Review of researches on strength model of foamed concrete

Compared with ordinary concrete, foamed concrete has the advantages of light weight, high fluidity, good thermal and sound insulation properties. It has been initially applied in the construction industry. As the most basic research content of foamed concrete, the strength prediction model has received widespread attention from scholars. In this paper, the existing foamed concrete strength prediction models were summarized. The development status of the traditional empirical model based on theory or experiment and the artificial intelligence model based on computer technology were analyzed, and the advantages and disadvantages of each model were described comprehensively. The following conclusions can be drawn: ①Porosity is an important factor affecting the density and compressive strength of foamed concrete. Considering pore structure and volume composition will be more convenient and accurate to predict the strength. ② As for gel-space ratio model, considering the contribution of different hydration products produced by the cementitious materials to the strength is helpful to improve the accuracy of the prediction model. ③ The artificial intelligence model overcomes the shortcomings of less factors in the empirical model and large amount of experiments with high cost, which makes the strength prediction more effective and precise. This paper provides a train of thought for further research on the strength prediction model of foamed concrete. The development of foamed concrete strength prediction model can provide theoretical guidance for the design of foamed concrete, and is of great significance for promoting the application of foamed concrete in various occasions.


Introduction
Foamed concrete is a new lightweight and environmentally friendly construction material.It is a kind of porous lightweight concrete, which is prepared by physical foaming, adding the foam into the paste made of cement-based cementitious materials, aggregates, admixtures and water，and then being mixed, casted and cured [1].It has the advantages of less aggregate consumption, light self-weight, high fluidity, good fire resistance, good thermal and sound insulation, etc.It has been paid more and more attention in the development of construction industry [2].Recently, foamed concrete has been used for trenches and cavern filling, mine and tunnel lining, abutment and surface works, slope support [3], etc.
Compressive strength of foamed concrete is one of its most important mechanical properties.Many papers focus on the factors that affect the compressive strength of foam concrete.It can be considered that density is the main factor affecting the strength of foam concrete [1, 4, 5].The compressive strength increases with the increase of density.Liu Zhongwei et al. [6] studied the influence of water cement ratio on the compressive strength and pore structure of foam concrete in detail, and believed
=  0 (1 − )  . 1 Where f c is compressive strength of foamed concrete (MPa), f o is the strength of the porous material (MPa), b is an empirical parameter, and n is the porosity.
Hoff [18] divided the volume composition of foamed concrete into the volume of water that can be evaporated, the volume of non-evaporable water, the volume of cement, the volume of air carried in, and the volume of foam.The porosity is defined as the ratio of the sum of air volume and evaporable water volume to the total volume, and 0.20 is taken as the average value of the ratio of water to cement by mass required for hydration.After mathematical derivation, the calculation equation of porosity is obtained: Where d c is density of concrete, k is water-to-cement ratio(by mass), P c is specific gravity of cement and y w is specific gravity of water.
Applying equation 2 into Balshin's model, the following model is obtained: =  0 *   (1 + 0.2  ) (1 + )    +  .3 Hoff used three experiments to verify this model.Different types of cement and water-to-cement ratio were used in the three sets of experiments.Different values of f o and b were obtained, all of which fit well with the above expression, with R² between 0.94 and 0.95.
Nambiar [19] took into account the influence of different additive materials and extended the calculation equation of porosity to: Where s w is the ratio of filler to cement by mass, s v the ratio of filler to cement by volume and k s is the ratio of water to solid materials by mass.Other parameters have the same meaning as Hoff's model.
Appling the ep.4 into Balshin's model, the strength model of foamed concrete considering filler is obtained.
=  0 (   (1 + 0.20  +   ) (1 +   )(1 +   )    )  .5 This equation has the same advantage as Hoff's model, that is, all parameters are easy to measure, which avoids the difficulty of porosity measurement in engineering practice and considers the contribution of any component to the volume.Kearsley and Wainwright [20] tested 27 sets of foamed concrete with different mix proportions.The design density of these foamed concrete is 1000kg/m3, 1250kg/m3, 1500kg/m3 and full (without foam).The ratio of fly ash to cement is 1, 2, 3 respectively and the water-to-binder ratio is about 0.3.
In Kearsley and Wainwright 's study, Balshin's model fitted well with the experimental data and the correlation coefficient was as high as 0.967.and the value of b is 3.6(b=3.6)Considering that the strength of concrete increases significantly from the 28th day to the 365th day after being casted, Kearsley added the factor of age  to Balshin's model and obtained the following expression: .174 (1 − ) 3.6 .6 Where  is age since casting (days);  is mature porosity (measured after 365 days).(In these experiments, the values of  0 and b are close to Hoff's conclusion.During the fitting process, it was found that two data points far away from the best fit line are foamed concrete with water-to-binder ratio of 0.40 and 0.60.This showed that different water-to-binder ratio will affect the value of empirical parameter in the model.
In previous studies, most of the progress of Balshin's models was carried out about porosity.Parameter  varied with the variation of the composition, mix proportions and other factors of foamed concrete in different studies.It was a parameter that completely dependent on experience, which brought uncertainty to the relationship between compressive strength and porosity.
Baozhen et al. [21] obtained that the exponent  of Balshin's model changed range from 1 to 2.2 in experiments and the strength change of low porosity concrete was more sensitive to porosity than that of high porosity concrete.
Y. Chen et al. [22] reported that parameter  was based on empirical regression and there was no theoretical method to determine it in previous studies.They concluded that the compressive strength of foamed concrete was controlled by its effective skeleton.The skeleton of foamed concrete is a fractal object.The establishment of foamed concrete fractal model can well explain the scaling phenomenon of the compressive strength of foamed concrete.The fractal model can accurately illustrate the relationship between compressive strength and porosity.The expression of parameter  was established by the fractal dimension D of foamed concrete.

𝑏 =
− 1 3(3 − ) .7 The fractal dimension D can be calculated from the scaling law of compressive strength.
Incorporating B into the Blashin model, when the water-cement ratio is 0.43 and 0.60 respectively, the calculated compressive strength of foamed concrete is in good agreement with the experimental data.Fractal model provides a simple and effective method for calculating the compressive strength of foamed concrete.Incorporating B into the Blashin model, when the water-cement ratio is 0.43 and 0.60 respectively, the calculated compressive strength of foamed concrete is in good agreement with the experimental data.Fractal model provides a simple and effective method for calculating the compressive strength of foamed concrete.
In 1953, Ryshkevitch [23] proposed the exponential function expression of the strength-porosity model of brittle materials：   =  0  ; .8 Where B is the slope of the    vs.  curve.This equation can be used to describe the relationship between porosity and strength of a single cementitious material.According to Chindaprasirt [24] , the relationship between strength and porosity of porous concrete can be well simulated by Ryshkevitch's model, and parameter  is independent of the strength of cementitious material.
According to the studies conducted by Li Haoran [25] ， when studying the effect of pore characteristics on the compressive strength of foamed concrete, it is far from enough to consider only the porosity.
Using Griffith fracture mechanics and composite material theory, on the basis of previous research results, Li Haoran obtained the relationship between critical stress and pore diameter of materials.The expression is following: Where   is critical stress of material,  is pore diameter, q is a parameter determined by Pore shape, E 0 is elastic modulus of material matrix and  0 is surface energy of material matrix fracture.

10
Where:  1 = 1 1: ,  is water-to-cement ratio.Putting . 10 substitute into Balshin's and Ryshkevitch's models and conducting the surface fitting will obtain the model of strength porosity of foamed concrete with consideration of pore size: = 1.04 .−3.67Where  is pore diameter,  is water-to-cement ratio and  is porosity.
According to the two equations above, it can be concluded that if the water-to-cement ratio decreases, the value of 1 1 +  ⁄ will increase.If the fluidity and viscosity of the paste are poorly controlled, the pore structure will deteriorate, making 1 r ⁄ smaller.Therefore, it is not advisable to only consider improving the matrix strength, while ignoring the reduction of the final strength caused by the change of pore size.This provides a reference for the mix proportions design of foamed concrete.
ZHANG Z et al. [26] investigated the strength and porosity models of geopolymer foamed concrete.Fitting the experimental data with Balshin's model and Ryshkevitch's model, the Ryshkevitch's model can better fit the strength of geopolymer foamed concrete, R² =0.92.
It was found that when the porosity was between 0.55 and 0.7, the model was in good agreement with the observed value.But the accuracy of Ryshkevitch's model decreased with the decrease of porosity.This may be because the effect of pore size distribution on strength is ignored in both models.Because the influence of large pores on the strength of concrete is greater than that of small pores, they divided the pores into those with diameter greater than 5.6 μm and those with diameter less than 5.6 μm.On the basis of Ryshkevitch's model, a new model was constructed as follows: =  ×   1 +  ×   2 .13 Where  and  are indicators of the effect of large and small pores on strength,  and  are empirical constants.The following strength model was obtained through experiments: = 61.55 ×  ;4.396 1 − 5.28 ×  ;280.15 2 .14 R² =0.97The results showed that the accuracy of the model was improved by the division of large and small pores.
In 1958, Schiller [27] summarized the strength porosity model of non-metallic brittle materials Where，  is a constant,  0 is critical volume fraction corresponding to zero strength.Schiller standardized the logarithmic relationship model and Ryshkevitch's exponential relation model and obtained the orthotropic equations: 0 =  ; ⁄ .16

⁄
Where   and   are the compressive strength in Ryshkevitch's and Schiller's models, respectively. 0 is the strength of the non-porous material.
It was found that there was a connection between the parameters of the two equations, that is, except for  = 0 and  =  0 , the strength of the foamed concrete calculated by these two equations were almost the same.The results can be seen clearly in the following figure.Hasselmann [28] proposed a model for the influence of spherical pores in glass on the strength and elastic modulus of glass in 1963: =  0 −    .18 Where  0 is the strength of non-porous material，  is a constant，and  is the porosity.M. RÖBler and I. Odler [29] investigated the relationship between the strength of cement based foamed concrete and four empirical models.It was found that porosity was the main factor determining the strength properties of cement paste regardless of the initial water-to-cement ratio and hydration degree.Among the four porosity-strength models, Hasselmann's model can better fit the strength.But he also pointed out that in the range of experimental research, the four models can all fit the strength of foamed concrete well.Youssef et al. [30] prepared foamed concrete with foam volume fraction between 0.37-0.72 and measured the compressive strength after 28 days.It was concluded that except for the linear model, the other three models can better match the strength of the foamed concrete.As shown in the figure below:

Gel-space model
In 1919, Duff Abrams [32] proposed the formula of compressive strength of concrete: Where  1 and  2 are constants,   ⁄ is water-to-cement ratio.This formula revealed that the strength of concrete decreased with the increase of water-to-cement ratio.The pore size of concrete depended on the pore that formed after the water evaporated, but inevitably there will be many gaps in concrete due to inadequate compaction or evaporation of free water.

COMSE-2023
Journal of Physics: Conference Series 2671 (2024) 012010 Feret [32] proposed a strength model for foamed concrete related to the volume of water, cement and air: Where  is a constant,  ，， are the volume of cement, water and air respectively.This formula considered the influence of air volume on the strength.However, due to the continuous hydration of cement, the volume of hydration products is different from the initial volume of cement.
Powers [31] proposed the concept of gel-space ratio.He suggested that the strength of concrete was related to the concentration of the hydration product in the available space(that is the gel-space ratio).The gel-space ratio is defined as the ratio of the volume of hydration product to the sum of the volume of hydration product and the volume of pores.Its strength prediction model is expressed as follows: = ()  eq.21 Where  is a constant and  is gel-space ratio.Based on the concept of gel-space ratio, Tam [32] considered the influence of hydration degree  on the strength, and modified the Feret's model to obtain: Where  3 is an empirical constant.When the degree of hydration α = 1, the model is consistent with Feret's model.According to eq.18, the main factors affecting the strength of foamed concrete are the degree of hydration, water-to-cement ratio and air-cement ratio.But the degree of hydration α is not easy to measure.
Based on .18, Tam deduced an inequality to determine the strength of foamed concrete with different mix proportions.He assumed that the compressive strength of the two kinds of foamed concretes satisfied the relationship:  1 >  2 .With the cement volume kept constant, the inequality can be obtained as follow: By comparing all( + )  ⁄ , the relative order of the strength of foamed concrete with different densities and different mix proportions can be accurately predicted.
Nambia et al. [19] considered the effect of fillers on the gel-space ratio of foamed concrete and expressed the total pore volume: Space = 1 −   −   (1 − ) .24 Take the volume of cement hydration products (gel volume) as 2.06   .Then the gel-space ratio can be expressed as: Where   is the volume of cement per m³ of concrete,   is the volume of the fillers per m³ of concrete and  is the degree of hydration which is assumed as 0.80.
Introduce .21 into the expression of Powers.Fit the strength of cement-sand foamed concrete and cement-sand-fly ash foamed concrete respectively.The following two strength models can be obtained: Cement-sand mix:   = 188.13 2.73 .26 R² =0.864 Cement-sand-fly ash mix: .74 .27 R² =0.798It can be seen from the exponent term of gel-space ratio that the cement-sand mixture depends more on the gel-space ratio.Nambia et al. thought that due to the participation of fly ash, the hydration products of fly ash were not considered in the air cement ratio, which led to the inaccurate prediction of cement-sand-fly ash mix.
Pichler [33] et al. used the engineering micromechanics method to model the elastic brittle strength, linking the mechanical properties with the microstructure of materials.The micromechanics model shows that the relationship between strength and gel-space ratio originates from the strengthening effect of unhydrated clinker particles, and cement paste can be regarded as hydrate foam reinforced by clinker particles.Based on this, a dimensionless empirical formula is proposed to simulate the change of uniaxial compressive strength of different components of cement paste.This is a quantitative prediction of strength in an algebraic way.The uniaxial compressive strength of cement paste is:  24, the uniaxial compressive strength model of cement paste related to the gel-space ratio can be obtained.The predicted results by using this model agree very well with those obtained from experiments.
On the basis of Pichler's research, Termkhajornkit [34] et al. proposed a concept of "beyond gel-space ratio".The author verified the traditional gel-space ratio model in C 3 S-C 3 A cement paste and C 3 S-C 2 S cement paste respectively, and obtained that the relationship between compressive strength and gel-space ratio of C 3 S-C 3 A cement paste (adding calcium sulfate in different proportions) was unique, expressed as follows: = 167( −  ) 5.42 eq.33 However, the relationship between strength and gel-space ratio of C 3 S-C 2 S paste is not unique anymore.For C 3 S: = 200( −  ) 4.55 eq.34 For C 3 S   = 127( −  ) 2.51 eq.35 This shows that the contribution of each hydration product to strength is different, that is, their contribution efficiency to strength is different.C-S-H plays a crucial role in the strength of hydration products.
The direction of a C-S-H needle is shown in the Fig4.The experimental data show that the model has a good prediction effect.This provides a new way to predict the compressive strength of concrete more accurately with the gel-space ratio model.In the next step, we can continue to study the contribution of hydration products of different cementitious materials to the strength of concrete.To study the influence of pozzolanic components such as fly ash, recycled clay brick powder and slag on the microcomposition of concrete material can improve the prediction ability of gel-space ratio model.

Other empirical models
Nambiar et al. [35] studied the influence of three factors on the density and compressive strength of foamed concrete, namely, the filler cement ratio, the replacement rate of fly ash on sand, and the total volume of foamed concrete, and used a second-order model considering curvature to fit the experimental data: Where   ( = 1,2, … , ) is quantitative variable,  0 and   are least square estimates of the regression coefficients.
The two response surface was determined by SAS software, and a series of prediction models for density and strength of foamed concrete with different mixture ratios at different ages were obtained.According to the response surfaces and contour plots of compressive strength of 7 days, 28 days and 90 days, the strength of foamed concrete is linearly changed with the amount of cement without adding fly ash, and it has a nonlinear change with the volume of foam.When the foam content is low, the strength is dominated by the amount of cement.When the foam content is high, the strength is dominated by porosity.With the addition of fly ash, the strength of foamed concrete rapidly between 28 days and 90 days.
This model can be used as a reference to design the mix proportion of the corresponding density and strength of foamed concrete, but this model has many coefficients and terms, which is not an easy model to express.
Lian et al. [36] take .35of elastic modulus: E = E 0 (1 − ) 3 .39 and .36 of fracture energy： =  0 (−) .40 into Griffith's fracture model .37: Where  0 is fracture energy at zero porosity,  is a constant, E 0 is the elastic modulus of the material at zero porosity  is porosity, σ is the stress at the fracture (Pa),  is the elasticity modulus(Pa),  is the fracture surface energy (J/m 2 ) and a is the half length of an internal crack (m).
The strength prediction model of porous concrete is obtained: Where m and n are new material constants for porous concrete.For the convenience of regression, let 2 0  0  = and make some mathematical transformations so that .38 can be regarded as a linear equation: = m 1 +  2 +  .Where ， = 2σ 、  1 = (1 − p)、  2 = p and c =  .After regression with experimental data, the empirical equation is obtained: = 5.96 1 − 10.01 2 + 10.61 R² =0.99 .43 The model can well describe the relationship between compressive strength and porosity of porous concrete.The advantage of this model is that it can predict the strength of foamed concrete without the strength of non-porous material.

Multiple linear regression (MLR)model
Regression analysis, as a statistical method, deals with the influence of one or more predictors on a related goal.In linear regression, the linear correlation of variables is determined.In the simple linear regression method, only one independent variable is used to predict the variable value of the target.If two or more independent predictors are used to predict the relevant target value, it is called multiple linear regression model (MLR model) [37] .The general form is as follows: Where is the target variable,   is the independent input variable of the model, and   is the partial regression coefficient.Least square method, the most widely used regression method, is used to select coefficients by minimizing the sum of squares of residuals.
ÖZTURAN [38] investigated the MLR of the compressive strength of concrete with five models including different influence factors, and obtained the MLR model which considered the cement, water, limestone with 5-10 mm particle size, limestone with 10-20 mm particle size, natural sand, crushed sand, water reducing admixture, slump, fresh density and 7-day compressive strength.The MLR model was more accurate for the prediction of the strength, R² =0.903.
Ashrafian et al. [39] considered the effects of foam, sand and cementitious material content and water cement ratio, sand to cement ratio and age on the compressive strength of foamed concrete.The MLR model for predicting the strength of foamed concrete was obtained.
Wang Kang [40] investigated the effects of hydrogen peroxide content, glass fiber content, the amount of vitrified microsphere and the dosage of ceramsite on the compressive strength and splitting strength of foamed concrete, and established the MLR model of compressive strength.
1 = 10.319− 0.013 1 + 0.027 2 + 0.001 3 + 0.002 4 R² =0.848 .46 Jing Zhang and Xiangdong Liu [41] investigated the effect of carbon nanotubes (CNTs) on the compressive strength of ultra-lightweight foamed concrete.He established a binary linear regression model and a ternary linear regression model for predicting the strength of six groups of ultra-lightweight foamed concrete with a density of 200-250 kg/m³ .The first bivariate linear regression model takes porosity and CNTs content as independent variables, the second bivariate linear regression model takes porosity and pore size as independent variables.The R² value of fitting results of different density bivariate linear regression model is between 0.86-0.98.The independent variables of the ternary linear regression model are porosity, pore size and CNTs content.The R² value of the fitting result is between 0.90-0.99.The three MLR models have good prediction accuracy for the strength of ultra-lightweight foamed concrete added to CNTs.
The MLR model is an easy application model.It takes the selected influence factors as independent variables to predict the compressive strength of foam coagulation.The general rule is that the more the selected independent variables, the more accurate the simulation results are.However, it is worth noting that the MLR fitting equation has no clear physical significance, nor can it explain the causal relationship between variables, it is only a regression of data.When using MLR model, it is necessary to ensure that there is a linear relationship between independent variables and dependent variables, otherwise the fitting accuracy will be poor or the fitting will be distorted.
Multiple nonlinear regressions is similar to multiple linear regression, y =

Artificial neural network (ANN)model
The human brain is composed of a large number of very complex neural networks.Because of the existence of these neural networks, when the information of the sensory organs is received by the brain, people can quickly reflect and learn.Using the learning and creating ability of the brain neural network, the abstract and simulated model of the brain neuron system is the artificial neural network model.
The ANN model consists of three layers: input layer, hidden layer and output layer.As shown in the Fig 5 below is a basic three-layer artificial neural network model [42] : The pattern of neurons in the hidden layer is as follows [43] : Neural networks are made up of simple processing units called neurons.Each neuron in each layer is connected to all the neurons in the next layer.Each node in the hidden layer adds the weighted input signals and the resulting sum is calculated by the activation function.The output of the artificial neuron can be expressed as [44] : In .44 ，(∑     + ) is activation function, represents the relationship between input and output of neurons. 1 ⋯   are  inputs,  1 ⋯   represent the connection weight of  inputs, ∑     is activation value, indicating the total input of neurons,  is output, and  is the threshold of neuron.The threshold is added so that the activation function image can move left and right, thus increasing the possibility of solving the problem.Activation function is the core of artificial neural network, and its ability to deal with problems is closely related to the activation function used.Sigmoid function is a most widely used activation function, and its expression is as follows: ANN model is widely used in predicting the compressive strength of concrete.ÖZTURAN et al. [38] established 5 system models including all or part of 9 factors including cement, water, limestone with different particle size, different types of sand and water reducing admixture, slump, fresh density and 7-day compressive strength, each of which established 6 ANN models with one and two hidden layers respectively.In these 60 models, they indicated that the accuracy of neural network model changes with the number of hidden layers and neurons.In this simulation, for all system models, the ANN model with two hidden layers can get higher precision than that with one hidden layer.The curve of compressive strength predicted by the obtained neural network model with the change of water-to-cement ratio is similar to the trend of actual data, so the author believes that the neural network prediction is of physical significance.
Yeh [45] established an ANN model with eight nodes (one hidden layer) to predict the compressive strength of high-performance concrete, and considered 8 factors such as cement, slag, fly ash content, etc., trained and tested with different data groups, the R² obtained was between 0.917-0.945.The accuracy of the regression equation is higher than that of the same data set only considering age and water binder / water-to-cement ratio.Ni Hong Guang et al. [46] have similar conclusions.
Nehdi et al. [47] selected cement content, water binder ratio, foam cementitious material ratio and fine sand cementitious material ratio as input variables.Sigmoid function  = Because there is no specific criterion to regulate the selection of network architecture, they selected the ANN model of two-layer hidden structure after simulation.The first hidden layer contains three neuron nodes, and the second hidden layer contains five neuron nodes.The ANN model is accurate to predict the compressive strength at 28 days.Compared with the existing Hoff's and Tam's model, the prediction errors are reduced by 106% and 47% respectively.
In order to improve the performance of the deep neural network model, Nguyen et al. [48] proposed the generalized high order deep neural network model (HO-DNN) of the neural network model.The cross-entropy cost function and the modified linear unit activation function are used to solve the problems of slow learning and gradient disappearance.The parameters considered are: density, water-to-cement ratio, sand cement ratio.Compared with the conventional artificial neural network (C-ANN) model and the second-order artificial neural network (SO-ANN) model, the HO-DNN model can break the barrier with a correlation coefficient R² of 0.99, which cannot be achieved by C-ANN and SO-ANN.
For the artificial neural network model, there is no final conclusion about the number of hidden layers and nodes.Most researchers use trial and error method to determine, which adds workload and uncertainty to the establishment of the model.Dong van Dao et al. [49] conducted research on this issue.On the basis of previous research, he selected 220 measured compressive strength values as input data, density, water-to-cement ratio and sand cement ratio as input variables, and established 420 ANN models with node numbers of the first hidden layer varying between 1-20 and the second hidden layer varying between 0-20.The research shows that increasing the number of nodes in the hidden layer can improve the accuracy of C-ANN to the training data set.In this study, the optimal ANN model is a model with 4 and 5 nodes in two hidden layers, R² =0.972.
However, this is still an empirical verification and discussion, and if there are too many layers or nodes, it will lead to over fitting or low efficiency.The theory of ANN model can be further studied to support the selection of hidden layers and nodes, which makes it possible to set up a more accurate ANN model.
Compared with the multiple regression model, the neural network model can easily add additional model parameters, thus making the predicted value less discrete [50] .However, the error in the artificial neural network model is multidimensional and may contain many local minimum.In order to avoid local minimum or slow convergence, a momentum factor is usually added to the weight adjustment.Momentum coefficient varies from 0.1 to 1, generally 0.5 [38] .

Support vector regression(SVR) model
Support vector machines (SVMs) were first proposed by Boser et al. in 1992.The basic principle of SVR is to project the data  in the real space into the high-dimensional space F, and then establish a linear function in the high-dimensional space to process the experimental data [51] .
So for a given data sample( In(Eq.43),  is the number of data samples;  is called the penalty factor, which is used to correct samples that exceed the error limit.On the right side of the equation, The modeling results are closely related to the selection of kernel function.If proper kernel function can be selected, the points in the high-dimensional feature space can be transformed into the low-dimensional space for calculation, which solves the dimension problem caused by the calculation in the high-dimensional space.The main requirement of kernel function is to be able to reflect the distribution characteristics of sample data as accurately as possible, and to select the appropriate kernel function is conducive to the accuracy of prediction.In the actual modeling process, the common kernel functions are： Linear function: (  , ) =    Polynomial function：(  , ) = (   + )  Radial basis function：(  , ) = (−|  − | 2 ) Sigmoid function：(  , ) = (  ,  + ) Azimi-pour [51] uses the above four functions as the kernel function to predict the compressive strength of self-compacting concrete with large amount of fly ash respectively.The results show that the radial basis kernel function has the best correlation, R² =0.94.The correlation of linear kernel function is poor, R² = 0.83.
Abd [52] used 111 samples to train the SVR model for predicting the strength of the foamed concrete.39 samples are used as the test group, and the above four functions are used as the kernel functions to establish the model.The results show that the SVR model with radial basis function as kernel function has the least mean square error and the least standard deviation, which is the best in training, testing and overall data set correlation.The correlation coefficient R 2 = 0.99 in the test group shows high prediction ability.
The support vector regression model is obviously superior to the traditional model in prediction accuracy, and has great potential in the prediction of foamed concrete strength.Compared with the neural network model, its main advantages are: it finally solves a convex quadratic programming problem.Theoretically, it can get the global optimal solution, avoid the local extremum problem in the neural network model, and avoid the influence of activation function as a priori information in the neural network model on the whole model.In the strength prediction of foamed concrete, radial basis function is used as a kernel function to achieve better prediction results.

Other artificial intelligence models
Yaseen et al. [53] used the model of extreme learning machine to predict the compressive strength of cellular concrete with good accuracy.The extreme learning machine model is a new model developed based on ANN model.Compared with ANN model, it can solve the regression problem faster because the weight in the hidden neurons is random.
Kiani et al. [54] established the strength prediction model of foamed concrete by using gene expression programming algorithm.

𝑒𝑞. 63
Where,  is cement content,  ⁄ is water binder ratio,  is foam volume.The advantage of gene expression programming is that it can generate reasonable prediction equations, but it does not need the form of preset equations.Compared with most artificial intelligence models (such as ANN model and SVR model), the prediction model cannot be expressed by formula, but it is more applied as a part of computer program, while Kiani's model has a clear expression form.
Ashrafian et al. [39] also proposed some artificial intelligence methods that can be used for prediction of foamed concrete, such as multivariate adaptive regression splines model, water cycle algorithm and so on.Some artificial intelligence models have not yet been applied to the strength prediction of foamed concrete, but they have been widely applied in other aspects of civil engineering, such as M5 model tree and neuro fuzzy inference system and other models are used to predict the mechanical properties of concrete with coarse recycled concrete aggregate [55] , the properties of glass fiber reinforced polymer [56] and the diffusion behavior of chloride ion in cement mortar [57] .

Conclusion and outlook
In this paper, empirical model and artificial intelligence model are summarized to predict the compressive strength of foamed concrete.The definition and application of each model are elaborated in detail, and the following conclusions are obtained: ①In the strength porosity model, only considering porosity cannot accurately predict the strength of foamed concrete.The effect of big pores on the strength is greater than that of small pores, and the accuracy of prediction can be improved to a certain extent by treating the big and small pores separately in the model.In engineering practice, the porosity is not an easy to measure amount, so the porosity is expressed by other easy to measure parameters, which can greatly facilitate the application of the formula in practice.
②The relation between strength and gel-space ratio comes from the strengthening effect of unhydrated clinker particles.The volume and chemical composition of hydrates produced by different cementing materials or auxiliary cementing materials are different, and the contributions of these different substances to compressive strength are also different.Subdivision of the contribution of each hydrate to the compressive strength from the microscopic point of view is conducive to improving the accuracy of the model.③MLR model is simple in structure and easy to apply.However, most of the influencing factors and strength are nonlinear, so the prediction accuracy is limited.In ANN model, the selection of hidden layers and hidden nodes is not conclusive, and the results obtained by different scholars are not the same.Different activation functions also affect the training and prediction of ANN model.SVR model avoids the problem of local extremum in ANN model, so that it can get the optimal solution in a limited range of data.
Based on the above conclusions, the following suggestions are offered for the study of foamed concrete.①The investigation of the microscopic mechanism affecting the strength of foamed concrete can provide a theoretical basis for the development of the model.②Replace the variables in the model with parameters that are easily measured in practice.③Improve the selection of prior conditions in ANN model and reduce the influence of prior conditions on strength prediction.④The strength prediction model is used to develop concrete with a high strength-density ratio, making it more in line with the requirements of environmental protection.

Figure 2 .
Figure 2. Comparison of four models.

Figure 3．
Figure 3．Comparison of three models for different constants.

Figure 4 . 36 Parameters
Figure 4. C-S-H orientation in space and Mohr-Coulomb failure criterion.It is assumed that as long as the C-S-H failure is the material failure, and the failure in one direction of C-S-H will lead to the complete failure of the material.According to the Mohr-Coulomb quasi-brittle failure criterion: max (,) {  (, ) + 1    (, )} ≤   eq.36 Parameters  and   are calculated by inverse analysis.  and   are the axial and shear stresses of a C-S-H in a given direction.Considering that the macro stress tensor applied on the cement paste is: ∑ = ∑  1 ⊗  1 max (uniaxial load in the direction of  1 ), the local stress of any C-S-H is calculated by the stress localization tensor (, ), then the average stress of the material is: σ(, ) = ∑ ∶ (, ) = ∑  1 ⊗  1 ∶ (, ) max =  0 (1 − ) 3.0 w/c=0.43，Therange of  0 in different ages :25.6Mpa-35.7 Mpa   =  0 (1 − ) 2.7 w/c=0.60，The range of  0 in different ages: 11.6Mpa-24.8Mpa--

Figure 5 .
Figure 5.The system used in the ANN model.

Figure 6 .
Figure 6.Operation mode of neurons in hidden layer.

1 1 :
− was selected as activation function to predict the density and 28 day compressive strength of foamed concrete.The value of  controls the slope of the linear part of the sigmoid function.The parameters  0 = 7.5 and  = 0.2 are determined by a series of numerical experiments.The activation function used in the ANN model is as follows:  = 15 ( 1 1 +  ;0.2 ) − 7.5 .50

Table 1．
Empirical model 0•  1 ⋀ 1  2 ⋀ 2……   ⋀  .47When the model is used to fit the data, the left and right sides of the equation are logarithmic, and 1,  1 ), ( 2,  2 )…( ,   ), the key of modeling is how to find a linear function in high dimensional space.()=   +  .51 In(Eq.42), the parameter  is called regression coefficient, and  is called regression deviation.The value of these two coefficients can be obtained by the principle of structural risk minimization.The structure risk function is defined as: