Estimation of compressive strength and slump of HPC concrete using neural network coupling with metaheuristic algorithms

Abstract

The compressive strength and slump of concrete have highly nonlinear functions relative to given components. The importance of predicting these properties for researchers is greatly diagnosed in developing constructional technologies. Such capacities should be progressed to decrease the cost of expensive experiments and enhance the measurements’ accuracy. This study aims to develop a Radial Basis Function Neural Network (RBFNN) to model the hardness features of High-Performance Concrete (HPC) mixtures. In this function, optimizing the predicting process via RBFNN will be aimed to be accurate, as the aim of this research, conducted with metaheuristic approaches of Henry gas solubility optimization (HGSO) and Multiverse Optimizer (MVO). The training phase of models RBHG and RBMV was performed by the dataset of 181 HPC mixtures having fly ash and superplasticizer. Regarding the results of hybrid models, the MVO had more correlation between the predicted and observed compressive strength and slump values than HGSO in the R² index. The RMSE of RBMV (3.7 mm) was obtained 43.2 percent lower than that of RBHG (5.3 mm) in the appraising slump of HPC samples, while, for compressive strength, RMSE was 3.66 MPa and 5 MPa for RBMV and RBHG respectively. Moreover, to appraise slump flow rates, the R² correlation rate for RBHG was computed at 96.86 % while 98.25 % for RBMV in the training phase, with a 33.30% difference. Generally, both hybrid models prospered in doing assigned tasks of modeling the hardness properties of HPC samples.

Keywords

Compressive strength slump flow multiverse optimization algorithm concrete hardness neural network

1 Introduction

Over the recent development in construction materials, there has been a significant increase in the use of high-performance concrete, known as HPC, in various structural applications such as bridges, masonry applications, pavement construction, high-rise buildings, residential buildings, and parking lots [1, 2]. To increase compressive strength, blast furnace slag, fly ash, and silica fume as additional cementitious materials can often be added to HPC [3 –6]. Regarding the high compressive strength and other beneficial properties of HPC, such as low diffusion and permeability, high resistance against attacking chemicals, high abrasion resistance, and high elasticity modulus, the prevalent usage of this high-performance material can be seen in projects around the world [7]. Compressive strength and slump flow rate features as hardness aspects of HPC should be considered in designing and controlling the quality of concrete mixes. However, they are influenced by many factors, such as mix ratios, material properties, environmental conditions and curing, and the age of HPC compounds [8]. In particular, apprising the mechanical aspects of HPC is deemed a problematic issue concerning the high complexity of computations. Existing highly nonlinearly relation among HPC components and concrete properties enters mathematical complications to model the hardness properties, including slump flow and compressive strength based on input data [9, 10]. Early predictions are paramount in planning to remove concrete formwork and reposition it into slabs. Using main data, predictive models have helped experts and engineers reduce costs and tensioning time. Traditional methods for predicting concrete characteristics employ linear and nonlinear regression procedures [11, 12]. Such methods utilize restricted experimental data to find out certain coefficients.

However, most regression-based approaches have illustrated that using exact regression equations is difficult. Additionally, some factors affecting the HPC hardness properties differ from conventional concrete. Thus, regression basis methods may not be suitable for predicting slump and compressive strength features [13]. Regular methods would not be suitable for analyzing complex nonlinear or uncertain materials, so many studies use artificial intelligence (AI) methods in hybrid and evolutionary systems to estimate the hardness properties of concrete compounds. On the other hand, using high-accuracy soft-based approaches to model the mechanical features of HPC samples has been adopted in a large number of researches [10 , 14–18]. In this regard, machine learning techniques such as Artificial Neural Networks (ANNs) [19 –22], the classification and regression tree (CART) [23], the chi-squared automatic interaction detector (CHAID) [24], linear regression (LR) [25, 26], support vector machines (SVMs) [27, 28], generalized linear regression (GENLIN) [29], and many other ways have been used as the predictive models.

In one research, Zhang et al. [29] surveyed the capability developed RBFNN in predicting the compressive strength of high-performance concrete, in which the RMSE of RBFNN was calculated at 2.079 and 2.427 MPa for training and testing, respectively. Abuodeh et al., with the aid of two deep machine learning techniques –Neural Interpretation Diagram (NID) and Sequential Feature Selection (SFS)–investigate the capability of an ANN model to model the compressive strength of 110 ultra-high performance concrete compounds. Including water, cement, silica fume, and fly ash, the ANN model computed the compressive strength values with a correlation coefficient of 80.1% and NMSE of 0.012 MPa [30]. Kaloop et al. [32] conducted research for investigation on the multivariate adaptive regression splines model (MARS) and gradient tree boosting machine (GBM) learning technique to predict the CS with acceptable results, the R² more than 0.9. Coupling metaheuristic algorithms with prediction models have also been conducted in many research studies in which model inputs were mostly optimized. Han et al. [31] used a machine learning bootstrap aggregation technique with an optimizing input stage to model the compressive strength rates of high-performance concrete mixtures. The outcomes indicated that the developed way was more efficient for optimizing input variables and could do desirable estimation than without variable optimization [31]. Kaloop et al. [32] researched to examine the gradient tree boosting machine (GBM) and multivariate adaptive regression splines model (MARS) to appraise the mechanical properties of HPC mixtures. A data set including 1030 HPC samples included ingredients among cement, blast furnace slag, water, fine aggregate, concrete age, and superplasticizer. Moreover, the joint MARS-GBM model showed the R² of 0.965 and MSE of 0.037 MPa, respectively) [32].

Therefore, a robust model for predicting HPC compressive strength should be developed to have optimal internal settings to reduce the errors and complexity of computations. Of course, such a model satisfies modeling requirements, but it must also be easy to work with and robust enough to model the uncertainties involved. Hybrid approaches are efficiently employed to predict the mechanical features of HPC in many studies today [17 , 33–36]. Notably, most of the articles have focused on coupling main prediction models with algorithms without modifying the key parameters. To this end, the present paper has aimed to model the hardness properties of 181 HPC samples, compressive strength, and slump flow rate using Radial Basis Function Neural Network (RBFNN).

Now, as the novelty of the research, coupling the two optimization algorithms, namely Henry Gas Solubility Optimization (HGSO) and Multiverse Optimization Algorithm (MVO), to RBFNN will be fulfilled to find the optimum number of neurons embedded in the hidden layer as well as an internal parameter of the main model. The abilities of these powerful solutions can be found in many types of research to appraise the dependent variables based on state ones [7 , 37–44].

Having metaheuristic algorithms involved in predicting the hardness features of concrete samples in the form of hybrid frameworks has raised the accuracy of modeling and curing the complexity of the RBF neural network. Finally, the evaluation indices will assess the performances of developed RBFNN models. Figure 1, has indicated the process of this research in apprising the mechanical features of HPC compounds.

Fig. 1

Schematic view of the present research processes.

2 Methodology

Estimating the special parameters highly correlated with other decision variables should be announced by checking training, validating, and testing points of view. The following section describes the preparation operation for the modeling process to understand better the interactions between input and dependent variables of Compressive Strength (CS) and Slump flow (SL). However, the definitions of optimizers and the main model of RBFNN are brought in the next part.

2.1 Preliminary dataset feeding models

The main step of initializing models to reproduce the mechanical features of CS and SL is preparing the data set feeding developed RBFNNs as models’ inputs. Inputs of models have mainly been divided into I) state variables (independent) that include ingredients of high-performance concrete mixtures; II) target variables (dependent) that mean the hardness properties of compressive strength and slump flow. The detailed data set of 181 HPC mixtures has been caught from a laboratory-based published article [45]. Subsequently, Table 1 illustrates some statistical indices of components used in HPC samples with various dosages.

Table 1
Inputs of hybrid models from a statistical viewpoint

Code CS SL W/B W F/T FA AE SF SP

Parameters Compressive strength Slump flow Water to Binder Water Fine to all aggregates Fly Ash Air entering Silica-Fume Superplasticizer

Unit MPa mm % Kg.m^–3 % % Kg.m^–3 % Kg.m^–3

Max 123 260 45 180 53 20 0.08 25 36.5

Min 38 95 18 140 35 0 0 0 1.89

Ave 74.17 202.73 31.17 162.13 42.15 5.8 0.03 6.44 10.93

St.dev 26.64 25.93 8.77 12.12 5.37 8.03 0.03 8.43 8.63

Median 63 205 30 160 42 0 0.038 0 8

Code	CS	SL	W/B	W	F/T	FA	AE	SF	SP
Unit	MPa	mm	%	Kg.m^–3	%	%	Kg.m^–3	%	Kg.m^–3
Max	123	260	45	180	53	20	0.08	25	36.5
Min	38	95	18	140	35	0	0	0	1.89
Ave	74.17	202.73	31.17	162.13	42.15	5.8	0.03	6.44	10.93
St.dev	26.64	25.93	8.77	12.12	5.37	8.03	0.03	8.43	8.63
Median	63	205	30	160	42	0	0.038	0	8

Based on the experimental reports, whole components were excluded from the silica fume experiment conducted in South Korea. Coarse aggregate particles were produced using granite crushed via a 2.7 special force of gravity and a fineness modulus of 7.20, having a maximum size of 19.00 mm. The concrete binder was considered Portland cement with the standard of type I ASTM. Moreover, the next HPC component of fine aggregate particles was generated from sand quartz in conditions of special gravity of 2.61 and fineness modulus of 2.94.

The silica fume and fly ash of class F elements in mixtures have been from Elkem, Norway. The superplasticizer used in mixtures playing the water-reducing agent was Naphthalene (C₁₀H₈) to lower the concrete water/binder ratio. For testing the target variable of the CS factor for 181 mixtures, cylinders with certain dimensions of 100×200 mm were operated and remodeled by passing 24 hours. In the next stage, the created samples were cured for 28 days in non-lime water with a degree of 20±3 °C with the standard of ASTM C 684-95. To gauge the air contents in HPC compounds, the experiment was done with the standard of ASTM C 231-91b. Additionally, testing of the slump rates was executed via standard ASTM C 143-90a immediately after the mixing process. Figure 2 exhibited the diagrams of all data as inputs to models.

Fig. 2

Ingredients data as inputs for models.

2.2 Henry’s gas solubility optimization algorithm (HGSO)

Henry’s Law of Physics [46] was first developed by J.W. Henry. The referred rule is designed according to the highest solute magnitude dissolved in one solvent at a certain assumed pressure and temperature [47].

Demonstrating the solubility of low soluble gas in a specific liquid is possible based on this law. Two determining items affecting solubility potential are temperature and pressure. However, gas solubility declines with enhancing temperature rate while directly affecting solids.

Additionally, increased pressure leads to increased solubility [48]. Hashemian et al. [46] introduced an algorithm considering gases and their solubility. This is taken into account in Fig. 2, in which increased pressure on the volume of gas is decreased [41].

The main steps of this algorithm are briefly set up as follows.

The gas molecules’ position and number are determined (generate initial population).

The same population clusters are created according to the features of the various gas types.

Each cluster’s cost is calculated, selecting and scoring the best to specify the best conditions.

Updating Henry’s rule coefficients.

H_{j} (t + 1) = H_{j} (t) \times e^{(- C_{j} (\frac{T^{θ} - T (t)}{T (t) \times T^{θ}}))}

(1)

T (t) = e^{(\frac{t}{iter})}

(2)

Wherein T^θ and C_j are a constant and random number between (0,1), alternatively, H_j shows the coefficient of Henry’s rule for j cluster, iter and t indicate the iteration number and temperature, respectively.

Upgrading the solubility rate with Equation (3)

S_{i, j} (t) = K \times H_{j} (t + 1) \times P_{i, j} (t)

(3)

Which K is a constant value, S_i,j illustrates the solubility value, and P_i,j (t) shows the gas i pressure within j cluster.

By the final stage, the first positions of the population are upgraded using Equation (4)

\begin{matrix} X_{i, j} (t + 1) = X_{i, j} (t) + F \times r \times γ \times \\ (X_{i, best} (t) - X_{i, j} (t)) + F \times r \times α \times \\ (S_{i, j} (t) \times X_{i, best} (t) - X_{i, j} (t)) \end{matrix}

(4)

γ = β \times e^{(- \frac{F_{best} (t) + ɛ}{F_{i, j} (t) + ɛ})} + ɛ

(5)

According to Equations (5) X_i,best indicates the gas with the best position near the answer within cluster j and X_best is the best gas molecule within the population; the position of gas i in cluster j has been denoted with X_i,j; the best cost within the population and i gas through cluster j are indicated via F_best and F_i,j; the factor of r, also shows an accidental value in the range of (zero to 1). α, β, and ɛ show, respectively, fixed numbers wherein β and α are one and epsilon (ɛ) is to be 0.05; finally, the parameter γ denotes the gases’ interaction capabilities.

The number of bad gases in terms of position to answer has been assigned to pass the trap at a local minimum.

N_{w} = N \times (rand (C_{2} - C_{1}) + C_{1})

(6)

Here, N shows the population factor. Constants are indicated via C₁ and C₂, which is determined to be, respectively, 0.1 and 0.2.

The bad position gases are updated using Equation (7).

G_{i, j} = G_{Min (i, j)} + r \times (G_{Max (i, j)} - G_{Min (i, j)})

(7)

Wherein G_Max and G_Min show, respectively, the high and low boundaries. G_i,j indicates the i gas within j cluster.

2.3 Multiverse optimizer (MVO) algorithm

The Multiverse Optimizer (MVO) algorithm is a physics-based optimization technique derived from the cosmic interactions of white, black, and wormhole worlds. In the rule of the multiverse, objects in two parallel galaxies can be transmitted from one universe to another using black and white holes, and objects in each universe can travel through wormholes [49]. Mathematically, all possible solutions and their decision variables resemble the universe or given existing objects. In addition, the objective function values of possible solutions are similar to the corresponding cosmic inflation rates of the world. Inflation is a universal property, so higher values indicate increased capacity. It can be calculated proportionally to the objective function. High-inflation universes tend to have white holes and send their objects to low-inflation universes with black holes. The flow of information from better solutions to worse solutions will cause radical changes in the universe, improve the average inflation rate of the population, and ensure MVO’s exploration capacity. White hole selection can be performed using the roulette wheel method [1] based on the normalized inflation rate. Wormholes ensure the exploitability of the metaheuristic algorithm through the objects’ accidental movement in the universe without considering its inflation rate. In this context, it is assumed that there will always be a wormhole tunnel between the universe and the best universe ever discovered. A formula can be defined as brought in Equation (8) [49]: $\begin{matrix} x_{ij} \\ = {\begin{matrix} {\begin{matrix} {BU}_{j} + TDR \times [rand () \times (U_{j} - L_{j}) + L_{j}] ra \\ nd () < 0.5 andrand () < WEP \\ {BU}_{j} - TDR \times [rand () \times (U_{j} - L_{j}) + L_{j}] \\ rand () ⩾ 0.5 andrand () < WEP \end{matrix} \\ x_{ij} \\ rand () ⩾ WEP \end{matrix} \end{matrix}$ (8)

Where rand () shows the accidental value from 0 to 1; x_ij is the decision variable j given the universe i; the lower and upper boundaries have been indicated, respectively, via L_j, U_j; also, the parameter of BU_j is the best universe recognized thus far given the decision variable j; WEP and TDR as, respectively, the probability of wormhole existence and the traveling distance magnitude keeping the balance between the exploitation and exploration capabilities have been variables in the range [0–1]. The WEP value has been linearly enhanced from its lowest to highest values over each iteration of the processing MVO algorithm, while TDR the value is decreased via Equation (9) [49]: $TDR = 1 - \frac{\sqrt[p]{t}}{\sqrt[p]{T}}$ (9)

Wherein t is the iteration number and T shows the highest number of epochs. Additionally, p is the coefficient of exploitation, so a higher value leads to more capability for exploitation. The pseudo-code of the MVO algorithm is given in the Algorithm 1.

Algorithm 1. pseudo-code of the MVO algorithm

Start

Generation of initial universe U randomly

For t = 1: Max_Iter

Determine WEP and TDR

Compute inflation rates of U

Construct a sorted universe (SU)

Compute normalized inflation rates (NIRs)

Save the best universe as the first universe

For i = 2: universe number

Black hole index I

For j = 1: objects number

If rand () < NIR (U_i)

White hole index = Roulette_wheel_selection(- NIR)

U(Black hole index,j)=SU(White hole index,j)

End if

If rand () < WEP

If rand () < 0 .

U (i, j) = BU_j + TDR × [rand () × (U_j - L_j) + L_j]

Else

U (i, j) = BU_j - TDR × [rand () × (U_j - L_j)

+L_j]

End if

End for

2.4 Preparing the radial basis function in developing models: RBHG and RBMV

Artificial Neural Networks (ANNs), the well-known branch of artificial intelligence (AI) that works based on human being neural network systems, have been used to predict the dependent variables based on the independent variables as state ones in the present study mechanical properties of concrete. The layers of the ANN structure, including input, hidden, and output, are interconnected and can be used for modeling several types of problems [50, 51]. The hidden and output layers of an ANN consist of central units called artificial neurons, where two major mathematical computations are performed. The first mathematical computation of the artificial neurons embedded in the hidden layer obtains the net value as the sum of the weighted inputs received from the nodes (neurons). Whereas the second mathematical computation places the activation function of Radial Basis, known as RBF, on the net value in which one hidden layer is considered. The resulting outcomes from the hidden layer are transferred to the output layer, utilizing a simple linear function [52]. A radial basis function neural network is a feed-forward type with a higher convergence speed than the common ANN [53]. To this end, Equation (10) has indicated the main structure of RBFNN in modeling the compressive strength and slump flow as outputs (Y) for HPC samples $Y = \sum_{i = 1}^{q} w_{i} φ (t - c_{i})$ (10)

Which t shows the network input and φ_i denotes the output of i^th node of the hidden layer. Also, w_i depicts the weights, which w₀ represents the output layer weight. c_i symbolize the prototype center of i^thkernel of the Radial basis function. The parameter σ is the spread rate.

The RBF feeds on some user-based data that, in the present research, they are dedicated to using optimization algorithms automatically in which a range of neuron numbers for the hidden layer and spread rate as arbitrary parameters are tried to generate slump and CS. Thereafter, the evaluating stage with an error index like MSE is choosing the best answers with the lowest costs and appropriate adjusting factors. To maximize the efficiency of the Network, the mentioned HGSO and MVO are employed to couple with RBFNN to reduce network complexity and cost. Therefore, by developing the prediction models and creating the RBHG and RBMV, the main estimation model of RBFNN can modify itself to find neurons’ numbers and spread value optimally. In this regard, Fig. 3 shows the schematic view of RBHG and RBMV frameworks to model the hardness properties of HPC samples. Generally, the accuracy of modeling with artificial intelligence was reassessed over several runs, and it was chosen by considering the errors of modeling with MSE/RMSE. However, by changing the neuron’s number, the cost function shifted to conditions with various cost modeling that all of them were reflected in errors indicated by MSE/RMSE.

Fig. 3

The conceptual view of hybrid models.

2.5 Assessment indicators to evaluate the performance of models

By coupling the main model of RBFNN with metaheuristic algorithms, RBHG and RBMV are capable of modeling the mechanical properties of HPC samples. In order to reach a comprehensive view of hybrid frameworks taking modeled outcomes into account, several indicators have been considered, as brought in Equations (11)–(15). $VAF = (1 - \frac{var (p_{n} - t_{n})}{var (t_{n})}) * 100$ (11) $MAE = \frac{1}{N} \sum_{n = 1}^{N} | p_{n} - t_{n} |$ (12) $\begin{matrix} OBJ = (\frac{n_{train} - n_{test}}{n_{train} + n_{test}}) \frac{{RMSE}_{train} + {MAE}_{test}}{R_{train}^{2} + 1} \\ + (\frac{2 n_{train}}{n_{train} + n_{test}}) \frac{{RMSE}_{test} - {MAE}_{test}}{R_{test}^{2} + 1} \end{matrix}$ (13) $RMSE = \sqrt{\frac{1}{N} \sum_{n = 1}^{N} {(p_{n} - t_{n})}^{2}}$ (14) $R^{2} = (\frac{\sum_{n = 1}^{N} (t_{n} - \bar{t}) (p_{n} - \bar{p})}{\sqrt{[\sum_{n = 1}^{N} {(t_{n} - \bar{p})}^{2}] [\sum_{n = 1}^{N} {(p_{n} - \bar{p})}^{2}]}})$ (15)

Where p_n has indicated the estimated magnitudes and t_n is the observed value; N shows the HPC samples’ number. Moreover, the number of HPC mixtures for the training and testing phases are shown via n_train and n_test.

3 Results and discussion

The present study’s main objective was to model the hardness properties of high-performance concrete samples of compressive strength and slump flow rate using hybrid models of RBHG and RBMV. To this end, HPC ingredients’ data as the models’ inputs trained the developed RBFNNs and tested models.

To investigate the relationship between constituents rather than each other, Fig. 4 has been introduced. With this respect, the highest rate for the R2 correlation coefficient of mentioned variables is recognized water to binder (W/B) for both technical properties having negative relation. Additionally, silica-fume and superplasticizer are factors with a higher positive relation to compressive strength, respectively, 0.908 and 0.0.945. However, for the slump flow case, the correlation rates have declined to the highest values of –0.37 for W/B, showing water to binder ratio (W/B) plays the main role, then fine aggregate to total aggregates ratio, then –0.328 for F/T.

Fig. 4

Correlation of inputs and targets used in models.

Based on the ingredients mixed in HPC samples, the role of each material is obvious with the aid of Fig. 5. For instance, except for superplasticizer and silica-fume, other ingredients have diminished the compressive strength rates, stability, and cohesion of concretes while playing same roles for slump factor. In this regard, scrutinizing the impacts of high-correlated ingredients has been surveyed in contour maps in Fig. 5 and the spreading condition of CS and SL values in sample number (Fig. 5(c)).

Fig. 5

Counter diagrams ingredients: a) water, superplasticizer, CS, b) water, superplasticizer, slump, c) compressive strength, and slump flow.

According to Fig. 5, the roles of superplasticizer and water for compressive strength (a) is clear that by enhancing water and lowering the superplasticizer, the compressive strength of HPC samples declines. In spite of the fact that as CS is reduced with increasing water amount, the SP can enhance the compressive strength rate. In the next map (b), considering SL with the impacts of the aforementioned elements of water and superplasticizer, the low slump rates have been seen in the low SP rates. The third map of Fig. 5(c) has been presented to show the interactions between CS and slump changes, besides showing the scattering condition of samples before modeling for training, validating, and testing stages.

The modeling processes were done, and the results of models from various aspects have been examined in the following sections. For this purpose, Fig. 6 has indicated the modeled values of CS and SL over the given target values.

Fig. 6

Target values and modeled outcomes of hybrid frameworks: a) CS modeling by RBHG, b) CS modeling by RBMV, c) SL modeling by RBHG, and d) SL modeling by RBMV.

According to the modeled values of CS and SL, both hybrid models have appeared to be powerful tools to simulate the hardness features with low bias that is perceived using gaps between the modeling green dashed line and target red/blue line. For compressive strength modeling, MVO (Fig. 6(b)) algorithm has been modified better than HGSO (Fig. 6(a)) algorithm. In samples #29 and #31, the difference between the performances of the two frameworks is seeable, in which the RBMV has reached the continuous target line while RBHG modeled compressive strength with more deviations. In overall view also, it is comprehendible that if considering the impact of phases on results, for the training stage, modeling results have overestimation, while in the test phase, we can mostly see underestimation trends. However, some cases in the training stage show underestimation. For example, samples #70 and #71 in RBHG have been modeled with CS values of 48.23 and 49.75 MPa, and these magnitudes have turned to 48.19 and 47.91 with discrepancies of 0.08 and 3.84 percent. The modeling trend for slump flow rates also can be seen with many cases of errors. In the training stage, the results are generated with more biased values for both models, and on the side of the testing stage, the smoothed error implies the efficient training effects reducing the errors in the test phase. Comparing the results of two models of RBHG and RBMV for slump prediction, the MVO algorithm has emerged better than the HGSO algorithm.

However, in some cases, RBHG has modeled the target values as more suitable than RBMV, like mixture #22, that earlier model with 4 percent difference appropriate models. Figure 7 illustrates the difference in the modeling process in light of the difference analysis among models. As seen in Fig. 7, modeling compressive strength (Fig. 7(a)) rates have been more different than modeling slump (Fig. 7(b)). Having harsh fluctuations in appraising CS has shown various optimizing mechanisms via HGSO and MVO metaheuristic algorithms to tune the prediction base model of RBFNN. At first look, the error domain in the vertical axis for CS is also seen from about –10% to+20%, while for the estimation of SL, this limit is confined to about –7% to+10%. In addition, in most cases, RBHG has appraised the hardness properties more than RBMV. Nevertheless, in many cases, two models are considered the same for both features and have appraised CS and SL values at the same rates.

Fig. 7

Difference percentage of modeling a) compressive strength and b) slump between two hybrid models of RBHG and RBMV.

Error distribution conditions and the rates of frequencies for both hybrid models are shown in Fig. 8. Based on Fig. 8, the highest bars of error frequency in the training phase compared to those in the testing phase for both models in appraising CS and SL, can be stemmed from better error controlling ability in training along with weak capabilities of optimizers in tunning RBFNN to generate no-biased results. However, in comparing the results between estimating CS and SL, the latter factor has been modeled with better modeling conditions with taller bars and a sharpened normal distribution curve of errors. From a modeling perspective, in the training phases, both frameworks have had similar performances in appraising SL despite the estimation of CS, with is done well via RBMV, that in the testing phase of modeling SL, also RBMV appeared more mighty than RBHG.

Fig. 8

The normal distribution curve of errors for modeling a) slump flow (train phase), b) slump flow (test phase), c) compressive strength (train phase), and d) compressive strength (test phase).

In the next stage, the error percentage of each HPC sample is investigated by comparing the outputs of models through Fig. 9. Based on the error limits in various phases of training and testing. We can see two opposite trends in which, for estimating compressive strength, the errors have been raised to their highest rates in the results of two hybrid models. Whereas, for the estimation of slump flow values, two models were conducted analogously and created the more smoothed plots in testing phases for the results of both developed models.

Fig. 9

Error rates of hybrid frameworks: a) CS modeling by RBHG, b) CS modeling by RBMV, c) SL modeling by RBHG, and d) SL modeling by RBMV.

In reproducing CS values, RBGH did a better job in the training stage, while RBMV could reduce the errors in the testing phase. Meanwhile, HGSO could not manage the modeling process in the testing phases as MVO has overcome the errors. For example, modeling CS of HPC mixtures by RBHG for samples #41 with –16.29%, #87 with –15.35%, and #165 with+20.07% deviation from target, were modeled with RBMV with error rates of same samples:11.28%, –11.34% and 17.23%, reducing errors by 5.35%, 4.74%, and 2.36%, respectively. In estimating the slump rate, the results have been improved in terms of error rates for the testing stage. RBHG modeled the SL factor of HPC samples with a maximum error domain of+14.60% to –9.21% (averagely 0.16%), while RBMV did this with a bias limit+10.44% to –8.27 (averagely 0.12%). Improving error rates by about 40 percent has implied the ability of the MVO algorithm to manage the errors against the HGSO algorithm.

The standard deviation and correlation coefficient factors for the hybrid models have been indicated using Taylor diagrams embedded in Fig. 10. For estimating both mechanical features of a slump and compressive strength, as shown in Fig. 10, RBHG was considered as a low-accuracy model estimating with lower correlation and remote standard deviation rate from its reference. However, estimating CS rates with a higher correlation employing RBMV with more difference to RBHG than estimating slump. In the final stage of assessing the results of two hybrid models, the outcomes of evaluative indices have been introduced in Fig. 11. According to Fig. 11, in the training phase, the correlation coefficient of R² was obtained at 96.60 percent for RBHG and 98.24 percent for RBMV, also, in the testing phase, MVO appeared to be better optimizer tunning the RBFNN with 1.57 percent difference in favor of RBMV. From the RMSE index point of view, RBMV could get a better result of 3.54 MPa with 39.27 percent in training. In the testing phase, RMSE calculated 3.59 MPa as the minimum rate for RBMV. On the other side, to appraise slump flow rates, the R² correlation rate for RBHG was computed at 96.86 percent while 98.25 percent for another model in the training phase, with a 33.30% difference. In the testing phase also, a difference of 50.80 percent was seen in favor of RBMV. In other studies, using an Artificial neural network model to appraise the compressive strength rates of HPC compounds, Bui et al. [54] conducted research that could reach the R² of 0.91 [54]. In another study, Han et al., Modeled the CS of concrete samples with an RMSE rate of 4.52 MPa [31]. Regarding the MAE indicator, RBMV obtained a 2.74 MPa error compared to RBHG, with 3.77 MPa with a 37.69% difference in the training phase. While for slump appraisal in the training stage, the difference was calculated at 35.97% in favor of RBHG. Considering OBJ as a comprehensive criterion for evaluating model performance, RBHG appeared weaker than RBMV in estimating slump and compressive strength rates with 43.02% and 46.00%, respectively.

Fig. 10

Taylor plots of developed RBFNN outcomes.

Fig. 11

Results of evaluative indices for assessing the performances of hybrid frameworks.

4 Conclusion

There has been a significant increase in high-performance concrete (HPC) over the recent development in construction materials in various structural applications such as bridges, masonry applications, pavement construction, high-rise buildings, residential buildings, and parking lots.

Compressive strength and slump flow rate features as hardness aspects of HPC should be considered in designing and controlling the quality of concrete mixes. To this end, influenced by many factors such as admixtures’ ratios, material properties, environmental conditions, and the age of HPC compounds, there have been several methods to measure the mechanical features of concrete samples. Using experimental approaches can be time-consuming and costly most of the time, with the previous laboratory experiments and results. Therefore, in this research, a robust Radial Basis Neural Network (RBFNN) model was developed to predict the mechanical properties of HPC samples, namely, slump flow and compressive strength. To increase the accuracy of RBFNN and reduce the errors and complexity of computations, two metaheuristic algorithms of Henry Gas Solubility Optimization (HGSO) and Multiverse Optimization Algorithm (MVO). Generated hybrid RBHG and RBMV were trained and tested via ingredients of HPC samples with various dosages. The results showed that both hybrid models appeared to be powerful tools to simulate the hardness features with low bias.

Generally, RBGH did a better job in the training stage in reproducing CS values, while RBMV could reduce errors in the testing phase. Meanwhile, HGSO could not manage the modeling process in the testing phases as MVO has overcome the errors. For estimating CS in the training phase, the correlation coefficient of R² was obtained at 96.60 percent for RBHG and 98.24 percent for RBMV. Also, in the testing phase, MVO appeared to be a better optimizer tunning the RBFNN with a 1.57 percent difference in favor of RBMV. From the RMSE index point of view, RBMV could get a better result of 3.54 MPa with 39.27 percent in training. In the testing phase, RMSE calculated 3.59 MPa as the minimum rate for RBMV. By reviewing the overall scores of two optimization algorithms in tunning the RBFNN to estimate the hardness properties of HPC samples, MVO appeared a better successor than HGSO considering metrics: R² and RMSE. In fact, the adjusting task of MVO has been done more accurately than HGSO in predicting the hardness properties of the slump and compressive strength.

While estimating CS, HGSO could empower the RBFNN to accurately model this factor of HPC samples. To sum up, based on the results of developed frameworks, RBFNN successfully modeled the hardness properties of HPC compounds by estimating the slump flow and compressive strength. Nevertheless, employing the hybrid models with desirable outcomes with smart software-based approaches can reduce the costs of physical experiments and simultaneously increase the accuracy of predicting mechanical features of crucial concrete material.

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Code	CS	SL	W/B	W	F/T	FA	AE	SF	SP
Parameters	Compressive strength	Slump flow	Water to Binder	Water	Fine to all aggregates	Fly Ash	Air entering	Silica-Fume	Superplasticizer
Unit	MPa	mm	%	Kg.m^–3	%	%	Kg.m^–3	%	Kg.m^–3
Max	123	260	45	180	53	20	0.08	25	36.5
Min	38	95	18	140	35	0	0	0	1.89
Ave	74.17	202.73	31.17	162.13	42.15	5.8	0.03	6.44	10.93
St.dev	26.64	25.93	8.77	12.12	5.37	8.03	0.03	8.43	8.63
Median	63	205	30	160	42	0	0.038	0	8