Integrated and optimized SVR analysis: Assessment of the preliminary and entire fracture energy of concrete

Abstract

Assessment of energy needed for crack growth in concrete structures has been an interesting topic since the use of fracture mechanics to concrete. However, experimental procedures need time, cost and efforts. Based on historical data, regression approaches were created using mechanical characteristics and mixed design factors to quantify the concrete preliminary (G_f) and whole (G_F) fracture energy. This work combined support vector regression (SVR) analysis with antlion optimization (ALO) and Harris Hawks optimization (HHO) approaches to build a hybridized SVR evaluation to fully comprehend G_f and G_F. Evaluation metrics demonstrate that both optimized ALO-SVR and HHO-SVR assessments could perform wonderfully throughout the estimation mechanism. Whenever the superior SVR investigation was contrasted to the literature, it was observed that the uniquely developed ALO-SVR regression also provides a reasonable boost in effectiveness, with benefits across the board. Finally, although the HHO-SVR technique has its particular capabilities in the simulating procedure, the ALO-SVR analysis seems to be highly reliable for determining G_f and G_F.

Keywords

Preliminary and entire fracture energy concrete SVR analysis metaheuristic optimization algorithms

1 Notations

G_f

Concrete preliminary fracture energy

G_F

greaterthan Concrete whole fracture energy

LEFM

greaterthan The linear elastic fracture mechanics

FPZ

greaterthan Fracture process zone

CBM

greaterthan Crack band model

SEM

greaterthan Size effect model

TPM

greaterthan The two-parameter model

ECM

greaterthan The effective crack model

WFM

greaterthan Work-of-fracture technique

ANN

greaterthan Artificial neural network

f _c

greaterthan Compressive strength

SVR

greaterthan support vector regression

ALO

greaterthan antlion optimization

HHO

greaterthan Harris Hawks optimization

d _max

greaterthan The largest aggregate size

greaterthan Curing time of samples

W/C

greaterthan ratio of water to cement

greaterthan Correlation coefficient

R ²

greaterthan Coefficient of determination

RMSE

greaterthan Root-mean-square error

RRSE

greaterthan Root Relative Squared Error

VAF

greaterthan Variance account factor

S/TA

greaterthan Sand to total aggregate ratio

CA/b

greaterthan Coarse aggregate to binder ratio

l _f

greaterthan Length of specimen ligament size (mm)

1 Introduction

It is currently clear that applying fracture mechanics ideas to the design of concrete samples in order to address all sorts of brittle failures results in long-lasting, cost-effective, and securer concrete structures [1]. Several empirical and experimental research [2–4] support this hypothesis. Moreover, past construction catastrophes highlight the need to consider fracture mechanics when designing concrete structures [5]. In spite of the reality that Griffith pioneered fracture mechanics in 1921 [6], it was regarded as nonsensical to concrete due to the absence of a specific fracture mechanics idea for concrete as a diverse medium compared to uniform materials like ceramics or steel [5]. Kaplan [7] used the linear elastic fracture mechanics (LEFM) idea to conduct the first laboratory test on concrete fracture mechanics. Kesler et al. [8] showed that the type of acute fractures using the fracture energy (G_f) was inadequate for concrete structures because of the existence in front of the crack tip of an inelastic area with exceptional size and entire micro-cracks, known as the fracture process zone (FPZ). Investigators determined that, at minimum, two fracture factors are necessary to characterize concrete fracture [1]. Various sorts of the study were conducted to produce the finest non-linear concrete fracture model based on FPZ in order to reduce LEFM’s concrete flaws [2]. There are two sorts of these systems [3]. At first, there’s the cohesive or fictitious crack form, as well as the crack band model (CBM), which may both be employed for numerical analysis. Second, the size effect model (SEM), the two-parameter model (TPM), the effective crack model (ECM), and the double-K model are examples of mathematical ideas. One of the most important aspects in characterizing a coherent crack model (G_F) is specific fracture energy. Utilizing the work-of-fracture technique (WFM)., specific fracture energy was described as the entire effort required to generate one unit region of a crack. Since the beam is broken into two parts, the fracture energy may be calculated by calculating the whole region under the force-movement curve and dividing the entire lost energy by the main ligament region (Fig. 1). The G_F obtained by the WFM method is reliant on the sample size and form; nevertheless, this disadvantage may be addressed by carefully determining the load-displacement curve’s tail and entire energy waste resources in the experiment [4]. First fracture energy (G_f), unlike G F, is independent of specimen size or form and is the region under the softening curve’s primary tangent [5]. The G_F and G_f fracture energies are materials’ two separate properties in reality. Based on the studies, the association between these two variables is weak since G_F number have a larger dispersion than G_f due to higher uncertainty in the tail of the softening curve compared to its beginning [6]. Based on Planas et al. [14], a rough estimate of the ratio among G_F and G_f might range between 2 and 2.5. These two meanings might be utilized to achieve different goals. G_F is appropriate and essential for quantifying energy consumption in total structure collapse and forecasting a structure’s whole post-peak softening force-movement curve, although G_f is adequate for estimating the structures’ greatest force and softening curves to their greatest level [7]. So far, a number of researchers have investigated the parameters which impact G_f in different types of concrete. G_f increases from 21.1 to 35.4 N/m when the largest aggregate size is increased from 4.75 to 19 mm, according to Jenq and Shah [2]. For concretes with W/C of 0.4 and 0.29, Shah et al. [16] discovered that G_f ranging from 20.6 to 37.5 N/m and from 36.7 to 62.3 N/m. In high-performance concrete, raising the W/C from 0.36 to 0.5 leads G_f to a decline from 46.3 to 40.4 N/m, based on Bharatkumar et al. [8]. Moreover, Beygi et al. [9] discovered that raising the W/C from 0.35 to 0.7 decreases the amount of G_f in SCC comprising limestone powder from 52.3 to 29.5 N/m. The variation of G_f has also been indicated to range from 66.6 to 99.9 N/m [10].

Fig. 1

G_f and G_F in the force-displacement curve [61].

By collecting the load-displacement graph, the concrete samples’ fracture energy under bending was previously calculated correctly. Nevertheless, from the aspect of experimentation, this is neither simple nor user-friendly [11]. Due to the requirement for a solid testing platform and apparatus, this testing approach is substantially more complex and time-consuming than compressive measurements used to estimate compressive strength (f_c). G_F and G_f, developers and engineers have intended to fill certain gaps in initial fracture energy predicting by using either a quantitative or analytical way to eliminate the time and cost required, as well as the necessary particular fracture energy measures. Different mathematical approaches for determining the fracture properties of concrete built utilizing several mixture ratios have been developed in later years, omitting real notched fracture samples [6]. These mathematical equations are often in the form of regressions and are according to actual data. On the other hand, traditional techniques could be more capable of describing the complicated non-linear structure of interacting factor correlations like concrete mix attributes and fracture characteristics. Moreover, utilizing a statistical technique to identify an appropriate regression equation in a complex non-linear system is hard due to the demand to incorporate methodologies and information. Consequently, developing novel, gratifying, easy, and accurate processes in contrast to current data analysis methodologies is essential [12]. Methods and soft calculating approaches have lately been used for a variety of objectives in order to further our knowledge of structure and material mechanics. Based on available empirical data, scientists attempted to construct strong, accurate, and inexpensive material attribute prediction models [13–16]. Various machine intelligence algorithms have been used to anticipate the detailed performance of structural materials [17–20]. A kriging surrogate, a response surface approach, and a dimension reduction projection are used to accomplish this [21–23]. Data from prior experiments and a combination of U-learning, Kriging, and K-means clustering were used to forecast concrete fracture energy. Combining Kriging with WFM’s U-learning function increases fracture energy forecasting over basic Kriging, K-means clustering, and previous links. When compared to various approaches, the SEM, K-means methods and half of the data collection produced better predicting results. The suggested Kriging pairing hypotheses have R² values ranging from 0.59 to 0.95, whereas previous understanding associations have R² values ranging from 0.14 to 0.69. The outcomes show that combining the Kriging approach, the U-learning function, and K-means clustering with a minimal number of past examination outcomes reduces examination time and cost while enhancing concrete fracture forecast quality [24]. Because concrete is a quasi-brittle material with a complicated non-linear fracture manner, it is essential to research and develop a fracture manner idea for it in order to create the finest and safest design feasible. It’s essential to give precise modelling of ineffective manners according to present experimental results and prediction approaches and equations [25] since it saves time and money (Table 1).

Table 1

The most related studies in literature

Reference	Input	Output	Model
[25]	f_c, d_max, CT, and W/C	G_f, G_F	ANN
[24]	f_c, d_max, CT, and W/C	G_f, G_F	A combination of Kriging surrogate method with U-learning function and K-means clustering
[62]	f_c, d_max, W/C and aggregate type	•critical stress intensity factor •critical crack tip opening displacement	ANN
[63]	$f_{c}, d_{\max}, \frac{W}{C}, \frac{S}{TA}, \frac{CA}{b}, and l_{f}$	G _F	ANN

1.1 Principal aim

Time, lab layouts, money, age-appropriate, and planning examinations are all required for high-quality research [26–29]. Concrete requires answers that are both time-saving and cost-effective. Human mistakes caused by a lack of skills and expertise, the scientist’s ineffectiveness throughout the examination, and equipment failures would squander money and time while yielding false findings. The assessment might be easier if you have the proper tools. Using simulation techniques to evaluate concrete fracture features according to prior data is helpful. Regression-based techniques offered in codes and publications as experimental equations cannot properly express the complex and non-linear structure of concrete’s features, notably fracture energy. It is impossible to ensure that users will accept regression-based connections in identifying fracture characteristics. Statistical assessment and several researchers have studied estimating methods for finding concrete fractures according to examination data.

To reach the aim function, a high-accuracy approximation strategy with a small amount of beginning input should be applied. To provide a stronger insight of G_f and G_F, this activity created a hybridized support vector regression (SVR) methodology. By combining the antlion optimization algorithm (ALO) and harris hawks optimization (HHO) techniques with SVR, platforms were applied to calculate the G_f and G_F in this case. A dataset of four inputs (f_c, d_max, CT, and W/C) and two outputs (G_f and G_F) was produced for this purpose by collecting actual records from previously published papers. Finally, by measuring and evaluating various statistical criteria, the effectiveness of SVR models has been compared across the board and with the publications.

1.2 Developed formulas for calculating the G_f and G_F

Applying the common statistical ways, a number of formulas for assessing G_f and G_F have been developed till the present. The followings are some of that:

From [6]: $G_{f} = α_{0} {(\frac{f_{c}}{0.051})}^{0.46} {(1 + \frac{d_{a}}{11.27})}^{0.22} {(\frac{w}{c})}^{- 0.3}$ (1) $G_{F} = 2.5 G_{f}$ (2) From [30]: $G_{F} = (0.0469 d_{\max}^{2} - 0.5 d_{\max} + 26) {(\frac{f_{c}}{10})}^{0.7}$ (3) From [31]: $G_{F} = 73 {(f_{cm})}^{0.18}$ (4) From [32]: $G_{F} = 10 {(d_{\max})}^{0.33} {(f_{c})}^{0.33}$ (5)

Terms in the Equations (1)–(5) are as follows, where G_f and G_F show the preliminary and whole fracture energy, respectively. f_c and f_cm are compressive strength (MPa) and the mean value of f_c (equal to: f_c + 8 (MPa)), respectively. Also, d_max, w/c, and α₀ present the highest aggregate size (mm), water to binder ratio, and for rounded aggregate (α₀ = 1) and for crushed aggregate (α₀ = 1.44), respectively.

2 Selecting input variables and gathering data

Finding appropriate input parameters is an essential milestone in using regression analysis to foresee the considered output (s). Voids, particle size, air gaps, cement type, aggregate ratio, and relative harshness of materials may all be utilised to estimate aimed output, in this case, the initial and total fracture energy of concrete [33–35]. Caused by a lack of comprehension of the methods in the documents, the above difficulties are disregarded. Additionally, the modulus of rupture and tensile strength appear to contribute to the evaluation of considered energy. The problem, nevertheless, is that these characteristics are complicated to calculate and are highly influenced by the scale of the object [6]. As a result, such factors are not included in the study because no data exists on their weight. W/C should always be given separately since examples of concrete with the same W/C but differing f_c have various fracture energies. Four characteristics are taken into consideration: compressive strength (f_c), the largest aggregate size (d_max), curing time of samples (CT), and the ratio of water to cement (W/C) associated with physical principles and the indicated pattern of fracture energy, along with the fundamental shape of recent systems.

This paper examined two separate data sets to compute the considered fracture energy of concrete, named G_f and G_F [7, 8, 10, 22, 36–54]. The following fractions were utilised for splitting in this application since they were often suggested percentages for the testing/training stages: 25% and 75%, respectively. The range of features needed to determine G_f (A series) and G_F (B series) is illustrated in Fig. 2. The description of characteristics for G_f and G_F simulation strategies are vividly depicted in Table 2.

Fig. 2

Distribution plots of attributes: A series for G_f, and B series for G_F.

Table 2

The properties of attributes related to G_f and G_F

Portion	Statistic	Independents								Dependent
		f _c		d _max		CT		W/C		G _f	G _F
		MPa		mm		Days		–		kN/m
		G _f	G _F	G _f	G _F	G _f	G _F	G _f	G _F
Train portion
	Max .	87.2	124	32	65	90	40	0.8	0.65	0.1	0.178
	Min .	12.6	3.01	2	2	1	0.56	0.2	0.28	0.0	0.035
	Median	39.9	38.5	12.7	10	28	28	0.5	0.5	0.0	0.0841
	Range	74.6	120.99	30.0	63	89.0	39.44	0.6	0.37	0.1	0.143
	Var .	213.66	296.26	38.68	61.092	178.136	115.579	0.012	0.0057	0.000154	0.00077
	Skew .	1.0547	1.642	0.252	3.557	3.0099	-1.089	-0.324	-0.789	1.986	0.828
Test portion
	Max .	110	90	20	20	90	36	0.7	0.7	0.1	0.1
	Min .	23.8	3	2	2	1	0.3	0.2	0.3	0.0	0.0
	Median	39.3	38.5	9	10.0	28	28.0	0.5	0.5	0.0	0.1
	Range	86.2	87	18.0	18	89.0	35.8	0.4	0.4	0.1	0.1
	Var .	343.77	328.989	42.67	21.49	325.741	137.485	0.00945	0.006	0.00019	0.00078
	Skew .	2.253	0.2173	0.3121	0.40435	2.1478	-0.9545	-0.776	-0.10812	1.807	0.0354

An association fraction defined as the Pearson correlation coefficient (PCC) was computed and clearly demonstrated to indicate the relationship between two attributes (Table 3). The quantities of PCC for G_f are shown in the A section of this table, and for G_F are depicted on the B part. Experts may be baffled as to how the described parameters affect the results if the PCC is significantly favourable or poor. PCC less than the acceptable ones imply that certain characteristics are most possibly caused by multicollinearity concerns, with 0.374 and 0.475 proposed as G_f and G_F bounds, respectively. With a magnitude of -0.719, the largest PCC for G_f is associated to the relation between W/C and f_c. In terms of G_F, the strongest negative PCC was found between W/C and f_c (-0.662), and the strongest positive PCC between G_F and f_c (0.539).

Table 3

The PCC values of variables for G_f and G_F

(A) G _f
		Inputs				Output
		f _c	d _max	CT	W/C	G _f
Inputs	f _c	1	-0.26495	-0.21828	-0.71902	0.36504
	d _max		1	0.14268	0.27708	0.37431
	CT			1	0.31448	0.07142
	W/C				1	-0.30232
Output	G _f					1
(B) G _F
		Inputs				Output
		f _c	d _max	CT	W/C	G _F
Inputs	f _c	1
	d _max	0.04409	1
	CT	0.47531	0.20671	1
	W/C	-0.66208	-0.03525	-0.10536	1
Output	G _F	0.53923	0.41351	0.24546	-0.31771	1

3 Used models and metaheuristic algorithms

3.1 Ant lion optimization algorithm (ALO)

Mirjalili [55] developed an ant lion optimization method according to the ant lions’ preying process. This method includes an accidental move exploration followed by agents’ accidental choice. ALO inspired us to utilize DG placement that had never been done before, according to the knowledge of the authors. Agents’ random moves, building snares, ants’ entrapment in snares, capturing prey, and rebuilding snares are the five major stages of preying. Local optima may be eliminated using the ALO optimizer’s random ant walks and roulette wheel. This part discusses the mathematical modelling of the ALO method.

3.1.1 Random walk of ants

Ants’ random move are presented by:

$\begin{matrix} X (t) = [0, cumsum (2 r (t_{1}) - 1), \\ cumsum (2 r (t_{2}) - 1), \dots, cumsum (2 r (t_{n}) - 1)] \end{matrix}$ (6)

Based on this equation, t stands for the random walk stage, n shows the iterations’ highest number, and cunsume is computing the cumulative sum. $r (t) = {\begin{matrix} 1 if rand > 0.5 \\ 0 ir rand < 0.5 \end{matrix}$ (7)

In this equation, rand stands for accidental value generator among [1,0]. Within the search area, a random walk may be performed using the formula below: $X_{i}^{t} = \frac{(x_{i}^{t} - a_{i}) \times (d_{i}^{t} - c_{i}^{t})}{(b_{i} - a_{i})} + c_{i}^{t}$ (8)

a_i is the random walk’s lowest number, and b_i is the random walk’s highest number of i_th variable. $d_{i}^{t}$ shows the highest of i_th variable, and $c_{i}^{t}$ shows the minimum of i_th variable in the t_th iteration.

The ant’s current position, as well as the accompanying fitness function matrix, are shown as follows: $M_{Ant} = (\begin{matrix} \begin{matrix} \begin{matrix} A_{1.1} \\ A_{2, 1} \end{matrix} & \begin{matrix} A_{1, 2} \\ A_{2, 2} \end{matrix} & \begin{matrix} \dots \\ \dots \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} \dots \\ \dots \end{matrix} & \begin{matrix} A_{1, d} \\ A_{2, d} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} ⋮ \\ A_{n, 1} \end{matrix} & \begin{matrix} ⋮ \\ A_{n, 2} \end{matrix} & \begin{matrix} ⋮ \\ \dots \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} ⋮ \\ \dots \end{matrix} & \begin{matrix} ⋮ \\ A_{n, d} \end{matrix} \end{matrix} \end{matrix})$ (9) $M_{OA} = [\begin{matrix} f ([A_{11}, A_{22}, \dots, A_{1 d} \\ f ([A_{21}, A_{22}, \dots, A_{2 d} \\ \begin{matrix} ⋮ \\ f ([A_{n 1}, A_{n 2}, \dots, A_{nd} \end{matrix} \end{matrix}]$ (10)

According to these equations, the suitability values of the position of the ants matrix M_Ant are represented by the M_OA matrix. if ants or ant lions are hiding in the search region, the associated position and profitability matrices are as follows: $M_{Antlion} = (\begin{matrix} \begin{matrix} \begin{matrix} A_{1, 1} \\ A_{2, 1} \end{matrix} & \begin{matrix} A_{1, 2} \\ A_{2, 2} \end{matrix} & \begin{matrix} \dots \\ \dots \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} \dots \\ \dots \end{matrix} & \begin{matrix} A_{1, d} \\ A_{2, d} \end{matrix} \end{matrix} \\ \begin{matrix} \begin{matrix} ⋮ \\ A_{n, 1} \end{matrix} & \begin{matrix} ⋮ \\ A_{n, 2} \end{matrix} & \begin{matrix} ⋮ \\ \dots \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} ⋮ \\ \dots \end{matrix} & \begin{matrix} ⋮ \\ A_{n, d} \end{matrix} \end{matrix} \end{matrix})$ (11) $M_{OAL} = [\begin{matrix} f ([A_{11}, A_{22}, \dots, A_{1 d} \\ \begin{matrix} f ([A_{21}, A_{22}, \dots, A_{2 d} \\ ⋮ \end{matrix} \\ f ([A_{n 1}, A_{n 2}, \dots, A_{nd} \end{matrix}]$ (12)

Where M_OAL matrix indicates the position of ant lions’ suitability values.

3.1.2 Building trap

A roulette wheel is utilized to increase the possibility of capturing ants. This process determines which ant lions are the fittest.

3.1.3 Trapping in ant lion pits

The trapping mathematical equations are: $\begin{matrix} c_{i}^{t} = {Antlion}_{j}^{t} + c^{t} \\ d_{i}^{t} = {Antlion}_{j}^{t} + d^{t} \end{matrix}$ (13)

$d_{i}^{t}$ and $c_{i}^{t}$ : Maximum and minimum of whole variables at t_th iteration

d^t and c^t: Maximum and minimum of whole variables for i_th ant

${Antlion}_{j}^{t}$ : Ant location of chosen ant lion

3.1.4 Sliding ants towards ant lions

Ant lions throw sand outwards to entice ants to approach them. The preceding action’s mathematical model is as follows: $\begin{matrix} C^{t} = \frac{c^{t}}{I} \\ D^{t} = \frac{d^{t}}{I} \end{matrix}$ (14)

T: Stands for the iterations’ highest number

t: Present iteration

I: $10^{w} \frac{t}{T}$

3.1.5 Catching prey and rebuilding the pit

The last stage of the ant lion-preying manner is trapping an ant that has reached the down of the pit and then updating its location to the most recent location using the equation below. ${Antlion}_{j}^{t} = {Ant}_{i}^{t} if : f ({Ant}_{i}^{t}) > f ({Antlion}_{j}^{t})$ (15)

3.1.6 Elitism

It is significant in the evolution method in order to keep the most proper answer. This could be simulated as: ${Ant}_{i}^{t} = \frac{R_{A}^{t} + R_{E}^{t}}{2}$ (16)

In this equation, $R_{A}^{t}$ , and $R_{E}^{t}$ show random walks close by the ant-lion, elite at i^th repetition. The flowchart of this algorithm is provided in Fig. 3.

Fig. 3

The flowchart of the ALO [55].

3.2 Harris hawks optimization (HHO)

The main ideas of the main HHO method will be discussed in this part [56]. The exploration and the exploitation phase of a population-based meta-heuristic method are often separated. The goal of the exploration step is to broaden the optimizer’s search scope so that the population created by search agents is as varied as feasible. The exploitation step is supposed to supply the method model with adequate increasing depth so that the search agents may find precise answers. Consequently, HHO’s simulation of the cooperative catch of hunt by Harris hawks is separated into two phases, as seen in Fig. 4. Particular information will be discussed in the subsections that follow.

Fig. 4

Description of each stage of HHO [56].

3.2.1 Exploration phase

The Harris hawks’ primary goal throughout the exploration step is to discover a hunt (the rabbit); they accidentally defend location spots within the group’s range or transfer to a novel site dependent on the residual search agents, as illustrated in Equation (17).

$X (t + 1) = {\begin{matrix} X_{rand} (t) - r_{1} | X_{rand} (t) - 2 r_{2} X (t) | q ⩾ 0.5 \\ (X_{rabbit} (t) - X_{m} (t)) - r_{3} (LB + r_{4} (UB - LB)) q < 0.5 \end{matrix}$ (17)

In this equation, q is the possibility that the present update technique will be chosen, r₁, r₂, r₃ and r₄ are accidental factors between [0, 1], t shows the iterations’ present number. LB and UB indicate the lower bound and upper bound of search agent’s every dimension, X_rand (t) show an accidental position in the population in the present iteration t, X (t + 1) shows the place of the hawk appearance in the subsequent iteration, X_m (t) indicates the mean location in the present population, and X_rabbit (t) shows the location where the hunt has the most possibility to happen, which is the location where the suitability value is the littlest after providing by the evaluation function. Note that Equation (1) shows the equation for the population’s mean location. Equation (18). $X_{m} (t) = \frac{1}{N} \sum_{i = 1}^{N} X_{i} (t)$ (18)

Based on Equation (18), the search agents’ number is represented by i while the iterations’ number is represented by t, and the i^th search agent’s location at the t^th iteration is represented by X_i (t).

3.2.2 Transition factor

The energy factor E determines how to transition from the exploration to the exploitation step in HHO. The hawk’s hunt is influenced by the shift of E. As a result, E is often referred to as the transfer factor in the present part. The factor E of the hunt has the following equation: $E = 2 E_{0} (1 - \frac{t}{T})$ (19)

According to this equation, the iterations’ present number is t, the iterations’ highest number is T, and the primary energy factor is denoted as E₀. This number ranges from 1 to 1 at random. When E is larger than 1, the method is in the exploration step that raises the search agent population’s variety; when E is less than 1, the method is in the exploitation step that is utilized to improve the quality of the answer.

3.2.3 Exploitation phase

The hawk conducts a magnificent a hunt for hunt at this point, increasing the chances of the hunt avoiding the hawk’s attack to 50%. The hawk’s siege strategy is classified into soft and hard besiege based on the factor E.

3.2.4 Hard besiege

The method is in a hard besiege condition when the factor E is less than 0.5. It may be separated into two tactics in this condition based on whether the hunt is able to avoid the hawk’s prey.

The hawk is able to detect the rabbit using the following equation when |E| and r are less than 0.5: $X (t + 1) = X_{rabbit} (t) - E | Δ X (t) |$ (20) $Δ X (t) = X_{rabbit} (t) - X (t)$ (21) where ΔX (t) is the difference among the present search agent’s position and the best value attained in the t^th iteration. The hawk searches for the rabbit using the given equation when |E|’s exact value is less than 0.5, but r’s exact value is more than 0.5: $X (t + 1) = {\begin{matrix} Y if F (Y) < F (X (t)) \\ Z if F (Z) < F (X (t)) \end{matrix}$ (22) $Y = X_{rabbit} (t) - E | J X_{rabbit} (t) - X_{m} (t) |$ (23) $J = 2 (1 - r_{5})$ (24) $Z = Y + S \times LF (D)$ (25)

Based on these equations, F (x) is the fitness function utilized to calculate the hawk’s location, J is utilized to alter the hunt’s movement step, D shows the search agent’s dimension, S shows an accidental vector of one row, and D columns, and r₅ is an accidental factor between [0, 1], and, the equation for Levy’s flight function, abbreviated as LF, is provided below: $LF (x) = \frac{u \times σ}{{| v |}^{β}}, σ = {(\frac{Γ (1 + β) \times sin (\frac{π β}{2})}{Γ (\frac{1 + β}{2}) \times β \times 2^{(\frac{β - 1}{2})}})}^{\frac{1}{β}}$ (26)

In this equation, v and u are accidental numbers among 1 and 0, β is a 1.5 constant, and Γ () is the gamma function. Figure 5 depicts the inexact flow of this stage.

Fig. 5

A general explanation of the hard besiege in HHO [56].

3.2.5 Soft besiege

The hawk’s search mode for a hunt is in the soft besiege step when the factor E is more than 0.5 but less than 1. At this point, the hawk’s search technique is separated into two phases, depending on whether the hunt is able to run away from the hawk’s siege.

Hawk’s search method is enhanced when |E| ⩾ 0.5 and r < 0.5, as below: $P = X_{rabbit} (t) - E | J X_{rabbit} (t) - X (t) |$ (27) $Q = P + S \times LF (D)$ (28) $X (t + 1) = {\begin{matrix} P if F (P) < F (X (t)) \\ Q if F (Q) < F (X (t)) \end{matrix}$ (29)

The expressions P and Q are somewhat different from the expressions Y and Z in Equations (23) and (25), as this is able to be seen in the equation, and the definitions of the other mathematical symbols are consistent with those in Equation (22).

The updated technique for the hawk’s location when |E| ⩾ 0.5 and r ⩾ 0.5 is below: $X (t + 1) = Δ X (t) - E | J X_{rabbit} (t) - X (t) |$ (30)

The definitions of every symbol in the statement have been highlighted above.

3.3 Support vector regression (SVR)

SVR is a regression analysis-based deployment of SVM. Despite the SVR’s utilization to forecasting concrete strength is not a novel notion, large-scale commercial implementations of the technique have yet to emerge. When compared to previous techniques, this model performs finer and is simpler to construct [58]. In order to a training data collection D ={ (x₁, y₁) , (x₂, y₂) , …, (x_n, y_n) }, where x_i is the input, and y_i is the result value, a linear function f (x) is gained as below that has the greatest ɛ deflection from the objective value y_i in ɛ _ SVR. $f (x) = w . x + b$ (31) In this equation, b stands for the bias, and w shows the weight vector. The issue is then ideally converted in the following way. ${\begin{matrix} Minimize \frac{1}{2} w^{2} \\ y_{i} - (w . x_{i}) - b ⩽ ɛ \\ (w . x_{i}) + b - y_{i} ⩽ ɛ \end{matrix}$ (32)

Two loose variables ξ_i and $ξ_{i}^{*}$ were established to cope with the optimization issue’s unviable limitations. ${\begin{matrix} Minimize \frac{1}{2} w^{2} + c \sum_{i = 1}^{n} (ξ_{i} + ξ_{i}^{*}) \\ y_{i} - (w . x_{i}) - b ⩽ ɛ + ξ_{i} \\ (w . x_{i}) + b - y_{i} ⩽ ɛ + ξ_{i}^{*} \\ ξ_{i}, ξ_{i}^{*} ⩾ 0 \end{matrix}$ (33)

According to the equation above, c stands for the penalty parameter, and c > 0.

A kernel function K (x_i, x_j) = ∅ (x_i) · ∅ (x_j) may be used to translate the sample to a higher-dimensional feature area and makes it linearly differentiable within this feature area for the nonlinear situation. Linear, polynomial, and radial basis functions (RBF) are often utilized kernel functions [59], and the RBF kernel function was employed in this research (Equation (34)). $K (x_{i}, x_{j}) = \exp (- g x_{i} - {x_{j}}^{2})$ (34)

Input variable space is affected by the support vectors during learning via kernel width (σ) and regulation (C) [60]. Inferring them may be a hard function if you decide your network to be more precise. Figures 6 and 7 show the flowchart and outline of the hybrid SVR analysis. The purpose of hybridizing metaheuristic algorithms to SVR network is to specify the optimal amount of SVR principa parameters.

Fig. 6

The ALO - SVR flowchart.

Fig. 7

The HHO - SVR flowchart.

4 Evaluation indices

So as to assess the models’ performance, some criteria were computed that are formalized as follows:

4.1 Correlation coefficient (R)

$R = \frac{\sum_{d = 1}^{D} (m_{d} - \bar{m}) (z_{d} - \bar{z})}{\sqrt{[\sum_{d = 1}^{D} {(m_{P} - m)}^{2}] [\sum_{d = 1}^{D} {(z_{d} - \bar{z})}^{2}]}}$ (35)

4.2 Coefficient of determination (R²)

$R^{2} = {(\frac{\sum_{d = 1}^{D} (m_{d} - \bar{m}) (z_{d} - \bar{z})}{\sqrt{[\sum_{d = 1}^{D} {(m_{P} - m)}^{2}] [\sum_{d = 1}^{D} {(z_{d} - \bar{z})}^{2}]}})}^{2}$ (36)

4.3 Root-mean-square error (RMSE)

$RMSE = \sqrt{\frac{1}{D} \sum_{d = 1}^{D} {(z_{d} - m_{d})}^{2}}$ (37)

4.4 Root Relative Squared Error (RRSE)

$RRSE = \sqrt{\frac{\sum_{d = 1}^{D} {(m_{d} - z_{d})}^{2}}{\sum_{d = 1}^{D} {(m_{d} - \bar{m})}^{2}}}$ (38)

4.5 Variance account factor (VAF)

$VAF = (1 - \frac{var (m_{D} - z_{D})}{var (m_{D})}) * 100$ (39)

Terms: m_d is the observations, $\bar{m}$ is the mean of observations, z_d is the simulations, $\bar{z}$ is the mean of simulations, and D is data set’s number

5 Results and discussion

5.1 Preliminary fracture energy (G_f) of concrete

Throughout this study, an empirical portion of the database was leveraged to examine the G_f utilizing paired SVR networks augmented by ALO and HHO to generate distinct suggestion networks. The ALO and HHO methodologies were chosen to emphasize the model’s critical features with the aim of improving the development’s modelling efficacy. As aforementioned, the purpose of hybridization was to provide the most proper solution for regulatory integers of SVR, named C, σ, and ɛ. Table 4 shows the suggested parameters for C, σ, and ɛ related to SVR using the ALO and HHO methods. This paper examines and discusses the advantages of employing blended SVRs to approximate concrete’s G_f, by integrating ALO and HHO methods with SVR. With the objective of generating the greatest effective adjuster identifiers in mind, the hybrid SVR principles were constructed. The real concrete was chosen from a random selection of research articles for the learning and assessment phases. Figure 8 exhibits a comparison of real and calculated G_f values, as well as the forecasted/recorded ratio histogram variation. The applicability of the existing techniques was evaluated by means of several metrics (R, R², RMSE, RRSE, and VAF) (Table 4).

Fig. 8

The performance of SVR for simulating G_f : a) HHO - SVR, b) ALO - SVR.

Table 4

The workability of the SVR models for G_f

Set	Index	Hyperparameter	Models		[24]
			HHO - SVR	ALO - SVR
		C	682.3	1086.3
		σ	0.652	0.892
		ɛ	0.102	0.105
Train	R		0.9185	0.9527
	R ²		0.8436	0.9077
	RMSE		0.0058	0.0044
	RRSE		0.4716	0.3563
	VAF		84.355	87.648
Test	R		0.9584	0.99	0.95
	R ²		0.9185	0.9801
	RMSE		0.0046	0.0026
	RRSE		0.3311	0.185
	VAF		91.804	96.667

The conclusions of the created indicators for the established analysis known as HHO - SVR and ALO - SVR will be contrasted in this portion of the research to see which model exceeds the others. Furthermore, a relevant comparison of the conclusions of this research with previously published ones was undertaken. With R values of 0.9185 and 0.9584 for HHO - SVR, and 0.9527 and 0.99 for ALO - SVR, respectively, for the learning and evaluating data segments, the findings of the estimations, which took into account appraisal aspects, demonstrate that both optimized HHO - SVR and ALO - SVR could astonishingly perform favourable productivity during the estimation methodology. R² scores in ALO - SVR analysis in the test data demonstrate the models’ high efficacy at 0.9801. In attempt to reveal the superior design, the produced signals should be studied and compared to one another. In the training sector, the ALO - SVR regression exhibits a decline in the RMSE when compared to the HHO - SVR, falling from 0.0058kN/m to 0.0044kN/m. Throughout the testing process, the index was reduced roughly by half, from 0.0046 kN/m to 0.0026kN/m. Other indicators, like RRSE, follow the same trend as RMSE, suggesting that the RRSE values for ALO - SVR were much lower compared to HHO - SVR, implying that the SVR optimized with ALO has further ability for predicting the G_f. In additional, the increasingly popular VAF index was computed to illustrate the computed competency (Table 3), where the bigger value, the more accurate. Within the learning segment, VAF index increased significantly from 84.355% for HHO - SVR to 87.648% for ALO - SVR, and within the tesing section, from 91.8% (HHO - SVR) to 96.67 % (ALO - SVR). In an effort to provide a reference point, the findings of this study were compared to those of a previous study [24]. The resulting ALO - SVR simulation has a major enhancement in effectiveness with an increment in R value from 0.95 to 0.99. As a result, although the HHO - SVR provides its own predicting capabilities, the SVR optimized with ALO seems to be rather reliable for G_f, according to the justifications.

5.2 Whole fracture energy (G_F) of concrete

Throughout this study, an empirical portion of the database was leveraged to examine the G_F utilizing paired SVR networks augmented by ALO and HHO to generate distinct suggestion networks. The ALO and HHO methodologies were chosen to emphasize the model’s critical features with the aim of improving the development’s modelling efficacy. As aforementioned, the purpose of hybridization was to provide the most proper solution for regulatory integers of SVR, named C, σ, and ɛ. Table 5 shows the suggested parameters for C, σ, and ɛ related to SVR using the ALO and HHO methods. This paper examines and discusses the advantages of employing blended SVRs to approximate concrete’s G_F, by integrating ALO and HHO methods with SVR. With the objective of generating the greatest effective adjuster identifiers in mind, the hybrid SVR principles were constructed. The real concrete was chosen from a random selection of research articles for the learning and assessment phases. Figure 9 exhibits a comparison of real and calculated G_F values, as well as the forecasted/recorded ratio histogram variation. The applicability of the existing techniques was evaluated by means of several metrics (R, R², RMSE, RRSE, and VAF) (Table 5).

Fig. 9

The performance of SVR for simulating G_F : a) HHO - SVR, b) ALO - SVR.

Table 5

The workability of the SVR models for G_F

Set	Index	Hyperparameter	System		[24]
			HHO - SVR	ALO - SVR
		C	658.55	1011.98
		σ	0.65	0.702
		ɛ	0.103	0.108
Train	R		0.9265	0.9363	0.91
	R ²		0.8584	0.8767
	RMSE		0.0105	0.0105
	RRSE		0.377	0.3725
	VAF		85.8154	87.6503
Test	R		0.9063	0.9336
	R ²		0.8214	0.8717
	RMSE		0.0119	0.0102
	RRSE		0.4277	0.3652
	VAF		81.9517	86.6796

The conclusions of the created indicators for the established analysis are known as HHO - SVR and ALO - SVR will be contrasted in this portion of the research to see which model exceeds the others. Furthermore, a relevant comparison of the conclusions of this research with previously published ones was undertaken. With R values of 0.9265 and 0.9063 for HHO - SVR, and 0.9363 and 0.9336 for ALO - SVR, respectively, for the learning and evaluating data segments, the findings of the estimations, which took into account appraisal aspects, demonstrate that both optimized HHO - SVR and ALO - SVR could astonishingly perform favourable productivity during the estimation methodology. In attempt to reveal the superior design, the produced signals should be studied and compared to one another. In the training sector, the ALO - SVR regression exhibits a identical in the RMSE when compared to the HHO - SVR, at 0.0105kN/m. Throughout the testing process, the index was reduced roughly by 20%, from 0.0119 kN/m to 0.0101kN/m. Other indicators, like RRSE, follow the same trend as RMSE, suggesting that the RRSE values for ALO - SVR were lower compared to HHO - SVR, implying that the SVR optimized with ALO has further ability for predicting the G_F. In additional, the increasingly popular VAF index was computed to illustrate the computed competency (Table 5), where the bigger value, the more accurate. Within the learning segment, VAF index increased slightly from 85.8154% for HHO - SVR to 87.6503% for ALO - SVR, and within the tesing section, from 81.9517% (HHO - SVR) to 86.6796% (ALO - SVR). In effort to provide a reference point, the findings of this study were compared to those of a previous study [24]. The resulting ALO - SVR simulation has a major enhancement in effectiveness with an increment in R value from 0.91 to 0.9363. As a result, although the HHO - SVR provides its own predicting capabilities, the SVR optimized with ALO seems to be rather reliable for G_F, according to the justifications.

6 Conclusion

In the concrete industry, cost-effective and time-saving solutions are essential. Human mistakes are caused by the researcher’s inefficacy throughout the operation, and technology failures would squander time and money while yielding wrong data. To provide a stronger insight of preliminary (G_f) and complete fracture energy (G_F), this activity created a hybridized support vector regression (SVR) methodology. By combining the antlion optimization algorithm (ALO) and harris hawks optimization (HHO) techniques with SVR, platforms were applied to calculate the G_f and G_F in this case. The purpose of hybridization was to provide the most proper solution for regulatory integers of SVR. A dataset of four inputs (f_c, d_max, CT, and W/C) and two outputs (G_f and G_F) was produced for this purpose with collecting actual records from previously published papers. The following are the main findings:

Regarding G_f, with R values of 0.9185 and 0.9584 for HHO - SVR, and 0.9527 and 0.99 for ALO - SVR, respectively, for the learning and evaluating data segments, the findings of the estimations, which took into account appraisal aspects, demonstrate that both optimized HHO - SVR and ALO - SVR could astonishingly perform favourable productivity during the estimation methodology. R² scores in ALO - SVR analysis in the test data demonstrate the models’ high efficacy at 0.9801. In the training sector, the ALO - SVR regression exhibits a decline in the RMSE when compared to the HHO - SVR, falling from 0.0058kN/m to 0.0044kN/m. Throughout the testing process, the index was reduced roughly by half, from 0.0046 kN/m to 0.0026kN/m. Another indicator, like RRSE, suggesting that for ALO - SVR were much lower compared to HHO - SVR, implies that the SVR optimized with ALO has the further ability for predicting the G_f. Within the learning segment, VAF index increased significantly from 84.355% for HHO - SVR to 87.648% for ALO - SVR, and within the testing section, from 91.8% (HHO - SVR) to 96.67 % (ALO - SVR). The resulting ALO - SVR simulation has a major enhancement in effectiveness with an increment in R value from 0.95 [24] to 0.99.

Turning to G_F, with R values of 0.9265 and 0.9063 for HHO - SVR, and 0.9363 and 0.9336 for ALO - SVR, respectively, for the learning and evaluating data segments, the findings of the estimations, which took into account appraisal aspects, demonstrate that both optimized HHO - SVR and ALO - SVR could astonishingly perform favourable productivity during the estimation methodology. Throughout the testing process, the RMSE index was reduced roughly by 20%, from 0.0119 kN/m to 0.0101kN/m. Other indicators, like RRSE, follow the same trend as RMSE, suggesting that the RRSE values for ALO - SVR were lower compared to HHO - SVR, implying that the SVR optimized with ALO has the further ability for predicting the G_F. Within the learning segment, VAF index increased slightly from 85.8154% for HHO - SVR to 87.6503% for ALO - SVR, and within the testing section, from 81.9517% (HHO - SVR) to 86.6796% (ALO - SVR). The resulting ALO - SVR simulation has a major enhancement in effectiveness with an increment in R value from 0.91 [24] to 0.9363.

As a result, although the HHO - SVR provides its own predicting capabilities, the SVR optimized with ALO seems to be rather reliable for G_f and G_F, according to the justifications, and could be recognized as the proposed model.

The future scope of this article was utilizing the developed models for the practical aims in the field of civil engineering. Because, the machine learning algorithms could reduce the cost and laborious efforts.

7 Funding

This work was supported by the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJQN202204016).

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Integrated and optimized SVR analysis: Assessment of the preliminary and entire fracture energy of concrete

Abstract

Keywords

1 Notations

1 Introduction

1.2 Developed formulas for calculating the G f and G F

3.1 Ant lion optimization algorithm (ALO)

3.1.1 Random walk of ants

3.1.3 Trapping in ant lion pits

3.2.4 Hard besiege

4.1 Correlation coefficient (R)

5.1 Preliminary fracture energy (G f ) of concrete

7 Funding

References

1.2 Developed formulas for calculating the G_f and G_F

5.1 Preliminary fracture energy (G_f) of concrete