@@ -6,9 +6,9 @@ The content of this chapter deals with numerical details of the coupled approach
An application of the developed approach is described in the next chapter.
The chapter is organized as follows.
\Cref{sec:lbm-mhfem:problem formulation} formulates the general problem and its special case with an analytical solution for numerical analysis.
\Cref{sec:lbm-mhfem:problem formulation} formulates the general problem and its special case with an analytical solution for convergence analysis.
Then, \cref{sec:lbm-mhfem:algorithm,sec:lbm-mhfem:interpolation,sec:lbm-mhfem:decomposition} provide details related to the coupled computational approach and its implementation.
The final \cref{sec:lbm-mhfem:numerical example} describes the results of the numerical analysis using the benchmark problem from the first section.
The final \cref{sec:lbm-mhfem:numerical example} describes the results of the experimental convergence analysis using the benchmark problem from the first section.
The scalar coefficients $N_{i,j}$, $m_i$, $r_{i,j}$, and $f_i$ in \cref{eq:NumDwarf} are approximated by projection into the space $S_0(\mathcal K_h)$, i.e.
@@ -564,7 +564,7 @@ Similarly to the above, we can replace the term $m_{i,K_\ell,E}^k$ with the comm
\right\} = 0
\end{equation}
for all $i \in\{1, \ldots, n \}$ and all interior faces $E \in\mathcal E_h^{\mathrm{int}}$ such that $E \in\mathcal E_{K_1}\cap\mathcal E_{K_2}$.
The aforementioned procedure where the mobility $m_{i,E}^{k,\mathrm{upw}}$ was eliminated from the balance equation no longer applies.
The aforementioned procedure, where the mobility $m_{i,E}^{k,\mathrm{upw}}$ was eliminated from the balance equation, no longer applies.
However, when the mobility $m_i$ does not depend on the unknown variables $Z_j$ and it is strictly positive, the scheme still results in a non-singular system of equations and the implicit upwind scheme may be advantageous for advection problems.
\subsection{Hybridization}
@@ -594,7 +594,7 @@ Overall, we have a system of $n \times (\# \mathcal K_h + \# \mathcal E_h)$ line
\end{equation}
where $\vec Z_{\mathcal K_h}^{k+1}=[\vec Z_K^{k+1}]_{K \in\mathcal K_h}$, $\vec Z_K^{k+1}=[Z_{j,K}^{k+1}]_{j=1}^n$, $\vec Z_{\mathcal E_h}^{k+1}=[\vec Z_F^{k+1}]_{F \in\mathcal E_h}$ and $\vec Z_F^{k+1}=[Z_{j,F}^{k+1}]_{j=1}^n$.
However, this system is unnecessarily large as we can use its block structure to further reduce the number of equations in the problem.
If the blocks $\matrix Q$, $\matrix R$ and $\vec G$ represent \cref{eq:dg:discrete}, then $\matrix Q$ is block-diagonal and contains $\#\mathcal K_h$ blocks of the size $n \times n$.
If the blocks $\matrix Q$, $\matrix R$, and $\vec G$ represent \cref{eq:dg:discrete}, then $\matrix Q$ is block-diagonal and contains $\#\mathcal K_h$ blocks of the size $n \times n$.
By computing $\vec Z_{\mathcal K_h}^{k+1}=\matrix Q^{-1}(\matrix R \vec Z_{\mathcal E_h}^{k+1}+\vec G)$, we can eliminate the unknowns $\vec Z_{\mathcal K_h}^{k+1}$ from the system.
In the context of MHFEM, the technique is called \emph{hybridization} or \emph{static condensation}\cite{brezzi:1991mixed}.
@@ -65,7 +65,7 @@ To the best of our knowledge, they represent unique ideas that push forward the
\item
To develop a scalable solver based on the lattice Boltzmann method for GPU-based supercomputers.
\item
To develop a mathematical model of vapor transport in air, implement a high-performance solver for the model, and validate it using experimentally measured data.
To develop a mathematical model of vapor transport by air flow above a soil surface, implement a high-performance solver for the model, and validate it using experimentally measured data.
\end{enumerate}
\section*{Achieved results}
@@ -96,7 +96,7 @@ The thesis presents the following novel results:
The solver benefits from native implementation of both LBM and MHFEM for GPU accelerators, which allows for efficient coupling between the methods.
Our results are included in the publication \cite{klinkovsky2022:WT}.
\item
\textbf{Mathematical model for vapor transport in air and its validation using experimental data.}
\textbf{Mathematical model for vapor transport in air flow and its validation using experimental data.}
The aforementioned coupled computational approach has been used to model transport of water vapor in the turbulent boundary layer above a disturbed soil surface.
While the porous medium below the surface is not simulated in this work, the interaction between soil and atmosphere such as evaporation is modeled using boundary conditions.
The model is compared both qualitatively and quantitatively to experimental data measured in three configurations resulting in different flow regimes.
In this chapter, we use the coupled LBM-MHFEM computational approach developed in \cref{chapter:LBM-MHFEM} to simulate vapor transport in air.
In this chapter, we use the coupled LBM-MHFEM computational approach developed in \cref{chapter:LBM-MHFEM} to simulate vapor transport in turbulent air flow above a soil surface.
The content is based on the paper \cite{klinkovsky2022:WT}.
The author developed the mathematical model and computational methodology, performed all simulations and compared the results with experimental data.
The experimental methodology was developed by Andrew Trautz and Tissa Illangasekare, the analysis of the results and overall integration of mathematical modeling and experimental work were performed collectively by all co-authors of the paper.
The experimental methodology was developed by Andrew C. Trautz and Tissa H. Illangasekare, the analysis of the results and overall integration of mathematical modeling and experimental work were performed collectively by all co-authors of the paper.
The chapter is organized as follows.
First, the motivation and introduction to the mathematical modeling of vapor transport in air is described in \cref{sec:WT:introduction}.
