attention is now paid on finding ways to simplify the modelling of the dynamic behaviour of structures [9–11].
Inevitable degradation suffered over years of use by masonry constructions necessitates the periodic assessment for structural integrity to ensure safety, and it can serve as an essential tool to validate the design of new constructions . However, even nowadays, it is a rather difficult task to find a reliable method that would encompass a variety of masonry materials with accuracy proximal to the experimental data . Physical experiments on masonry structures itself are prohibitively expensive which requires significant consumption of material and time. Thus, the amount of available experimental data of masonry structures is sparse and warrants the use of various computational analysis approaches.
The discontinuous and increasingly nonlinear nature of masonry under stress, especially after crack initiation, complicates structural analysis. Broadly, studies of masonry behaviour fall into two approaches, homogeneous and heterogeneous . A homogeneous approach regards masonry as a composite material, and it is used in macromodelling to study the overall structural response . It simplifies calculations, but macromodelling cannot properly predict the local behaviour. The heterogeneous approach is used in micromodelling where every masonry component, unit, and mortar are modelled individually. Hence, it is more detailed and realistic but complicated and time-consuming. A homogenization approach stands as an intermediate between macro- and micromodelling that considers masonry as a heterogeneous structure divided into periodic cells. It allows the user to extract a representative element to describe the whole structure [16, 17].
Initially, constitutive models with continuum representations were utilized for engineering modelling of masonry yielding acceptable estimations. However, the inhomogeneous composition is better simulated using discontinuous micromodelling approaches, and thus such techniques were frequently used in the past two decades [18–20]. Even so, collating the multitude of compositional and structural variables inside the finite/discrete element method (F/DEM) to solve equations of motions for material deformation, contact point interactions, failure load, and crack propagation remains a daunting task. Many researchers turned to the use of artificial intelligence (AI) techniques to overcome these issues. Among such techniques, neural networks (NNs) have progressively gained popularity due to its ability to solve a wide variety of problems at lower computational costs and simplified approach. However, only a few studies incorporate the use of NNs for the approximation of masonry behaviour in general. Recently, NNs have been successfully used in civil engineering to solve a variety of problems [21–30].
The analysis of panels under biaxial bending is amongst the earliest applications of NNs for the prediction of masonry behaviour , which showed the ability of NNs to solve complex nonlinear problems. A multilayer perceptron NN  and a radial basis function NN  were able to predict creep deformations in masonry structures showing a relatively small prediction error. Garzón-Roca et al. used NN and fuzzy logic to estimate the axial load and compressive strength of masonry made of clay bricks and cement mortar [34, 35]. Asteris and Plevris employed a NN to approximate failure for masonry under biaxial stress. They proposed a computational procedure for approximating failure curves in 3D achieving improved prediction performance and providing valuable information about the influence of different loading angles [36, 37]. Cascardi et al. proposed an artificial NN model to predict the in-plane shear strength of masonry panels strengthened by Fibre Reinforced Polymer systems based on a large experimental database [38, 39].
Another (AI) technique that has gained popularity in resolving problems of masonry structures analysis is cellular automata (CA). CA method for masonry panels is first mentioned by Zhou G. C. in 2002. They applied CA techniques to improve the standard finite element (FE) method used to calculate failure load, proposing a concept of similar zones and strength/stiffness corrector . Zhang et al. applied NN to predict the cracking patterns of masonry walls loaded vertically at different orientations using the CA model and experimental data of the recorded cracking patterns . By harnessing the advantages of CA and finite element analysis (FEA), Huang et al. developed a method for predicting the failure load of masonry wall panels relying on generalized strain-energy density (GSED) extracted from the so-called “base” panel . And later on, Huang et al. used the combination of CA and GSED to map cracking patterns of laterally loaded masonry wall panels with openings based on displacements of CA cells calculated from the FE method and maximum correlation coefficient . In our previous work, a combined NN and CA approach also showed the capacity to effectively predict the crack propagation of masonry panels with openings based on panels’ configuration information .
This study is focused on the homogenization approach through the investigation of the synergistic effect of NN and a modified CA method to predict failure load of masonry panels, with and without openings, subjected to lateral loading relying on panels’ geometric configuration. An opening-centric modified CA method is combined with a NN that uses a backpropagation algorithm and two hidden layers for modelling. To the best of our knowledge, this is the first report on a combined CA and NN approach for predicting the failure load of masonry panels with openings. The proposed method is verified through comparison with FEA, yield line theory (YL), and GSED methods.
2. Materials and Methods
As the geometry variation has a significant effect on the unreinforced masonry resistance to lateral loading , masonry panels were modelled as a structure of nine cells defined by the length-height ratio of the cell relative to the openings using the CA theory. Then, the NN is trained on the obtained CA information and wallette strength (WS) parameter to predict the failure load for the panels.
2.1. Experimental Data
The experimental data was randomly divided into three groups: training, testing, and validation by 70, 15, and 15 percent, respectively.
It should be noted that while the amount of the available training data was limited, it is comparable to experimental training data sets used by other researchers in the field [38, 47], which proved to be enough to showcase the method for the current problem.
2.2. Cellular Automata
CA can be described as a model of a spatially extended decentralized system made up of a number of individual cells. Each cell is in a specific state which changes over time depending on the state of its local neighbours. CA is a collection of cells that each adapts one of a finite number of states. Single cells change in states by following a local rule that depends on the environment of the cell .
Herein, a new generalized CA model was developed using the concept of eight-neighbourhood Moore CA. In this opening-centric model, the opening of a panel is imparted with the role of the central cell surrounded by discretized neighbouring cells (Figure 1).
In this method, the CA model for masonry panels is described as a pattern of 3 × 3 cells. State values for cells on that model are formed to consider the size effect and the effect of length-height ratio (Figure 2). It describes the percentage of the solid area of the panel that the cell occupies. The dimensions and size of a cell on the CA model relate to the position of an opening. Hence, for the state values for every cell of a solid masonry, the panel is considered equal as shown in equation (1) (Figure 2(a)). But state values for cells of masonry panels with an opening relate to dimensions of a cell and are calculated by equation (2). Openings and areas outside the CA model have state values equal to zero, such as in Figures 2(b) and 2(c) and in Figure 2(d).
where and are the panel’s length (m) and height (m), respectively, and and are the relative length (m) and height (m) of the CA cell , respectively.
The size effect for masonry panels is taken into consideration in this way. The information on CA model parameters for masonry panels with openings is given in Table 2.
2.3. Neural Network Application
Artificial neural networks are often described as being a simplified model of a human brain. They consist of neuron layers where different neurons of different layers are connected through applied synaptic weights . The structure of a simple neuron for a backpropagation NN is shown in Figure 3. There the input is transmitted through a connection where it multiplies its strength by the scalar weight . The bias of the —neuron is added to the weighted input by the summing function. Then, the activation (transfer) function φ uses this sum as its argument. Backpropagation NN usually uses a sigmoid function as the activation function.
The weights and biases of NN are adjustable scalar parameters. The central principle of NN is that they can reach some desired behaviour by adjusting weights and biases. And, the network can be trained to solve a particular problem. This ability to solve complicated nonlinear problems incorporating multiple parameters and variables makes NN well-suited for the highly anisotropic properties displayed by masonry structures. In the backpropagation training algorithm, the output values calculated by the activation function are compared with the desired output or acceptable convergence. This forms a predefined error function, and the error is then fed back through the network . The algorithm adjusts the weights to reduce the value of the error function in response to this information . This training process repeats until the network reaches a state where the error of the calculations is minimized.
The convergence process is expected to be relatively slow due to the nature of the stochastic gradient descent algorithm used in backpropagation . Diverse techniques were proposed to improve this situation (solve this problem) [53–55]. However, the most popular solutions are based on “minibatches,” where the network’s training set is divided into small subsets, and each one of them is iterated through an epoch . Generally, the state-of-the-art software packages and implementations are optimized to use these options . However, the choice of the training method is constrained by the nature of the problem that needs to be solved with available computational resources . Nonetheless, the difference in speed for small-medium size problems, such as the case of this research, is often marginal and can be ignored with the development of computing technology.
The initialization of weights is a crucial step for backpropagation. The size of the gradient changes relative to the size of the weights. If weights are too small, the gradient becomes too small as well (vanishing gradient problem) and weights can never reach the optimal global minima. This significantly slows down the learning if not prevents the network from learning at all. On the other hand, if the weights are too large, then it leads to an exploding gradient problem when the network keeps learning on the large weights and gradient keeps getting larger, never reaching the convergence point. But if all weights are initialized to zeros, the neurons learn the same features during training and evolve symmetrically unable to learn different features . There are diverse methods to set the initial weights, such as Xavier initialization and He initialization, that change the randomly calculated weights depending on the activation function [60–62]. However, no determined rule would work for all problems. But, stochastic initialization of weights that follows the standard normal distribution is the most used approach. Therefore, both vanishing gradients and exploding gradients are rarely a problem for large networks and modern backpropagation techniques .
The NN in this study is trained on two nonlinear layers using feed-forward backpropagation and the efficient Levenberg–Marquardt training algorithm [64–66]. The activation functions for the first hidden layer is the hyperbolic tangent sigmoid function, which returns a matrix of elements in the interval , shown in equation (3). The second hidden layer uses the logarithmic-sigmoid activation function (equation (4)). The result’s range of this function varies from 0 to 1. And, the linear activation function is used in the output layer. The number of learning cycles was 1,000:
The most efficient configuration in time/accuracy comparison for the problem was achieved when the number of neurons in the first hidden layer is 11 and neurons in the second hidden layer are 5 (Figure 4).
2.4. Input, Training, and Output Data
Eleven parameters are used as the training data for the NN. Firstly, the panel's configuration coefficient is considered to describe every calculated CA cell as the part of a masonry panel with a given geometry:where is the panel’s configuration coefficient, and are the panel’s length (m) and height (m), respectively, and is the panel’s thickness (m).
Next, the input parameters are wallet strength () and state values from equations (1)–(2). Both von Neumann and Moore neighbourhood discretization strategies were evaluated, and the latter was found to be more accurate for the given data set and all results presented hereafter are based on it.
The backpropagation algorithm requires the input data to be less than 1 which is suitable for the sigmoid activation function. There are different methods to normalize the input data for backpropagation neural networks [67–69].
It was decided to normalize parameter through multiplication on using several optimization trial experiments.
The output data in the proposed method are vectors of failure load. Those vectors were normalized to match the interval using the min-max normalisation method:where is the output failure load, is the experimental failure load, and and are minimum and maximum failure load equal to 0 and 100, respectively.
The topology of NN is shown in Figure 4.
Table 3 contains the example of input data for panels with an opening SB02, Panel 3, and ART06.
3. Results and Discussion
Initially, a thousand training epochs were set for the NN to validate the rationality of the devised approach. However, the training was stopped after the best performance was reached at 363 epochs. The performance, in this case, was calculated by the mean square error for the training output compared with the experimental data.
The results of the training process achieved performance and training regression and are shown in Figure 5.
The predicted results of the NN were compared with the FEA and yield line theory (YL) to evaluate the performance of the proposed method. The YL prediction was built according to the British Standard Institution BS 5628 by Chong V. L. The FEA prediction was also made by Chong V.L. using 8 × 8 FE mesh. The comparison between predictions made by different methods and experimental failure loads are listed in Table 4. Moreover, the accuracy of the different methods can be seen in Table 5.
As the micromacro scale variation in masonry structure from one panel to the next makes it a tremendously difficult task to model it analytically, there is a large discrepancy in NN, FEA, and YL predictions in comparison to experimental data. In this research, two comparative models that fundamentally approach analytical load modelling from different points of scale were used.
YL theory briefly approaches the panel (slab) load prediction from a macroscale level by assuming that the panel behaves like a perfectly plastic structure and develops positive and negative yield lines under an applied overload. It also permits the determination of the ultimate load for a defined collapse mechanism. While the results from the YL method provide reasonable experimental agreement, they tend to overestimate the failure strength. Another drawback of this method is its inability to precisely define the position of openings .
In comparison, FEA approaches the load prediction problem from the microscale level by allowing the division of panel into many discretized units (elements) and tuning their properties to illustrate the local behaviour. FEA requires the definition of each element’s material properties, loading, geometry, location, and relationship with neighbouring elements. However, for heterogeneous structures such as masonry panels, such exhaustive elemental definitions can lead to an exponential increase in the required computing power while offering little flexibility for adapting to different panels.
In this work, we have tried to bridge the gap between FEA and YL modelling techniques by combining the cellular automata discretization and the adaptable learning ability of ANNs. CA allowed the accurate definition of panel structures and opening locations, while the macroscopic panel properties such as wallet strength, length, height, and thickness, are fed as input training data for the NN.
Further evaluation of the method was performed by using different statistical parameters such as the root-mean-squared error (RMSE), the mean absolute percentage error (MAPE), the coefficient of determination (), and the integral absolute error (IAE). In theory, the closer the values of RMSE, MAPE, and IAE parameters to zero, the higher the accuracy of the proposed NN. Besides, the closer the values to 1, the greater are the similarities between predicted and experimental failure loads. Parameters RMSE, MAPE, , and IAE were calculated by equations (7)–(10). Table 6 shows the comparison between FEA, proposed NN, and YL methods based on these statistical parameters. The comparison is calculated for 47 panels for which the YL results are given in Tables 4 and 5 and for all 55 panels studied here for FEA and NN:where is the number of tested panels, is the experimental failure load, is the predicted failure load, and is the average experimental failure load.
It was observed that for more than 75% of the panels the percentage error was smaller than 20%. Especially for the panels with an opening where all panels had an error deviation below 20%. The low accuracy of the prediction for a few panels can be explained by the fact that prediction in this case significantly depends on the WS parameter, limited training data, and possible variations in the experimental process. Overall, it is safe to assume that the optimized NN was able to outperform FEA and YL.
Figure 6 compares the results of NN and FEA prediction to the experimental data. While both methods can offer comparable accuracy for panels with low experimental failure loads, the error variation range for masonry panels with failure loads more than 10 kN/m2 is lower and significantly more accurate using the NN prediction.
In addition, the proposed method is compared with the GSED method proposed by Huang and based on CA and FEA . Huang et al. used the strain-energy density of a known (base) panel and a new (to be predicted) panel and a criterion for matching zone similarities to calculate failure load for masonry panels.
Although the results from GSED and NN give a good prediction for failure load, a closer inspection reveals that in all cases except one, the percentage error is smaller for NN as compared with GSED results (Table 7).
An additional drawback of the GSED method is that its accuracy depends on the base panel used for prediction. Hence, it is crucial to have experimental information for a panel with a similar condition to the predicted panel. Besides, the NN prediction can be calculated regardless of the configuration of the new (predicted) panel. NN can be trained on available experimental data to calculate failure load for panels with any configuration.
In this study, the backpropagation NN with two hidden layers is developed to accurately predict the failure load of masonry panels subjected to lateral loading based on panels’ configuration and WS. An additional input parameter, panels’ configuration coefficient , calculated through length, height, and thickness of the panel, is introduced. This parameter together with WS and discretized cell locations is used as the input training data for the neural network. The modified CA technique is used to realize the state values of the cells where the central cell representing the opening in the panel is fixed. The most efficient configuration for the algorithm programmed for the NN was achieved when using the tangent-sigmoid activation function for the first hidden layer and logarithmic-sigmoid activation function for the second hidden layer with 11 and 5 neurons in the layers, respectively. It is shown that NNs can predict failure load for masonry panels based only on the panel’s configuration information. But, the accuracy and comprehensiveness of the network deeply depend on the training data. The results were compared with other well-established methods, and judging by RMSE, MAPE, , and IAE statistical parameters NN prediction excels the prediction accuracy of FEA and YL analysis by every parameter. While GSED and NN offered effective failure load prediction capability, the NN is found to be more accurate and general in its applicability.
Even though the proposed method has only been used to validate the experimental data used in this research, an increase in the training data for NN should allow for further refinement of its achieved results. Anyway, the proposed method can provide an additional tool for predicting failure loads and complement other methods like FEA and GSED for in-depth structural analysis. An inherent advantage of the neural modelling method over traditional numerical techniques is that it implicitly identifies and extracts the different characteristics of the panels, including their nonlinear material properties, without requiring explicit and rigorous mathematical expressions, enabling a simplification in numerical modelling and computational cost of masonry.
The code for neural network and relative weights and biases used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors would like to thank the Chinese Scholarship Council (CSC) for the PhD scholarship.
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