# A Finite Element Method for Nonlinear and Fully Coupled Transient Hydrothermal Transfer
## 6 Conclusions  
The constitutive equations governing the moisture content and temperature are usually highly nonlinear. For the majority of applications, the coupling diffusion equations are not amenable to analytical solutions. As such, a numerical method is essential. This paper constructs a finite element numerical technique to study the time-dependent equations that govern the coupled phenomenon of heat and moisture diffusion. The following conclusions can be drawn:  
1. The proposed numerical code includes the nonlinear coupling between the temperature and moisture. 2. The code can consider the temperature- and moisture-dependent material properties. 3. The code can study the transient heat and moisture conduction. The finite difference is employed to resolve the time-dependent portion of the formulation. Finite element discretization applies only to the space variables. 4. Numerical examples are given to validate the proposed computer code. Accuracy of the proposed model is confirmed. 5. Taking into account fully coupling of moisture and heat transfer is essential. As such, all existing theoretical solutions neglecting the coupling are not exact and even completely wrong. 6. Although numerical results are presented for two-dimensional problems, the extension to three dimensions is obvious and straightforward. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{\(tt_{\textbf{0}}\),\(t_{\textbf{0}}\),\(t_{\textbf{0}}\)} & \multirow{2}{*}{\(\textbf{r}lb\)} & \(\boldsymbol{T}\) & \multirow{2}{*}{\(\boldsymbol{T}\)} & \multirow{2}{*}{\(\boldsymbol{Relative}\)} & \multirow{2}{*}{\(\boldsymbol{H/(\gamma T_{\textbf{0}})}\)} & \multirow{2}{*}{\(\boldsymbol{H/(\gamma T_{\textbf{0}})}\)} & \multirow{2}{*}{\(\boldsymbol{Relative}\)} \\  & & **Element** & **(Analytic)** & **Difference (\%)** & & **Element** & **(Analytic)** & **Difference (\%)** \\ \hline \multirow{8}{*}{0.01} & 0.0 & 0.2837 & 0.2828 & 0.32 & 0.3060 & 0.3051 & 0.29 \\ \cline{2-8}  & 0.2 & 0.3206 & 0.3185 & 0.66 & 0.3459 & 0.3436 & 0.67 \\ \cline{2-8}  & 0.4 & 0.4247 & 0.4225 & 0.52 & 0.4581 & 0.4557 & 0.53 \\ \cline{2-8}  & 0.6 & 0.5851 & 0.5833 & 0.31 & 0.6274 & 0.6234 & 0.64 \\ \cline{2-8}  & 0.8 & 0.7833 & 0.7823 & 0.13 & 0.6626 & 0.6604 & 0.33 \\ \cline{2-8}  & 1.0 & 1.000 & 1.0000 & 0 & 0.0000 & 0.0000 & 0 \\ \hline \multirow{8}{*}{0.02} & 0.0 & 0.6550 & 0.6528 & 0.34 & 0.7066 & 0.7042 & 0.34 \\ \cline{2-8}  & 0.2 & 0.6736 & 0.6710 & 0.39 & 0.7266 & 0.7237 & 0.40 \\ \cline{2-8}  & 0.4 & 0.7247 & 0.7224 & 0.32 & 0.7785 & 0.7752 & 0.43 \\ \cline{2-8}  & 0.6 & 0.8014 & 0.7998 & 0.20 & 0.8061 & 0.8029 & 0.40 \\ \cline{2-8}  & 0.8 & 0.8968 & 0.8960 & 0.09 & 0.5902 & 0.5894 & 0.14 \\ \cline{2-8}  & 1.0 & 1.000 & 1.0000 & 0 & 0.0000 & 0.0000 & 0 \\ \hline \multirow{8}{*}{0.03} & 0.0 & 0.8290 & 0.8272 & 0.22 & 0.8940 & 0.8920 & 0.22 \\ \cline{2-8}  & 0.2 & 0.8374 & 0.8355 & 0.23 & 0.9014 & 0.8989 & 0.28 \\ \cline{1-1} \cline{2-8}  & 0.4 & 0.8609 & 0.8593 & 0.19 & 0.9071 & 0.9044 & 0.27 \\ \cline{1-1} \cline{2-8}  & 0.6 & 0.8979 & 0.8968 & 0.12 & 0.8309 & 0.8297 & 0.14 \\ \cline{1-1} \cline{2-8}  & 0.8 & 0.9466 & 0.9460 & 0.06 & 0.5260 & 0.5258 & 0.04 \\ \cline{1-1} \cline{2-8}  & 1.0 & 1.000 & 1.0000 & 0 & 0.0000 & 0.0000 & 0 \\ \hline \end{tabular}
\end{table} 
Table 4: Comparison of finite element results and the analytical results (Equation (27)) for sudden change of surface temperature from 0 to \(T_{0}\). **Author Contributions:** Conceptualization, B.W. ; Investigation, J.L. All authors have read and agreed to the published version of the manuscript. **Funding:** This research received no external funding. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding authors. **Acknowledgments:** [PERSON] was supported by the Jinling Institute of Technology Promising Talent Hatching Project.