[web] Added heatconduction benchmarks.

web/config.toml

@@ -33,6 +33,12 @@ news = "news"
 [[menu.benchmarks]]
   name = "Elliptic"
   identifier = "elliptic"
+  weight = 1
 [[menu.benchmarks]]
   name = "Small Deformations"
   identifier = "small-deformations"
+  weight = 2
+[[menu.benchmarks]]
+  name = "Heatconduction"
+  identifier = "heatconduction"
+  weight = 3

web/content/docs/benchmarks/heatconduction/heatconduction-dirichlet.md

++++
+author = "Tianyuan Zheng"
+date = "2017-02-15T14:54:02+01:00"
+title = "Heatconduction (Dirichlet)"
+weight = 121
+project = "Parabolic/T/1D_dirichlet/line_60_heat.prj"
+
+[menu]
+  [menu.benchmarks]
+    parent = "heatconduction"
+
++++
+
+{{< project-link >}}
+
+## Equations
+
+We consider homogeneous parabolic equation here:
+$$
+\begin{equation}
+\rho C_\textrm{p}\frac{\partial T}{\partial t} + \nabla\cdot(\lambda \nabla T) = 0 \quad \text{in }\Omega
+\end{equation}$$
+w.r.t boundary conditions
+$$
+\eqalign{
+T(x) = g_D(x) &\quad \text{on }\Gamma_D,\cr
+k{\partial T(x) \over \partial n} = g_N(x) &\quad \text{on }\Gamma_N,
+}
+$$
+where $T$ could be temperature, the subscripts $D$ and $N$ denote the Dirichlet- and Neumann-type boundary conditions, $n$ is the normal vector pointing outside of $\Omega$, and $\Gamma = \Gamma_D \cup \Gamma_N$ and $\Gamma_D \cap \Gamma_N = \emptyset$.
+
+## Problem specification and analytical solution
+
+We solve the Parabolic equation on a line domain $[60\times 1]$ with $k = 3.2$ and $\rho C_\textrm{p} = 2.5e10^6$ w.r.t. the specific boundary conditions:
+
+$$
+\eqalign{
+T(x) = 274.15 &\quad \text{on } (x=0) \subset \Gamma_D,\cr
+k {\partial T(x) \over \partial n} = 0 &\quad \text{on } (x=60) \subset \Gamma_N.
+}
+$$
+
+The solution of this problem is
+$$
+\begin{equation}
+T(x,t) = T_\textrm{b} \textrm{erfc}(\frac{x}{\sqrt{4\alpha t}}),
+\end{equation}
+$$
+
+where $T_\textrm{b}$ is the boundary temperature, $\textrm{erfc}$ is the complementary error function and $\alpha = \lambda/(C_p \rho)$ is the thermal diffusivity.
+
+## Input files
+
+The main project file is [line_60_heat.prj](https://github.com/ufz/ogs-data/blob/master/Parabolic/T/1D_dirichlet/line_60_heat.prj). It describes the processes to be solved and the related process variables together with their initial and boundary conditions. It also references the mesh and geometrical objects defined on the mesh.
+
+As of now a small portion of possible inputs is implemented; one can change:
+ - the mesh file
+ - the geometry file
+ - introduce more/different Dirichlet boundary conditions (different geometry or other constant values)
+ - introduce more/different Neumann boundary conditions (different geometry or other constant values)
+
+The geometries used to specify the boundary conditions are given in the `line_60_heat.gml` file.
+
+The input mesh `line_60_heat.vtu` is stored in the VTK file format and can be directly visualized in Paraview for example.
+
+## Results and evaluation
+
+The result, written in the `.vtu` file, can be visualized with Paraview, for example.
+
+Loading the `line_60_heat_pcs_0_ts_65_t_5078125.000000.vtu` file in Paraview and Plotting over line. Compared to the analytical solution 'temperature_analytical.vtu', the results are very good:
+
+{{< img src="../validation-1.png" >}}

web/content/docs/benchmarks/heatconduction/heatconduction-neumann.md

++++
+date = "2017-02-15T14:54:12+01:00"
+title = "Heatconduction (Neumann)"
+weight = 122
+project = "Parabolic/T/1D_neumann/line_60_heat.prj"
+author = "Tianyuan Zheng"
+
+[menu]
+  [menu.benchmarks]
+    parent = "heatconduction"
+
++++
+
+{{< project-link >}}
+
+## Equations
+
+We consider homogeneous parabolic equation here:
+$$
+\begin{equation}
+\rho C_\textrm{p}\frac{\partial T}{\partial t} + \nabla\cdot(\lambda \nabla T) = 0 \quad \text{in }\Omega
+\end{equation}$$
+w.r.t boundary conditions
+$$
+\eqalign{
+T(x, t=0) = T_0,\cr
+T(x) = g_D(x) &\quad \text{on }\Gamma_D,\cr
+k{\partial T(x) \over \partial n} = g_N(x) &\quad \text{on }\Gamma_N,
+}
+$$
+where $T$ could be temperature, the subscripts $D$ and $N$ denote the Dirichlet- and Neumann-type boundary conditions, $n$ is the normal vector pointing outside of $\Omega$, and $\Gamma = \Gamma_D \cup \Gamma_N$ and $\Gamma_D \cap \Gamma_N = \emptyset$.
+
+## Problem specification and analytical solution
+
+We solve the Parabolic equation on a line domain $[60\times 1]$ with $k = 3.2$ and $\rho C_\textrm{p} = 2.5e10^6$ w.r.t. the specific boundary conditions:
+
+$$
+\eqalign{
+k {\partial T(x) \over \partial n} = 2 &\quad \text{on } (x=0) \subset \Gamma_N.\cr
+k {\partial T(x) \over \partial n} = 0 &\quad \text{on } (x=60) \subset \Gamma_N.
+}
+$$
+
+The solution of this problem is
+
+TODO: is not rendered correct
+
+$$
+\begin{equation}
+T(x,t) = \frac{2q}{\lambda}\left((\frac{\alpha t}{\pi})^{\frac{1}{2}} e^{-x^2/4\alpha t} - \frac{x}{2}\textrm{erfc}( \frac{x}{2\sqrt{\alpha  t}})\right),
+\end{equation}
+$$
+
+where $T_\textrm{b}$ is the boundary temperature, $\textrm{erfc}$ is the complementary error function and $\alpha = \lambda/(C_p \rho)$ is the thermal diffusivity.
+
+## Input files
+
+The main project file is `line_60_heat.prj`. It describes the processes to be solved and the related process variables together with their initial and boundary conditions. It also references the mesh and geometrical objects defined on the mesh.
+
+As of now a small portion of possible inputs is implemented; one can change:
+ - the mesh file
+ - the geometry file
+ - introduce more/different Dirichlet boundary conditions (different geometry or other constant values)
+ - introduce more/different Neumann boundary conditions (different geometry or other constant values)
+
+The geometries used to specify the boundary conditions are given in the `line_60_heat.gml` file.
+
+The input mesh `line_60_heat.vtu` is stored in the VTK file format and can be directly visualized in Paraview for example.
+
+## Results and evaluation
+
+The result, written in the `.vtu` file, can be visualized with Paraview, for example.
+
+Loading the `line_60_heat_pcs_0_ts_65_t_5078125.000000.vtu` file in Paraview and Plotting over line. Compared to the analytical solution 'temperature_analytical.vtu', the results are very good:
+
+{{< img src="../validation_neumann-1.png" >}}