diff --git a/doc/content/index.md b/doc/content/index.md index 3aa6611e..4d6c5ef6 100644 --- a/doc/content/index.md +++ b/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). diff --git a/doc/content/verification/bibfile.bib b/doc/content/verification/bibfile.bib new file mode 100644 index 00000000..72766bdd --- /dev/null +++ b/doc/content/verification/bibfile.bib @@ -0,0 +1,16 @@ +@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)} +} +@article{longhurst2005verification, + title={Verification and validation of the tritium transport code TMAP7}, + author={Longhurst, Glen R and Ambrosek, James}, + journal={Fusion science and technology}, + volume={48}, + number={1}, + pages={468--471}, + year={2005}, + publisher={Taylor \& Francis} +} diff --git a/doc/content/verification/figures/val-1a_comparison.png b/doc/content/verification/figures/val-1a_comparison.png new file mode 100644 index 00000000..b3463aac Binary files /dev/null and b/doc/content/verification/figures/val-1a_comparison.png differ diff --git a/doc/content/verification/figures/val-1b_comparison_dist.png b/doc/content/verification/figures/val-1b_comparison_dist.png new file mode 100644 index 00000000..ae252155 Binary files /dev/null and b/doc/content/verification/figures/val-1b_comparison_dist.png differ diff --git a/doc/content/verification/figures/val-1b_comparison_flux.png b/doc/content/verification/figures/val-1b_comparison_flux.png new file mode 100644 index 00000000..cf7e7ab4 Binary files /dev/null and b/doc/content/verification/figures/val-1b_comparison_flux.png differ diff --git a/doc/content/verification/figures/val-1b_comparison_time.png b/doc/content/verification/figures/val-1b_comparison_time.png new file mode 100644 index 00000000..a81aea65 Binary files /dev/null and b/doc/content/verification/figures/val-1b_comparison_time.png differ diff --git a/doc/content/verification/figures/val-1c_comparison_time.png b/doc/content/verification/figures/val-1c_comparison_time.png new file mode 100644 index 00000000..5a390ced Binary files /dev/null and b/doc/content/verification/figures/val-1c_comparison_time.png differ diff --git a/doc/content/verification/figures/val-1d_comparison_diffusion.png b/doc/content/verification/figures/val-1d_comparison_diffusion.png new file mode 100644 index 00000000..30c7aec7 Binary files /dev/null and b/doc/content/verification/figures/val-1d_comparison_diffusion.png differ diff --git a/doc/content/verification/figures/val-1d_comparison_trapping.png b/doc/content/verification/figures/val-1d_comparison_trapping.png new file mode 100644 index 00000000..e7b95099 Binary files /dev/null and b/doc/content/verification/figures/val-1d_comparison_trapping.png differ diff --git a/doc/content/verification/val-1a.md b/doc/content/verification/val-1a.md new file mode 100644 index 00000000..b6c5bb37 --- /dev/null +++ b/doc/content/verification/val-1a.md @@ -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 diff --git a/doc/content/verification/val-1b.md b/doc/content/verification/val-1b.md new file mode 100644 index 00000000..57a32377 --- /dev/null +++ b/doc/content/verification/val-1b.md @@ -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 diff --git a/doc/content/verification/val-1c.md b/doc/content/verification/val-1c.md new file mode 100644 index 00000000..a6070d77 --- /dev/null +++ b/doc/content/verification/val-1c.md @@ -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 diff --git a/doc/content/verification/val-1d.md b/doc/content/verification/val-1d.md index e5bde788..e65acdb6 100644 --- a/doc/content/verification/val-1d.md +++ b/doc/content/verification/val-1d.md @@ -1,13 +1,94 @@ # val-1d -This reproduces the val-1d verification test from TMAP4. We examine two -different regimes, one where diffusion is the rate-limiting step, and one where -trapping is the rate-limiting step. +# Permeation Problem with Trapping + +## Test Description + +This validation problem is taken from [!cite](longhurst1992verification). It models permeation through a membrane with a constant source in which traps are operative. The breakthrough time may have one of two limiting values depending on whether the trapping is in the effective diffusivity or strong-trapping regime. A trapping parameter is defined by: + +\begin{equation} + \label{eqn:zeta} + \zeta = \frac{\lambda^2 \nu}{\rho D_o} exp \left( \frac{E_d - \epsilon}{kT} \right) + \frac{c}{\rho} +\end{equation} + +where + +$\lambda$ = lattice parameter + +$\nu$ = Debye frequency ($\approx$ $10^{13} \; s^{-1}$) + +$\rho$ = trapping site fraction + +$D_o$ = diffusivity pre-exponential + +$E_d$ = diffusion activation energy + +$\epsilon$ = trap energy + +$k$ = Boltzmann's constant + +$T$ = temperature + +$c$ = dissolved gas atom fraction + +The discriminant for which regime is dominant is the ratio of $\zeta$ to c/$\rho$. If $\zeta$ > c/$\rho$ then the effective diffusivity regime applies, and the permeation transient is identical to the standard diffusion transient but with the diffusivity replaced by an effective diffusivity. + +\begin{equation} +\label{eqn:Deff} + D_eff = \frac{D}{1 + \frac{1}{\zeta}} +\end{equation} + +In this limit, the breakthrough time, defined as the intersection of the steepest tangent to the diffusion transient with the time axis, will be + +\begin{equation} +\label{eqn:tau_be} + \tau_{b_e} = \frac{l^2}{2 \; \pi^2 \; D_eff} +\end{equation} + +where $l$ is the thickness of the slab and D is the diffusivity of the gas through the material. The permeation transient is then given by + + +\begin{equation} +\label{eqn:Jp} + J_p = \frac{c_o D}{l} \Bigg\{ 1 + 2 \sum_{m=1}^{\infty} \left[ (-1)^m \exp \left( -m^2 \frac{t}{2 \; \tau_{b_e}} \right) \right] \Bigg\} +\end{equation} + + +[!cite](longhurst2005verification) where $\tau_{b_e}$ is defined in [eqn:tau_be] + +In the deep-trapping limit, $\zeta$ < c/$\rho$, and no permeation occurs until essentially all the traps have been filled. Then permeation rapidly turns on to its state value. The breakthrough time is given by + +\begin{equation} +\label{eqn:tau_bd} + \tau_{b_d} = \frac{l^2 \rho}{2 \; c_o \; D} +\end{equation} + +where $c_o$ is the steady dissolved gas concentration at the upstream (x = 0) side. + +Using TMAP8 we examine these two different regimes, one where diffusion is the rate-limiting step, and one where trapping is the rate-limiting step. The upstream-side starting concentration of 0.0001 atom fraction, a diffusivity of 1 $m^2$/s, a trapping site fraction of 0.1, $\lambda^2 = 10^{-15} \; m^2$, and a temperature of 1000 K is considered. + ## Diffusion-limited +For the effective diffusivity limit, we selected $\epsilon/k = 100 K$ to give $\zeta = 90.48 c/\rho$. The comparison results are presented in [val-1d_comparison_diffusion]. + +!media figures/val-1d_comparison_diffusion.png + style=width:50%;margin-bottom:2% + id=val-1d_comparison_diffusion + caption=Permeation history of a slab subject to effective-diffusivity limit trapping. + ## Trapping-limited +For the deep trapping limit we took $\epsilon/k = 10000 K$ to give $\zeta = 0.04533 c/\rho$. The comparison results are presented in [val-1d_comparison_trapping]. + +!media figures/val-1d_comparison_trapping.png + style=width:50%;margin-bottom:2% + id=val-1d_comparison_trapping + caption=Permeation transient in a slab subject to strong trapping. + + + + ### Notes The trapping test features some oscillations in the solution for whatever @@ -23,3 +104,5 @@ summarized below: The oscillations in the permeation graph go away with increasing fineness in the mesh and in `dt`. + +!bibtex bibliography diff --git a/doc/content/verification/val-list.md b/doc/content/verification/val-list.md new file mode 100644 index 00000000..35843f99 --- /dev/null +++ b/doc/content/verification/val-list.md @@ -0,0 +1,8 @@ +# List of validation cases + +| Case | Title | +| - | - | +| val-1a | [Depleting Source Problem](val-1a.md) | +| val-1b | [Diffusion Problem with Constant Source Boundary Condition](val-1b.md) | +| val-1c | [Diffusion Problem with Partially Preloaded Slab](val-1c.md) | +| val-1d | [Permeation Problem with Trapping](val-1d.md) | diff --git a/test/tests/val-1a/analytical.csv b/test/tests/val-1a/analytical.csv new file mode 100644 index 00000000..0467dc74 --- /dev/null +++ b/test/tests/val-1a/analytical.csv @@ -0,0 +1,47 @@ +time(s),frac_rel +0,0 +1,6.00E-07 +2,0.000262 +3,0.002399 +4,0.007873 +5,0.016781 +6,0.028554 +7,0.042499 +8,0.06218 +9,0.074568 +10,0.091823 +11,0.109482 +12,0.127337 +13,0.145233 +14,0.163059 +15,0.180733 +16,0.198197 +17,0.21541 +18,0.232343 +19,0.248977 +20,0.265301 +21,0.281307 +22,0.296991 +23,0.312353 +24,0.327395 +25,0.342119 +26,0.356529 +27,0.37063 +28,0.384426 +29,0.397923 +30,0.411126 +31,0.424042 +32,0.436677 +33,0.449035 +34,0.461122 +35,0.472945 +36,0.484509 +37,0.49582 +38,0.506883 +39,0.517703 +40,0.528286 +41,0.538637 +42,0.54876 +43,0.558662 +44,0.568346 +45,0.577818 \ No newline at end of file diff --git a/test/tests/val-1a/comparison_val-1a.py b/test/tests/val-1a/comparison_val-1a.py new file mode 100644 index 00000000..63d0a606 --- /dev/null +++ b/test/tests/val-1a/comparison_val-1a.py @@ -0,0 +1,34 @@ +import csv +import matplotlib.pyplot as plt +import numpy as np +from matplotlib import gridspec +import pandas as pd + +#=============================================================================== + +fig = plt.figure(figsize=[6.5,5.5]) +gs = gridspec.GridSpec(1,1) +ax = fig.add_subplot(gs[0]) + +tmap_sol = pd.read_csv("./val-1a_csv.csv") +analytical_sol = pd.read_csv("./analytical.csv") +tmap_time = tmap_sol['time'] +tmap_fr = tmap_sol['rhs_release'] +analytical_time = analytical_sol['time(s)'] +analytical_fr = analytical_sol['frac_rel'] + +ax.plot(tmap_time,tmap_fr,label=r"TMAP8",c='tab:gray') +ax.plot(analytical_time,analytical_fr,label=r"Analytical",c='k', linestyle='--') + + +ax.set_xlabel(u'Time(s)') +ax.set_ylabel(r"Fractional release") +ax.legend(loc="best") +ax.set_xlim(left=0) +ax.set_xlim(right=45) +ax.set_ylim(bottom=0) +plt.grid(b=True, which='major', color='0.65', linestyle='--', alpha=0.3) + +ax.minorticks_on() +plt.savefig('comparison.png', bbox_inches='tight'); +plt.close(fig) diff --git a/test/tests/val-1b/comparison_val-1b.py b/test/tests/val-1b/comparison_val-1b.py new file mode 100644 index 00000000..3a27b21f --- /dev/null +++ b/test/tests/val-1b/comparison_val-1b.py @@ -0,0 +1,100 @@ +import csv +import matplotlib.pyplot as plt +import numpy as np +from matplotlib import gridspec +import pandas as pd +from scipy import special + +#========= Comparison of concentration as a function of time =================== + +fig = plt.figure(figsize=[6.5,5.5]) +gs = gridspec.GridSpec(1,1) +ax = fig.add_subplot(gs[0]) + +tmap_sol = pd.read_csv("./gold/u_vs_t.csv") +tmap_time = tmap_sol['time'] +tmap_conc = tmap_sol['conc_point1'] + +analytical_time = tmap_time +x = 0.2 +C_o = 1 +D = 1 +analytical_conc = C_o * special.erfc( x / (2 * np.sqrt(D*analytical_time))) +ax.plot(tmap_time,tmap_conc,label=r"TMAP8",c='tab:gray') +ax.plot(analytical_time,analytical_conc,label=r"Analytical",c='k', linestyle='--') + + +ax.set_xlabel(u'Time(s)') +ax.set_ylabel(r"Normalized specie concentration") +ax.legend(loc="best") +ax.set_xlim(left=0) +ax.set_xlim(right=45) +ax.set_ylim(bottom=0) +plt.grid(b=True, which='major', color='0.65', linestyle='--', alpha=0.3) + +ax.minorticks_on() +plt.savefig('val-1b_comparison_time.png', bbox_inches='tight'); +plt.close(fig) + + +#============ Comparison of concentration as a function of distance ============ + +fig = plt.figure(figsize=[6.5,5.5]) +gs = gridspec.GridSpec(1,1) +ax = fig.add_subplot(gs[0]) + +tmap_sol = pd.read_csv("./gold/u_vs_x.csv") +tmap_distance = tmap_sol['x'] +tmap_conc = tmap_sol['u'] + +analytical_distance = tmap_distance +time = 25 +C_o = 1 +D = 1 +analytical_conc = C_o * special.erfc( analytical_distance / (2 * np.sqrt(D * time))) + +ax.plot(tmap_distance,tmap_conc,label=r"TMAP8",c='tab:gray') +ax.plot(analytical_distance,analytical_conc,label=r"Analytical",c='k', linestyle='--') + +ax.set_xlabel(u'Distance (m)') +ax.set_ylabel(r"Normalized specie concentration") +ax.legend(loc="best") +ax.set_xlim(left=0) +ax.set_xlim(right=50) +ax.set_ylim(bottom=0) +plt.grid(b=True, which='major', color='0.65', linestyle='--', alpha=0.3) + +ax.minorticks_on() +plt.savefig('val-1b_comparison_dist.png', bbox_inches='tight'); +plt.close(fig) +#================== Comparison of flux as a function of time =================== + +fig = plt.figure(figsize=[6.5,5.5]) +gs = gridspec.GridSpec(1,1) +ax = fig.add_subplot(gs[0]) + +tmap_sol = pd.read_csv("./gold/u_vs_t.csv") +tmap_time = tmap_sol['time'] +tmap_flux = tmap_sol['flux_point2'] + +analytical_time = tmap_time +x = 0.5 +C_o = 1 +D = 1 +analytical_flux = C_o * np.sqrt(D/(np.pi * analytical_time)) * \ + np.exp( x / (2 * np.sqrt(D * analytical_time))) +ax.plot(tmap_time,tmap_flux,label=r"TMAP8",c='tab:gray') +ax.plot(analytical_time,analytical_flux,label=r"Analytical",c='k', linestyle='--') + + +ax.set_xlabel(u'Time(s)') +ax.set_ylabel(r"Diffusive flux") +ax.legend(loc="best") +ax.set_xlim(left=0) +ax.set_xlim(right=45) +ax.set_ylim(bottom=0) +plt.grid(b=True, which='major', color='0.65', linestyle='--', alpha=0.3) + +ax.minorticks_on() +plt.savefig('val-1b_comparison_flux.png', bbox_inches='tight'); +plt.close(fig) diff --git a/test/tests/val-1b/gold/u_vs_t.csv b/test/tests/val-1b/gold/u_vs_t.csv index f7b98255..17e08707 100644 --- a/test/tests/val-1b/gold/u_vs_t.csv +++ b/test/tests/val-1b/gold/u_vs_t.csv @@ -1,52 +1,52 @@ -time,point -0,0 -1,0.88273211356153 -2,0.91806516381586 -3,0.93394299841698 -4,0.94319642173673 -5,0.94937211590339 -6,0.95385710419842 -7,0.95730610658033 -8,0.9600688763038 -9,0.96234976575734 -10,0.96427665787963 -11,0.96593406541669 -12,0.96738036851543 -13,0.96865739813358 -14,0.96979605801011 -15,0.97081977555309 -16,0.97174670426263 -17,0.97259117762818 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type = Diffusion @@ -21,6 +28,16 @@ [../] [] +[AuxKernels] + [./flux_x] + type = DiffusionFluxAux + diffusivity = ${fparse 1.0} + variable = flux_x + diffusion_variable = u + component = x + [../] +[] + [BCs] [./left] type = DirichletBC @@ -48,11 +65,16 @@ [] [Postprocessors] - [point] + [conc_point1] type = PointValue variable = u point = '.2 0 0' [] + [flux_point2] + type = PointValue + variable = flux_x + point = '.2 0 0' + [] [] [Executioner] diff --git a/test/tests/val-1c/comparison_val-1c.py b/test/tests/val-1c/comparison_val-1c.py new file mode 100644 index 00000000..8a88b41d --- /dev/null +++ b/test/tests/val-1c/comparison_val-1c.py @@ -0,0 +1,43 @@ +import csv +import matplotlib.pyplot as plt +import numpy as np +from matplotlib import gridspec +import pandas as pd +from scipy import special + +#========= Comparison of concentration as a function of time =================== + +fig = plt.figure(figsize=[6.5,5.5]) +gs = gridspec.GridSpec(1,1) +ax = fig.add_subplot(gs[0]) + +tmap_sol = pd.read_csv("./gold/val-1c_csv.csv") +tmap_time = tmap_sol['time'] +tmap_conc0 = tmap_sol['point0'] +tmap_conc10 = tmap_sol['point10'] +tmap_conc12 = tmap_sol['point12'] + +analytical_sol = pd.read_csv("./analytical.csv") +analytical_time = analytical_sol['time'] +analytical_conc0 = analytical_sol['point0'] +analytical_conc10 = analytical_sol['point10'] +analytical_conc12 = analytical_sol['point12'] + +ax.plot(tmap_time,tmap_conc0,label=r"TMAP8 x=0",c='tab:blue') +ax.plot(analytical_time,analytical_conc0,label=r"Analytical x=0",c='tab:blue', linestyle='--') +ax.plot(tmap_time,tmap_conc10,label=r"TMAP8 x=10",c='tab:red') +ax.plot(analytical_time,analytical_conc10,label=r"Analytical x=10",c='tab:red', linestyle='--') +ax.plot(tmap_time,tmap_conc12,label=r"TMAP8 x=12",c='k') +ax.plot(analytical_time,analytical_conc12,label=r"Analytical x=12",c='k', linestyle='--') + + +ax.set_xlabel(u'Time(s)') +ax.set_ylabel(r"Normalized specie concentration") +ax.legend(loc="best") +ax.set_xlim(left=0) +ax.set_ylim(bottom=0) +plt.grid(b=True, which='major', color='0.65', linestyle='--', alpha=0.3) + +ax.minorticks_on() +plt.savefig('val-1c_comparison_time.png', bbox_inches='tight'); +plt.close(fig) diff --git a/test/tests/val-1c/gold/val-1c_csv.csv b/test/tests/val-1c/gold/val-1c_csv.csv new file mode 100644 index 00000000..baece9b3 --- /dev/null +++ b/test/tests/val-1c/gold/val-1c_csv.csv @@ -0,0 +1,2002 @@ +time,point0,point10,point12 +0,0,0,0 +0.05,1,0.40253463036945,4.8667561107043e-05 +0.1,1,0.44506728109067,0.00027710978988594 +0.15,1,0.45839753113382,0.00086748607446549 +0.2,1,0.46523018866654,0.0019886006173187 +0.25,1,0.46953559757592,0.0037479416665834 +0.3,1,0.47256111852499,0.0061780752970588 +0.35,0.99999999999999,0.47483534557135,0.0092485143728364 +0.4,0.99999999999996,0.47662483030683,0.012887567557683 +0.45,0.99999999999985,0.4780802608965,0.017003334932399 +0.5,0.99999999999953,0.47929402714806,0.021499296334811 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+0,0 @@ -#!/usr/bin/env python - -import numpy as np -import matplotlib.pyplot as plt - -numerical_data = np.loadtxt("val-1c_csv.csv",delimiter=',',skiprows=1) - -analytical_data = np.loadtxt("analytical.csv",delimiter=',',skiprows=1) - -numerical_time = numerical_data[1:,0] -analytical_time = analytical_data[:,0] - -plt.plot(numerical_time,numerical_data[1:,1], - label='numerical, x = 0',color='blue',linestyle='-') -plt.plot(numerical_time,numerical_data[1:,2], - label='numerical, x = 10',color='red',linestyle='-') -plt.plot(numerical_time,numerical_data[1:,3], - label='numerical, x = 12',color='green',linestyle='-') -plt.plot(analytical_time,analytical_data[:,1], - label='analytical, x = 0',color='blue',linestyle='--') -plt.plot(analytical_time,analytical_data[:,2], - label='analytical, x = 10',color='red',linestyle='--') -plt.plot(analytical_time,analytical_data[:,3], - label='analytical, x = 12',color='green',linestyle='--') -plt.legend() -plt.show() diff --git a/test/tests/val-1d/comparison_val-1d.py b/test/tests/val-1d/comparison_val-1d.py new file mode 100644 index 00000000..79da086e --- /dev/null +++ b/test/tests/val-1d/comparison_val-1d.py @@ -0,0 +1,96 @@ +import csv +import matplotlib.pyplot as plt +import numpy as np +from matplotlib import gridspec +import pandas as pd +from scipy import special + +#============ Comparison of permeation as a function of time =================== + +num_summation_terms = 1000 + +N_o = 3.16e22 # For convenience lattice density chosen as 3.16e22 atom/m^3 + # Verification and Validation of Tritium Transport code TMAP7 + # Glen Longhurst & James Ambrosek, Fusion Science & Technology + # 2017 + +lambdaa = 3.162e-8 # lattice parameter (m) || lambda is a python keyword +nu = 1e13 # Debye frequency (1/s) +rho = 0.1 # trapping site fraction +D_o = 1 # diffusivity pre-exponential (m^2/s) +Ed = 0 # diffusion activation energy + +k = 1.38064852e-23 # Boltzmann's constant(m^2-kg / sec^2-K) +T = 1000 # temperature (K) +epsilon_k_ratio = 100 +epsilon = k * epsilon_k_ratio #epsilon: trap energy +c = 0.0001 # dissolved gas atom fraction +zeta = ((lambdaa**2) * nu * np.exp((Ed - epsilon) / (k * T)) / (rho * D_o)) + (c / rho) + +D = D_o * np.exp((epsilon - Ed) / (k * T)) # diffusivity (m^2/s) +D_eff = D / (1 + (1/zeta)) # Effective diffusivity (m^2/s) + +l = 1 # slab thickness (m) +c_o = c +tau_be = l**2 / (2 * (np.pi)**2 * D_eff) + +def summation_term(num_terms, time): + sum = 0.0 + for m in range(1, num_terms): + sum += (-1)**m * np.exp(-1 * m**2 * time / (2*tau_be)) + return sum + +c_o = c +analytical_time = np.linspace(0.0001, 3.0, 500) +Jp = N_o * (c_o * D / l) * (1 + 2 * summation_term(num_summation_terms, analytical_time)) + +fig = plt.figure(figsize=[6.5,5.5]) +gs = gridspec.GridSpec(1,1) +ax = fig.add_subplot(gs[0]) + +analytical_permeation = Jp +ax.plot(analytical_time,analytical_permeation,label=r"Analytical",c='k', linestyle='--') + +tmap_sol = pd.read_csv("./gold/val-1d-diffusion_out.csv") +tmap_time = tmap_sol['time'] +tmap_perm = tmap_sol['scaled_outflux'] +ax.plot(tmap_time,tmap_perm,label=r"TMAP8",c='tab:gray') + +ax.set_xlabel(u'Time(s)') +ax.set_ylabel(u"Permeation (atom/m$^2$s)") +ax.legend(loc="best") +#ax.set_xlim(left=0) +ax.set_ylim(bottom=0) +plt.grid(b=True, which='major', color='0.65', linestyle='--', alpha=0.3) + +ax.minorticks_on() +plt.savefig('val-1d_comparison_diffusion.png', bbox_inches='tight'); +plt.close(fig) + +#========================= Trapping limited ==================================== +tau_bd = l**2 * rho/ (2 * c_o * D) # breakthrough time +analytical_time = [tau_bd, tau_bd] +analytical_sol = [0, 3.2e18] # Adding numbers according to the range of axis + # to show the line for analytically calculated + # breakthrough time. + +tmap_sol = pd.read_csv("./gold/val-1d-trapping_out.csv") +tmap_time = tmap_sol['time'] +tmap_perm = tmap_sol['scaled_outflux'] + +fig = plt.figure(figsize=[6.5,5.5]) +gs = gridspec.GridSpec(1,1) +ax = fig.add_subplot(gs[0]) +ax.plot(tmap_time,tmap_perm,label=r"TMAP8",c='tab:gray') +ax.plot(analytical_time,analytical_sol,label=r"Analytical breakthrough time",c='k', linestyle='--') + +ax.set_xlabel(u'Time(s)') +ax.set_ylabel(u"Permeation (atom/m$^2$s)") +ax.legend(loc="best") +#ax.set_xlim(left=0) +#x.set_ylim(bottom=0) +plt.grid(b=True, which='major', color='0.65', linestyle='--', alpha=0.3) + +ax.minorticks_on() +plt.savefig('val-1d_comparison_trapping.png', bbox_inches='tight'); +plt.close(fig) diff --git a/test/tests/val-1d/gold/val-1d-diffusion_out.csv b/test/tests/val-1d/gold/val-1d-diffusion_out.csv new file mode 100644 index 00000000..46209a9f --- /dev/null +++ b/test/tests/val-1d/gold/val-1d-diffusion_out.csv @@ -0,0 +1,302 @@ +time,outflux,scaled_outflux +0,0,0 +0.01,7.9652711428312e-12,25187780.407861 +0.02,3.1165200852448e-12,9855059.813561 +0.03,9.9714227410311e-12,31531632.991689 +0.04,5.9065447876632e-11,186776759.27549 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