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.
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.
This simple example should demonstrate the use of the time dependent 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 Dirichlettype 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.
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This article was written by Thomas Fischer. If you are missing something or you find an error please let us know.
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Last revision: July 5, 2024
Commit: [T] Are times equivalent for tiny time increments 4bf1ab6
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