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Investigation of mechanical properties of high-performance concrete via optimized neural network approaches


In this paper, an artificial intelligence approach has been employed to analyze the slump and compressive strength (CS) of high-performance concrete (HPC), focusing on its mechanical properties. The importance of assessing these critical concrete characteristics has been widely acknowledged by experts in the field, leading to the development of innovative methods for estimating parameters that typically require laboratory testing. These intelligent techniques improve the accuracy of mechanical property predictions and reduce the resource-intensive and costly nature of experimental work. The radial basis function neural network (RBFNN) is the foundational model for predicting the mechanical attributes of various HPC mixtures. To fine-tune the RBFNN’s performance in replicating the mechanical properties of HPC samples, two optimization algorithms, namely the Golden Eagle Optimizer (GEO) and Dynamic Arithmetic Optimization Algorithm (DAOA), have been employed. In this manner, both RBGE and RBDA models were trained using a dataset comprising 181 HPC samples that included superplasticizers and fly ash. The results show that DAOA has significantly improved the base model’s predictive capability, achieving a higher correlation with a value R 2 of 0.936 when estimating slump. Furthermore, RBDA exhibited a more favorable root mean square error (RMSE) in predicting compressive strength compared to RBGE, with a notable 16% difference. Ultimately, both integrated models demonstrated their effectiveness in accurately modeling the mechanical properties of HPC.


Around the world, there are many more places where large-scale concrete construction is taking place. In general, related industries and businesses will be duplicated due to global trends in the construction industry towards reinforced concrete structures, the construction of tall buildings, and the development of construction techniques [1]. The safety and durability of cast concrete is a fundamental issue when using much concrete for construction. To address these issues, a lot of work has gone into developing high-performance concrete \(({\text{HPC}})\). \({\text{HPC}}\) is made to offer properties that are matched to workability, strength, longevity, and durability for particular material sets, uses, and exposure conditions [2,3,4].

\({\text{HPC}}\) can be used for structures in harsh environments, including prefabricated buildings, highways, bridges, sidewalks, and nuclear structures [5,6,7]. The main difference between conventional concrete and HPC is the use of particular chemical and mineral admixtures. The water content and porosity of the paste of hydrated cement will both decrease with the addition of some chemicals. It is not advisable to use high doses of chemical admixtures to reduce the water content to too low levels. The effectiveness of admixtures like superplasticizers, however, largely depends on the surrounding temperature as well as the fineness and chemistry of the cement. In place of cement, mineral admixtures can be used as pozzolanic and fine-filling materials. This strengthens and densifies the hydrated cement’s microstructure. Incorporating fly ash or slag into concrete allows for a slow setting and subsequent hardening if durability is a top concern [8, 9]. Additionally, mineral mixtures are typically produced industrially, so such applications at reasonable costs can result in significant economic benefits. In light of this, it is possible to produce concrete using a superplasticizer and cement replacement materials to produce cost-effective construction materials with increased strength, workability, and durability [7, 10, 11].

Researchers have paid particular attention to determining the concrete slump flow \(({\text{SL}})\) and compressive strength \(({\text{CS}})\) factors as the mechanical properties, reflecting the quality of the materials. These procedures are primarily carried out through empirical experiments, and particular tools are used to assess the mentioned concrete features accurately. However, physical laboratory procedures are considered time-consuming and expensive, and some tools might not be available. As a result, experts are working to estimate the correlation between the \({\text{SL}}\) and \({\text{CS}}\) of \({\text{HPC}}\) and the components of mixtures using algorithms and formulas [12,13,14]. Zhou et al. [15] examined the impact of aggregates on the \({\text{CS}}\) of high-performance concrete. In another study, Duval and Kadri [16] examined the impact of silica-fume on compressive strength of \({\text{HPC}}\) using an empirical formulations and models.

Different coefficients of regression have been produced by the experimental formulas used to evaluate the \({\text{SL}}\) or \({\text{CS}}\) of concretes in order to show the effects of different admixtures. As a result, the prediction processes of such formulas are uncertain, such as the relationship between the \({\text{CS}}\) of concrete and highly nonlinear ingredients. Civil science fields have received highly accurate results from predictions made using artificial intelligence \(({\text{AI}})\) and machine learning (ML) techniques, particularly \({\text{HPC}}\) with multiple components as opposed to traditional types. Over the past two decades, various \({\text{ML}}\) algorithms with different mechanisms, such as decision trees, have been developed [7, 17]; artificial neural network (ANN) [6, 18] ensemble algorithm (EA) [19], and support vector machine (SVM) [20,21,22] have demonstrated that models working with \({\text{ML}}\) approaches have better results compared to traditional ways in term of accuracy and time.

For predicting the ultimate strength of rectangular and square piles, Moodi et al. \((2022)\) used \({\text{ML}}\)-based techniques such as radial basis function neural network (RBNN), multi-layer perceptron \(({\text{MLP}}),\) and support vector regression \(({\text{SVR}})\). The correlation index of R 2 for the \({\text{MLP}}\), \({\text{RBF}}\), and \({\text{SVR}}\) procedures was calculated using experimental data from \(463\) samples, and it was \(0.970\), \(0.970\), and \(0.91,\) respectively [23].

\(SVR\) technique was used by Saha et al. \((2020)\) to identify the properties of freshly poured and hardened self-compacting concrete \(({\text{SCC}}).\) The exponential radial basis function \(({\text{ERBF}})\) and \({\text{RBF}}\), two different kernel functions, were used to create the \({\text{SVR}}\) model. \({\text{SVR}}-{\text{ERBF}}\) outperformed \({\text{SVR}}-{\text{RBF}}\) in the training and testing phases after collecting 115 experimental samples with fly ash, fine aggregate, water-powder ratio, coarse aggregate and superplasticizer, and binder content as input parameters. Results showed a correlation coefficient of \(0.965\), \(0.954\), \(0.979\), and \(0.9773\) for the predicted slump flow, L-box ratio, \({\text{V}}\)-funnel, and \({\text{CS}}\), respectively [24].

In order to achieve this, the current paper aims to model the \({\text{CS}}\) and SL of \({\text{HPC}}\) mixtures using \({\text{RBFNN}}\). For information on feeding inputs, \(181\) \({\text{HPC}}\) samples taken from relevant literature gave information on the components of mixtures and the desired levels of \({\text{CS}}\) and slump. Additionally, the main features of the \({\text{RBFNN}}\), namely the neurons, and spread, were tuned by two powerful optimization algorithms as the novelty of the current research. The algorithms that optimize these hyperparameters are Golden Eagle Optimizer (GEO) and Dynamic Arithmetic Optimization Algorithm (DAOA). Integrating RBF with GEO and DAOA enhances predictive accuracy by fitting complex patterns in concrete mix design parameters for compressive strength and slump. GEO efficiently explores the solution space, emulating golden eagles’ hunting strategy. DAOA’s adaptability accommodates varying concrete conditions, ensuring model effectiveness amidst changes. The approach ensures robust, generalizable predictions, reduces overfitting, and accelerates convergence, which is vital for real-time decision-making in construction. Optimal resource utilization and iterative refinement capabilities further optimize the model for maximum accuracy and efficiency.


Radial basis function neural network

The \({\text{RBFNN}}\) was first presented by Broomhead and Lowe [25] and recognized as a feedforward network trained via a supervised training algorithm. The input layer, hidden layer, and output layer are the three layers that make up the \({\text{RBF}}\), as depicted in Fig. 1. There are numerous \({\text{RBF}}\) of various types, including sigmoid, polynomial, inverse polyquadratic, and Gaussian functions. One of the useful functions given by the spread rate and center is the Gaussian type. The first section of a neural network, the input layer, contains nodes without any processes, and the number of input layer neurons equals the number of variables [26]. The hidden layer, which is the second section, resembles a calculator. In order to form the answers within the predefined curves and find the best solutions, it contains a radial function. In order to perform a nonlinear mapping of input values, the hidden layer obtains a data set from the input layer. The inputs’ distance from a specific center point can be calculated using the symmetrical-based function used in this platform. With the concentrations of produced data using neurons of the hidden layer as a straightforward regression process in the output part, \({\text{RBFNN}}\) on the input nodes can be applied to the output layer.

Fig. 1
figure 1

Structure of \({\text{RBFNN}}\)

The \({\text{RBF}}\) stages can be started with (a) assigning input vector (\(x\)) and the center (\({c}_{i}\)) and their radial distance (\({d}_{i}\)) for the nodes embedded in the hidden layer as well as the outcomes (\({h}_{i}\)) appraised by the network of \(G\) using relations presented through Eqs. (1) and (2):

$${d}_{i}=\Vert x-{c}_{i}\Vert$$
$${h}_{i}=G({d}_{i}\times {\sigma }_{i})$$

where \(\sigma\) is the node width of the hidden layer, and \(G\) denotes the \({\text{RBF}}\). Consequently, the results can be presented using Eq. (3):


in which, in the hidden layer, the number of layers equals one, and \({w}_{i}\) shows the weight among the neurons of output and hidden layers.

Dynamic arithmetic optimization

Two novel accelerator functions have been incorporated into the foundational arithmetic optimization algorithm version to enhance efficiency. The dynamic version, which controls the ratio of exploration to exploitation behavior, modifies the candidate solutions and search phase during the optimization process. What sets DAOA apart is its ability to operate without requiring any preliminary adjustments to its parameters compared to the current state-of-the-art metaheuristic. The DAOA pseudo-code is shown in Algorithm 1, while the subsequent section delves into a detailed discussion of its novel dynamic features.

figure a

Algorithm 1. Pseudo-Code of DAOA

Dynamic accelerated function for DAOA

The arithmetic optimization algorithm’s dynamic component relies heavily on the essential role played by the dynamic accelerated function (DAF) during the search process. When using the AOA, it is necessary to fine-tune the initial \({\text{min}}\) and \({\text{max}}\) values of the accelerated function. However, employing an algorithm devoid of internally adjustable parameters is preferable, given that DAF is substituted with a fresh descending function. This adjustment factor in the optimization algorithm is presented as follows:

$$DAF=({\frac{{Iter}_{max}}{Iter})}^{\alpha }$$

In this context, “Iter”represents the ongoing iteration count, “Itermax”, signifies the upper limit for iterations, and the value of “α” remains a constant. The function undergoes a reduction with each successive iteration within the algorithm.

Dynamic DAOA candidate solution

The following dynamic qualities created for potential solutions in the DAOA are shown in this section. There are two main stages of metaheuristic algorithms: exploration and exploitation. Achieving a balanced balance between these stages is essential to the algorithm’s performance. During the optimization process, each solution in the suggested dynamic adaptation which places a high emphasis on maximizing exploration and exploitation constantly adjusts its positions by making reference to the best-obtained solution. Equation (5) in the fundamental version is replaced with Eq. (6) in the dynamic candidate solution \(({\text{DCS}})\) function.

$${x}_{i,j}\left({C}_{iter}+1\right)=\left\{\begin{array}{c}best{(x}_{j})\div (DCS+\epsilon )\times (({UB}_{j}-{LB}_{j})\times \mu +{LB}_{j})), r2<0.5\\ best ({x}_{j})\times DCS\times (({UB}_{j}-{LB}_{j})\times \mu +{LB}_{j})) Otherwise\end{array}\right.$$
$${x}_{i,j}\left({C}_{iter}+1\right)=\left\{\begin{array}{c}{best (x}_{j})-DCS\times (({UB}_{j}-{LB}_{j})\times \mu +{LB}_{j})), r3<0.5\\ {best (x}_{j})+DCS\times (({UB}_{j})\times \mu +{LB}_{j})) , Otherwise\end{array}\right.$$

Introducing the \({\text{DCS}}\) function directly responds to the decreasing proportion of candidate solutions. Its value continually reduces during each iteration, adhering to this established pattern.

$$DCS\left(t+1\right)=DCS(t)\times 0.99$$

Golden eagle optimization

Golden eagles have a special relationship with humans, holding sacred positions in beliefs and being seen as signs of fortunate events. They hunt in Kazakhstan and Kyrgyzstan, using a unique spiral-shaped cruising and hunting motion. They balance their propensity to cruise and attack, making extensive circles around their territory. They alert other eagles to their best catch and continue to hunt throughout the flight, using both cruising and attacking strategies.

The golden eagle’s balance between exploration and exploitation is reflected in its flight pattern. A metaheuristic algorithm, GEO, is developed based on this spiraling pattern, segmenting ROD images for precise examination and disease diagnosis. Consider a hypothetical \({\text{RGB}}\) image with dimensions \(M*N\). The image element \((pixel)\) at \((x, y)\) is therefore equal to:


Assuming \(T\) is the experimental image’s grey level and that the overall grey values are \(0, 1, 2, 3,..., T-1\), indicated by \(R\), as follows:

$$F\left(x,y\right)\in {R}^{\forall }(x,y)\in picture$$

The following is the definition of the image’s standardized histogram (bar chart):

$$J=\{j0.j1,\dots jR1\}$$

The equation above can be expressed as follows using the geometrically active multi-contours method:

$$J\left(Th\right)=j0\left({th}_{1}\right)+j1\left({th}_{2}\right),\dots ,jR-1\left({th}_{k-1}\right)$$

\(Th *\) stands for the threshold of choice. The GEO technique uses the DRLS method to extract data from preprocessed images, requiring fewer initial parameters. The data is normalized, used as training data for a vector machine model, and compared to expert observational images. The first step involves calculating image similarity metrics like GEOccard, Dice, FPR, and FNR by the articles. The mathematical formula is shown below:

$$Jaccard \left({I}_{g},{I}_{m}\right)={I}_{g}\cap {I}_{m}/{I}_{g}\cup {I}_{m}$$
$$Dice \left({I}_{g},{I}_{m}\right)=2\left({I}_{g}\cap {I}_{m}\right)/\left|{I}_{g}\right|\cup \left|{I}_{m}\right|$$
$$FPR \left({I}_{g},{I}_{m}\right)= \left({I}_{g}/{I}_{m}\right)/ \left({I}_{g}\cup {I}_{m}\right)$$
$$FNR \left({I}_{g},{I}_{m}\right)= \left({I}_{m}/{I}_{g}\right)/ \left({I}_{g}\cup {I}_{m}\right)$$

Additionally, the following formulas are used to calculate the image’s statistical values, including sensitivity, specificity, and accuracy:


\({T}_{N}\), \({T}_{P}\), \({F}_{N}\), and \({F}_{P}\) stand for true negative, true positive, false negative, and false positive, respectively. \({I}_{g}\) is equal to GT. \({I}_{m}\) is the extracted region.

Data gathering

The current study uses an experimental data set including \(181 {\text{HPC}}\) mixes [27] with constituents: ratio of water to binder, fine aggregates to coarse aggregates ratio, fly ash, air entraining agent, and additive of superplasticizer which Fig. 2 has indicated symbol-line plot for the input and output. It is important to remember that the \({\text{SL}}\) and \({\text{CS}}\) measured magnitudes were performed on concrete that was \(28\) days old. Table 1 provides a general summary of the inputs to the models, including constituents (state variables) and geotechnical characteristics of \({\text{CS}}\) and slump flow.

Fig. 2
figure 2

Line symbol plot for the input and output

Table 1 Summary statistical report of model inputs

Assessing the developed hybrid models

Several metrics have been used to investigate the \({\text{RBDA}}\) and \({\text{RBEO}}\) performance to estimate the slump and \({\text{CS}}\) rates of \({\text{HPC}}\) mixes; they are introduced in Eqs. (20) through (24), where \({p}_{n}\) represents predicted values and \({t}_{n}\) represents measured values, and \(N\) represents the number of samples. Also, \({n}_{train}\) and \({n}_{test}\) represent the number of concrete compounds for the training and testing steps, respectively.

$${R}^{2}={\left(\frac{{\sum }_{n=1}^{N}\left({t}_{n}-\overline{t }\right)\left({p}_{n}-\overline{p }\right)}{\sqrt{\left[{\sum }_{n=1}^{N}{\left({t}_{n}-\overline{p }\right)}^{2}\right]\left[{\sum }_{n=1}^{N}{\left({p}_{n}-\overline{p }\right)}^{2}\right]}}\right)}^{2}$$

Here, \({t}_{n}\) shows the measured numbers of CS and SL, and the means are indicated via \(\overline{t }\); the estimated values have been indicated with \({p}_{n}\) with mean of \(\overline{p }\). The number of HPC mixtures for the training and testing phases is shown by n train and n test, alternatively.

Results and discussions

The primary objectives of the current study revolved around modeling the mechanical properties of \({\text{HPC}}\) samples. By integrating the optimization techniques employed in this research with the \({\text{RBFNN}}\) model, two distinct models known as \({\text{RBDA}}\) and \({\text{RBGE}}\) were developed, showcasing their ability to predict the \({\text{CS}}\) and SL of \({\text{HPC}}\) mixtures with remarkable accuracy. Both models underwent comprehensive evaluation in terms of their performance in predicting \({\text{CS}}\) and \({\text{SL}}\) from multiple perspectives. Table 2 presents the results of developed models using the RBF to predict CS and HPC slump. The table comprises evaluation metrics such as R 2, RMSE, MAE, VAF, and OBJ. The HPC features differentiate the models, the specific RBF-based model (RBGE or RBDA), and the evaluation phase (train, test, or all, denoting a combined evaluation).

Table 2 The result of developed models for RBF

RBGE and RBDA models are evaluated in the training and testing phases to predict CS. In the training phase, RBDA demonstrates a higher R 2 (0.928) than RBGE (0.911), indicating better accuracy. In the testing phase, RBDA maintains a slightly higher R 2 (0.922) and lower RMSE (7.411) compared to RBGE (R 2 = 0.898, RMSE = 8.506). Overall, considering all data, RBDA consistently shows a marginally superior R 2 (0.923) and lower RMSE (7.380) compared to RBGE (R 2 = 0.896, RMSE = 8.551). Additionally, the OBJ values for RBDA in the combined evaluation are 8.64, suggesting optimization efficiency.

Similarly, RBGE and RBDA models undergo evaluation in the training and testing phases for predicting slump flow. RBDA exhibits a slightly higher R 2 and lower RMSE than RBGE across the training, testing, and combined datasets. Notably, in the testing phase, RBDA achieves a higher R 2 (0.936) and lower RMSE (7.382) compared to RBGE (R 2 = 0.915, RMSE = 8.504). The overall evaluation reiterates RBDA’s marginally superior performance, with a higher R 2 (0.933) and lower RMSE (7.360) compared to RBGE (R 2 = 0.909, RMSE = 8.549). The OBJ values for RBDA in the combined evaluation are 8.59, indicating efficient optimization. These results demonstrate that RBDA exhibits slightly better accuracy and lower error metrics than RBGE in predicting CS and slump.

Figure 3 illustrates the relationship between observed and projected CS and SL values in HPC using data points. It compares two models: RBDA, which combines an RBF model with DAOA optimization, and RBGE, which pairs an RBF model with GEO optimization, for predicting CS and SL values. The graph includes an R 2 evaluation, positioning training, validation, and testing data points around a central reference line (Y = X) for linear regression. Two boundary lines (Y = 0.9X and Y = 1.1X) indicate potential deviations from the central line, highlighting possible accuracy issues. The analysis shows that the RBDA hybrid model consistently outperforms the RBGE model in terms of R 2 and RMSE, especially in the training phase for CS prediction and the testing phase for SL prediction.

Fig. 3
figure 3

Correlation between the measured and predicted values

The additional examination of Fig. 4, which contrasts the R 2, RMSE, and MAE metrics of different models, strengthens the observation that the RBDA hybrid model’s predictions closely align with the actual test outcomes. Within the context of this figure, a notable trend can be observed: the lines connecting the R 2 values of the RBDA across three distinct phases are positioned near the edges of the triangle. In sharp contrast, the lines representing the error values of the RBDA for these phases are clustered in the central area of the triangle. This particular distribution pattern serves as a compelling signal of the model’s impressive precision.

Fig. 4
figure 4

Comparison for the developed models is based on metrics

In Figs. 5 and 6, the error percentages for CS and SL prediction in HPC are depicted in radial and line plots for both the RBGE and RBDA models. In the CS plots, maximum errors are recorded as approximately (− 0.25, 0.35) during the testing phase and (− 0.2, 0.3) in the training phase for the RBGE model. As for the SL plots, maximum errors of approximately (− 0.2, 0.35) during the testing phase and (− 0.25, 0.22) in the training phase are observed for the RBGE model.

Fig. 5
figure 5

The error percentage for the hybrid models is based on the radial plot

Fig. 6
figure 6

The line plot of errors among the developed models


This paper employs an artificial intelligence approach to model the \({\text{SL}}\) and \({\text{CS}}\) of \({\text{HPC}}\), focusing on their mechanical properties. Recognizing the significance of accurately estimating these crucial concrete characteristics, experts have emphasized the need for novel, more efficient procedures that reduce reliance on laboratory experiments. These intelligent methods can enhance the precision of mechanical property predictions and reduce the associated physical energy and experimental costs. In this pursuit, the \({\text{RBFNN}}\) is the foundational model for predicting the mechanical properties of \({\text{HPC}}\) mixtures. Furthermore, two optimization algorithms, namely the \({\text{GEO}}\) and the \({\text{DAOA}}\), are employed to fine-tune the RBFNN’s operations in replicating the mechanical properties, specifically \({\text{CS}}\) and \({\text{SL}}\), of \({\text{HPC}}\) samples. The results obtained from both models in predicting \({\text{CS}}\) and \({\text{SL}}\) are similar, but the performance of the \({\text{DAOA}}\) optimizer is demonstrated to be superior when coupled with the \({\text{RBFNN}}\). For instance, in the estimation of \({\text{CS}}\), the \({\text{RBDA}}\) framework achieved an R 2 coefficient of \(0.928\) during the training phase, which is \(2.74\)% higher than that of \({\text{RBGE}}\). In the testing phase, the correlation coefficients were calculated at \(0.922\) for \({\text{RBDA}}\) and \(0.898\) for the other model, affirming the effectiveness of the training stage in reducing error rates. The error margins for slump predictions range from \(-15\) to \(+20\mathrm{\%},\) while for \({\text{CS}}\), they span \(\pm 40\mathrm{\%}\). Although \({\text{RBDA}}\) exhibited weaker performance in the testing phase when estimating \({\text{CS}}\), error fluctuations became more pronounced during testing. However, \({\text{RBDA}}\)’s estimated \({\text{SL}}\) values were superior to those of \({\text{RBGE}}\), with the highest errors observed in the testing phase when appraising \({\text{SL}}\) and the training phase when estimating \({\text{CS}}\).

In conclusion, the results confirm the capabilities of both frameworks to simulate \({\text{CS}}\) and \({\text{SL}}\), representing mechanical properties at acceptable levels. In most cases, \({\text{DAOA}}\) proves to be a highly accurate optimizer for fine-tuning \({\text{RBFNN}}\) compared to \({\text{GEO}}\). Utilizing such intelligent methods instead of costly experimental approaches can significantly improve the cost-effectiveness of research endeavors, especially in future studies where these models can be employed for sensitivity analyses of concrete mixture constituents.

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I would like to take this opportunity to acknowledge that there are no individuals or organizations that require acknowledgment for their contributions to this work.


This work was supported by the Jilin City Social Science Association, research on the promotion strategy of ecological livable city in Jilin City based on collaborative governance theory (Project No.: 2224).

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All authors contributed to the study conception and design. Data collection, simulation, and analysis were performed by “ Xuyang Wang and Rijie Cong”. The first draft of the manuscript was written by “ Xuyang Wang”and all authors commented on previous versions of the manuscript. All authors have read and approved the manuscript.

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Wang, X., Cong, R. Investigation of mechanical properties of high-performance concrete via optimized neural network approaches. J. Eng. Appl. Sci. 71, 73 (2024).

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