Moisture uptake within a semifinite region
This benchmark test is taken from Skerget et al. It reproduces the normative benchmark described in [CEN2007]. 
1. Problem description
Proposed in Annex A of the norm BS EN 15026:2007 [CEN2007], this normative benchmark test is designed for the validation of heat and moisture tranfer software.
The test deals with thick single homogeneous material in equilibrium with a constant surrounding climate. The material is perfectly airtight. At a certain time the temperature and the relative humidity undergoes a step change. The objective is to calculate the moisture and temperature distribution after 7 days, 30 days and 365 days.
 Geometry

The domain diemsions are not specified in [CEN2007] but it should be taken large enough to be represented as a semifinite region. As in [Škerget2014], we consider a [\(12\text{m}\times2\text{m}\)] wall. Here is the description of the benchmark’s test geometry:
 Boundary Conditions

We have \(\Omega = [0,12]\times [0,2]\). We write \(\partial\Omega=\partial\Omega_D\cup\partial\Omega_N\) where :

\(\partial\Omega_D\) is the left border of the domain on which Dirichlet condition are applied,

\(\partial\Omega_N\) is composed of the three other borders, on which homogenous Neunamm conditions stand.

 Initial condition


Initial condition: \(\phi = 50\%\), \(T=20^{\circ}C\)

After the step change: \(\phi = 95\%\), \(T=30^{\circ}C\)

 General data
 Material data


Water retention curve


Sorption isotherm

Vapour diffusion
with \(R = 8.314~J/\mathrm{mol} \cdot K\) is the universal gas constant.
We use \(\delta_p\) as it is given in [SKER20014]. The formula given in [CEN2007] leads to erronous solution. 

Liquid water permeability

Porosity equal maximum point of moisture storage function.

Thermal conductivity

Heat capacity for dry material
The value of \(C_{p,\mathrm{w}}\) is get from this page.
2. Results from [SKER2014]
 Time and space discretization


Two nonuniform nonsymmetric meshes of \(M = 100\times 2\) and \(M = 200 \times2\) macroelements were used, each with two aspect ratios \(R_x1 = 40\) and \(R_x1 = 80\) between the largest and the smallest boundary element, thus four different discretization models were considered.

Two timeintegration schemes were considered, e.g. constant and linear approximation of functions over each timestep.

The timedependent analysis was performed by running the simulation from the initial state with a time step value of \(\Delta t = 0.01 \) day, \(\Delta t = 0.1\) day and \(\Delta t = 1.0\) day, applying constant approximation of functions over each time step, and \(\Delta t = 1.0\) day for the linear approximation.

 Results with constant time interpolation

The following figures show the griddependence study when the computational mesh is refined from \(100\times2\) to \(200\times2\) with aspect ratios \(R_x=40\) and \(R_x=80\) for the constant time model. The moisture and temperature distributions were plotted for time increment \(\Delta t1=1.0\) day at time instants \(7\), \(30\) and \(365\) days.
Time step dependence for the time constant model was studied on the finest grid \(200\times2\) \(R_x=80\) and by varying the time step from \(0.01\) to \(1.0\) day. The computational results are shown the following figures:
 Results with linear time interpolation

The following figures show the griddependence study when the computational mesh is refined from \(100 \times2\) to \(200\times2\) with aspect ratios \(R_x=40\) and \(R_x=80\) for the linear time model. The water content and temperature distributions are plotted for time increment \(\Delta t=1.0\) day at the time instants \(7\), \(30\) and \(365\) days.
3. Variationnal problem
Let \(X = M = \left\{v\in H^1(\Omega)\,\middle\,v_{\partial\Omega_D}=0\right\}\).
From what was done here, the problem become :
At time \(n+1\), the variationnal problem is : Find \((T^{n+1},\phi^{n+1})\in X\times M\) such as
with :

\(a(T,v) = \displaystyle\int_\Omega\left(\rho C_{p}\right)^n_{\mathrm{eff}} \frac{T^{n+1}}{\Delta t} v + \int_\Omega \left(k^n_{\mathrm{eff}} \nabla T^{n+1} \right) \cdot \nabla v\)

\(b(T,v) = \displaystyle \int_\Omega L^n_{\mathrm{v}} \delta^n_{\mathrm{p}} p^n_{\mathrm{sat}} \left(\nabla\phi^{n+1}\right) \cdot \nabla v + \int_\Omega \left(L^n_{\mathrm{v}} \delta^n_{\mathrm{p}} \left( \nabla p^n_{\mathrm{sat}}\right) \phi^{n+1} \right) \cdot \nabla v  \int_{\partial\Omega_N} \left(L^n_{\mathrm{v}} \delta^n_{\mathrm{p}} \left( \nabla p^n_{\mathrm{sat}}\right) \phi^{n+1} \cdot n \right) v\)

\(c(T,q) = 0\)

\(d(T,v) = \displaystyle\int_\Omega \left(\xi^n \frac{\phi^{n+1}}{\Delta t}\right)q + \int_\Omega\left(\xi^n D^n_{\mathrm{w}} \nabla \phi^{n+1}\right)\cdot\nabla q + \int_\Omega \left(\delta^n_{\mathrm{p}} (\nabla\phi^{n+1}) p^n_{\mathrm{sat}}\right)\nabla q + \int_\Omega \left(\delta_p\phi^{n+1}\nabla p^n_\mathrm{sat}\right)\nabla q  \int_{\partial\Omega_N} \left(\delta_p\phi^{n+1}\nabla p^n_\mathrm{sat}\cdot n\right)q\)

\(f(v) = \displaystyle\int_\Omega\left( \left(\rho C_{p}\right)^n_{\mathrm{eff}} \frac{T^n}{\Delta t} + Q^{n+1} \right) v\)

\(g(q) = \displaystyle\int_\Omega \left(\xi^n\frac{\phi^n}{\Delta t}+G^{n+1}\right) q\)
This variationnal problem gives a discretized problem : find \((T_h,\phi_h)\in X_h\times M_h\) such as
Where \(A,B,C\) are the matrix associated to the bilinear forms and \(F,G\) are the vectors associated to the linear form on the bases of the spaces \(X_h\) and \(M_h\).
4. Command
The configurations files (cfg, json and geo) can be found in the repository github src/cases/moistureuptake.
The command to execute this simulation is :
mpirun np 8 ./feelpp_hm_heat_moisture configfile cases/moistureuptake/moistureuptake.cfg modfile "\$cfgdir/moistureuptake.json" hm.export={604800,2592000,31536000}
5. Resolution
The results of the temperature and the water content at \(t=7\) days, \(t=30\) days and \(t=365\) days are given of the following figures. The minimal and maximal values from [CEN2007] are also plotted.
The simulation above was made with a timestep of 1 day. With smaller timesteps, the results stay the same for both temperature and water content, as shown above :
References

[CEN2007] EN 15026, Hygrothermal performance of building components and building elements  Assessment of moisture transfer by numerical simulation, CEN, 2007.

[HAM2002] C.E. Hagentoft, HAMSTAD – Final report: methodology of HAMmodeling, Report R02:8, Gothenburg, Department of Building Physics, Chalmers University of Technology, 2002.

[Kumaran1994] Kumaran, M. K., Mitalas, G. P. and Bomberg, M. T. (1994), 'Fundamentals of transport andstorage of moisture in building materials and components', Trechsel, H. R. (ed), Moisture control in buildings.

[KUN1995] Künzel H, Simultaneous Heat and Moisture Transport in Building Components, PhD thesis, Fraunhofer Institute of Building Physics, 1995.

[Kunzel2004] Kunzel, H.M., Holm, A., Zirkelbach, D. and Karagiozis, A.N., Simulation of indoor temperature and humidity conditions including hygrothermal interactions with building envelope Solar Energy 78 (2005) 554–561.

[Mendes2019] Mendes, N., Chhay, M., Berger, J. and Dutykh, D., Numerical methods for diffusion phenomena in building physics, Springer Nature Switzerland AG 2019.

[Neymark2002] Neymark, J. and Judkoff, R. International energy agency building simulation test and diagnostic method for heating, ventilation, and airconditioning equioement models (HVAC BESTEST). Volume 1: Cases E100E200 Technical Report NREL/TP55030152.

[Škerget2014] Škerget, L. and Tadeu, A. BEM numerical simulation of coupled heat and moisture flow through a porous solid Engineering Analysis with Boundary Elements, 2014.

[Straube2002] Straube J. F., Moisture in Buildings, ASHRAE Jornal, January 2002.