Preliminary Optimization of Steady-State and Dynamic Thermal Performance of 3D Printed Foamed Concrete

Abstract

The integration of Foamed Concrete (FC) into 3D Concrete Printing (3DCP) processes facilitates the design of energy-efficient building envelopes. However, strategies for optimizing material porosity and printing topology to balance winter and summer performance remain underexplored. This study presents a 2D numerical thermal analysis of an innovative 3D-printed building envelope block characterized by sinusoidal internal partitions. Through a parametric variation in porosity (ranging from 10% to 50%) and internal geometry (amplitude and period of the partitions), 45 distinct configurations were simulated. Performance was evaluated by calculating the steady-state thermal transmittance (U) and the periodic thermal transmittance (Y_ie) under dynamic climatic conditions. The results demonstrate that porosity is the governing parameter; increasing porosity from 10% to 50% reduces U by 31% and, contrary to traditional assumptions for massive structures, also improves Y_ie by 12.3%. These outcomes are physically driven by the drastic reduction in thermal conductivity, which overcompensates for the loss of thermal mass, leading to a net reduction in overall thermal diffusivity. While internal topology plays a secondary role, its optimization allows for fine-tuning dynamic damping without compromising insulation. The study confirms that 3D printing with foamed concrete enables the overcoming of the traditional trade-off between insulation and thermal inertia. High-porosity configurations (50%) with optimized internal topology emerge as the most effective solution, simultaneously guaranteeing beneficial steady-state and dynamic thermal performance for sustainable buildings.

1. Introduction

In recent years, the traditionally conservative construction industry has witnessed a significant acceleration towards digitalization and automation. In this context, Digital Fabrication with Concrete (DFC) and, more specifically, 3D Concrete Printing (3DCP), have emerged as promising technologies for the realization of large-scale structures [1]. Interest in Additive Manufacturing (AM) has grown exponentially due to the substantial advantages it offers, including reduced construction times, containment of labor costs, decreased dependence on conventional construction equipment, and, crucially, the elimination of traditional formwork, allowing for unprecedented geometric freedom [2]. Concurrently, the demand for energy-efficient building envelopes has renewed interest in advanced cementitious materials that offer intrinsic thermal performance. Foamed Concrete (FC) stands out as a versatile material, characterized by a cellular structure obtained by entrapping air bubbles within the cementitious matrix [3]. With a variable density typically ranging from 200 to 1900 kg/m³, FC offers unique properties that combine structural capacity with excellent thermal insulation and fire resistance [3,4]. The integration of FC into 3DCP processes facilitates the creation of monolithic wall elements that may eliminate the need for additional insulation layers, significantly simplifying the construction process and enhancing both the economic and ecological sustainability of the building [3]. However, despite its recognized potential, thermal research on printable foamed concrete is still in a relatively early stage. Most existing studies have focused predominantly on optimizing mix designs to ensure printability, flowability, and structural stability during the layer-by-layer deposition process, or on characterizing mechanical properties as a function of density [3,4]. Conversely, the thermal behavior of these printed envelopes remains less explored. Accurate characterization of thermal properties in such heterogeneous materials is complex, with divergent hypotheses existing regarding heat transfer modelling. While some simplified approaches consider the material as a homogeneous medium, advanced research highlights the necessity of accounting for void distribution and microscopic heat transfer phenomena, including intra-pore radiation and convection, which vary with density and temperature [5,6,7]. Beyond material properties, the geometric freedom offered by AM allows for a radical redesign of internal morphology to maximize thermal resistance. Optimizing air cavity configurations within walls is a known strategy for reducing heat transfer, though it introduces a trade-off between thermal and mechanical performance [8]. Leveraging formwork-free 3D printing enables the realization of complex internal structures, such as sinusoidal partitions, specifically proposed to mitigate convective heat transfer and minimize thermal bridges [9]. Despite these advancements, there remains a gap in research dedicated to the integrated optimization of mixture porosity and geometric printing layouts (infill patterns) specifically targeting dynamic thermal performance. Designing elements for passive thermal regulation requires balancing conflicting physical requirements: minimizing steady-state thermal transmittance ($U$) for winter heat retention while maximizing thermal inertia to reduce periodic thermal transmittance ($Y_{ie}$) for summer comfort. This study addresses this gap by performing a 2D-numerical thermal analysis on an innovative 3D-printed foamed concrete building envelope block (standard dimensions 50 × 12 × 20 cm). Through parametric variation in density (and consequently conductivity) and internal topology, and utilizing an exhaustive search algorithm, this work aims to identify optimal design configurations. The objective is to resolve the trade-off between winter heat conduction and summer thermal wave damping, thereby defining optimal solutions that guarantee high insulation performance and contribute significantly to the development of sustainable, energy-efficient buildings realized via additive technologies.

2. Materials and Methods

This study investigates the thermal performance of an innovative building envelope block characterized by sinusoidal internal partitions. The complex internal geometry is made feasible through 3D concrete printing (3DCP) technology, which allows for formwork-free manufacturing of non-standard shapes [8]. To assess the energy efficiency of the proposed component, a 2D numerical analysis was conducted to determine the thermal transmittance (U) and the periodic thermal transmittance (Y_ie), subject to specific boundary conditions and geometric constraints.

2.1. Geometric Configuration and 2D Parametric Design

The reference element for this study is a modular building block with overall dimensions of 500 mm × 120 mm × 200 mm (L × H × W), designed to be compatible with standard construction modules. Given the computational complexity of fully simulating dynamic heat transfer in 3D geometries and the necessity of performing an extensive parametric sweep, the analysis was simplified to a two-dimensional (2D) cross-section (see Figure 1).

The printing process imposes specific geometric constraints; specifically, a constant width of 40 mm (s) was assumed for the external walls and internal partitions, consistent with the nozzle dimensions of large-scale construction printers [10]. To enhance thermal performance, the internal topology replaces traditional rectangular cavities with two sinusoidal partitions. Preliminary evaluations indicated that alternative continuous patterns, e.g., zigzag, yield equivalent thermal performance, provided they maintain comparable cavity dimensions and do not create a direct connection with the external walls. This configuration is generated using 3D printing technology, which allows for the continuous deposition of non-standard geometries without formwork. The geometry of the 2 internal sinusoidal septa is governed by the following harmonic function:

$$f\left(x\right)=A_1\cdot \sin \left({\displaystyle \frac{2\pi \cdot x}{P_1}}+{\varphi }_1\right)$$

(1)

where:

• $x$ represents the coordinate along the length of the block ($L$).
• $A_1$ is the amplitude of the sine wave, determining the transverse extension of the partition.
• $P_1$ is the period, controlling the frequency of the internal oscillations.
• ${\varphi }_1$ is the phase shift, set to 0.5 to ensure geometric symmetry and printability.

A critical aspect of the thermal analysis is the accurate evaluation of air convection heat transfer. While radiation is modelled using surface-to-surface radiosity, natural convection requires a specific approach to balance accuracy and computational efficiency. Instead of solving the full Navier–Stokes equations for every geometric iteration, a preliminary set of 2D Computational Fluid Dynamics (CFD) simulations was conducted in the cross-section of the concrete block, i.e., x-z plane, resembling the benchmark rectangular cavity scenario. The height of the cavity was set to be equal to H, and the width was spanned from the minimum to the maximum values of the air hollow shown in Figure 1.

These preliminary simulations were performed under the specific boundary conditions of a differentially heated cavity (steady-state) to characterize the convective flow regimes. From these results, an effective thermal conductivity of air ($k_{air,eff}$) was derived. This parameter effectively lumps the convective heat transfer mechanism into a conductive term, allowing the air domains to be modelled as solid materials with enhanced conductivity. This methodology ensures that the impact of natural convection is carefully accounted for while significantly reducing the computational cost of the optimization process. The values of $k_{air,eff}$ derived from the preliminary simulations were interpolated using a piecewise cubic Hermite interpolating polynomial, as shown in Figure 2. This function describes $k_{air,eff}$ as a continuous function of temperature, ensuring a smooth transition of properties during the dynamic thermal analysis.

To assess the validity of the assumption adopted for the effective thermal conductivity, a comparison was performed with a fully conjugate heat-transfer model accounting for natural convection, in which the continuity and momentum equations are solved under the Boussinesq approximation. The computational domain is a cube with a side length of 120 mm, selected to be representative of the effective block height H and of the maximum air-cavity width associated with the sinusoidal profile geometry. A temperature difference of $\mathsf{\Delta }T$ = 30 °C was imposed between two opposite faces of the cube by defining Dirichlet boundary conditions of $T_c$ = 26 °C and $T_h$ = 56 °C, which resembles typical summer wall temperatures. The results are reported in Figure 3, which shows the isothermal contours obtained from both the conjugate and the simplified models. As expected, the temperature fields differ between the two cases, since the simplified model is purely diffusive and does not capture the flow-induced distortion of the isotherms associated with natural convection. The figure also provides a quantitative analysis of how the heat rate changes when the applied temperature difference changes from 1 to 30 °C. Despite the differences in the temperature distributions, the heat rates predicted by the two models are in close agreement over the entire temperature range considered. This indicates that the proposed effective thermal conductivity $k_{air,eff}$ appropriately compensates for the impact of natural convection on the temperature field when computing the global heat transfer, thereby supporting the validity of the adopted interpolation for the conditions investigated.

2.2. Thermophysical Properties of Printable Foamed Concrete

The material investigated in this study is a lightweight foamed concrete, which is suitable for extrusion-based additive manufacturing. The mix design is characterized by a binder composition of 80% Portland cement and 20% fly ash, with a water-to-cement ratio (w/c) of 0.4, consistent with formulations analyzed by Wei et al. [5]. To define the constitutive laws for the numerical model, experimental data from the literature regarding dry density ($\rho$), porosity ($\epsilon$), and thermal conductivity ($k$) were used.

Table 1. Physical properties of foamed concrete mixes [5].

Mix	Target Density (kg/m3)	Dry Density (kg/m3)	Porosity (%)	Average Air-Void Diameter (mm)	Thermal Conductivity (W/(m·K))
A	1900	1870	0	-	0.5
B	1700	1636	12.51	0.104	0.423
C	1500	1461	21	0.113	0.363
D	1300	1201	32.54	0.122	0.282
E	1000	948	47.33	0.173	0.217
F	800	757	57.87	0.263	0.165
G	600	570	68.4	0.59	0.124
H	500	453	73.67	0.7	0.091
I	400	374	78.9	0.8	0.08
J	300	252	84.20	0.956	0.065

summarizes the physical properties of foamed concrete mixes with varying densities, which serve as the basis for the regression analysis performed in this study.

Based on these data, high-accuracy fitting functions ($R^2\approx 0.999$) were derived to model the material properties continuously across the parametric sweep, as shown in Figure 4.

The relationship between dry density and porosity follows a linear trend:

$${\rho }_{eff}=-18.80\cdot \epsilon +1852.2$$

(2)

The effective thermal conductivity ($k_{eff}$) was modelled as a quadratic function of density to capture the non-linear insulation effect of the air voids:

$$k_{eff}=6.05\cdot {10}^{-8}\cdot {\rho }^2+1.45\cdot {10}^{-4}\cdot \rho +0.021$$

(3)

Alternatively, expressing thermal conductivity directly as a function of porosity ($\epsilon$), the relationship is described by the following equation:

$$k_{eff}=2.55\cdot {10}^{-5}\cdot {\epsilon }^2-7.37\cdot {10}^{-3}\cdot \epsilon +0.504$$

(4)

These constitutive laws model the effective thermal conductivity as a scalar property, assuming the material to be macroscopically isotropic. Although the layer-by-layer extrusion process can induce anisotropy, the stochastic characteristic of the pore structure [5] dominates the heat transfer mechanism in low-density foamed concrete, justifying this approximation.

To accurately model the thermal energy storage capacity, the specific heat capacity ($c$) of the hardened material was determined analytically based on the mixture’s phase composition, following the methodology proposed by Bentz et al. [7]. The effective specific heat of the dry mass ($c_{eff}$) calculated as the weighted sum of the specific heat capacities of its individual constituents, accounting for the binder and chemically bound water:

$$c_{eff}=\sum _i{m}_i\cdot c_i$$

(5)

where $m_i$ represents the mass fraction of each constituent and $c_i$ represents their respective specific heat capacities. For the specific mixture investigated in this study, the calculation assumes a binder composition of 80% cement and 20% fly ash. Adopting specific heat values of 0.74 J/gK for cement and 0.72 J/gK for fly ash [7], and accounting for a residual chemically bound water content of approximately 25% by mass (with $c_{p,wa} \approx 2.2$ J/gK) as described by Van Breugel [11], the effective specific heat of the dry mass ($c_{dry}$) was computed. Consequently, a $c_{dry}$ value of 1003 J/(kg·K) is obtained and assumed constant across the investigated density range. This assumption relies on the definition of specific heat as a mass-dependent property; since the mass of the air entrapped in the pores is negligible compared to the cementitious matrix, the specific heat capacity of the composite per unit mass is effectively determined by the phase composition of the solid concrete, rendering the influence of porosity negligible [7]. In the parametric analysis, the porosity of the foamed concrete was varied within a range of 10% to 50%. Porosities exceeding 50% were excluded from the investigation based on the findings of Kearsley and Wainwright [12]. Their research demonstrates that compressive strength in foamed concrete exhibits an exponential decay with increasing porosity; beyond the 50% threshold, the mechanical properties degrade drastically, rendering the material unsuitable for self-supporting structural applications.

2.3. Numerical Simulation Setup and Boundary Conditions

A two-dimensional conjugate (conduction-radiation) heat-transfer model was developed to simulate heat transfer through the wall and the adjacent air layer. This 2D assumption focuses on the heat transfer through the standard cross-section, essentially neglecting three-dimensional thermal bridging effects at the block corners. The computational domain was divided into a solid region, representing concrete, and an air region. In the solid domain, heat transfer was modelled exclusively by diffusion, while in air cavities, by diffusion and radiation. Fluid motion was neglected, and no momentum or mass conservation equations were solved; instead, the effect of convection within the air cavities was incorporated through a temperature-dependent effective thermal conductivity in the air domain. Winter conditions were modelled under steady-state assumptions, whereas summer conditions were treated as transient in order to capture the diurnal thermal response of the system. For winter simulations, steady-state heat conduction was solved in the air and solid domains according to:

$$\nabla \cdot \left(k_{air,eff}\left(T\right)\hspace{0.17em}\nabla T_{air}\right)=0$$

(6)

$$\nabla \cdot \left(k_s\hspace{0.17em}\nabla T_s\right)=0$$

(7)

where $T$ denotes temperature, $k_{air,eff}\left(T\right)$ is the effective thermal conductivity of air, $k_s$ is the thermal conductivity of concrete, and subscripts $air$ and $s$ refer to air and solid, respectively. For summer simulations, transient heat transfer was considered in both domains, and the unsteady energy equations were solved:

$${\rho }_{air}{c}_{p,air}{\displaystyle \frac{\partial T_{air}}{\partial t}}=\nabla \cdot \left(k_{air,eff}\left(T\right)\hspace{0.17em}\nabla T_{air}\right)$$

(8)

$${\rho }_s{c}_s{\displaystyle \frac{\partial T_s}{\partial t}}=\nabla \cdot \left(k_s\hspace{0.17em}\nabla T_s\right)$$

(9)

where $\rho$ and $c_p$ are the density and specific heat capacity of the corresponding medium. At the air-concrete interface, continuity of temperature and heat flux was imposed:

$$T_{air}=T_s$$

(10)

$$-k_{air,eff}\left(T\right)\hspace{0.17em}\nabla T_{air}\cdot \mathit{n}=-k_s\hspace{0.17em}\nabla T_s\cdot \mathit{n}$$

(11)

with $\mathit{n}$ denoting the unit normal vector at the interface. Third–type (Robin) thermal boundary conditions were applied on both the internal and external wall surfaces, using a constant heat transfer coefficient $h$. The general form of the boundary condition is:

$$-k_{eff}\left(T\right)\hspace{0.17em}\nabla T\cdot \mathit{n}=h(T-T_{\infty })$$

(12)

where $\mathit{n}$ is the outward normal vector and $T_{\mathrm{\infty }}$ is the ambient temperature at the boundary. The applied boundary conditions differ between winter and summer simulations, as described below. In particular, winter simulations represent a Mediterranean climate [13] and were conducted under steady-state conditions. Constant ambient temperatures were imposed, i.e., internal ambient temperature ($T_{int}$ = 20 °C) and external ambient temperature ($T_{out}$ = 2 °C) [13]. The heat transfer coefficients at the internal and external surfaces were kept constant throughout the simulations, and equal to h_int = 7.69 W/(m² K), while h_out = 25 W/(m² K) [13].

Summer simulations were performed under transient conditions to account for the daily variation in the external temperature. The internal ambient temperature was kept constant, while the external ambient temperature followed a sinusoidal law:

$$T_{out}\left(t\right)=30+8\cdot \mathrm{c}\mathrm{o}\mathrm{s}\left({\displaystyle \frac{2\pi (t-15)}{24}}\right)$$

(13)

where $T_{out}$ is expressed in °C and $t$ is the time in hours. This formulation represents a diurnal temperature cycle with a maximum occurring at 15:00. A constant heat transfer coefficient was applied at both the internal and external boundaries. The indoor temperature is fixed to a value $T_{int}$ = 26 °C. Radiative heat transfer was also included in the model to perform an accurate prediction of the heat transfer of the system, particularly under summer conditions and within enclosed air cavities. At the external surface, solar radiation can represent a heat flux of the same order of magnitude as convective heat transfer, and neglecting it would lead to a significant underestimation of the wall surface temperature and of the resulting heat flux entering the structure. Moreover, due to the nonlinear dependence of radiative heat transfer on temperature, radiation directly affects the transient thermal response of the wall and the phase shift of heat propagation through the envelope. Inside the air cavities, radiative heat transfer between facing surfaces can be comparable to, or even exceed, conductive heat transfer through the enclosed air layer, especially when temperature differences are moderate and convective motion is weak or suppressed. On the external surface, a time-dependent solar radiative flux was imposed as:

$$G_{out}\left(t\right)=750\cdot \left\{\max \left[0, sin\left({\displaystyle \frac{\pi \left(mod\left(t,24\right)-6\right)}{12}}\right)\right]\right\}$$

(14)

where $G_{\mathrm{o}\mathrm{u}\mathrm{t}}$ is expressed in W m⁻². The absorbed solar flux is determined assuming a solar absorptivity coefficient ${\alpha }_R = 0.9$ for the external concrete surface. Within the internal air cavities, radiative heat exchange between surfaces was modelled using a surface-to-surface (S2S) formulation under the grey-body assumption, with a constant emissivity ${\epsilon }_R=$0.90 assigned to all radiating surfaces.

The computational grid of 16700 triangular elements was selected to ensure grid-independent results. A mesh sensitivity analysis was performed by progressively refining the grid until variations in the temperature field and heat flux became negligible. For transient summer simulations, a fixed time step of 5 s was adopted. This value was chosen as a compromise between numerical stability, temporal resolution of the boundary condition, and computational efficiency. Steady–state winter simulations did not require time discretization.

2.4. Performance Metrics and Data Post-Processing

To identify the geometric configuration that maximizes energy efficiency, a comprehensive parametric sweep was conducted. The study aims to simultaneously minimize the steady-state thermal transmittance and the periodic thermal transmittance.

$$F\left(X\right)=\left\{\min \left(U\right),\min \left(Y_{ie}\right)\right\}$$

(15)

The design space is defined by the variation in three key parameters: the amplitude ($A_1$) and period ($P_1)$ of the sinusoidal partition, and the material porosity $\left(\epsilon \right)$. As detailed in

Table 2. Parametric sweep domains.

Parameter	Range	Unit
Amplitude (${\mathit{A}}_1$)	[0.01, 0.02, 0.03]	m
Period (${\mathit{P}}_1$)	[0.01, 0.05, 0.1]	m
Porosity ($\mathit{\epsilon })$	[10, 20, 30, 40, 50]	%

, the combination of these discrete values results in a total of 45 distinct configurations simulated via COMSOL Multiphysics v6.0.

2.4.1. Thermal Transmittance ($U$)

Under winter conditions, characterized by a steady-state thermal regime, the system was simulated with a constant temperature difference between the indoor and outdoor environments. The steady-state thermal transmittance ($U$) was computed by processing the heat flux density ($\dot{q}$) and the surface temperatures obtained from the thermal analysis. The calculation was performed using the following relationship [13], implemented via a dedicated script:

$$U ={\displaystyle \frac{\dot{q}}{T_{int}-T_{out}}}$$

(16)

where: $\dot{q}$ represents the average heat flux density passing through the block’s cross-section ($W/m^2$) and $T_{int}$ and $T_{out}$ are the internal and external ambient temperatures recorded at the internal and external surface boundaries, respectively (°C).

This approach ensures that the calculated U-value accurately reflects the thermal resistance of the composite geometry, inclusive of the conductive and convective effects within the air cavities and air-voids given by the porosity.

2.4.2. Periodic Thermal Transmittance ($Y_{ie}$)

To rigorously quantify the dynamic thermal performance under summer conditions, the simulations subjected the external surface to a sinusoidal temperature profile with a 24 h period. Unlike simplified peak-to-peak evaluation methods, this study employed a robust statistical fitting approach to extract the harmonic amplitudes from the time-domain simulation data, in compliance with the dynamic characterization principles of ISO 13786 [14]. Simulations were run for three consecutive daily cycles until a periodic steady state was achieved. The regression analysis was then performed on the final complete 24 h cycle. The time-series data for the internal heat flux ($\dot{q}\left(t\right)$) and the outdoor ambient temperature difference ($T_{out}\left(t\right)$) were extracted. To isolate the fundamental 24 h frequency component and ensure accuracy, a sinusoidal regression model was fitted to the data using the ordinary least squares method. The fitting function for a generic variable $y\left(t\right)$ was defined as:

$$y\left(t\right)= C + A\cdot sin\left(\omega t\right)+ B\cdot cos\left(\omega t\right)$$

(17)

where $\omega = 2\pi /24 rad/h$ corresponds to the angular frequency of the diurnal cycle. The amplitude ($A_{mp}$) of the oscillation was then derived from the regression coefficients $A$ and $B$:

$$A_{mp}= \sqrt{A^2+ B^2}$$

(18)

Finally, the periodic thermal transmittance ($Y_{ie}$) was calculated as the ratio of the amplitude of the heat flux wave to the amplitude of the temperature difference wave:

$$Y_{ie}={\displaystyle \frac{A_{mp}\left(\dot{q}\right)}{A_{mp}\left(T_{out}\right)}}$$

(19)

This methodology characterizes the envelope’s thermal inertia, quantifying its ability to attenuate the incoming heat wave during the cooling season.

3. Results

The numerical analysis was performed on a dataset of 45 unique configurations, generated by permuting the geometric parameters ($A_1$, $P_1$) and the material porosity ($\epsilon$). The results highlight a synergistic relationship between density reduction and thermal performance, contradicting the traditional trade-off between insulation and inertia often observed in massive structures.

3.1. Influence of Porosity on Thermal Performance

Material porosity proves to be the governing parameter for the envelope’s overall energy performance, influencing both steady-state insulation and dynamic thermal damping. As illustrated in Figure 5, a monotonic improvement in both performance metrics is observed as the porosity increases from 10% to 50%.

The steady-state thermal transmittance ($U)$ exhibits a sharp decline, dropping from an average of 0.95 W/(m²·K) at 10% porosity to 0.66 W/(m²·K) at 50% porosity. This corresponds to a total improvement of approximately 31%. This enhancement is physically driven by the drastic reduction in the thermal conductivity ($k_{eff}$) of the foamed concrete matrix. Based on the material constitutive laws, increasing porosity from 10% to 50% causes $k_{eff}$ to decrease by 54% (from 0.433 to 0.199 W/m·K), significantly outweighing the geometric contributions to the total thermal resistance. Unexpectedly, for a lightweight solution, the dynamic performance also improves with higher porosity. The periodic transmittance ($Y_{ie}$) decreases from an average of 0.87 W/(m²·K) to 0.76 W/(m²·K), representing a 12.3% improvement in thermal damping capability. This behaviour challenges the traditional assumption that higher mass is strictly required for thermal inertia. The physical explanation lies in the reduction in thermal diffusivity $(\alpha =k/(\rho \cdot c\left)\right)$. While the density ($\rho$) decreases by 45% across the investigated range (from around 1850 to 912 kg/m³), the thermal conductivity ($k_{eff}$) drops even faster (54%), leading to a net reduction in thermal diffusivity of approximately 16% (from $2.59\times {10}^{-7}$ to $2.18\times {10}^{-7}$ m²/s). A lower diffusivity implies a slower propagation of the thermal wave through the material thickness. This results in a reduced thermal penetration depth, making the wall effectively “thicker” relative to the 24 h heat wave and thus enhancing its attenuation capacity, despite the reduced thermal mass. This finding highlights the unique advantage of foamed concretes in 3D printing: they allow for the simultaneous optimization of insulation and inertia without the weight penalty typical of massive construction.

Decoupling Mass and Thermal Conductivity Effects

To unequivocally confirm that the improvement in dynamic performance is driven by the reduction in thermal conductivity rather than geometric factors or mass changes, a theoretical control analysis was conducted. In a constant conductivity scenario, the porosity was varied (reducing density and thermal mass) while the thermal conductivity constant ($k_{eff}=const$). The results, presented in Figure 6, reveal a diametrically opposed trend compared to the material behaviour. When conductivity is decoupled from porosity, the periodic thermal transmittance ($Y_{ie}$) increase significantly as porosity increases. Specifically, increasing porosity from 10% to 50% under constant conductivity results in an increase in $Y_{ie}$ from 1.75 to 2.59 W/(m²·K).

This deterioration is physically governed by thermal diffusivity ($\alpha$). In the constant conductivity scenario, the reduction in density ($\rho$) leads to a direct increase in diffusivity, accelerating the propagation of the thermal wave through the medium and reducing the damping effect. Conversely, in the realistic model, the drastic drop in conductivity ($k_{eff}$) overcompensates for the loss of mass, resulting in a net reduction in diffusivity.

This comparison isolates the physical mechanisms, demonstrating that the insulation capability is the predominant factor defining the dynamic performance of lightweight foamed concretes, overriding the traditional loss of thermal inertia associated with mass reduction.

3.2. Influence of Internal Geometry

While material porosity drives the macroscopic performance trends, the optimization of the internal topology allows for crucial fine-tuning of the dynamic response ($Y_{ie}$), offering an additional margin for improvement without compromising the steady-state insulation ($U$). The analysis at the optimal porosity of 50% reveals a complex interplay between the partition period ($P_1$) and amplitude ($A_1$).

The period of the sinusoidal partition acts as the primary geometric driver. Increasing $P_1$ from 0.01 m to 0.1 m (i.e., reducing the number of internal chambers) consistently improves dynamic performance. In fact, at 50% porosity, shifting from a short-period partition ($P_1$ = 0.01) to a long-period one ($P_1$ = 0.1) reduces the average $Y_{ie}$ from 0.773 to 0.751 W/(m²·K). This corresponds to a 2.9% improvement in thermal damping. A physical interpretation could be that wider cavities reduce the density of the solid web connections per unit length of the wall, thereby minimizing the conductive thermal bridges that facilitate heat transfer. The influence of amplitude is non-linear and dependent on the chosen period. In configurations with narrow cavities ($P_1=0.01$), increasing the amplitude from 0.01 m to 0.03 m (“wavier” partitions) worsens the dynamic performance, increasing $Y_{ie}$ by approximately 3.2% (from 0.761 to 0.785 W/(m²·K)). This is likely due to the increased length of the conductive solid path, as shown in Figure 7. However, this trend inverts at the optimal period ($P_1=0.1$). In this specific configuration, increasing the amplitude to $A_1=0.03$ yields the absolute lowest periodic transmittance of the entire dataset (0.748 W/(m²·K)). While flatter partitions ($A_1=0.01$) are generally more robust across different frequencies, the global optimum for thermal inertia is achieved by combining the widest possible period with the highest amplitude, maximizing the phase-shift capability of the internal structure. Regarding the steady-state thermal transmittance ($U$), the impact of internal geometry is notably less pronounced compared to its effect on dynamic performance (approximately 0.654–0.659 W/(m²·K)), yet distinct physical trends remain observable. At the optimal porosity of 50%, the variation in $U$ across all geometric permutations is contained within a very narrow range, confirming that steady-state insulation is predominantly governed by the material’s conductivity rather than geometry, as presented in Figure 7.

Nevertheless, larger periods ($P_1=0.1$ m) tend to yield slightly lower U-values compared to shorter periods, as the reduction in the number of internal septs decreases the overall area available for conduction. Conversely, variations in amplitude ($A_1$) have a negligible effect on static insulation in high-porosity blocks. This stability contrasts with denser configurations (e.g., 10% porosity), where increasing the amplitude in short-period partitions ($P_1=0.01$ m) leads to a slight penalty in insulation due to the elongation of the heat transfer path. Consequently, the geometric design can be freely optimized to maximize dynamic damping ($Y_{ie}$) without risk of compromising the steady-state thermal performance.

3.3. Optimal Configuration and Trade-Off Analysis

The multi-objective analysis demonstrates that the optimization of the 3D-printed block does not require a significant compromise between winter insulation and summer thermal inertia, at least regarding the variables considered in this study. As evidenced by the scatter plot in Figure 8, the solution space does not exhibit a wide, convex Pareto frontier typical of conflicting objectives. Instead, the optimal solutions for both metrics converge decisively in the bottom-left region of the design space, corresponding to the configurations with the highest porosity $\left(\epsilon = 50\%\right)$. Within this optimal cluster, a micro-trade-off can be observed based on the amplitude of the internal partitions ($A_1$), while the period is fixed at its optimal value ($P_1= 0.1$ m).

The absolute minimum steady-state thermal transmittance is achieved by the configuration with $\epsilon =50\%,$ $P_1=0.1$ m, and $A_1=0.01$ m. This specific geometry yields a $U$ of 0.654 W/(m²·K) and a corresponding $Y_{ie}$ of 0.753 W/(m²·K). The mechanism behind this performance lies in the lower amplitude ($A_1=0.01$ m), which minimizes the curvilinear length of the internal septs, thereby slightly reducing the solid conduction paths compared to wavier geometries. Conversely, the absolute minimum periodic thermal transmittance is achieved by the configuration with $=50\%,$ $P_1=0.1$ m, and $A_1=0.03$ m. This setup results in a $U$ of 0.655 W/(m²·K) and a $Y_{ie}$ of 0.748 W/(m²·K). Here, the higher amplitude ($A_1=0.03$ m) increases the effective thermal mass interaction length and enhances the convective damping within the wider cavities, providing a marginal benefit to inertia.

Comparing these two extremes reveals that the penalty for prioritizing one objective over the other is negligible. Choosing the best steady-state configuration causes only a 0.60% increase in $Y_{ie}$ compared to the dynamic optimum. Similarly, selecting the best dynamic configuration results in only a 0.15% increase in $U$ relative to the static optimum. Since the performance deviation is consistently less than 1%, the typical design conflict between insulation and mass is effectively resolved. Consequently, a robust design strategy for this 3D-printed element should prioritize maximizing porosity up to the 50% limit, which serves as the primary driver for energy efficiency. Furthermore, adopting a large period ($P_1=0.1$ m) is recommended to consistently improve dynamic response, while the amplitude ($A_1$) can be treated as a free parameter to optimize for printability, material usage, or aesthetic requirements, given its practically negligible impact on thermal performance.

4. Discussion and Conclusions

The results of this study challenge the conventional heuristic often applied to massive building envelopes, which speculates a dichotomy between lightweight insulation (for steady-state winter performance) and heavy thermal mass (for dynamic summer comfort). In traditional construction, high thermal inertia is typically achieved by increasing density. However, the foamed concrete configurations analyzed herein demonstrate that dynamic performance can be improved simultaneously with static insulation by maximizing porosity. Specifically, increasing porosity from 10% to 50% yielded a simultaneous improvement in both steady-state thermal transmittance ($\mathit{U}$), which decreased by 31%, and periodic thermal transmittance (${\mathit{Y}}_{\mathit{i}\mathit{e}}$), which improved by 12.3%.

The physical mechanism driving this synergistic behaviour is the non-linear relationship between porosity and thermal diffusivity ($\mathit{\alpha }$). As porosity increases, the material undergoes a density-conductivity decoupling: while the thermal mass ($\mathit{\rho }$) decreases linearly, the effective thermal conductivity (${\mathit{k}}_{\mathit{e}\mathit{f}\mathit{f}}$) drops quadratically and much more rapidly due to the insulating power of the air voids. Consequently, the thermal diffusivity decreases, slowing heat wave propagation. This confirms that for lightweight cellular concretes, the resistance term ($\mathit{R}$) in the thermal circuit is far more influential than the capacitance term ($\mathit{C}$). These findings align with recent experimental research on advanced lightweight concretes. Notably, Zhang et al. [15] demonstrated that ultra-lightweight aerogel foamed concrete can achieve a thermal time lag of 9 h, twice that of ordinary dense concrete, thanks to the drastic reduction in thermal diffusivity. Similarly, Zlateva et al. [16] classify foam concrete as a “transitional” material where the insulation capability dominates the thermal behavior compared to structural density. Our results confirm that this phenomenon also applies to 3D-printable foamed concrete mixtures: the reduction in effective thermal conductivity (${\mathit{k}}_{\mathit{e}\mathit{f}\mathit{f}}$) exceeds the loss of thermal mass ($\mathit{\rho },{\mathit{c}}_{\mathit{p}}$), lowering thermal diffusivity ($\mathit{\alpha }$). Practically, this suggests the effective strategy for 3D-printed envelopes in Mediterranean climates is to push the material formulation towards the lowest structurally viable density. While material properties define the macro-performance, the internal topology offers a degree of freedom unique to additive manufacturing, acting as a “fine-tuning” variable. The optimization of the sinusoidal partitions revealed that minimizing conductive thermal bridges is more critical than maximizing the total developed surface area. Configurations with larger periods (${\mathit{P}}_{\mathbf{1}}=\mathbf{0.1}$ m) consistently outperformed high-frequency partitions due to the reduction in solid web connections per unit length. Furthermore, the amplitude (${\mathit{A}}_{\mathbf{1}}$) shown a non-linear effect: wavier partitions (${\mathit{A}}_{\mathbf{1}}=\mathbf{0.03}$ m) become beneficial only when combined with large periods, allowing for convective damping without creating excessive conductive bridges.

A significant outcome of the multi-objective analysis is the resolution of the traditional design trade-off. The Pareto front collapsed into a cluster of optimal solutions at high porosity, where the deviation between the best $\mathit{U}$ and best ${\mathit{Y}}_{\mathit{i}\mathit{e}}$ configurations is less than 1%. This simplifies the design process: maximizing porosity does not penalize summer comfort but enhances it. Consequently, the internal pattern design can be driven by other constraints—such as printability or mechanical buckling resistance—without compromising energy efficiency.

Despite these promising trends, the present study relies on numerical assumptions. The reduction in the domain to a 2D cross-section neglects three-dimensional thermal bridging effects at corners or connections. Additionally, the fluid dynamics within air cavities were modelled using an effective thermal conductivity approach; while computationally efficient, this simplifies the transient nature of natural convection. Finally, this study did not perform a coupled thermo-mechanical analysis. Future research must focus on verifying the structural integrity of these optimized, high-porosity geometries and validating the numerical findings through experimental hot-box testing on full-scale 3D-printed prototypes.