_{1}

^{*}

Construction 3D printing is changing construction industry, but for its immaturity, there are still many problems to be solved. One of the major problems is to study materials for construction 3D printing. Because printed buildings are very different from traditional buildings, there are special requirements for printing materials. Based on environmental and cost considerations, the recycled concrete as printing material is a perfect choice. In order to study and develop the construction 3D printing materials, it is necessary to predict the properties of them. As one of the most effective artificial intelligence algorithms, artificial neural network can deal with multi-parameter and nonlinear problems, and it can provide useful reference to predict the performance of recycled concrete for 3D printing. However, since there are many types and parameters for neural network, it is difficult to select the optimal neural network with excellent prediction performance. In this paper, by comparing different types of neural networks and statistically analyzing the distribution of the root-mean-square error (RMSE) and the coefficient of determination (R2) of these neural networks, we can determine the best performance among four neural networks and finally select the suitable one to predict the performance of 3D printing concrete.

3D printing is the method of converting virtual 3D model from digital file to physical object, which is achieved through the use of additive process that objects are created by laying down successive layers of materials until an object is created. Advances in 3D printing technology can significantly change and improve many fields such as energy use, waste reduction, product availability, medicine, art, and construction [

The printing materials play a very important role in 3D printing and should have basic features such as rapid hardening. Many studies have shown that the strength and stability of current printing materials are poor, which hinders the application of 3D printing in large-scale models or building construction. Sungwoo Lim described a concrete printing process with a practical example in designing and manufacturing a concrete component called Wonder Bench [

Yi Wei Tay pointed that recycled materials as well as industrial waste such as fly-ash and slag should be to used in construction 3D printing. Also, functional materials such as fiber reinforced concrete (FRC) and highly ductile engineered cementitious composite (ECC) may be beneficial to 3D concrete printing, since they have high tensile and flexural strength than conventional concrete and these properties are very crucial for stability of concrete structure [

In order to protect environment, it is necessary to recycle solid waste. At the same time, the increase of costs and reduction of natural resources make humans turn to make use of waste. The use of recycled concrete aggregates from construction waste as substitute for non-renewable natural aggregates in concrete has been considered to increase the sustainable use of resources in construction industry and reduce the impact on the environment. However, since there are changes in reproduction, the mechanical properties obtained by using recycled aggregate concrete (RAC) differ from the natural aggregate concrete (NAC).

Recycled aggregates are substantially different in composition and properties compared with natural aggregates, causing it hard to predict the performance of recycled aggregate concrete and design optimal mix proportions. Artificial neural network (ANN) has good potential to be used as a tool for predicting the compressive strength of recycled aggregate concrete [

Togay Ozbakkaloglu presented new empirical models for prediction of the mechanical properties of recycled aggregate concrete (RAC) using gene expression programming (GEP) technique. The assessment results indicate that the predictions of proposed models are close with the test results and the new models provide improved estimates of the mechanical properties of RACs compared to existing models [

Recycled aggregate concrete and construction 3D printing are promising fields of research in the future. It is of great significance to apply them to green and sustainable construction by using the recycled aggregate concrete as 3D printing material. In this way, recycled aggregate concrete can be used, which not only disposes with construction waste, but also saves costs. In addition, using construction 3D printing technology has improved efficiency and shortened construction time. However, the performance of recycled concrete is not easy to determine, and will be affected by many other factors, preventing from applying in construction 3D printing. Therefore, this paper proposes a method by using artificial neural networks to predict compressive strength of recycled concrete. By comparing the characteristics of different types neural networks, the influence of various parameters on their performance and the distribution of evaluation indexes are analyzed, and finally an optimal neural network with best predicting performance is selected.

The neural networks have unique advantages for solving multi-parameters, nonlinear problems. Therefore, it is a good reference for the prediction performance of recycled concrete, and then guides the design of construction 3D printing materials. However, due to the variety of neural networks and the great differences between them, it is very important to choose appropriate neural network and determine the suitable parameters so as to achieve the most accurate prediction. From the numerical experiments in this paper, we can see that different neural networks are very different and some of them are extremely sensitive to the value of parameters. This article selects the data about the compressive strength of recycled concrete from several papers and tries to find the best neural network to predict the compressive strength. At first, this paper briefly introduces four kinds of neural networks, then uses numerical experiments to determine the optimal performance of each network, and finally chooses the best-performing neural networks. There are many factors needed to be considered such as the value of relevant error, the statistical distribution of the error, the accuracy of the prediction. After above analysis, the prediction performance of the recycled 3D printing concrete based on neural network method can be determined.

In general, the main factors affecting the performance of neural network are the number of neurons and layers in the hidden layer. Therefore, in same training function, the influence of neurons and layers in the hidden layer on the training effect is mainly investigated. Firstly, set up multiple neural networks with different parameters. The number of hidden layers is 1 to n and the number of hidden layer neurons is 1 to 100. Secondly, run these programs and obtain evaluation indicators of predicting performance such as root mean squared error (RMSE) and coefficient of determination (R2) which are presented in this paper as statistical evaluations for errors while training and testing the models. Finally, the RMSE and R2 values of each group are statistically analyzed and their histograms and probability distribution function curves were presented. The distribution of these data was shown in the figure. Taking into account the errors and accidental factors in the calculation process, the algorithm is set to run three times and then averaged.

R M S E = 1 n ∑ i = 1 n ( t i − o i ) 2 (1)

R 2 = ( n ∑ t i o i − ∑ t i ∑ o i ) 2 ( n ∑ t i 2 − ( ∑ t i ) 2 ) ( n ∑ o i 2 − ( ∑ o i ) 2 ) (2)

Here t is the target value, o is the output value and n is the number of all collected data. Root mean square error (RMSE) is the root mean square value of both predicted value and actual value, which is used to evaluate predicting performance. This quantity is especially useful when variables are positive and negative. If RMSE values increase, the performance of models will reduce. Coefficient of determination (R2) shows the fitness level of defined function on data set. If the values of R2 are above 0.7 and closer to 1, this shows that predicted results are closer to experimental results.

Error back propagation (BP) is one of the most commonly used methods and is a supervised learning neural network. The principle of it involves using the steepest gradient descent method to achieve approximation. There are three layers in BP: input layer, hidden layer, and output layer. The two nodes in each adjacent layer are called links and are directly connected. Each link has a weight that indicates the degree of relationship between the two nodes. Choosing the input I_{i} of the input layer neuron i, the weight W_{ji} from the input layer neuron i to the hidden layer neuron j, and the threshold K_{j} of the hidden layer neuron j, the output H_{j} of the hidden layer neuron j is calculated by

H j = f ( ∑ i w j i ⋅ I i + K j ) (3)

f ( x ) = 1 1 + exp ( − x ) (4)

where f usually denotes sigmoid function.

Impact of neuron numbers on predictive performance. From

Impact of hidden layers on predictive performance. The operating time of two layers neural network is much longer than single layer neural network. As the number of hidden layers increases, the running time becomes longer but RMSE does not decrease. While the value of R2 increases slightly, but the effect is not obvious. The characteristics of multiple layers neural network are similar to those of single layer neural network.

Impact of cascading on predictive performance. Cascading refers to the connections between different layers. There are not only connections between adjacent layers but also connection between output layers and hidden layers. Whether the BP neural network adopts cascaded form has little difference on RMSE and R2. Considering the cascade will increase operation time and there is no direct relationship between multi-layers and multi-nodes on prediction accuracy. In summary, this paper gives priority to use single-layer non-cascaded BP neural network.

After calculating BP neural network with different parameters, multiple sets of RMSE and R2 data can be obtained. Analyzing the relationships and distribution rules of these data not only help to select the best performance neural network, but also discover the rules in these data and improve the algorithm.

The distribution of RMSE. From the

The distribution of R2. The

Elman neural networks can be divided into four layers: input layer, hidden layer, connection layer and output layer. The connection layer is used to remember the output of the previous moment in hidden layer and can be regarded as a one-step time delay operator. Based on basic structure of BP neural network, the output of hidden layer is connected with the output of hidden layer through its own delay and storage, making it sensitive to data of historical conditions. The feedback network has improve the ability to handle dynamic information and the storage of the internal state has the function of mapping dynamics, so that the system can adapt to the time-varying conditions.

Suppose there are n inputs, m outputs and r neurons in hidden layer and connecting layer, the weight from input layer to hidden layer is w_{1} while the weight between connecting layer and hidden layer is w_{2} and the weight from hidden layer to output layer is w_{3}, u(k − 1) represents the inputs of neural network, x(k) represents the outputs of the hidden layerm x_{c}(k) represents the outputs of the connecting layer, and y(k) represents the outputs of neural network. Then

x ( k ) = f ( w 2 x c ( k ) + w 1 ( u ( k − 1 ) ) ) (5)

x c ( k ) = x ( k − 1 ) (6)

y ( k ) = g ( w 3 x ( k ) ) (7)

In which f represents the transfer function of hidden layer and sigmoid function is commonly used and can be defined as

f ( x ) = 1 1 + exp ( − x ) (8)

g is the transfer function of output layer and it is usually a linear function. Elman network uses BP algorithm to calculate the weight values and the error of the network is

E = ∑ k = 1 m ( t k − y k ) 2 (9)

Impact of neuron numbers on predictive performance. From

Impact of hidden layers on predictive performance. The operating time of the two layers neural network is much longer than single layer neural network. The more layers are, the longer the running time is. As the number of hidden layers of neurons increases, the running time becomes longer and the values of RMSE and R2 do not change significantly. The characteristics of multiple layers neural network are similar to those of the single-layer neural network. Increasing the number of hidden layer neurons and the number of layers will increase the computation time, and at the same time, this will not improve the accuracy of predicting performance greatly, in some cases it may even result in bad impact. In summary, this paper gives priority to use single-layer Elman neural network.

After calculating Elman neural network with different parameters, multiple sets of RMSE and R2 data can be obtained. Analyzing the relationships and distribution rules of these data not only help to select the best performance neural network, but also discover the rules in these data and improve the algorithm.

The distribution of RMSE. From

The distribution of R2. From

Generalize regression neural network (GRNN) is a type of neural network which

is widely used for continuous function mapping. The main function of GRNN is to estimate the linear or nonlinear regression of variables. In other words, the network only gives the training vector x and calculates the most likely value of the output y. Specifically, the network calculates the joint probability density function (PDF) of x and y. Then the expected value of output y for given input vector x is calculated by

E [ y | x ] = ∫ − ∞ ∞ y f ( x , y ) d y ∫ − ∞ ∞ f ( x , y ) d y (10)

One of the important advantages of GRNN is very simple and quick training procedure. Another attractive feature is that, unlike BP neural network, GRNN does not converge to a local minimum. In addition, the training process of GRNN algorithm is more effective than BP neural network. The input layer is fully connected to the pattern layer, with one neuron for each pattern. It calculates the pattern function in the expression

h i = exp ( − D i 2 2 σ 2 ) (11)

D i 2 = ( x − u i ) T ( x − u i ) (12)

where σ denotes the smoothing parameter, x is input of network and u_{i} is training vector. The summation layer has two units, N and D. The first unit calculates the sum weight of output in hidden layer. The weight of the second unit is equal to 1 and the sum of the individual index items is h_{i}. Finally, the output unit divides N by D to calculate the prediction result.

For probabilistic neural network, we mainly study the influence of spread speed on prediction performance. The spread value range is (0, 1]. The initial value of spread rate is set to 0.01 and the increment is 0.01, which gradually increases to 1. Then get 100 increments of substeps, plot it in a graph and examine the effect of incremental substeps on RMSE and R2. From

After calculating GRNN neural networks with different parameters, multiple sets of RMSE and R2 data can be obtained. Analyzing the relationships and distribution rules of these data not only help to select the best performance neural network, but also discover the rules in these data and improve the algorithm.

The distribution of RMSE. According to

The distribution of R2. According to

The RBF neural network generally consists of three layers: input layer, hidden layer and output layer. The input layer feeds input data to each node of hidden layer. The node of hidden layer is very different from other neural networks because each node represents data cluster centered on a specific point of given radius. Each node in hidden layer calculates the distance from input vector to its own center. The calculated distance is transformed by basis functions and the result is the output from the node. The output of node is multiplied by a constant value and fed to the output layer. The output layer contains only one node that sums the output of previous layers and produces the final output value.

The calculation of RBF neural network follows below process. When the network receives the k dimensional input vector X, the network uses the following formula to calculate scalar value

Y = f ( X ) = w 0 + ∑ i = 1 m w i r ( D i ) (13)

where w_{0} is the bias, w_{i} is the weight parameter, m is the number of nodes in the hidden layers of RBF neural network. In this paper, the Gaussian function r(D_{i}) is used as RBF, as shown below

r ( D i ) = exp ( − D i 2 / σ 2 ) (14)

where σ is the radius of cluster represented by center node, D_{i} represents the distance between input vector X and all the data centers. It is clear that r(D_{i}) will return values between 0 and 1. Usually, the Euclidean norm is used to calculate distance, but other methods can also be used. The Euclidean norm is calculated by

D i = ∑ j = 1 k ( x j − c j i ) 2 (15)

where c is a cluster center for any of the given nodes in the hidden layer.

Complex nonlinear systems such as foreign exchange rate data are often difficult to model by using linear regression methods. Unlike regression, neural networks are nonlinear, and their parameters are determined by several learning techniques and search algorithms (such as error back propagation and steep gradient algorithms). The main drawback of BP neural network is that the learning process is slow and time consuming. In addition, they often get stuck at local minimum value. However, RBF neural network overcomes the above problems and has good performance because these parameters to be trained are in hidden layer of the network. Determining these values is a solution to linear problem and is obtained by interpolation. Therefore, these parameters were found to be much faster than BP neural network. In addition, RBF neural networks can often implement training data sets with near-perfect precision without trapping in local minimum.

Impact of neuron numbers on predictive performance. For RBF neural network, from

Impact of hidden layers on predictive performance. The operating time of single layer neural network is much shorter than multiple layers neural network. The more layers, the longer the running time is. As the number of neurons in hidden layers increases, the running time becomes longer and the values of

RMSE and R2 do not change significantly. The characteristics of multiple layers neural network are similar to those of single-layer neural network. Increasing the number of hidden layer neurons and the number of layers will increase the computation time, and at the same time, this will not improve the accuracy of predicting performance greatly, in some cases it may even result in bad impact. In summary, this paper gives priority to use single-layer RBF neural network.

After calculating RBF neural networks with different parameters, multiple sets of RMSE and R2 data can be obtained. Analyzing the relationships and distribution rules of these data not only help to select the best performance neural network, but also discover the rules in these data and improve the algorithm.

The distribution of RMSE. From

The distribution of R2. From

Comparing the RMSE values of various neural networks, it can be found from

RMSE | The type of neural network | |||
---|---|---|---|---|

BP | RBF | GRNN | EIMAN | |

Min | 8.2318 | 3.2434e-08 | 0 | 4.3202 |

Max | 21.0021 | 11.7075 | 11.6021 | 9.3928 |

Mean | 12.6381 | 2.4178 | 7.9106 | 6.3501 |

Median | 12.1435 | 5.4922e-08 | 8.6534 | 6.4726 |

Std | 2.7324 | 3.3543 | 3.2529 | 0.8690 |

R2 | The type of neural network | |||
---|---|---|---|---|

BP | RBF | GRNN | EIMAN | |

Min | 0.3354 | 0.2724 | 0.1266 | 0.5299 |

Max | 0.7468 | 1 | 1 | 0.8960 |

Mean | 0.6207 | 0.9098 | 0.5345 | 0.7750 |

Median | 0.6252 | 1 | 0.5568 | 0.7692 |

Std | 0.0598 | 0.1602 | 0.2763 | 0.0605 |

Based on above analysis, it can be seen that there are many differences between different neural networks. At the same time, there are many factors that affect the performance of the neural network. The various neural networks should be analyzed and compared with different parameters. This paper mainly studies the influence of different factors on prediction performance and the distribution of evaluation indexes. For recycled concrete, its performance is affected by many factors, so it is of great significance to determine its performance and provide reference for use in the 3D printing process of construction. Therefore, this paper studies four kinds of neural networks, compares the curves and distributions of the performance evaluation indicators such as RMSE and R2, and considers running time and convergence. Finally, it is recommended to apply RBF neural network to predict the performance of recycled concrete and provide references for choosing construction 3D printing materials.

Tan, K. (2018) Predicting Compressive Strength of Recycled Concrete for Construction 3D Printing Based on Statistical Analysis of Various Neural Networks. Journal of Building Construction and Planning Research, 6, 71-89. https://doi.org/10.4236/jbcpr.2018.62005