diff --git a/web/content/docs/benchmarks/liquid-flow/PVCDBC_1_ts_2.png b/web/content/docs/benchmarks/liquid-flow/PVCDBC_1_ts_2.png new file mode 100644 index 0000000000000000000000000000000000000000..dca595d3cb0ad3e56db17ce6b6a5c4f96e20cbff Binary files /dev/null and b/web/content/docs/benchmarks/liquid-flow/PVCDBC_1_ts_2.png differ diff --git a/web/content/docs/benchmarks/liquid-flow/primary-variable-constrain-dirichlet-boundary-condition.pandoc b/web/content/docs/benchmarks/liquid-flow/primary-variable-constrain-dirichlet-boundary-condition.pandoc new file mode 100644 index 0000000000000000000000000000000000000000..23cc48060e2045d974fdff90e0a8cfe8092cebde --- /dev/null +++ b/web/content/docs/benchmarks/liquid-flow/primary-variable-constrain-dirichlet-boundary-condition.pandoc @@ -0,0 +1,66 @@ ++++ +date = "2020-06-18T11:33:45+01:00" +title = "Primary variable constraint Dirichlet-type boundary condition" +weight = 171 +project = "Parabolic/LiquidFlow/SimpleSynthetics/PrimaryVariableConstraintDirichletBC/cuboid_1x1x1_hex_1000_Dirichlet_Dirichlet_1.prj" +author = "Thomas Fischer" + +[menu] + [menu.benchmarks] + parent = "liquid-flow" + ++++ + +{{< data-link >}} + +## Problem description + +We start with the following parabolic PDE: +$$ +\left( c \rho_R + \phi \frac{\partial \rho_R}{\partial p}\right) \frac{\partial +p}{\partial t} - \nabla \cdot +\left[ \rho_R \frac{\kappa}{\mu} \left( \nabla p + \rho_R g \right) \right] +- Q_p = 0. +$$ +where + +- $c$ ... a constant that characterizes the storage as a consequence that the + solid phase is changing +- $\rho_R$ ... the density +- $p$ ... pressure +- $t$ ... time +- $\kappa$ ... the intrinsic permeability tensor of the porous medium +- $\mu$ ... is the pressure dependent dynamic viscosity +- $Q_p$ ... source/sink terms + +In order to obtain a unique solution it is necessary to specify conditions on +the boundary $\Gamma$ of the domain $\Omega$. + +The benchmark at hand should demonstrate the primary variable constraint +Dirichlet-type boundary condition. Here, the size of the sub-domain, the +Dirichlet-type boundary condition is defined on, is variable and changes +according to a condition depending on the value of the primary variable. + +$$ +\Gamma^\ast_D = \{ x \in \mathbb{R}^d, x \in \Gamma_D, \text{Condition}(p(x)) \} +$$ + +## Examples: + +On the left (x=0) and right side (x=1) of the domain $\Omega = [0,1]^3$ the +usual Dirichlet-type boundary conditions are set, i.e., +$$ +p = 1, \quad x=0 \qquad \text{and} \qquad p = 1\quad x=1 +$$ +The initial condition $p_0$ is set to zero. Additionally, a primary variable +constraint Dirichlet-type boundary condition (PVCDBC) is specified: +$$ +p = -0.1, \quad \text{for}\quad z = 1 \quad \text{and}\quad p(x,y,1) > 0. +$$ +At the beginning of the simulation the PVCDBC is inactive. Because of the +'normal' Dirichlet-type boundary conditions the pressure is greater than zero +after the first time step and the PVCDBC is activated in the second time step. +The effect is depicted in the figure: + +{{< img src="../PVCDBC_1_ts_2.png" >}} +