[webdocu] TimeDependentHeterogeneousParameter docu.

web/content/docs/benchmarks/liquid-flow/time-dependent-heterogeneous-source-term-and-boundary-conditions.pandoc

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+date = "2019-08-02T11:33:45+01:00"
+title = "Liquid flow with time dependent boundary conditions and source term"
+weight = 171
+project = "Parabolic/LiquidFlow/TimeDependentHeterogeneousSourceTerm/TimeDependentHeterogeneousSourceTerm.prj"
+author = "Thomas Fischer"
+
+[menu]
+  [menu.benchmarks]
+    parent = "liquid-flow"
+
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+
+{{< data-link >}}
+
+## Motivation
+
+In real world examples the boundary conditions or source terms can vary over time
+and can be heterogeneous in space. This behaviour can be modelled using the
+TimeDependentHeterogeneousParameter for boundary conditions or source terms.
+
+## Specification in OGS project file
+
+In the parameter specification section of the project file it is possible to add
+a parameter type with the type `TimedependentHeterogeneousParameter`.
+```
+<parameter>
+    <name>ParameterForSourceTerm</name>
+    <type>TimeDependentHeterogeneousParameter</type>
+    <time_series>
+        <pair>
+            <time>0</time>
+            <parameter_name>parameter_for_timestep1</parameter_name>
+        </pair>
+        <pair>
+            <time>1</time>
+            <parameter_name>parameter_for_timestep2</parameter_name>
+        </pair>
+        ...
+        <pair>
+            <time>end_time</time>
+            <parameter_name>parameter_for_end_time</parameter_name>
+        </pair>
+    </time_series>
+</parameter>
+```
+Of course, the referenced parameters for the particular time steps have to be
+defined also. Values of the parameter are piecewise linear interpolated.
+
+
+## Example
+
+This simple example should demonstrate the use of the time depenendent
+heterogeneous parameter. We start with homogeneous parabolic problem:
+$$
+\begin{equation}
+s\; \frac{\partial p}{\partial t} + k\; \Delta p = q(t,x) \quad \text{in }\Omega
+\end{equation}
+$$
+w.r.t boundary conditions
+$$
+\eqalign{
+p(t, x) = g_D(t, x) &\quad \text{on }\Gamma_D,\cr
+k\;{\partial p(x) \over \partial n} = g_N(x) &\quad \text{on }\Gamma_N,
+}$$
+
+The example the domain $\Omega = [0,1]^2$ is a square. On the left
+($x=0$) side and the right ($x=1$) side time dependent Dirichlet-type boundary
+conditions are set. Until half of the simulation time high pressure values are
+set on the left side and low pressure values on the right side. In the second
+half of the simulation there are low pressure values on the left side and high
+pressure values on the right side. Additionally, the source term $q$ acts in
+the first quarter as a source, in the second quarter as a sink, in the third
+quarter as a source, and in the last quarter as a sink again.
+
+## Results
+
+<video width="838" height="762" controls>
+<source src="../TimeDependentHeterogeneousBoundaryConditionsAndSourceTerm.mp4" type="video/mp4" />
+</video>
+<p>
+<strong>Download Video:</strong>
+<a href="../TimeDependentHeterogeneousBoundaryConditionsAndSourceTerm.mp4">"MP4"</a>
+</p>