Automatic generation of high-fidelity representative volume elements and computational homogenization for the determination of thermal conductivity in foamed concretes

Abstract

Foamed concretes are highly porous materials with excellent insulation properties. Their thermal conductivity is strongly dependent on the pore structure, characterized by the porosity as well as the shape and size distribution of pores. To define the representative volume element (RVE) of foamed concretes with high fidelity, we develop NRGene, an automatic generator of cubic samples with spherical air inclusions whose number and size obey a given distribution. We compute the effective thermal conductivity tensor for a given RVE using finite-element-based computational homogenization. The hi-fi RVE of foamed concretes may contain hundreds of millions of finite elements, making it better suited for benchmarking than for everyday engineering applications. Then, having the hi-fi model as benchmark, we propose a simplified model consisting of the hi-fi model called with a truncated histogram as input, which is largely cheaper while keeping a satisfactory accuracy. We model 21 different foamed concretes with a wide range of porosities and different compositions of the cement paste. Further, we demonstrate that it is the volume of all the pores of a given size, and not its quantity, what affects the effective thermal conductivity of foamed concretes.

Appendices

Appendix I: Scheme of the mesh generation algorithm

To generate the voxelized RVE of a foamed concrete, NRGene takes the histogram $H=\{D_1,n_1,\dots ,D_Q,n_Q\}$ (with $D_q<D_{q+1}$) and the porosity p as input data. The user-defined mesh generation setup includes the initial guess $V_\text {\tiny {RVE}}^{(0)}$ for the RVE volume, the porosity tolerance $\varepsilon _\text {por}$, the refinement parameter $n_\text {ref}$ (ratio between the minimum pore size $D_1$ and the maximum voxel size $h_\text {max}$), free or forced periodicity ($\text {PER}=0$ or $\text {PER}=1$, respectively), and to free or control the pore volume loss ($\text {PVL}=0$ or $\text {PVL}=1$, respectively). In case $\text {PVL}=1$, the tolerance $\varepsilon _\text {\tiny {PVL}}$ and the maximum number of tries $n_\text {\tiny {PVL}}$ are also needed; note that $\text {PVL}=0$ is implied by adopting $\varepsilon _\text {\tiny {PVL}}=\infty$.

The full procedure is summarized below.

Appendix II: Static condensation

Following Toro et al. [41], we use the static condensation technique to convert the semi-definite system (26) into an equivalent but positive definite system. To this end, let us firt identify all the couples $({\textbf{x}}_\alpha ,{\textbf{x}}_\beta )$ of opposite boundary nodes in the finite element mesh of the RVE. Then, let us order the indices $\alpha$ and $\beta$ in the sets ${\mathcal {M}}$ and ${\mathcal {S}}$ of indices of the “master” and “slave” boundary nodes, respectively. Either ${\mathcal {M}}$ or ${\mathcal {S}}$ or both have repeated indices, but one can manage to order the indices in such a way that one of these sets has no repeated indices; let ${\mathcal {M}}$ be such set. Being ${\mathcal {M}}\cup {\mathcal {S}}$ the set of indices of the boundary nodes, let ${\mathcal {I}}$ be the set of $N_\text {int}$ indices of those nodes strictly inside the RVE. Further, master and interior nodes are the “free” nodes, whose indices are grouped in the set ${\mathcal {F}}$ with dimension $N_\text {free}=N_\text {int}+N_\text {con}$.

Then, the semi-definite linear algebraic system (26), that is

$$\begin{aligned} \begin{bmatrix} {\textbf{K}}&{}{\textbf{P}}^T\\ {\textbf{P}}&{}{\textbf{0}}_{N_\text {con}\times N_\text {con}} \end{bmatrix}\begin{bmatrix} {\textbf{T}}\\ \mathbf {\Lambda }\end{bmatrix}=\begin{bmatrix} {\textbf{0}}_{N_\text {nod}\times 1}\\ {\textbf{R}}\end{bmatrix}, \end{aligned}$$

is rewritten as

$$\begin{aligned} {\textbf{K}}_{{\mathcal {F}}{\mathcal {F}}}{\textbf{T}}_{{\mathcal {F}}}+{\textbf{K}}_{{\mathcal {F}}{\mathcal {S}}}{\textbf{T}}_{{\mathcal {S}}}={\textbf{0}}_{N_\text {free}\times 1}, \end{aligned}$$

(35)

$$\begin{aligned} {\textbf{K}}_{{\mathcal {F}}{\mathcal {S}}}^T{\textbf{T}}_{{\mathcal {F}}}+{\textbf{K}}_{{\mathcal {S}}{\mathcal {S}}}{\textbf{T}}_{{\mathcal {S}}}={\textbf{0}}_{N_\text {con}\times 1}, \end{aligned}$$

(36)

where ${\textbf{K}}_{{\mathcal {F}}{\mathcal {F}}}$ is the $N_\text {free}\times N_\text {free}$-matrix whose ij component is $K_{{\mathcal {F}}_i {\mathcal {F}}_j}$, ${\textbf{K}}_{{\mathcal {F}}{\mathcal {S}}}$ is the $N_\text {free}\times N_\text {con}$-matrix whose ij component is $K_{{\mathcal {F}}_i {\mathcal {S}}_j}$, ${\textbf{K}}_{{\mathcal {S}}{\mathcal {S}}}$ is the $N_\text {con}\times N_\text {con}$-matrix whose ij component is $K_{{\mathcal {S}}_i {\mathcal {S}}_j}$, ${\textbf{T}}_{\mathcal {F}}$ is the unknown $N_\text {free}\times 1$-vector with components $T_{{\mathcal {F}}_i}$, and ${\textbf{T}}_{\mathcal {S}}$ is the $N_\text {con}\times 1$-vector with components $T_{{\mathcal {S}}_i}$, ${\mathcal {F}}_i$ and ${\mathcal {S}}_i$ denoting the i-th entry of the sets ${\mathcal {F}}$ and ${\mathcal {S}}$, respectively. The vector ${\textbf{T}}_{\mathcal {S}}$ is defined by the linear equality constraints (23) as

$$\begin{aligned} {\textbf{T}}_{\mathcal {S}}={\textbf{C}}_{{\mathcal {S}}{\mathcal {F}}}{\textbf{T}}_{\mathcal {F}}-{\textbf{R}}, \end{aligned}$$

(37)

where ${\textbf{C}}_{{\mathcal {S}}{\mathcal {F}}}$ is the highly sparse $N_\text {con}\times N_\text {free}$-matrix with components

$$\begin{aligned} C_{ij}&={\left\{ \begin{array}{ll} 1&{}\text {if }({\textbf{x}}_{{\mathcal {S}}_i},{\textbf{x}}_{{\mathcal {F}}_j})\text { is a couple of opposite boundary nodes,}\\ 0&{}\text {otherwise.} \end{array}\right. } \end{aligned}$$

(38)

With ${\textbf{T}}_{\mathcal {S}}$ defined by the Eq. (37), the FEM heat conduction Eqs. (35) and (36) take the form

$$\begin{aligned} ({\textbf{K}}_{{\mathcal {F}}{\mathcal {F}}}+{\textbf{K}}_{{\mathcal {F}}{\mathcal {S}}}{\textbf{C}}_{{\mathcal {S}}{\mathcal {F}}}){\textbf{T}}_{{\mathcal {F}}}={\textbf{K}}_{{\mathcal {F}}{\mathcal {S}}}{\textbf{R}}, \end{aligned}$$

(39)

$$\begin{aligned} ({\textbf{K}}_{{\mathcal {F}}{\mathcal {S}}}^T+{\textbf{K}}_{{\mathcal {S}}{\mathcal {S}}}{\textbf{C}}_{{\mathcal {S}}{\mathcal {F}}}){\textbf{T}}_{{\mathcal {F}}}={\textbf{K}}_{{\mathcal {S}}{\mathcal {S}}}{\textbf{R}}, \end{aligned}$$

(40)

Finally, premultiplying both sides of the last Eq. by ${\textbf{C}}_{{\mathcal {S}}{\mathcal {F}}}^T$ and summing the resulting equation to Eq. (39), we arrive to the statically condensed version of the localization problem:

$$\begin{aligned} {\textbf{K}}^*{\textbf{T}}_{\mathcal {F}}= {\textbf{R}}^*, \end{aligned}$$

(41)

with

$$\begin{aligned} {\textbf{K}}^*&={\textbf{K}}_{{\mathcal {F}}{\mathcal {F}}}+{\textbf{K}}_{{\mathcal {F}}{\mathcal {S}}}{\textbf{C}}_{{\mathcal {S}}{\mathcal {F}}} +{\textbf{C}}_{{\mathcal {S}}{\mathcal {F}}}^T{\textbf{K}}_{{\mathcal {F}}{\mathcal {S}}}^T+{\textbf{C}}_{{\mathcal {S}}{\mathcal {F}}}^T{\textbf{K}}_{{\mathcal {S}}{\mathcal {S}}}{\textbf{C}}_{{\mathcal {S}}{\mathcal {F}}},\end{aligned}$$

(42)

$$\begin{aligned} {\textbf{R}}^*&=({\textbf{K}}_{{\mathcal {F}}{\mathcal {S}}}+{\textbf{C}}_{{\mathcal {S}}{\mathcal {F}}}^T{\textbf{K}}_{{\mathcal {S}}{\mathcal {S}}}){\textbf{R}}. \end{aligned}$$

(43)

Now, ${\textbf{K}}^*$ exhibits the valuable properties of symmetry and positive definiteness.

Once determined ${\textbf{T}}_{\mathcal {F}}$ after solving Eq. (41) and ${\textbf{T}}_{\mathcal {S}}$ by Eq. (37), and knowing the respective sets of indices ${\mathcal {F}}$ and ${\mathcal {S}}$, it is straightforward to build the original nodal vector ${\textbf{T}}$.

Appendix III: Material data for the modeled foamed concretes

All the foamed concretes modeled in this work were produced by Batool in the framework of her PhD thesis [17]. The reference foamed concrete Ref-$\rho$ (with $\rho =$ 400, 600 or 800 referring to the cast density in kg/m$^3$) was prepared by adding a pre-formed foam produced by a synthetic foaming agent to a slurry made of HE-type Portland cement with 0.69 water-to-cement ratio. The foamed concretes labeled FA10-$\rho$, SF10-$\rho$, and MK10-$\rho$ were prepared in a similar way as Ref-$\rho$ except for the use of fly ash (FA), silica fume (SF) or metakaolin (MK) to replace 10% in mass fraction of Portland cement; this fraction increases to 20% for FA20-$\rho$, SF20-$\rho$, and MK20-$\rho$.

For each one of these foams, the thermal conductivity of the solid phase is assumed to be equal to that of the corresponding hardened cement paste at the hydration age of 300 days, listed in Table 1.

Table 1 Thermal conductivity of hardened cement pastes at the hydration age of 300 days [17]

Finally, Figs. 14, 15, 16 and 17 show the porosity and pore size distribution for all the modeled concrete foams.

Fig. 14

Pore size distribution and porosity p of the Ref-$\rho$ foamed concretes, with $\rho =$ 400, 600, or 800 (cast density in kg/m$^3$)

Fig. 15

Pore size distribution and porosity p of the FA10-$\rho$ and FA20-$\rho$ foamed concretes, with $\rho =$ 400, 600, or 800 (cast density in kg/m$^3$)

Fig. 16

Pore size distribution and porosity p of the SF10-$\rho$ and SF20-$\rho$ foamed concretes, with $\rho =$ 400, 600, or 800 (cast density in kg/m$^3$)

Fig. 17

Pore size distribution and porosity p of the MK10-$\rho$ and MK20-$\rho$ foamed concretes, with $\rho =$ 400, 600, or 800 (cast density in kg/m$^3$)