Application `feelpp_hm_heat_moisture` devlopment

As is it coded yet, the application `feelpp_hm_heat_moisture` can only be used for one of the previous benchmarks at a time (drying-layer, and moisture-uptake) : the values of the parameters are distinguished with macros, and we have to build again if we want to chang e, which is not a good strategy !

The idea is now to adapt it to make it work for all types of problems we want to model, using a JSON file.

1. Complete variationnal problem

We considere the problem of heat-moisture transport (cf here) :

\begin{align} \left(\rho C_{p}\right)_{\mathrm{eff}} \frac{\partial T}{\partial t}+\nabla \cdot\left(-k_{\mathrm{eff}} \nabla T-L_{\mathrm{v}} \delta_{\mathrm{p}} \nabla\left(\phi p_{\mathrm{sat}}\right)\right)=Q \tag{E_t}\label{eq:T} \\ \quad \xi \frac{\partial \phi}{\partial t}+\nabla \cdot\left(-\xi D_{\mathrm{w}} \nabla \phi-\delta_{\mathrm{p}} \nabla\left(\phi p_{\mathrm{sat}}\right)\right)=G \tag{E_\phi}\label{eq:phi} \end{align}

This system of equations is satisfied on the domain $\Omega$. We add those boundary counditions :

\begin{align} T &= d_T & \text{on}\,\Gamma_{T,D}\\ \phi &= d_\phi &\text{on}\,\Gamma_{\phi,D}\\ \nabla T\cdot n &= N_T &\text{on}\,\Gamma_{T,N}\\ \nabla\phi\cdot n &= N_\phi &\text{on}\,\Gamma_{\phi,N}\\ \end{align}

We have $\partial\Omega=\Gamma_{T,D}\cup\Gamma_{T,N}$, and $\partial\Omega=\Gamma_{\phi,D}\cup\Gamma_{\phi,N}$.

We make the usual time derivate approximation : $\dfrac{\partial T}{\partial t}\approx\dfrac{T^{n+1}-T^n}{\Delta t}$

Let $v$ be a test function associated to $T$. The first equation $\ref{eq:T}$ become, after making a partial integration :

\begin{align} \int_\Omega (\rho C_p)_\mathrm{eff}^n\dfrac{T^{n+1}}{\Delta t}v &+ \int_\Omega(k_\mathrm{eff}^n\nabla T^{n+1})\nabla v + \int_{\Gamma_{T,N}}\left(-k_\mathrm{eff}^n\nabla T^{n+1}\cdot\mathbf{n}\right)v + \int_{\Gamma_{T,D}}\left(-k_\mathrm{eff}^n\nabla T^{n+1}\cdot\mathbf{n}\right)v\\ &+ \int_\Omega L_v^n\delta_\mathrm{p}^np_\mathrm{sat}^n\nabla\phi^{n+1}\nabla v + \int_{\Gamma_{\phi,N}}\left(-L_v^n\delta_\mathrm{p}^np_\mathrm{sat}^n\nabla\phi^{n+1}\cdot\mathbf{n}\right)v + \int_{\Gamma_{\phi,D}}\left(-L_v^n\delta_\mathrm{p}^np_\mathrm{sat}^n\nabla\phi^{n+1}\cdot\mathbf{n}\right)v \\ &+ \int_\Omega L_v^n\delta_\mathrm{p}^n\nabla p_\mathrm{sat}^n \phi^{n+1}\nabla v + \int_{\Gamma_{\phi,N}}\left(-L_v^n\delta_\mathrm{p}^n\nabla p_\mathrm{sat}^n\phi^{n+1}\cdot\mathbf{n}\right)v + \int_{\Gamma_{\phi,D}}\left(-L_v^n\delta_\mathrm{p}^n\nabla p_\mathrm{sat}^n\phi^{n+1}\cdot\mathbf{n}\right)v\\ &=\int_\Omega \left((\rho C_p)_\mathrm{eff}^n\dfrac{T^n}{\Delta t} + Q^{n+1}\right)v \end{align}

As $v=0$ on $\Gamma_{T,D}$ and on $\Gamma_{\phi,D}$, and also there are the two Neimann conditions, this equation become :

$a(T^{n+1},v)+b(\phi^{n+1},v)=f(v)$

with :

• $a(T^{n+1},v) = \displaystyle\int_\Omega (\rho C_p)_\mathrm{eff}^n\dfrac{T^{n+1}}{\Delta t}v + \int_\Omega(k_\mathrm{eff}^n\nabla T^{n+1})\nabla v$

• $b(\phi^{n+1},v) = \displaystyle\int_\Omega L_v^n\delta_\mathrm{p}^np_\mathrm{sat}^n\nabla\phi^{n+1}\nabla v + \int_\Omega L_v^n\delta_\mathrm{p}^n\nabla p_\mathrm{sat}^n \phi^{n+1}\nabla v - \int_{\Gamma_{\phi,N}}\left(L_v^n\delta_\mathrm{p}^n\nabla p_\mathrm{sat}^n\phi^{n+1}\cdot\mathbf{n}\right)v$

• $f(v) = \displaystyle\int_\Omega \left((\rho C_p)_\mathrm{eff}^n\dfrac{T^n}{\Delta t} + Q^{n+1}\right)v + \int_{\Gamma_{T,N}} k_\mathrm{eff}^n N_T^{n+1} v + \int_{\Gamma_{\phi,N}} L_v^n\delta_\mathrm{p}^n\nabla p_\mathrm{sat}^n N_\phi^{n+1} v$

Doing the same process with a test function $q$ associated to $\phi$, the equation $\ref{eq:phi}$ become :

$d(\phi^{n+1},q)=g(q)$

with :

• $d(\phi^{n+1},q) = \displaystyle\int_\Omega\xi^n\frac{\phi^{n+1}}{\Delta t}q + \int_\Omega\xi^nD_\mathrm{w}^n\nabla\phi^{n+1}\nabla q + \int_\Omega\delta_\mathrm{p}^n\nabla\phi^{n+1}p_\mathrm{sat}^n\nabla q + \int_\Omega\delta_\mathrm{p}^n\phi^{n+1}\nabla p_\mathrm{sat}^{n}\nabla q - \int_{\Gamma_{N,\phi}}\left(\delta_\mathrm{p}^n\phi^{n+1}\nabla p_\mathrm{sat}^n\cdot\mathbf{n}\right)q$

• $g(q) = \displaystyle\int_\Omega\left(\xi^n\dfrac{\phi^n}{\Delta t} + G^{n+1}\right)q + \int_{\Gamma_{N,\phi}}\left(\xi^nD_\mathrm{w}^n+\delta_\mathrm{p}^np_\mathrm{sat}^n\right)N_\phi^{n+1}q$

2. Model properties

As explained in the Feel Tutorial Dev, the same set of equation allow solving a lot of different problems. That is why we define a model in Feel.

Models are represented by a class `ModelProperties` which contains sub-classes corresponding to the different sections of a JSON file. For example, I will show how to do it with boundary conditions, but the process is quite similar.

This object is stored as a member of the class `HeatMoisture`, named `props_`.

3. Example of usage for boundary conditions

``````"BoundaryConditions":
{
"field":                    (1)
{
"BC_type":              (2)
{
"marker":           (3)
{
"expr":"value"  (4)
}
}
}``````
 1 `field` is the field on what the boundary condition is applied (`phi` for relative humidity or `T` for heat) 2 `BC_type` is the type of boundary condition applied to the field (`Dirichlet` or `Neumann`) 3 `marker` is the name of the physical entity in the geo file, on which the condition is applied 4 `value` is the expression of the value of the condition applied to the field. For Dirichlet condition, it is equal to the value above it, and for Neumann condition, it is the value of $f$ such as $\frac{\partial \phi}{\partial n}=f$.

The boundary conditions are used in the code at the definition of the forms, in the functions `update` of the two classes (see section 5.3 of the project report)

We distinguich three type of physics used in the model :

• `heat`, meaning that we solve only the heat problem

• `moisture`, meaning that we solve only the moisture problem

• `heat-moisture`, combining the two problems

We define the physics in the JSON file in a section `Materials`, on associated markers, allowing to combine physics in a more complex geometry :

``````"Materials":
{
"Omega":
{
"physics":["moisture"],
"markers":["Omega"],
}
}``````

Let’s focus on the implementation in C++ of the boundary conditions. Here is the (compacted) code of the `update` function in the class `Heat`.

``````void update( double t, PhiT const& phi, double dt )
{
this->a_.zero() ;
this->l_.zero() ;

for ( auto const& [name, mat] : this->props_.materials() ) (1)
{
if ( mat.hasPhysics( "heat" ) || mat.hasPhysics( "heat-moisture" ) ) (2)
{
// don't depend on boundary conditions
this->a_ += integrate( _range = markedelements( this->mesh(), mat.meshMarkers() ),
_expr = ... );
this->l_ += integrate( _range = markedelements( this->mesh(), mat.meshMarkers() ),
_expr = ... );

if ( mat.hasPhysics( "moisture" ) || mat.hasPhysics( "heat-moisture" ) ) (2)
{
this->l_ += integrate( _range = markedelements( this->mesh(), mat.meshMarkers() ),
_expr = ... );
}

// depend on the boundary conditions
for (auto const& [bcid, bc] : this->props_.boundaryConditions2().flatten()) (3)
{
if (bcid.type() == "Neumann")
{
if (bcid.field() == "T")    // for heat
this->l_ += integrate( _range = markedfaces( this->mesh(), bc.markers() ), _expr = ... );
if ( bcid.field() == "phi" && (mat.hasPhysics( "moisture" ) || mat.hasPhysics( "heat-moisture" )) )  // for moisture
this->l_ += integrate( _range = markedfaces( this->mesh(), bc.markers() ), _expr = .... );
}
}

for (auto const& [bcid, bc] : this->props_.boundaryConditions2().flatten()) (4)
{
if (bcid.type() == "Dirichlet" && bcid.field() == "T")
{
auto Ts = expr( bc.expression() ) ;
Ts.setParameterValues({"t",t}) ;
this->a_ += on( _range=markedfaces(this->mesh(), bc.markers()), _rhs=this->l_, _element=T_, _expr=Ts) ;
}

}
}
}
}``````
 1 We begin by making a loop on all the materials. 2 If this material has the physic `heat` (or `heat-moisture`), we apply the conditions (and idem for the `moisture` term). 3 We make a loop on all the boundary conditions, adding in a first time the terms of Neumann conditions 4 Finally, we make a second loop on the boundary conditions, because we have to finish by adding the Dirichlet conditions.

4. Difficulties encountered

With all those modifications, the application work on the first try on the case drying-layer involving only the moisture transport process, but didn’t worked on moisture-uptake : after a few steps, the linear solver of Feel++ didn’t converge, and therefore the results were totally false !

When we try to compare the values of the coefficients got from this simulation and the one got on the application when it was working (without the json file, it correspond to this this state in the history), we see that atfer the first step there are mistakes. The main hypothesis of such error is because we use here expression depending on other expression : for instance, `K` depends on `w`, and `w` depends on `phi`. There is a fonctionnality in Feel++ allowing to do refactoring, which is still in development : we have to be in the branch `features/refactoring` of the repository `feelpp`.

And, of course, there were parenthesis mistakes that made the solutions false ! Once corrected, all worked quite correctly !

5. Documentation of the application

The documentation of the application `feelpp_hm_heat_moisture` can be found here.

6. Test on several physics

We watch here a created case, with no real correspondance in physic. It is a mix between the two previous cases, moisture-uptake and drying-layer. The geometry is represented on figure 1 : on the left domain $\Omega_M$, the physic `moisture` of the drying-layer case is applied, and on the right one $\Omega_{HM}$, the physic `heat-moisture` is applied. Moreover, we add a Dirichlet condition on the edge $\Gamma_\mathrm{mid}$ with $\phi=0.95$.

Figure 1. Geometry of the case

In figure 2 is represented the water content after a simulation of 7 days.

Figure 2. Water content (kg/m³)

What we see is not relevant in physic, but we figure that the results of the application are coherent.