Developing support vector regression and radial basis function neural networks in optimized holistic frameworks to estimate hardness properties of concrete

Abstract

The hardness properties of constructional materials should be investigated as important factors in assessing the performance over the operation period. Two tests are performed to determine the stiffness characteristic, including slump and compressive strength (CS). They must be considered to examine efficiency, durability, and resistance to pressure. Due to the structure’s susceptibility and usage in dams, bridges, etc., high-performance concrete must have an appropriate set of these tests. There are two soft-based and laboratory methods for performing these tests. The laboratory method is not economical in terms of cost and time, and artificial intelligence (AI) is used to reduce the aforementioned factors. Models and optimizers use software-based methods to help reduce errors and increase model accuracy. So, The main purpose of this research has been introducing novel ways of coupling an ensemble model with optimizers by adjusting some internal parameters. In this article, two models, the Radial Basis Function Neural network and Support Vector Regression were combined and coupled with General Normal Distribution Optimization (GNDO) and Archimedes optimization algorithm (AOA) into the two frameworks of SVRRBF-AOA and SVRRBF-GNDO. As a result, the hybrid model of SVRRBF-AOA could perform well by obtaining R² and RMSE of 0.9915 and 2.71 for the slump and 0.9845 and 3.34 for CS, respectively.

Keywords

High-performance concrete slump compressive strength support vector regression Radial basis function generalized normal distribution optimization archimedes optimization algorithm

1 Introduction

In the early 1970 s, High-Performance Concrete (HPC) was the third-generation concrete material. Analogized to second-generation concretes like high-strength concrete (HSC), HPC not only has higher compressive strength (CS) however other significant properties, including high modulus, high flowability, high flexural strength, high durability, low permeability, etc. [1, 2]. These mechanical property improvements have created HPCs broadly utilized in long-run construction applications, especially in skyscrapers, long-span bridges, tunnels, and road construction [3, 4]. A key technology in the HPC manufacturing method is to achieve the highest possible density [5 –7].

Therefore, the main rules that discriminate HPC production compared to regular concrete are (i) the lower aggregate sizes usage, (ii) the auxiliary’s addition of cementitious materials like fly ash, and (iii), most importantly, the superplasticizers’ application for decreasing the water/binder ratio [8, 9]. Manufacturing HPC requires careful selection and control of components and their proportions. Many studies in the literature have suggested a mixed design process for HPC. Producing HPCs with increased mechanical properties is the main goal of work to acquire the combinations and relative proportions of the components [10, 11].

The HPC’s compressive strength (CS) is of particular interest to researchers as it is a significant mechanical property that reflects the material’s quality. Current practice is largely according to laboratory experiments to get the expected CS as a function of all components. This experimental method is very time-consuming, expensive, and always needs some equipment that may not be available. As a result, investigators have attempted to offer several formulations that show functional correlations between HPC CS and several corresponding mechanical parameters or properties [12 –14]. Zhou et al. [15] studied the aggregates’ effect on HPC compressive strength and supposed that it could be estimated by several formulas, except for the case of very high and very low modulus aggregates. Duval and Kadri [16] studied the silica fume’s influence on HPC CS and gave a predictive model with a correlation coefficient of 0.991.

Concrete workability is a function of the relative sizes of the different concrete mix components. The slump test is one of the tests that measure parameters close to providing and workability proper information. This is the most common measuring concrete consistency process, which can be used in the laboratory and field. The slump test is derived by measuring the drop from the collapsed fresh concrete’s top. More information about concrete workability can be achieved by obeying the shape of the concrete slump [17].

Concrete testing should be performed to measure slumps in any construction type to ensure that it has the desired workability and strength. Nonetheless, investigators have studied the property parameters that influence the slump value of high-strength concrete. The proportions of the concrete mix constituents containing cement, blast furnace slag, coarse aggregate, fine aggregate, fly ash, and superplasticizer are believed to affect workability and are defined according to the desired concrete properties. In addition, engineers often experiment with different mix ratios to get concrete with the suitable and desired workability, the cost of concrete production, and a time-consuming process that leads to wasted material [18 –21].

As suggested by a brief review by Rafiei [22], great efforts have been made in recent decades to solve civil engineering issues using intelligent computer algorithms. Data-driven methods have been utilized for analyzing structural behavior [23, 24]. Researchers developed predictive models with the ultimate goal of minimizing prediction errors compared to data collected from experiments in estimating material properties. Ni and Wang suggested a multilayer feedforward neural network to estimate the concrete’s CS [25]. The process was used to handle the non-linear relationship between concrete strength and input properties. Rafiei et al. employed a computer intelligence-based classification algorithm to solve the concrete mix design issue considering the necessary constraints and a non-linear optimization algorithm [26]. The same author also presented neural networks and statistical models for estimating concrete properties according to input parameters [27].

Furthermore, Radial basis function (RBF) neural networks are well-known for pattern recognition and approximation. For these applications, the pattern dimensionality is often undersized. As Moody and Darken [28] pointed out, “RBF neural networks are best done for learning to piecewise or approximate continuous real-valued maps with sufficiently small input dimensions.” A neural RBF classifier design is presented to process a small training set of high-dimensional feature vectors [29, 30].

Accordingly, the present research intends to predict HPC compressive strength and slump properties using ensemble results of models Radial Basis Function Neural Network (RBFNN) [31, 32], and Support Vector Regression (SVR) [33, 34] that are coupled with generalized normal distribution optimization (GNDO) [35] and Archimedes optimization algorithms (AOA) [36] in the hybrid frameworks. In fact, As a novelty of this research, coupling type of matheuristic optimization algorithms with main model as an ensemble framework with having separate elements are designed to find the optimum values that finally will be used to model hardness properties of high performance concrete. Optimizers can reduce the error rates of modeling hardness properties of HPC compounds by enhancing the capabilities of main models RBFNN and SVR by tuning them. There are, however, several ways to appraise geomechanical characteristics of concrete. Solely usage of main models are examples of common ways to estimate CS and SL also using alone optimizers. In previous versions we can review the researches with ensemble models. In light of designing comprehensive mechanism to estimate desired results of compressive strength and slump rates with lowest error, a combination of main models in form of ensemble one with optimization algorithms was considered as the fundamental axe of this article.

Ghasemi Rad et al., conduct a research to simulate the geometric characteristics of gravity dams. To achieve the optimum result for concreting amount, Monte Carlo simulation was employed in couple with GNDO matheuristic algorithm. Consequently, with proposing an efficient model for the probabilistic design of gravity dams utilizing a novel developed reliability based design optimization method the usage of GNDO-SVR comparing with SVR boosted the estimated R² from 0.85 to 0.99 [35]. Khodadadi and Mirjalili used Generalized Normal Distribution Optimization (GNDO) algorithm to design the truss structures with optimal weight. To evaluate the GNDO algorithm, three benchmark truss optimization problems are considered with frequency constraints. Numerical data show GNDO’s reliability, stability, and efficiency for structural optimization problems than other meta-heuristic algorithms [37]. Ge et al., in an investigation of pile settlement in a practical transportation project modeled the subsidence of piles using support vector regression based on many pile settlement samples. To increase the capability of SVR, Archimedes optimization algorithms (AOA) was chosen to be coupled with main model. Finally, the values of RMSE for SVRAOA and SVRGOA were obtained at 0.55 and 0.59, alternatively, and MAE showed the values of 0.52 and 0.56, respectively. The R² of the model SVR AOA exhibited the magnitude 0.994, that was 0.10% higher than that of SVR-GOA [38]. This creative procedure has rarely been applied in appraising the stiffness aspects of concrete, to which Fig. 1 shows the overall process used in the research.

Fig. 1

Overall view of key stages to model the hardness properties.

2 Materials and methodology

2.1 Data gathering

In the present study, 189 HPC mixtures, given from published literature [39], are used, that Table 1 shows some statistical aspects of concrete mixtures. Fig. 2 has provided the diagrams of components of HPC samples plus accumulative curves. In the prediction task of the slump and compressive strength for HPC samples, the input data are constituted of the weight of fine aggregate to total aggregate weight ratio (S/A), water to binder ratio (W/B), the slump flow (SL), fly-ash (FA), silica-fume (SF), air-entraining agent (AE), water budget of HPC samples (W), and superplasticizer (SP).

Table 1
Summary statistical report of model inputs

Code Unit Max Min Ave St. Dev. Median

SL (mm) 260 95.0 202.73 25.93 205.0

AE (kg/m³) 0.08 0.00 0.03 0.03 0.038

SP (kg/m³) 36.5 1.89 10.93 8.63 8.00

W (kg/m³) 180 140 162.13 12.12 160.0

FA (%) 20 0.00 5.80 8.03 0.00

W/B (%) 45 18.0 31.17 8.77 30.00

SF (%) 25 0.00 6.44 8.43 0.00

S/A (%) 53 35.0 42.15 5.37 42.0

Code	Unit	Max	Min	Ave	St. Dev.	Median
SL	(mm)	260	95.0	202.73	25.93	205.0
AE	(kg/m³)	0.08	0.00	0.03	0.03	0.038
SP	(kg/m³)	36.5	1.89	10.93	8.63	8.00
W	(kg/m³)	180	140	162.13	12.12	160.0
FA	(%)	20	0.00	5.80	8.03	0.00
W/B	(%)	45	18.0	31.17	8.77	30.00
SF	(%)	25	0.00	6.44	8.43	0.00
S/A	(%)	53	35.0	42.15	5.37	42.0

Fig. 2

Frequency of ingredients used to produce HPC samples.

It is worth noting that the experiments were done in South Korea, and after 28 days, slump measurements of the HPC samples were also recorded. The slump tests were met immediately after compounding based on the ASTM C 143-90a standard. Measuring the incoming air into the mixture was done according to the standard of ASTM C 231-91b. Moreover, coarse aggregates with a grain fineness factor of 7.2 and a special density of 2.7 were produced from crushed granite stones with size19 mm. Based on ASTM Type I standards, cement was selected to be Portland. The fine silica sand aggregates with 2.61 specific gravity and 2.94 as fineness coefficient were produced. A naphthalene superplasticizer was employed to reduce water by controlling HPC’s water-to-binder ratio (W/B). The class F fly ash plus the silica fume were produced in Norway.

2.2 Developing hybrid-ensemble models

This paper aims to indicate the importance of missing data management and hyperparameter initialization. Therefore, two popular machine learning algorithms are used to handle HPC issues through two optimizers in civil engineering and data mining. Specifically, the main models are (i) Support Vector Regression (SVR) used in regression applications, (ii) Radial Basis Function Neural Network (RBFNN), also well-documented as the deep forwarding network; basically a specific deep artificial function; as well as generalized normal distribution optimization (GNDO) and Archimedes optimization algorithms (AOA) are considered as the determining successors for models. Cost function of models play main role in determining the parameters embedding in that are calculated using optimizers based on some tries to find optimum rate of each regulatory variable. In fact, optimizers receive the feedbacks of model as outputs that are compared with target values of practical experiments. Extracting the logical relationships between dedicated main model variables with final errors comparing results, performance of each determined value of regulatory factors in models by optimization algorithms will be assigned.

As a matter of fact, the main models of SVR and RBFNN were used as ensemble models of SVR-RBFNN, to which assigned metaheuristic optimization algorithms were used to tune the internal settings of SVR and RBFNN. Consequently, to better understand the present study’s overall processes, Fig. 3 is presented.

Fig. 3

Flowchart of the present study to model the hardness properties.

2.3 Support vector regression (SVR)

In the 1990 s, SVM was a commonly supervised machine learning model primarily generated via Vapnik and his colleagues at AT&T Bell Labs [40 –42]. The SVM’s key idea is to map input vectors into a high-dimensional attribute distance employing pre-selected non-linear kernel functions called hyperparameters. Hyperparameters are initialized and fixed parameters before training the machine learning model. In the attribute distance, a linear decision surface is formed by properties that guarantee a high degree of generalizability for learning machines [41]. SVMs are broadly utilized and have achieved high performance in regression and classification applications. When SVM is used in regression applications, it is called Support Vector Regression (SVR) [33].

The ɛ-SVR is used to handle HPC regression issues. Particularly obtained training data ${(x_{1}, y_{1}), (x_{2}, y_{2}), \dots, (x_{n}, y_{n})} \in x \times ℝ$ ., here x shows the input features’ space. The ɛ-SVR’s target is to find function f (x), which has at most ɛ deviation from the actual target value y_i to every n sample of the training data. Suppose f is the form’s linear function: $\begin{matrix} {\begin{matrix} y_{j, t} + R sin (d) | {By}_{best} - y_{j, t} | if rand < 0.5 \\ y_{j, t} + R cos (d) | {By}_{best} - y_{j, t} | if rand ⩾ 0.5 \end{matrix} \\ f (x) = (m, x) + t t \in ℝ \end{matrix}$ (1)

Here, (m, x) represents the point function. m shows desired to smooth the function f via minimizing the norm in the following: $\begin{matrix} y_{j, t + 1} = \\ \min . \frac{1}{2} {∥ m ∥}^{2} \\ subjectto | y_{i} - (m, x) - t | ⩽ ɛ \end{matrix}$ (2)

An issued example that can be solved in 2D space indicate in Fig. 4. Nevertheless, solving equation 2 satisfies all pairs (x_i, y_i) is not always achievable because of the data’s large amount in the real problem, and as indicated in Fig. 5, some data points are just outside the support range bounded with ɛ. Model errors should be allowed, leading to the idea presented by Vapnik and Cortes to use “soft margins” in SVM. This is completed via way of means of presenting slack variables $(ξ_{j}, ξ_{j}^{*})$ to handle illegal conditions. Therefore, SVR can be developed as follows: $\begin{matrix} \min . \frac{1}{2} {∥ m ∥}^{2} + U \sum_{j = 1}^{n} (ξ_{j} + ξ_{j}^{*}) \\ subjectto {\begin{matrix} y_{i} - ((m, x) + t) ⩽ ɛ \\ ξ_{j}, ξ_{j}^{*} ⩾ 0 \end{matrix} \end{matrix}$ (3)

Fig. 4

Liner issues with standard SVM in 2D space.

Fig. 5

Non-liner issues by standard SVM in 2D space.

The consistent U > 0 defines the trade-off between the quantity and the flatness of the characteristic f as much as which deviations large than are tolerated. This links to deal with a so called -insensitive loss function |ξ|_ɛ. ${| ξ |}_{ɛ} = {\begin{matrix} 0 if | ξ | ⩽ ɛ \\ | ξ | - ɛ otherwise \end{matrix}$ (4)

An employing example of “soft margins” to deal with a regression issue in 2D space is indicated in Fig. 5. It should be noted that the parameters of ξ, ɛ and U are assigned to be adjusted via optimizers.

In fact, employing “soft margins” allows some error when training models with f s linear form, this feature is not always available. To solve this problem, one can create the SVR algorithm non-linear, which can be done using a kernel function to transform the original data from the low dimensional vector distance to the higher dimensional vector where the f’s linear form can be obtained. In addition, a hyperparameter, the kernel function, can be a polynomial, a linear basis, a sigmoid function, or a radial basis function (RBF).

In HPC regression applications, it should be noted that several SVR extensions have been suggested and achieved high performance, where the authors focus on modifying the architecture of the model [43, 44].

2.4 Radial basis function (RBF)

RBF neural networks have a fixed 3-layer structure containing an input, a hidden, and an output layer. The input layer affords network input. Hidden layers remap the input statistics to be linearly detachable. The output layer plays linear separation. For a given structure, designing an RBF network usually includes 3 steps: 1) locating the proper network size, 2) locating desirable preliminary parameters, and 3) Training the network.

Figure 6 suggests an RBF network’s overall shape with N inputs, J hidden units, and Q outputs.

The simple computation for the RBF network consists of the following:

a) The input layer’s calculation

The input vector Y is weighted with the input weights J^hat the hidden unit l’s input: $J_{j} = [Y_{1} J_{1, j}^{h}, Y_{2} J_{2, j}^{h}, \dots, Y_{n} J_{n, j}^{h}, \dots, Y_{N} J_{N, j}^{h}]$ (5)

Fig. 6

RBFNN with N inputs, J hidden units, and Q outputs.

In Equation (4), n, j, and Y_n donate the input’s indication, the indication of the hidden unit, and the n^thinput. Furthermore, $J_{n, j}^{h}$ defines the input weight between hidden unit j and input n.

b) Hidden layer calculation

The j as the output hidden unit is determined in the following: $δ_{j} (Y) = exp (- \frac{{∥ x - e_{j} ∥}^{2}}{ρ_{j}^{2}})$ (6)

where, δ_jshows the hidden unit’s activation function, j is commonly got as a Gaussian function. Also, e_j and ρ_j Indicates the hidden unit’s center and the width of the hidden unit j, respectively.

c) Output layer calculation

The network output q is determined as follows: $x_{q} = \sum_{j = 1}^{n} w_{jq} δ j (Y) + b_{j}$ (7)

Where, q indicates the output index. In addition, jq and δj (Y) show the output weights between unit j of hidden and unit q of output and the output of bias weight, alternatively.

To increase the capability of model in generating accurate results the number of neurons in hidden layer in which the calculation are processing as well as ρ_j parameter as hidden unit’s center (or spread value) are the variables for adjusting by optimization algorithms.

2.5 Generalized normal distribution optimization (GNDO)

The normal distribution theory inspires GNDO. The normal distribution is also known as the Gaussian distribution, which is a very significant tool for describing natural phenomena. A normal distribution can be determined in the following. Suppose a random variable x follows a probability distribution by scale parameter δ and position parameter μ, and its probability density function can be represented in the following: $f (x) = \frac{1}{\sqrt{2 π δ}} . \exp (- \frac{{(x - μ)}^{2}}{2 δ})$ (8)

Then x shows a normal random variable, and the distribution is normal. Based on Equation (7), a normal distribution contains two variables: the scale parameter δ. And the position parameter μ. The location parameter μ and the scale parameter δ are employed to alternatively represent the mean and the random variable’s standard variance.

Local exploitation refers to finding the best solutions in the explore space, which consists of everyone’s current positions. Based on the relationship between the distribution of individuals and the normal distribution in the population, a generalized normal distribution model can be constructed for optimization as follows: $v_{i}^{t} = μ_{i} + δ_{i} \times η, i = 1, 2, 3, \dots, N$ (9)

Here, $v_{i}^{t}$ shows the tracking vector of the ith individual at time t, μ_i shows the generalized mean position of the ith individual, δ_i shows the generalized standard variance, and η indicates the penalty factor. Moreover, μ_i, δ_i and η can be determined as follows: $μ_{i} = \frac{1}{3} (x_{i}^{t} + x_{best}^{t} + M)$ (10) $δ_{i} = \sqrt{\frac{1}{3} [{(x_{i}^{t} - μ)}^{2} + {(x_{best}^{t} - μ)}^{2} + {(M - μ)}^{2}]}$ (11) $η = {\begin{matrix} \sqrt{- log (λ_{1})} \times cos (2 π λ_{2}), if a ⩽ b \\ \sqrt{- log (λ_{1})} \times cos (2 π λ_{2} + π), if a > b \end{matrix}$ (12)

Here, a, b, λ₁, and λ₂ show random numbers between 0 and 1, $x_{best}^{t}$ Indicates the current best position, and M shows the current population mean location. Also, M can be calculated as follows: $M = \frac{\sum_{i = 1}^{N} x_{i}^{t}}{N}$ (13)

Global search is about searching a language area worldwide to find promising areas. The global scan in GNDO is according to three randomly chosen people, which can be represented as follows: $v_{i}^{t} = x_{i}^{t} + β \times (| λ_{3} | \times v_{1}) + (1 - β) \times (| λ_{4} | \times v_{2})$ (14)

Here, λ₃ and λ₄ show two random numbers subject to a standard normal distribution, β is the fitting parameter, show a random number between 0 and 1, and v₁ and v₂ indicate two tracking vectors. Also, v₁ and v₂ can be calculated as follows: $v_{1} = {\begin{matrix} x_{i}^{t} - x_{p 1}^{t}, if f (x_{i}^{t}) < f (x_{p 1}^{t}) \\ x_{p 1}^{t} - x_{i}^{t}, otherwise \end{matrix}$ (15) $v_{2} = {\begin{matrix} x_{p 2}^{t} - x_{p 3}^{t}, if f (x_{p 2}^{t}) < f (x_{p 3}^{t}) \\ x_{p 3}^{t} - x_{p 2}^{t}, otherwise \end{matrix}$ (16)

Here, p1, p2, and p3 show three random integers chosen from 1 to N. Fig. 7 depicts the framework of the proposed GNDO algorithm.

Fig. 7

The flowchart of the GNDO algorithm.

2.6 Archimedes optimization algorithm (AOA)

AOA can be viewed as a global optimization algorithm because it involves both exploitation and search methods. Algorithm 2 shows the pseudocode of the suggested algorithm. Includes population evaluation, population initialization, and parameter update. Mathematically, the AOA procedure is detailed in the following.

Initialize all objects’ positions as: $P_{i} = {lb}_{i} + rand \times ({ub}_{i} - {lb}_{i}); i = 1, 2, \dots, N$ (17)

Here, P_i shows the ith object in N objects’ population, ub_i and lb_i indicates the explore space’s upper and lower bounds, alternatively.

Initialize volume (vol) and density (den) for each i^th object in Equation (17): ${den}_{i} = rand, {vol}_{i} = rand$ (18)

Here, rand is a dimensional vector that randomly develops a number between [0, 1]. Finally, initialize the acceleration (acc) of the i^thobject with Equation (19): ${acc}_{i} = {lb}_{i} + rand \times ({ub}_{i} - {lb}_{i})$ (19)

Evaluate the initial population and select the object with the best fitness score. Assign x_best, den_best, vol_best, and acc_best.

The density and object’s volume i at repetition t + 1 are updated in Equation (20): $\begin{matrix} {den}_{i}^{t + 1} = {den}_{i}^{t} + rand \times ({den}_{best} - {den}_{i}^{t}) \\ {vol}_{i}^{t + 1} = {vol}_{i}^{t} + rand \times ({vol}_{best} - {vol}_{i}^{t}) \end{matrix}$ (20)

Here, den_best and vol_best donate the best object-related density and volume found so far, and the rand is a uniformly distributed random number.

First, there is a collision between the objects; after a certain time, the objects try to reach the state of equilibrium. This is implemented in AOA using the transfer operator T F, which converts explore from exploration to exploitation, determined with Equation (21): $TF = \exp (\frac{e - e_{\max}}{e_{\max}})$ (21)

Here, till it reaches 1 the transmit T F gradually improves over time. eande_max are the number of repetitions and the maximum number of repetitions, alternatively. Also, the density reduction factor b AOA helps in searching from global to local. Decrease over time with Equation (22): $b^{t + 1} = exp (\frac{e - e_{\max}}{e_{\max}}) - (\frac{e}{e_{\max}})$ (22)

Here, b^t+1 reduces over time, providing an opportunity to focus on an already recognized promising area. In AOA, note that this variable’s proper handling maintains the balance between exploitation and search.

If TF≤0.5, the collision between objects occurs, update the object’s acceleration for repetition t + 1 and choose a random material (mr) with Equation (23): ${acc}_{i}^{t + 1} = \frac{{den}_{mr} + {vol}_{mr} \times {acc}_{mr}}{{den}_{i}^{t + 1} \times {vol}_{i}^{t + 1}}$ (23)

Here, den_i, vol_i, and acc_i are the object i’s density, volume, and acceleration. While den_mr, vol_mr, and acc_mr are the density, volume, and acceleration of (mr). It is important to mention that TF≤0.5 ensures search during a third of the repetition. Apply a value other than 0.5 changes scan exploit demeanor.

If TF > 0.5, there is no collision between objects, update the object’s acceleration for repetition t + 1 with Equation (24): ${acc}_{i}^{t + 1} = \frac{{den}_{best} + {vol}_{best} \times {acc}_{best}}{{den}_{i}^{t + 1} \times {vol}_{i}^{t + 1}}$ (24)

Here, acc_best is the best object’s acceleration.

Normalize the acceleration to compute the percent change with Equation (25): ${acc}_{i - norm}^{t + 1} = t \times \frac{{acc}_{i}^{t + 1} - \min (acc)}{\max (acc) - \min (acc)} + r$ (25)

Here, t and r are set to 0.9 and 0.1and the normalization range and, alternatively. The ${acc}_{i - norm}^{t + 1}$ defines the step percentage that each agent changes. If object i is far from the global optimum, the acceleration value is high, which means that the object is in the search section; otherwise in the recovery section. This indicates how the search moves from the search phase to the exploitation section. Normally, the acceleration factor starts with a decrease over time and a large value. while moving away from local solutions, this helps search agents move towards the best global solution. However, it’s worth noting that there may be some explore agents that require more time to stay in the search section than usual. Therefore, AOA strikes a balance between search and exploitation.

If T F≤0.5 (scan phase), the i^th object’s position for the next repetition t + 1 in Equation (26): $x_{i}^{t + 1} = x_{i}^{t} + C_{1} \times rand \times {acc}_{i - norm}^{t + 1} \times b \times (x_{rand} - x_{i}^{t})$ (26)

Here, C₁ shows constantly equal to 2. On the other hand, if T F > 0.5 (exploit phase), objects update their positions with Equation (27): $\begin{matrix} x_{i}^{t + 1} = x_{best}^{t} + F \times C_{2} \times rand \\ \times {acc}_{i - norm}^{t + 1} \times b \\ \times (T \times x_{rand} - x_{i}^{t}) \end{matrix}$ (27)

Here, C₂ indicates a constant equal to 6. T improves with time and is directly proportional to the transfer operator and is defined by T = C₃×T F. T increases with time in the range [C₃×0.3, 1] and initially decreases some percentage from the best position. Starting with a low percentage as this will result in a large difference between the current position and the best position, so the random walk step size will be high. As the search progresses, this percentage will gradually increase until the difference between the best position and the current position decreases that the mentioned stages are depicted as pseudo code in Fig. 8. This results in a proper balance between exploitation and search. F is the flag to change movement’s direction with Equation (28):

Fig. 8

Pseudo code of Archimedes Optimization Algorithm (AOA).

$F = {\begin{matrix} + 1 if O ⩽ 0.5 \\ - 1 if O > 0.5 \end{matrix}$ (28)

Here, O = 2 × rand - C₄.

2.7 Evaluation criteria

Regarding the performance’s validating the of the machine learning models described in the paper, the linear correlation coefficient (R²), mean square error (RMSE), mean absolute error (MAE), and mean absolute percent error (MAPE). The calculations for these performance evaluators are: $R^{2} = {(\frac{\sum_{d = 1}^{D} (n_{d} - \bar{n}) (c_{d} - \bar{c})}{\sqrt{[\sum_{d = 1}^{D} {(n_{P} - n)}^{2}] [\sum_{d = 1}^{D} {(c_{d} - \bar{c})}^{2}]}})}^{2}$ (29) $RMSE = \sqrt{\frac{1}{D} \sum_{d = 1}^{D} {(c_{d} - n_{d})}^{2}}$ (30) $MAE = \frac{1}{D} \sum_{d = 1}^{D} | c_{d} - n_{d} |$ (31) $MAPE = \frac{1}{N} \sum_{n = 1}^{N} | \frac{m_{n} - d_{n}}{m_{n}} | \times 100$ (32)

Here, m_n is the actual value, $\bar{m}$ is the mean of actual value, d_n and $\bar{d}$ shows the predicted and mean of predicted value, alternatively. In addition, N donates the dataset’s numbers.

3 Results and discussion

Modeling includes two training and testing sections. The data set is 189, 70% of the samples belong to the training and 30% for the test section. In addition, the general state of the process is shown graphically in Fig. 1. Results for two attachment models, containing SVRRBF-AOA and SVRRBF-GNDO, as shown in Table 2. This comparison was made between the two attachment models using indices R², RMSE, MAE, and MAPE. The ideal conditions for both the R² scorers are close to 1, and generally higher values for the R² metric indicate better conditions. On the other side, the minimum value is ideal for RMSE, MAE, and MAPE evaluators. SVRRBF-AOA and SVRRBF-GNDO correspond to 0.9486 and 0.9428 for the slump, respectively, moreover 0.9665 and 0.984 for CS, respectively for the R² data in the training section of the two hybrid models. The difference between the models is 3% for slump and nearly 2% for CS. The SVRRBF-AOA slightly outperforms the SVRRBF-GNDO. Additionally, the test section of R² is 0.9686 and 0.9915 for SVRRBF-GNDO and SVRRBF-AOA, alternatively. The RMSE associated with SVRRBF-GNDO equals 8.9635 and SVRRBF-AOA, with the lowest value in the training phase being RMSE = 7.2744 belonging to slump, for CS 5.15 and 3.37 of GNDO and AOA, respectively AOA outperforming on this metric. In addition, for MAE error, the GNDO-related test part yielded a minimum value of 1.55 and 3.92 and AOA had the lowest MAE = 0.91 and MAE = 2.52 for slump and CS, respectively. As mentioned earlier, the model should have the lowest value for MAPE. The lowest MAPE belongs to GNDO MAPE = 2.74 and MAPE = 6.03 which is related to slump and CS, alternatively. AOA had the lowest value which is equal to 1.875 and 3.519 for slump and CS, alternatively. Generally, AOA was able to form a better combination with SVRRBF.

Table 2
The results obtained from the hybridized models

Metric Phase Model

SVRRBF-GNDO SVRRBF-AOA SVRRBF-GNDO SVRRBF-AOA

Slump Compressive Strength

R ² Train 0.9486 0.9728 0.9665 0.984

Test 0.9686 0.9915 0.9734 0.9845

RMSE Train 8.9635 7.2744 5.1509 3.3763

Test 4.0172 2.7101 4.8021 3.3476

MAE Train 5.0216 3.382 4.3979 2.5293

Test 1.5584 0.9175 3.9278 2.5662

MAPE Train 2.7473 1.9047 6.4071 3.6611

Test 3.1552 1.875 6.0399 3.5193

Metric	Phase	Model
R ²	Train	0.9486	0.9728	0.9665	0.984
	Test	0.9686	0.9915	0.9734	0.9845
RMSE	Train	8.9635	7.2744	5.1509	3.3763
	Test	4.0172	2.7101	4.8021	3.3476
MAE	Train	5.0216	3.382	4.3979	2.5293
	Test	1.5584	0.9175	3.9278	2.5662
MAPE	Train	2.7473	1.9047	6.4071	3.6611
	Test	3.1552	1.875	6.0399	3.5193

Figure 9 shows the dispersion plot of measured and predicted values of the slump and compressive strength. The measured points are marked in red in each figure. The most ideal situation is that the measured points are placed on the predicted points. In a general view, it can be seen that both models of the respective shape have similar behavior, but in a more detailed view, it is clear that the SVRRBF-AOA combined model has been able to perform relatively better. The points in Figs. 9(a) and 9(b) belong to slump and are in the range of 170 to 240. But on the other hand, for Figs. 9(c) and (d), the samples are more scattered. In both models, the samples have been underestimated and overestimated. In general, it can be seen that the combined SVRRBF-AOA model performed better than SVRRBF-GNDO in the figure below for both slump and CS.

Fig. 9

The dispersion plot of measured and predicted values of the slump and compressive strength a) SVRRBF-GNDO (slump), b) SVRRBF-AOA (slump), c) SVRRBF-GNDO (CS), and d) SVRRBF-AOA (CS).

Figure 10 indicates the correlation of the accuracy of the current model. Figures 10(a) and (b) belong to slump on the other side Fig. 10(c) and 10-d determine the CS. Based on the two evaluators RMSE and R², the corresponding figure determines that the points are continuous for high R² and lie on the X = Y line. However, increasing the RMSE pushes the points further apart. The 6-a associated with SVRRBF-GNDO indicates the spread of the points, which is the reason for the low R² and high RMSE. On the other hand, Fig. 10(b) refers to the SVRRBF-AOA hybrid model, with a correspondingly higher R² percentage, lower RMSE than SVRRBF-GNDO, points closer to each other, and the X = Y line. In addition, the difference between the linear fit and the center line shows suitable model performance and the smallest difference belongs to the SVRRBF-AOA. On the other side, Figs. 10(c) and (d) define the CS. Fig. 10(c) in relation to SVRRBF-GNDO, can see that both the RMSE and R² metrics decreased and increased significantly, alternatively. The difference in R² for the SVRRBF-GNDO in CS is about 2% compared to the slump, and the difference in RMSE is 35%. Furthermore, in Fig. 10-d, the result of coupled SVRRBF-AOA model is RMSE = 3.3677 MPa and R² = 98.41%, so there is much less filling of points and most of the points lie on the center line. Approximating the slump and CS of AOA models, the difference in R² is over 1% and the RMSE is nearly 46%. Comparing the two models of SVRRBF-AOA and SVRRBF-GNDO, in general, can conclude that the performance of AOA is more satisfactory.

Fig. 10

The correlation plot between the measured and the predicted values of the slump and compressive strength by hybrid models.

Figure 11 shows the percentage error plot of the model generated in the training and testing sections. Forasmuch as 70% of the sample is in the training section and 30% is in the test section, so more errors are expected from the training phase. The most ideal situation is an error close to 0%. Figures 11(a) and (b) are related to slumping. Figure 11(a) is for SVRRBF-GNDO, where the 50th sample yielded the largest error at about 44.35%, but the error decreased in the applicable model test phase, reaching the largest error at about 6%, which is associated with the 182nd sample. Figure 11(b) is specific to SVRRBF-AOA, with the highest model training error of 45.46% obtained on sample 53. The Sample errors in the DT test phase have decreased significantly, reaching 4.9%. If compare the two models in slump form, can see that the GNDO error is slightly smaller than AOA in the training phase but in the test phase, AOA was able to have the lowest error percentage. Thus, the error difference between the models is about 2.5% in the training phase, and in the test phase is about 33%. Moreover, Fig. 11(c) and 11(d) support the coupled SVRRBF-GNDO and SVRRBF-AOA. Comparing Figs. 11(a) and (b), the maximum error is 17.79% in Fig. 11(c), which shows a 60% reduction in error compared to the slump. Also, in the test section, the largest error is 22.5%, unlike the training phase, over 100% had increased compared to the slump. Also, in Fig. 11(d), the largest error was 13.8% in the training phase and increased to 21.5% in the test phase. Comparing SVRRBF-AOA in CS and slump, the error was reduced by 69% in the training phase but increased by over 100% in the test phase. In addition, In the comparison of the two models, in the general case, AOA was able to have a better combination with SVRRBF for slump and CS.

Fig. 11

The error percentage diagram of generated models a) SVRRBF-GNDO (slump), b) SVRRBF-AOA (slump), c) SVRRBF-GNDO (CS), and d) SVRRBF-AOA (CS).

Figure 12 indicates histograms of the error density percentages for the hybrid models. Figure 12 is plotted as a function of sampling frequency and percent error. Comparing AOA and GNDO in Fig. 12(a) which is related to slump, can see that the pattern of AOA follows the bell pattern to some scope, while the pattern of SVR is slightly flat. In addition, the percent error for AOA and GNDO have a similar demeanor and close to zero percent, ranging from -5 to 5 for most sample rates. On the other hand, it is assigned to the two coupled models for CS in Fig. 12(b). Compared to Fig. 12(a), the error percentage has increased and is more extensive. Comparing SVRRBF-GNDO and SVRRBF-AOA, one can realize that the normal distribution of both models is flat but AOA is slightly sharper than GNDO. The maximum frequency for AOA is 71 and GNDO is 51. Overall, Fig. 12 shows that AOA is stronger than GNDO in CS and slump form.

Fig. 12

The histogram of error percentage density of two present models a) Slump, b) Compressive strength.

4 Conclusion

The main rules that discriminate HPC production compared to regular concrete are (i) the lower aggregate sizes usage, (ii) the auxiliary’s addition cementitious materials like fly ash, and (iii) most importantly, the superplasticizers’ application for decreasing the water/binder ratio. The HPC’s compressive strength (CS) is of particular interest to researchers as it is a significant mechanical property that reflects the material’s quality. Current practice is largely according to laboratory experiments to get the expected CS as a function of all components. Concrete workability is a function of the relative sizes of the different concrete mix components. The slump test is one of the tests that measure parameters close to providing and workability proper information. This is the most common measuring concrete consistency process, which can be used both in the laboratory and in the field. The slump test is derived by measuring the drop from the collapsed fresh concrete’s top. More information about concrete workability can be achieved by obeying the shape of the concrete slump. The article intends to predict HPC compressive strength and slump by using two models, Radial Basis Function (RBF) and Support Vector Regression (SVR), moreover, two optimizers include generalized normal distribution optimization (GNDO), Archimedes optimization algorithm (AOA). In addition, predicting is done in the framework of SVRRBF-GNDO and SVRRBF-AOA. Two combined models are compared with each other via evaluators. The ideal conditions for both the R² scorers are close to 1, and generally higher values for the R² metric indicate better conditions. On the other side, the minimum value is ideal for RMSE, MAE, and MAPE evaluators. SVRRBF-AOA and SVRRBF-GNDO correspond to 0.9486 and 0.9428 for the slump, respectively, moreover 0.9665 and 0.984 for CS, respectively for the R² data in the training section of the two hybrid models. The difference between the models is 3% for slump and nearly 2% for CS. The SVRRBF-AOA slightly outperforms the SVRRBF-GNDO. Additionally, the test section of R² is 0.9686 and 0.9915 for SVRRBF-GNDO and SVRRBF-AOA, alternatively. The RMSE associated with SVRRBF-GNDO equals 8.9635 and SVRRBF-AOA, with the lowest value in the training phase being RMSE = 7.2744 belonging to slump, for CS 5.15 and 3.37 of GNDO and AOA, respectively AOA outperforming on this metric. In addition, for MAE error, the GNDO-related test part yielded a minimum value of 1.55 and 3.92 and AOA had the lowest MAE = 0.91 and MAE = 2.52 for slump and CS, respectively. As mentioned earlier, the model should have the lowest value for MAPE. The lowest MAPE belongs to GNDO MAPE = 2.74 and MAPE = 6.03 which is related to slump and CS, alternatively. AOA had the lowest value which is equal to 1.875 and 3.519 for slump and CS, alternatively. Generally, AOA was able to form a better combination with SVRRBF. The main purpose of this research has been introducing novel ways of coupling ensemble models with optimizers by adjusting some internal parameters. Generally, using such smart frameworks will increase the efficiency of appraising geomechanical features of aggregates as well as reducing the errors instead of doing experiments with physical practices with high costs in terms of time, energy and financial aspects.

Footnotes

Acknowledgments

This work was sponsored in part by 2021 University Science and Education Development Fund Project (PhD Research Initiation Project) Research of an Experimental Platform for Mechanism Analysis and Intelligent Control of Pressure Fluctuations in the Earth Pressure Balance Chamber of Compound Soil Compaction Shield Tunneling. (Z202206004).

References

Van Dao

, et al., A sensitivity and robustness analysis of GPR and ANN for high-performance concrete compressive strength prediction using a Monte Carlo simulation, Sustainability 12(3) (2020), 830.

Neville

and Aïtcin

P.-C.

, High performance concrete—An overview, Mater Struct 31(2) (1998), 111–117. doi: 10.1007/BF02486473

Huang

, Jiang

, Wang

, Zhu

and Afzal

, Prediction of long-term compressive strength of concrete with admixtures using hybrid swarm-based algorithms, Smart Struct Syst 29(3) (2022), 433–444.

Nawy

E.G.

, Fundamentals of high-performance concrete. John Wiley & Sons, 2000.

Cherian

, Determining the amount of earthquake displacement using differential synthetic aperture radar interferometry (D-InSAR) and satellite images of Sentinel-1 A: A case study of Sarpol-e Zahab city, Adv Eng Intell Syst 1(01) (2022).

Nurlan

, A novel hybrid radial basis function method for predicting the fresh and hardened properties of self-compacting concrete, Adv Eng Intell Syst 1(01) (2022).

Cheng

, Kitchen

and Daniels

, Novel hybrid radial based neural network model on predicting the compressive strength of long-term HPC concrete, Adv Eng Intell Syst 1(02) (2022).

Huchante

S.R.

, Chandupalle

, Ghorpode

V.G.

and TC

V.R.

, Mix design of high performance concrete using silica fume and superplasticizer, Pan 18(1.8) (2014), 100.

Chang

P.-K.

and Peng

Y.-N.

, Influence of mixing techniques on properties of high performance concrete, Cem Concr Res 31(1) (2001), 87–95.

10.

Zhang

and Afzal

, Prediction of the elastic modulus of recycled aggregate concrete applying hybrid artificial intelligence and machine learning algorithms, Struct Concr 2021.

11.

H.-B.

, Le

T.-T.

, Vu

H.-L.T.

, Tran

V.Q.

, Le

L.M.

and Pham

B.T.

, Computational hybrid machine learning based prediction of shear capacity for steel fiber reinforced concrete beams, Sustainability 12(7) (2020), 2709.

12.

Chan

Y.N.

, Luo

and Sun

, Compressive strength and pore structure of high-performance concrete after exposure to high temperature up to 800°C, Cem Concr Res 30(2) (2000), 247–251. doi: 10.1016/S0008-8846(99)00240-9

13.

Rashid

M.A.

, Mansur

M.A.

and Paramasivam

, Correlations between Mechanical Properties of High-Strength Concrete, J Mater Civ Eng 14(3) (2002), 230–238. doi: 10.1061/(ASCE)0899-1561(2002)14:3(230)

14.

Ramezanianpour

A.A.

, Pilvar

, Mahdikhani

and Moodi

, Practical evaluation of relationship between concrete resistivity, water penetration, rapid chloride penetration and compressive strength, Constr Build Mater 25(5) (2011), 2472–2479. doi: 10.1016/j.conbuildmat.2010.11.069

15.

Zhou

F.P.

, Lydon

F.D.

and Barr

B.I.G.

, Effect of coarse aggregate on elastic modulus and compressive strength of high performance concrete, Cem Concr Res 25(1) (1995), 177–186. doi: 10.1016/0008-8846(94)00125-I

16.

Duval

and Kadri

, Influence of Silica Fume on the Workability and the Compressive Strength of High-Performance Concretes, Cem Concr Res 28(4) (1998), 533–547. doi: 10.1016/S0008-8846(98)00010-6

17.

Agrawal

and Sharma

, Prediction of slump in concrete usingartificial neural networks, World Acad Sci Eng Technol 45 (2010), 25–32.

18.

Laskar

A.I.

, Correlating slump, slump flow, vebe and flow tests to rheological parameters of high-performance concrete, Mater Res 12 (2009), 75–81.

19.

Masoumi

, Najjar-Ghabel

, Safarzadeh

and Sadaghat

, Automatic calibration of the groundwater simulation model with high parameter dimensionality using sequential uncertainty fitting approach, Water Supply 20(8) (2020), 3487–3501. doi: 10.2166/ws.2020.241

20.

Amlashi

A.T.

, Abdollahi

S.M.

, Goodarzi

and Ghanizadeh

A.R.

, Soft computing based formulations for slump, compressive strength, and elastic modulus of bentonite plastic concrete, J Clean Prod 230 (2019), 1197–1216. doi: 10.1016/j.jclepro.2019.05.168

21.

Öztas

, Pala

, Özbay

, Kanca

, Çaglar

and Bhatti

M.A.

, Predicting the compressive strength and slump of high strength concrete using neural network, Constr Build Mater 20(9) (2006), 769–775. doi: 10.1016/j.conbuildmat.2005.01.054

22.

Adeli

, Four decades of computing in civil engineering, in CIGOS 2019, Innovation for sustainable infrastructure, Springer, 2020, pp. 3–11.

23.

Nguyen

T.N.

, Lee

, Nguyen-Xuan

and Lee

, A novel analysis-prediction approach for geometrically nonlinear problems using group method of data handling, Comput Methods Appl Mech Eng 354 (2019), 506–526.

24.

Lee

, Ha

, Zokhirova

, Moon

and Lee

, Background information of deep learning for structural engineering, Arch Comput Methods Eng 25(1) (2018), 121–129.

25.

H.-G.

and Wang

J.-Z.

, Prediction of compressive strength of concrete by neural networks, Cem Concr Res 30(8) (2000), 1245–1250. doi: 10.1016/S0008-8846(00)00345-8

26.

Rafiei

M.H.

, Khushefati

W.H.

, Demirboga

and Adeli

, Novel Approach for Concrete Mixture Design Using Neural Dynamics Model and Virtual Lab Concept, ACI Mater J 114(1) (2017).

27.

Rafiei

M.H.

, Khushefati

W.H.

, Demirboga

and Adeli

, Supervised Deep Restricted Boltzmann Machine for Estimation of Concrete, ACI Mater J 114(2) (2017).

28.

Moody

and Darken

C.J.

, Fast learning in networks of locally-tuned processing units, Neural Comput 1(2) (1989), 281–294.

29.

M.J.

, Wu

, Lu

and Toh

H.L.

, Face recognition with radial basis function (RBF) neural networks, IEEE Trans Neural Networks 13(3) (2002), 697–710.

30.

Chen

, Orthogonal Least Sequares Learning Algorithm for Radial Basis Function Networks, IEEE Trans Neural Networks 4 (1991), 246–257.

31.

Bors

A.G.

and Pitas

, Median radial basis function neural network, IEEE Trans Neural Networks 7(6) (1996), 1351–1364.

32.

Chen

, Fei

and Yuan

, High-performance Concrete Strength Prediction Model Based on the Radial Basis Function Neural Network of Human Cerebral Cortex, NeuroQuantology 16(5) (2018).

33.

Smola

A.J.

and Schölkopf

, A tutorial on support vector regression, Stat Comput 14(3) (2004), 199–222. doi: 10.1023/B:STCO.0000035301.49549.88

34.

Kavitha

, Varuna

and Ramya

, Acomparative analysis on linear regression and support vector regression, in 2016 online international conference on green engineering and technologies (IC-GET), 2016, pp. 1–5.

35.

Rad

M.J.G.

, Ohadi

, Jafari-Asl

, Vatani

, Ahmadabadi

S.A.

and Correia

J.A.F.O.

, GNDO-SVR: An efficient surrogate modeling approach for reliability-based design optimization of concrete dams, in Structures 35 (2022), 722–733.

36.

Hashim

F.A.

, Hussain

, Houssein

E.H.

, Mabrouk

M.S.

and Al-Atabany

, Archimedes optimization algorithm: a new metaheuristic algorithm for solving optimization problems, Appl Intell 51(3) (2021), 1531–1551.

37.

Khodadadi

and Mirjalili

, Truss optimization with natural frequency constraints using generalized normal distribution optimization, Appl Intell 52(9) (2022), 10384–10397. doi: 10.1007/s10489-021-03051-5

38.

, Li

and Yang

, Support Vector Machine to Predict the Pile Settlement using Novel Optimization Algorithm, Geotech Geol Eng (2023). doi: 10.1007/s10706-023-02487-5

39.

Lim

C.-H.

, Yoon

Y.-S.

and Kim

J.-H.

, Genetic algorithm in mix proportioning of high-performance concrete, Cem Concr Res 34(3) (2004), 409–420.

40.

Vapnik

, The nature of statistical learning theory. Springer science & business media, 1999.

41.

Cortes

and Vapnik

, Support-vector networks, Mach Learn 20(3) (1995), 273–297.

42.

Vapnik

, Golowich

and Smola

, Support vector method for function approximation, regression estimation and signal processing, Adv Neural Inf Process Syst 9 (1996).

43.

Chou

J.-S.

and Pham

A.-D.

, Enhanced artificial intelligence for ensemble approach to predicting high performance concrete compressive strength, Constr Build Mater 49 (2013), 554–563. doi: 10.1016/j.conbuildmat.2013.08.078

44.

Chou

J.-S.

, Chong

W.K.

and Bui

D.-K.

, Nature-inspired metaheuristic regression system: programming and implementation for civil engineering applications, J Comput Civ Eng 30(5) (2016), 4016007.