Documentation for validation cases 1-3 idaholab#12

Author: singhgp4321

doc/content/index.md

@@ -27,3 +27,9 @@ effectively changing the concentration of the trapped specie from being measured
 in #/volume to k#/volume where 'k' indicates kilo. After this transformation,
 the numerical concentrations of the trapped and mobile species are on the same
 order of magnitude.
+
+## [Validation](verification/val-list.md)
+
+Several problems originally developed for the TMAP4 code have been used for the
+validation of the TMAP8 code. These validation cases can be found
+[here](verification/val-list.md).

doc/content/verification/bibfile.bib

@@ -0,0 +1,6 @@
+@techreport{longhurst1992verification,
+  title={Verification and Validation of TMAP4},
+  author={Longhurst, GR and Harms, SL and Marwil, ES and Miller, BG},
+  year={1992},
+  institution={EG and G Idaho, Inc., Idaho Falls, ID (United States)}
+}

doc/content/verification/figures/val-1a_comparison.png

@@ -0,0 +1,48 @@
+# val-1a
+
+# Depleting Source Problem
+
+## Test Description
+
+This validation problem is taken from [!cite](longhurst1992verification). The model consists of an enclosure containing a finite concentration of atoms which are allowed to diffuse into a SiC layer over time. No solubility or trapping effects are included. The fractional release from the outside of the shell in a depleting source model in a slab geometry is given by:
+
+\begin{equation}
+    FR = 1.0 - \sum_{n=1}^{\infty} \frac{2\ L sec \ \alpha_{n} - \exp\left(\frac{-\alpha_{n}^2 D T}{l^{2}}\right)}{L(L+1) + \alpha_n^{2}}
+\end{equation}
+
+where
+
+\begin{equation}
+    L = \frac{lA}{V \phi}
+\end{equation}
+
+\begin{equation}
+    \phi = \frac{source \ concentration}{layer \ concentration}
+\end{equation}
+
+where the layer concentration is that at the interface with the source ($\phi$ is constant in time),
+
+    $A$ = surface area
+
+    $V$ = source volume
+
+    $l$ = layer thickness
+
+and the $\alpha_n$ are the roots of
+
+\begin{equation}
+    \alpha_n = \frac{L}{tan \ \alpha_n}
+\end{equation}
+
+## Results
+
+
+[val-1a_comparison] shows the comparison of the TMAP8 calculation and the analytical solution. There is good agreement between the two plots.
+
+
+!media figures/val-1a_comparison.png
+    style=width:50%;margin-bottom:2%
+    id=val-1a_comparison
+    caption=Comparison of TMAP8 calculation with the analytical solution
+
+!bibtex bibliography

doc/content/verification/figures/val-1b_comparison_dist.png

@@ -0,0 +1,72 @@
+# val-1b
+
+# Diffusion Problem with Constant Source Boundary Condition
+
+This validation problem is taken from [!cite](longhurst1992verification). Diffusion of tritium through a semi-infinite SiC layer is modeled with a constant
+source located on one boundary. No solubility or traping is included. The
+concentration as a function of time and position is given by:
+
+\begin{equation}
+C = C_o \; erfc \left(\frac{x}{2\sqrt{Dt}}\right)
+\end{equation}
+
+Comparison of the TMAP8 results and the analytical solution is shown in
+[val-1b_comparison_time] as a function of time at
+x = 0.2 mm. For simplicity, both the diffusion coefficient and the initial
+concentration were set to unity. The TMAP8 code predictions match very well with
+the analytical solution.
+
+!media figures/val-1b_comparison_time.png
+    style=width:50%;margin-bottom:2%
+    id=val-1b_comparison_time
+    caption=Comparison of concentration as function of time at x\=0.2m calculated
+     through TMAP8 and analytically
+
+As a second check, the concentration as a function of position at a given time
+t = 25s, from TMAP8 was compared with the analytical solution as shown in
+[val-1b_comparison_dist]. The predicted concentration profile from TMAP8 is in
+good agreement with the analytical solution.
+
+!media figures/val-1b_comparison_dist.png
+    style=width:50%;margin-bottom:2%
+    id=val-1b_comparison_dist
+    caption=Comparison of concentration as function of distance from the source
+    at t\=25sec calculated through TMAP8 and analytically
+
+Finally, the diffusive flux ($J$) was compared with the analytic solution where the
+flux is proportional to the derivative of the concentration with respect to x and
+is given by:
+
+\begin{equation}
+J = C_o \; \sqrt{\frac{D}{t\pi}} \; exp \left(\frac{x}{2\sqrt{Dt}}\right)
+\end{equation}
+
+The flux as given by Equation (?) is compared with values calculated by TMAP8 in
+Table ?. The diffusivity, D, and the initial concentration, C$_o$, were both
+taken as unity, and the distance, x, was taken as 0.5 in this comparison.
+TMAP8 initially under predicts but the results match well subsequently. Comparison
+results are shown in []
+
+!media figures/val-1b_comparison_flux.png
+    style=width:50%;margin-bottom:2%
+    id=val-1b_comparison_flux
+    caption=Comparison of flux as function of time at x\=0.5m calculated through
+    TMAP8 and analytically
+
+### Notes
+
+The trapping test features some oscillations in the solution for whatever
+reason. In order for the oscillations to not take over the simulation, it seems
+that the ratio of the **inverse of the Fourier number** must be kept
+sufficiently high, e.g. `h^2 / (D * dt)`. Included in this directory are three
+`png` files that show the permeation for different `h` and `dt` values. They are
+summarized below:
+
+- `nx-80.png`: `nx = 80` and `dt = .0625`
+- `nx-40.png`: `nx = 40` and `dt = .25`
+- `nx-20.png`: `nx = 20` and `dt = 1`
+
+The oscillations in the permeation graph go away with increasing fineness in the
+mesh and in `dt`.
+
+!bibtex bibliography

doc/content/verification/figures/val-1b_comparison_flux.png

@@ -0,0 +1,39 @@
+# val-1c
+
+# Diffusion Problem with Partially Preloaded Slab
+
+This validation problem is taken from [!cite](longhurst1992verification). Diffusion of tritium through a semi-infinite SiC layer is modeled with an initial
+loading of 1 atom/m{^3} in the first 10 m of a 2275-m slab. Solubility is unity
+and no traping is included. The analytical solution is given by:
+
+\begin{equation}
+C = \frac{C_o}{2} \left( erf \left( \frac{h-x}{2} \sqrt{Dt} \right) + erf \left( \frac{h+x}{2\sqrt{Dt}}) \right) \right)
+\end{equation}
+
+
+where h is the thickness of the pre-loaded portion of the layer.
+
+At the surface (x = 0) the concentration is given by:
+
+\begin{equation}
+C = \frac{C_o}{2} \; erf \left( \frac{h}{2\sqrt{Dt}} \right)
+\end{equation}
+
+while at x = h its value is described by
+
+\begin{equation}
+C = \frac{C_o}{2} \; erf \left( \frac{h}{\sqrt{Dt}} \right)
+\end{equation}
+
+A comparison of the mobile species concentration values at x = 0 m, 10 m and
+12.5 m calculated through TMAP8 and analytically is shown in
+[val-1c_comparison_time]. The TMAP8 calculations are found to be in good agreement
+with the analytical solution.
+
+!media figures/val-1c_comparison_time.png
+    style=width:50%;margin-bottom:2%
+    id=val-1c_comparison_time
+    caption=Comparison of concentration as function of time at x\=0 m, 10 m and 12 m
+    calculated through TMAP8 and analytically
+
+!bibtex bibliography