Multi-Objective Optimization Design of Foamed Cement Mix Proportion Based on Response Surface Methodology

Abstract

Foam cement, as a building insulation material, encounters a major problem in practical application, which is the difficulty in achieving a balance between its strength and insulation performance. To achieve multi-objective optimization of foamed cement mix design, this study first determined the optimal ranges of nano-silica aerogel (NSA), foaming agent, and polypropylene (PP) fiber dosage through single-factor experiments. Then, response surface methodology (RSM) was employed to construct a quadratic polynomial regression model, systematically investigating the influence of different NSA contents, foaming agent contents, and PP fibers contents on the thermal conductivity and compressive strength of foamed cement. Finally, the optimal mix ratio was further predicted and experimentally validated. The results demonstrate that the regression model developed using RSM exhibits high accuracy and reliability. The correlation coefficients R² of the regression models established by the response surface method are 0.9756 and 0.9684, respectively, indicating good prediction accuracy. The optimized mix ratio was determined as follows: NSA content, 9.548%; foaming agent content, 0.533%; and PP fiber content, 0.1%. Under this mix, the model predicted a thermal conductivity of 0.123 W/(m·K) and a 28-day compressive strength of 1.081 MPa. Experimental verification confirmed that the errors between predicted and measured values for all performance indicators were within 5%, demonstrating the high reliability of the predictive model. This study provides support for the practical application of foam cement as a thermal insulation material in construction projects and offers guidance for optimizing its mixture composition.

1. Introduction

Research indicates that approximately 40% of global energy consumption is attributed to the construction sector [1,2]. The energy efficiency of a building largely depends on its thermal insulation properties, and employing high-performance insulation materials can significantly lower energy usage in buildings [3,4,5]. Current building insulation materials commonly suffer from inadequate thermal insulation performance and insufficient mechanical strength. Therefore, there is an urgent need to enhance both the thermal insulation properties and structural strength of building envelopes. Nowadays, more and more nanoparticles are used in concrete [6,7,8,9,10]. Nano-SiO₂ aerogel has an extremely low thermal conductivity and is a stable and environmentally friendly insulation material. Therefore, it can be combined with foam cement. Nano-SiO₂ aerogel foamed cement, an innovative thermal insulation material, has gained increasing popularity in energy-efficient construction owing to its superior properties, including lightweight structure, excellent thermal insulation, fire resistance, and ease of application compared to conventional insulation materials. [11,12,13,14]. In their study, Pan et al. [15] introduced silica aerogel (AG) into foam concrete to prepare aerogel foam concrete (AGFC) with improved insulation performance, effectively reducing the thermal conductivity of the foam concrete by approximately 24%. Wu et al. [16] developed two aerogel-enhanced foam concrete composites by incorporating lightweight and ultra-lightweight adiabatic aerogel powders. Their study systematically examined how varying aerogel densities influence the foam concrete’s thermal conductivity, compressive strength, and bulk density. The foaming agent plays a crucial role in determining foamed concrete characteristics, as it directly governs the material’s insulation capacity, dry density, and mechanical strength [17,18,19]. Rodhia et al. [20] investigated the performance of foam concrete prepared with recycled fine aggregates from construction and demolition waste as a replacement for natural sand in their experiments, using a foaming agent mixed with water at different dilution ratios. Maglad et al. [21] investigated the interaction mechanisms between cement particles and anionic foaming agents, systematically examining the effects of four types of anionic surfactants on the fresh properties, mechanical behavior, microstructure, and transport characteristics of lightweight foamed concrete. Fibers have been widely adopted as reinforcement additives in construction materials [22,23]. When incorporated into foamed cement matrices, these randomly oriented three-dimensional fibers can effectively transform the material’s failure mode from brittle fracture to elastic-plastic deformation, thereby enhancing key mechanical properties including compressive strength, tensile capacity, and structural ductility [24]. Xu et al. [25] successfully prepared fiber-reinforced superhydrophobic foamed concrete by incorporating polypropylene (PP) fibers of three different lengths at dosage rates of 0.3%, 0.6%, 0.9%, and 1.2%, respectively, using a superhydrophobic concrete system modified with calcium stearate, polydimethylsiloxane, and hydroxypropyl methyl cellulose. Although these methods have accumulated certain experience in engineering practice, their inherent flaws have gradually become apparent. Firstly, the single-variable testing method cannot effectively reveal the coupling mechanism of multiple parameters, resulting in the risk of local optimization in the optimization scheme; secondly, accurately capturing the extreme points of material properties using the conventional analysis methods is difficult. More critically, seeking the global optimal ratio scheme under multi-objective optimization conditions has become a technical bottleneck that the current industry urgently needs to overcome.

Response surface methodology (RSM) is an efficient experimental design approach that enables the construction of mathematical models for multivariate interactions through systematic limited testing, allowing for precise prediction of optimal process parameters. This provides an effective solution to the aforementioned challenges. In recent years, RSM has demonstrated significant advantages in building materials science, particularly achieving breakthrough progress in the optimization of composite material formulations [26,27,28]. For instance, Wang Jingwen et al. [29] applied RSM to systematically analyze the synergistic influence mechanism of basalt fiber content and foaming ratio on the mechanical properties of foam concrete, successfully establishing a multi-objective optimization model. Similarly, Zhang et al. [30] employed this method to co-optimize the gradation curve and chemical admixture dosage of recycled aggregate pervious concrete, obtaining an ideal combination of material properties. Therefore, establishing relevant models based on RSM for multi-objective optimization represents a crucial pathway to transcend the empirical design limitations of existing mix proportions.

Polypropylene (PP) fiber , foaming agent, and nano-silica aerogel (NSA) content were chosen as factor factors in this investigation, while thermal conductivity and compressive strength after 28 days were used as response targets. To identify the ideal ranges of important parameters, single-factor experiments were initially carried out. Then, using the Box–Behnken design (BBD) and the RSM framework, with the goals of having a lower thermal conductivity and higher compressive strength, a quadratic polynomial regression model was created to methodically examine the interaction impacts of several parameters on foamed cement performance. The results are intended to support the practical use of foamed cement as a thermal insulation material in building engineering and serve as a guide for optimizing their mix composition.

2. Materials and Methods

2.1. Materials

As a cementitious material, regular Portland cement was employed.

Table 1. Chemical constituent of Portland cement (weight percentage).

SiO2	Al2O3	Fe2O3	CaO	MgO	SO3	Loss on Ignition
24.99	8.26	4.03	51.42	3.71	2.51	3.31

and

Table 2. Physical property parameters of Portland cement.

Setting Time/min		Compressive Strength/MPa		Flexural Strength/MPa
Initial setting	Final setting	3 d	28 d	3 d	28 d
172	234	27.2	54.5	5.5	8.7

display the cement’s chemical makeup and physical characteristics, respectively. The primary ingredient in the foam stabilizer was calcium stearate; the blowing agent was an animal protein foaming agent. ZhongNing Technology Co. (Shenzhen, China) manufactured the nano-SiO₂ aerogel. This test used nano-SiO₂ aerogel paste, which may be added directly for use and has been disseminated in water quality. The PP fiber is produced by Linxiang Building Materials Co., Ltd. (Changsha, China).

2.2. Single-Variable Experimental Approach

This study adopted a single-factor experimental approach to systematically investigate the influence of three key parameters—NSA, foaming agent, and PP fiber content—on the performance of foamed cement through controlled variable methodology. During the experimental process, only the dosage of the studied factor was varied while maintaining other components constant. Thermal insulation performance and 28-day compressive strength characteristics of each sample group were measured accordingly. Plot the correlation curves between experimental factors and response variables to identify the optimal dosage range for each parameter.

2.3. Response Surface Methodology Experimental Design

After determining the optimal ranges for NSA, foaming agent, and PP fiber content, the experimental design and data analysis were conducted using the Box–Behnken method in Design Expert 13 software based on Response Surface Methodology. The three elements served as input variables, denoted as A (NSA content), B (blowing agent dosage), and C (PP fiber content), while thermal conductivity (Y₁) and 28-day compressive strength (Y₂) of foamed cement were selected as response variables. The tested ranges were 6–8% for NSA content (A), 0.46–0.66% for foaming agent dosage (B), and 0.1–0.3% for PP fiber content (C). Three levels were assigned for each factor, with the center point coded as 0, while the minimum and maximum values were coded as −1 and +1, respectively. The coded levels of all parameters are presented in

Table 3. Coded level design of mix proportion parameters.

Code	Influence Factor	Code Level
		−1	0	1
A	NSA content/%	6	8	10
B	Foaming agent content/%	0.46	0.56	0.66
C	PP fiber content/%	0.1	0.2	0.3

. The experimental design comprised 17 randomized test points. To enhance result reliability, five replicate tests were performed at the center point (8, 0.56, 0.2) to evaluate experimental error. Figure 1 displays the test point configuration for the implemented Box–Behnken design with three variables. The experimental results were analyzed using second-order response surface methodology for functional fitting, with the second-order polynomial function expression shown in Equation (1) [31,32].

$$\mathrm{Y}={\mathrm{\beta }}_0+{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{(\mathrm{\beta }}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}})+{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{(\mathrm{\beta }}_{\mathrm{i}\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}^2)+{\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\sum }_{\mathrm{j}=1}^{\mathrm{n}}({\mathrm{\beta }}_{\mathrm{i}\mathrm{j}}{\mathrm{x}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{j}})+\mathrm{\epsilon }$$

(1)

where $\mathrm{Y}$ is the response value, ${\mathrm{\beta }}_0$ is the constant term; ${\mathrm{x}}_{\mathrm{i}}{,\mathrm{x}}_{\mathrm{j}}$ are independent variables, ${\mathrm{\beta }}_{\mathrm{i}}$ is the coefficient of the linear term, ${\mathrm{\beta }}_{\mathrm{i}\mathrm{i}}$ is the coefficient of the quadratic term, ${\mathrm{\beta }}_{\mathrm{i}\mathrm{j}}$ is the coefficient of the interaction term, $\mathrm{\epsilon }$ is the random error, and n is the number of variables.

ANOVA must be performed on the regression model to establish term significance, while model validity is determined using the coefficient of determination (${\mathrm{R}}^2$) and ${\mathrm{R}}_{\mathrm{a}}^2$ [33].

$${\mathrm{R}}^2=1-{\displaystyle \frac{{\mathrm{S}}_{\mathrm{r}}}{{\mathrm{S}}_{\mathrm{m}}+{\mathrm{S}}_{\mathrm{r}}}}$$

(2)

$${\mathrm{R}}_{\mathrm{a}}^2=1-{\displaystyle \frac{{\mathrm{S}}_{\mathrm{r}}/{\mathrm{D}}_{\mathrm{r}}}{({\mathrm{S}}_{\mathrm{m}}+{\mathrm{S}}_{\mathrm{r}})/({\mathrm{D}}_{\mathrm{m}}+{\mathrm{D}}_{\mathrm{r}})}}$$

(3)

where ${\mathrm{S}}_{\mathrm{r}}$ represents residual sum of squares, ${\mathrm{S}}_{\mathrm{m}}$ represents regression sum of squares, ${\mathrm{D}}_{\mathrm{r}}$ is residual degrees of freedom, and ${\mathrm{D}}_{\mathrm{m}}$ is regression degrees of freedom.

Following successful response surface modeling, numerical optimization methods enable determination of ideal factor level settings. The current investigation implements desirability function methodology for simultaneous response optimization, with single-response desirability (${\mathrm{d}}_{\mathrm{i}}$) and overall desirability ($\mathrm{D}$) expressed as [34]:

$${\mathrm{d}}_{\mathrm{i}}=\left\{\begin{array}{c}0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{Y}}_{\mathrm{i}}\le \mathrm{L}\\ {\hspace{1em}\left[{\displaystyle \frac{{\mathrm{Y}}_{\mathrm{i}}-\mathrm{L}}{\mathrm{U}-\mathrm{L}}}\right]}_1^{{\mathrm{W}}_{\mathrm{i}}},\hspace{1em}\hspace{1em}\mathrm{L}<{\mathrm{Y}}_{\mathrm{i}}<\mathrm{U}\\ 1,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{Y}}_{\mathrm{i}}\ge \mathrm{U}\end{array}\right.$$

(4)

$$\mathrm{U}={\mathrm{d}}_1^{{\mathrm{W}}_1}{\mathrm{d}}_2^{{\mathrm{W}}_2}$$

(5)

where ${\mathrm{Y}}_{\mathrm{i}}$ is the predicted value of the $\mathrm{i}$th response, $\mathrm{L}$ and $\mathrm{U}$ are the lower and upper limits of the target range, respectively, and ${\mathrm{W}}_{\mathrm{i}}$ is the weight coefficient. The value of $\mathrm{D}$ ranges from 0 to 1, with values closer to 1 indicating higher overall desirability and greater reliability of the resulting optimal parameter combination.

2.4. Sample Preparation

The experimental procedure, illustrated in Figure 2, involved the following steps: All components were precisely weighed according to predetermined ratios. The dry ingredients (cement, foam stabilizer, nano-SiO₂ aerogel (NSA), and polypropylene fibers) were initially dry-mixed in a mortar mixer for 2 min. Subsequently, two-thirds of the designated water content was gradually added and mixed for an additional 2 min to achieve a homogeneous NSA-cement slurry. The remaining water was then incorporated with 3 min of mixing. Simultaneously, the blowing agent was mixed with water at a 1:40 ratio and processed using a high-velocity mixer (2 min blending → 15 s standing → 1 min final mixing) to eliminate wall-adhered foam. The prepared foam was immediately (<1 min) combined with the cement slurry. The composite mixture was cast into molds, manually vibrated, and sealed with plastic film. Following 48 h of ambient curing, specimens were demolded and transferred to a 28-day standard curing period prior to testing.

2.5. Experimental Methods

In accordance with the JG/T 266-2011 “Foamed Concrete” standard [35], thermal conductivity testing was conducted using triplicate specimens (300 mm × 300 mm × 30 mm) for each mixture formulation. Prior to analysis, all samples underwent surface grinding and were conditioned in a forced-air drying oven at 60 °C to achieve consistent testing conditions. The mass was measured at 4-h intervals until the difference between consecutive measurements fell below 1g, indicating that the specimens had reached absolute dryness. Immediately after drying is completed, place it in the JTRG-III thermal conductivity tester. This tester is produced by Beijing Century Jian Tong Technology Co., Ltd(Beijing, China). The space where the test block is placed is completely sealed. Then, adjust the temperatures of the hot plate and the cold plate, and take readings when the heat flow stabilizes.

Compressive strength testing followed the JG/T 266-2011 “Foamed Concrete” standard [35]. After 28 days of standard curing, compressive strength was determined using 40 mm × 40 mm × 40 mm cubic specimens. The dry density range of the test blocks obtained is 400–600 kg/m³. Six specimens were tested for each group, with the average value representing the compressive strength of that mix proportion group.

3. Results and Discussion

3.1. Single-Factor Analysis

3.1.1. Effect of NSA Content

The influence of different NSA contents on foamed cement properties is shown in Figure 3. As illustrated in Figure 3a, the thermal conductivity of foamed cement gradually decreased with increasing NSA content. When the NSA content went up from 0% to 10%, thermal conductance reduced to 0.186 W/(m·K), representing a 30.9% decrease. The reason for this lies in the fact that NSA has a short distance between its fine particles, which is conducive to hindering heat conduction. Furthermore, the nano-silica aerogel’s (NSA) porous architecture contains micropores that effectively inhibit gaseous and thermal conduction, thereby enhancing its insulating properties [36,37]. Figure 3b reveals that the compressive strength of foamed cement showed an original rise followed by a reduction with higher NSA content. At 6% NSA content, the 28-day compressive strength showed a 39.6% improvement. This enhancement occurs because NSA particles act as fillers that densify the cement matrix by occupying voids and pores, thereby improving structural compactness and stability [38,39]. However, excessive NSA content (>6%) led to particle agglomeration, which weakened the bond within the cementitious matrix and created a looser matrix-phase structure, ultimately compromising the mechanical performance of the foamed specimens.

3.1.2. Effect of Foaming Agent Content

The influence of different dosages of foaming agent on the performance of foamed cement is shown in Figure 4. As can be seen from Figure 4a, with the increase in foaming agent dosage, the thermal conductivity of foamed cement gradually decreases. When the foaming agent dosage increases from 0.26% to 0.66%, the thermal conductivity of foamed cement reduces to 0.121 W/(m·K), a decrease of approximately 31.02%. This is because the increased dosage of foaming agent leads to higher bubble density in the cementitious slurry, and these bubbles form a uniformly distributed non-interconnected pore architecture within the cement matrix, resulting in significantly enhanced porosity of the foamed cement. These pores hinder the heat transfer within the material, thereby greatly reducing its thermal conductivity. As shown in Figure 4b, the compressive strength of foamed cement gradually decreases with the increase in foaming agent dosage. When the foaming agent dosage is 0.66%, the 28-day compressive strength of foamed cement decreases by 73.6%. This is because these introduced pores create microstructural discontinuities and defect sites, particularly compromising the material’s flexural and compressive strength characteristics.

3.1.3. Effect of PP Fiber Content

Figure 5 demonstrates the effect of varying polypropylene fiber dosage on the properties of foamed cement . As can be seen from Figure 5a, with the increase in PP fiber dosage, the heat conduction coefficient of foamed cement first enhances and then decreases. The experimental results demonstrate an optimal polypropylene (PP) fiber content of 0.4%, achieving a thermal conductivity of 0.162 W/(m·K). While PP fibers inherently exhibit low thermal conductivity [40,41], their incorporation in nano-silica aerogel foamed cement creates complex thermal effects. At low concentrations, dispersed fibers may establish thermal bridges, whereas higher concentrations (0.4%) modify the composite’s microstructure, potentially increasing density while decreasing porosity, thereby extending heat transfer pathways. As illustrated in Figure 5b, the compressive strength follows a characteristic rise-and-fall pattern with increasing fiber content. The initial strength enhancement stems from the three-dimensional fiber network’s crack-arresting capability. However, beyond the critical 0.2% threshold, excessive fibers induce microstructural heterogeneity, ultimately compromising both flexural and compressive performance [42,43]. This transition corresponds with the weakening reinforcement efficiency observed at elevated fiber concentrations.

3.2. Response Surface Modeling and Analysis

3.2.1. Model Development and Analysis of Variance

Table 4. Design and results of response surface methodology experiments.

Sample No.	Factor and Level			Thermal Conductivity/W/(m·K)		28d Compressive Strength/MPa
	A/%	B/%	C/%	Tested Level	Predicted Level	Tested Level	Predicted Level
1	8 (0)	0.46 (−1)	0.3 (1)	0.128	0.129	1.08	1.06
2	6 (−1)	0.66 (1)	0.2 (0)	0.116	0.115	0.8	0.9
3	8 (0)	0.56 (0)	0.2 (0)	0.118	0.116	0.99	1.02
4	6 (−1)	0.56 (0)	0.1 (−1)	0.131	0.131	1.22	1.21
5	10 (1)	0.46 (−1)	0.2 (0)	0.136	0.138	1.31	1.29
6	8 (0)	0.66 (1)	0.3 (1)	0.103	0.104	0.65	0.68
7	8 (0)	0.56 (0)	0.2 (0)	0.12	0.121	0.93	0.94
8	10 (1)	0.56 (0)	0.3 (1)	0.126	0.128	1.05	1.08
9	8 (0)	0.46 (−1)	0.1 (−1)	0.138	0.137	1.36	1.37
10	8 (0)	0.66 (1)	0.1 (−1)	0.106	0.105	0.66	0.68
11	8 (0)	0.56 (0)	0.2 (0)	0.114	0.113	0.87	0.87
12	10 (1)	0.56 (0)	0.1 (−1)	0.119	0.117	0.98	0.97
13	8 (0)	0.56 (0)	0.2 (0)	0.117	0.119	0.89	0.91
14	10 (1)	0.66 (1)	0.2 (0)	0.112	0.112	0.78	0.77
15	6 (−1)	0.56 (0)	0.3 (1)	0.123	0.124	1.01	1.02
16	8 (0)	0.56 (0)	0.2 (0)	0.119	0.120	0.82	0.83
17	6 (−1)	0.46 (−1)	0.2 (0)	0.142	0.140	1.31	1.29

presents a comparative analysis between the measured and predicted values of thermal conductivity and 28-day compressive strength for foamed cement based on the experimental design outlined in Table 3. The data demonstrate strong agreement between experimental and predicted values.

The experimental results presented in Table 4 were analyzed through regression modeling with Design-Expert 13, generating distinct response surface equations for each performance parameter: thermal conductivity and 28-day compressive strength.

$${\mathrm{Y}}_1=0.1176-0.0024\mathrm{A}-0.0134\mathrm{B}-0.0018\mathrm{C}+0.0005\mathrm{A}\mathrm{B}+0.0038\mathrm{A}\mathrm{C}+0.0018\mathrm{B}\mathrm{C}+0.0074{\mathrm{A}}^2+0.0014{\mathrm{B}}^2-0.0003{\mathrm{C}}^2$$

(6)

$${\mathrm{Y}}_2=0.9000-0.0275\mathrm{A}-0.02713\mathrm{B}-0.0538\mathrm{C}-0.0005\mathrm{A}\mathrm{B}+0.0700\mathrm{A}\mathrm{C}+0.0675\mathrm{B}\mathrm{C}+0.138{\mathrm{A}}^2+0.0113{\mathrm{B}}^2+0.0263{\mathrm{C}}^2$$

(7)

The accuracy of the developed regression models was evaluated through analysis of variance (ANOVA), with detailed results presented in

Table 5. Variance analysis of thermal conductivity model.

Source Code	Sum of Squares	Df	Mean Square	F-Value	p-Value	Significance
Model	0.0018	9	0.0002	31.12	<0.0001	Significance
A	0.000	1	0.0000	6.95	0.0336	Significance
B	0.0014	1	0.0014	220.42	<0.0001	Significance
C	0.000	1	0.0000	3.77	0.0932	Not significance
AB	1.6 × 10−6	1	1.6 × 10−6	0.1540	0.7064	Not significance
AC	0.0001	1	0.0001	8.66	0.0216	Significance
BC	0.0000	1	0.0000	1.89	0.2119	Not significance
A2	0.0002	1	0.0002	35.99	0.0005	Significance
B2	8.8 × 10−6	1	8.8 × 10−6	1.36	0.2812	Not significance
C2	3.7 × 10−7	1	3.7 × 10−7	0.0584	0.8160	Not significance
Residual	0.0000	7	6.49 × 10−6	-	-	-
Lack of fit	0.0000	3	8.08 × 10−6	1.53	0.3376	Not significance
Cor total	0.0019	16	-	-	-

and

Table 6. ANOVA evaluation of the 28 d compressive strength regression model.

Source Code	Sum of Squares	Df	Mean Square	F-Value	p-Value	Significance
Model	0.7433	9	0.0826	23.81	0.0002	Significance
A	0.0060	1	0.0060	1.74	0.2281	Not significance
B	0.5886	1	0.5886	169.73	<0.0001	Significance
C	0.0231	1	0.0231	6.66	0.0364	Significance
AB	0.0001	1	0.0001	0.0288	0.8700	Not significance
AC	0.0196	1	0.0196	5.65	0.0491	Significance
BC	0.0182	1	0.0182	5.26	0.0556	Not significance
A2	0.0811	11	0.0811	23.37	0.0019	Significance
B2	0.0005	1	0.0005	0.1537	0.7067	Not significance
C2	0.0029	1	0.0029	0.8366	0.3908	Not significance
Residual	0.0243	7	0.0035	-	-
Lack of fit	0.0079	3	0.0026	0.6402	0.6280	Not significance
Cor total	0.7676	16	-	-	-

. The ANOVA assessed the predictive capability by examining the statistical significance of both the overall regression model and its constituent terms, including a comprehensive analysis of p-value and F-values for lack-of-fit, linear, quadratic, and interaction terms. This evaluation determined both model reliability and the significant effects of individual factors on response variables. From a statistical perspective, the F-value is negatively correlated with the p-value—that is, the larger the F-value, the smaller the p-value, reflecting a stronger model significance. If the p-value is <0.05, it indicates a highly significant model, and the regression model is reliable; otherwise, it indicates that the regression model does not have statistical significance. Notably, in lack-of-fit testing, p-values reflect the significance of discrepancy between experimental data and the model; values exceeding 0.05 indicate good model–data agreement.

Statistical analysis of the thermal conductivity regression model (Table 5) revealed highly significant results (p < 0.0001, F = 31.12), demonstrating excellent model fit. The non-significant lack of fit term (p = 0.3376 > 0.05) confirmed minimal model error and good agreement with experimental data. Factor significance testing indicated that both NSA content and foaming agent concentration (p < 0.05) significantly influenced thermal conductivity, whereas PP fiber content showed no statistically significant effect (p > 0.05). This indicates that the NSA and foaming agent dosages have a significant impact on the thermal conductivity of the foamed cement, while the influence of the PP fiber dosage is not significant. The two-factor interaction analysis shows that only the interaction between the NSA dosage and the PP fiber dosage is significant. The degree of influence of the quadratic term can also be judged for significance based on the p value. The single-factor quadratic effect has the greatest influence on the slump, which is the NSA dosage, while the quadratic effects of the foaming agent dosage and the PP fiber dosage are not significant.

As shown in Table 6, the p value of the quadratic polynomial regression equation for the 28-day compressive strength is 0.0002, and the F value is 23.81, indicating extremely strong significance; the p value of the lack of fit term is 0.6280, which is >0.05, reflecting that the model’s lack of fit is not prominent and the fitting accuracy is good. The single-factor analysis indicated that there were significant differences in the degree of influence of each factor on the 28-day compressive strength. The dosage of the foaming agent showed a highly significant effect, followed by the dosage of PP fiber. The dosage of NSA did not show a significant influence. In the two-factor interaction, the interaction between the dosage of NSA and the dosage of PP fiber was very significant, while the interactions between the dosage of NSA and the dosage of foaming agent, as well as between the dosage of foaming agent and the dosage of PP fibers, were not significant. The quadratic effect of NSA dosage had the best influence on the 28-day compression strength, while the quadratic effects of the dosage of blowing agent and the dosage of PP fibers were not significant.

As shown in

Table 7. Results of model reliability tests.

Regression Model	R2	Ra2	Rp2	C.V./%	Adeq Precision
Y1	0.9756	0.9443	0.7741	2.09	19.1244
Y2	0.9684	0.9277	0.8025	5.99	15.3048

, the comprehensive evaluation metrics for model reliability include the correlation coefficient (${\mathrm{R}}^2$), adjusted coefficient (${\mathrm{R}}_{\mathrm{a}}^2$), predicted coefficient (${\mathrm{R}}_{\mathrm{p}}^2$), coefficient of variation (C.V.), and signal-to-noise ratio (Adeq Precision). Specifically, ${\mathrm{R}}^2$ indicates the agreement between predicted and experimental values, with higher values representing greater predictive accuracy. A smaller difference between ${\mathrm{R}}_{\mathrm{a}}^2$ and ${\mathrm{R}}_{\mathrm{p}}^2$ suggests better regression equation fitting and more stable model performance. Additionally, when the C.V. is less than 10% and the Adequate Precision exceeds 4, the experimental data demonstrate high reliability and precision. The results in Table 7 reveal that for thermal conductivity and 28-day compressive strength, the ${\mathrm{R}}^2$ values between experimental and predicted values were 0.9756 and 0.9684, respectively; the ${\mathrm{R}}_{\mathrm{a}}^2$ values were 0.9443 and 0.9277, respectively; the C.V. values were 2.09% and 5.99%, respectively; and the Adequate Precision values were 19.1244 and 15.3048, respectively. These results collectively demonstrate that the regression models for both thermal conductivity and compressive strength exhibit excellent predictive performance, with strong agreement between model fits and experimental data, indicating high reliability.

The studentized residuals, obtained by dividing raw residuals by their standard deviation, serve as a statistical measure to assess the normality assumption of residual distribution. Figure 6 presents the studentized residual plots, where the residual data points for both thermal conductivity and 28-day compressive strength are uniformly distributed on both sides of the reference line. This distribution pattern confirms that the ANOVA results of the regression models exhibit excellent fitting accuracy, thereby validating the significant relationships between the regression model equations and the three key parameters: NSA, foaming agent, and PP fiber content.

As illustrated in Figure 7, the comparative analysis between experimental and predicted values for both thermal conductivity and 28-day compressive strength demonstrates a linear distribution of data points along the 45° reference line with excellent agreement. This alignment confirms the high predictive reliability of the developed regression models for both thermal and mechanical properties.

3.2.2. Response Surface Analysis

The 3D response surface and the corresponding contour map can quantitatively characterize the influence mechanism of the interaction between two factors on the thermal conductivity and 28-day compressive strength of foamed cement. The interaction strength between factors is reflected in the contour shape: significant for elliptical, minimal for circular. Meanwhile, the density of the intersection points of the contour lines with the coordinate axes can reflect the degree of influence of each factor on the response value: the denser the intersection points, the more significant the influence of that factor.

Figure 8 presents the 3D response surface and contour plots, illustrating the pairwise interactive effects on thermal conductivity. The figure displays the influence of the interaction between the other two factors on the thermal conductivity when the third factor is at the medium-level coding 0. From Figure 8a, it can be seen that when the content of PP fibers is 0.1%, as the content of NSA increases, the decrease in the thermal conductivity of the foamed cement is relatively small, while as the content of the foaming agent increases, the thermal conductivity of the foamed cement shows a significant decreasing trend. At the same time, according to Table 5, the P value of the interaction between the dosage of NSA and the dosage of the foaming agent is 0.7064, which is greater than 0.05, indicating that the interaction between the dosage of NSA and the dosage of PP fibers on the thermal conductivity is not significant. Figure 8b demonstrates that when the dosage of the foaming agent is constant at 0.56% and the dosage of NSA is greater than 8%, the thermal conductivity increases with the increase in the dosage of NSA and PP fibers. At the same time, according to Table 5, the P value of the interaction between the dosage of NSA and the dosage of the PP fiber is 0.0216, which is less than 0.05, indicating that the interaction between the content of NSA and the dosage of PP fibers on the thermal conductivity is relatively significant. From Figure 8c, it can be known that when the content of NSA is 8% and the content of PP fibers increases, the thermal conductivity of the foamed cement does not change significantly, while the increase in the dosage of the foaming agent has a more pronounced effect on the thermal conductivity of the foamed cement. Therefore, the interaction effect between the dosage of the foaming agent and the dosage of PP fibers is not significant. This conclusion is consistent with the statistical test results of the P value obtained from the model regression analysis.

In summary, among the three factors of NSA, foaming agent, and PP fiber content, the significance ranking of the variables affecting the thermal conductivity is: foaming agent > NSA > PP fiber content.

Figure 9 shows the 3D response surface plot and the corresponding contour plot of the pairwise interaction effects of the compressive strength at 28 days. The figure displays the influence of the mutual effect between the other two factors on the 28-day compressive strength when the third factor is at the medium-level coding of 0. From Figure 9a, it can be known that when the dosage of PP fibers is 0.1%, as the dosage of NSA increases, the reduction in the 28-day compressive strength of the foamed cement is relatively small, while as the dosage of the foaming agent increases, the 28-day compression strength of the foamed cement shows a significant decreasing trend. At the same time, according to Table 6, the P value of the interaction between NSA and the dosage of the foaming agent is 0.8700, which is greater than 0.05, indicating that the interaction between NSA and the dosage of foaming agent has no significant influence on the 28-day compressive strength. From Figure 9b, it can be seen that when the dosage of the foaming agent is constant at 0.56% and the dosage of NSA is greater than 8%, the 28-day compressive strength increases with the increase in the dosage of NSA and PP fibers. At the same time, according to Table 6, the P value of the interaction between NSA content and the dosage of the PP fiber is 0.0491, which is less than 0.05, reflecting that the mutual effect between NSA content and the content of PP fibers has a significant effect on the coefficient of heat conductivity. From Figure 9c, it can be known that when the content of NSA is 8%, as the content of PP fibers increases, the 28-day compressive strength of the foamed cement slightly increases, while the increase in the content of the foaming agent has a more pronounced effect on the 28-day compressive strength of the foamed cement. Therefore, the interaction effect between the dosage of the foaming agent and the content of PP fibers is not significant. This conclusion is consistent with the statistical test results of the P value obtained from the model regression analysis.

In summary, among the three factors of NSA, foaming agent, and PP fiber content, the significance ranking of the variables affecting the 28-day compressive strength is: foaming agent > PP fiber > NSA content.

3.3. Multi-Objective Optimization and Model Validation

The desirability function distribution was generated through multi-objective optimization analysis using Design-Expert 13 software (Figure 10). This visualization reveals the correlation between factor levels and desirability values, offering valuable information about parameter–performance relationships. The optimization results identified an ideal composition with 9.548% nano-silica aerogel (NSA) content, 0.533% foaming agent dosage, and 0.1% polypropylene (PP) fiber content.

Figure 11 presents the slope analysis results, visualizing the predicted performance metrics for all response variables at the optimized mixture composition. The predicted values are as follows: thermal conductivity is 0.123 W/(m·K), and a 28-day compressive strength is 1.081 MPa. The comprehensive optimization effect is shown in Figure 12.

To validate the reliability of the optimization results, performance tests were conducted based on the optimal mix proportion.

Table 8. Comparative analysis between RSM model predictions and empirical results.

	Thermal Conductivity/W/(m·K)	28 d Compressive Strength/MPa
Predicted value	0.123	1.081
Measured value1	0.119	1.125
Measured value2	0.121	1.118
Measured value3	0.125	1.131

presents a comparative analysis between laboratory-measured results and model-predicted values. The data reveal minor discrepancies between predicted and experimental values: the average error was 2.2% for thermal conductivity and 3.88% for 28-day compressive strength. Critically, all performance metrics exhibited errors below 5%, confirming the high reliability of the predictive model.

4. Discussion

The method proposed in this study for optimizing the mix ratio of foam cement aims to simultaneously enhance its thermal insulation and strength. The unit price of NSA is relatively high but the dosage is relatively small, resulting in a slight increase in cost. Additionally, this technology is fully compatible with the existing processes and does not require the addition of new equipment. The “high performance—low cost—easy implementation” characteristics make this mix ratio suitable for large-scale construction applications. The practical significance of this research lies in the development of a new type of optimized foam cement formula. Through the use of various admixtures, the key problem of difficult balance between strength and thermal insulation performance in building insulation materials has been successfully addressed. This research not only provides a scientific method for optimizing the performance of building materials, but also holds significant engineering value for achieving the goal of energy conservation and consumption reduction in buildings.

Subsequent studies plan to employ advanced computer vision techniques to quantitatively analyze the relationship between microstructure and performance. Two types of deep learning models will be particularly introduced: (1) DeepLabv3+: a semantic segmentation architecture based on dilated convolution and spatial pyramid pooling, which can accurately identify multi-scale pore morphology and is particularly suitable for distinguishing closed pores from connected pore structures; (2) EfficientNet: a convolutional neural network that optimizes computational efficiency through compound scaling, capable of efficiently counting fiber distribution orientation and local aggregation features [44,45]. By coupling SEM image analysis with these algorithms, quantitative prediction models for porosity/fiber orientation parameters and thermal conductivity, compressive strength can be established, providing microscopic mechanism explanations for material design.

5. Conclusions

(1) The quadratic polynomial regression model developed using Response Surface Methodology (RSM) effectively characterizes the nonlinear effects of NSA content, foaming agent dosage, and PP fiber dosage on both the thermal conductivity and 28-day compressive strength of foamed cement. The model demonstrates high accuracy and reliability, with correlation coefficients (${\mathrm{R}}^2$) reaching 0.9756 and 0.9684 for thermal conductivity and compressive strength, respectively.

(2) The addition amount of foaming agent has the most significant impact on the thermal conductivity and 28-day compressive strength (p < 0.0001). Increasing the foaming agent addition amount can effectively reduce the thermal conductivity of foamed cement; however, the loss of compressive strength needs to be considered, while the interaction effect between the NSA addition amount and the PP fiber addition amount has a significant impact on the thermal conductivity and 28d compressive strength.

(3) RSM optimization identifies the optimal mix proportion as 9.548% NSA content, 0.533% foaming agent dosage, and 0.1% PP fiber dosage. Under this formulation, the model predicts a coefficient of heat conductivity of 0.123 W/(m·K) and a 28-day compressive strength of 1.081 MPa.

(4) Experimental validation confirms strong agreement between predicted and measured values, with all errors remaining below 5%. This outcome robustly validates the model’s reliability for practical applications.