Equations

This problem is a Poisson problem with Neumann boundary conditions on two regions of the boundary and homogeneous Neumann or Robin boundary conditions elsewhere.

On the computation domain $\Omega$ , the problem reads:

$\left\{\begin{array}{ll} \nabla\cdot(\sigma\nabla u) = 0 & \text{on }\Omega,\\ \partial_{\bm{n}} u = \frac{1}{\left|\Gamma^+\right|} & \text{on} \Gamma^+,\\ \partial_{\bm{n}} u = -\frac{1}{\left|\Gamma^-\right|} & \text{on} \Gamma^-,\\ \sigma\partial_{\bm{n}} u = \mu(\bm{x}) u &\text{on }\partial\Omega\setminus\{\Gamma^+\cup\Gamma^-\}.\\ \end{array}\right.$

If $\mu(\bm{x})=0$ on all boundary sites except electrodes (i.e. if the boundary condition is homogeneous Neumann), the problem is ill-posed. In that case, a null-mean constraint is added to guarantee the unicity of the solution.

Unknowns

CEPS ID	Name	Math symbol	Unit
0	`Electric potential`	$u(\bm{x})$	$\si{\milli\volt}$

Tissue properties

$\sigma(\bm{x})$ is the conductivity tensor,
$\mu(\bm{x})$ is the Robin coefficient and the boundary condition. It is set to 0 by default (thus changing the BC to homogeneous Neumann). It can be defined as a piecewise constant function of region attributes.
The domains $\Gamma^+$ and $\Gamma^-$ are the anode and cathode, respectively. They can be defined using mesh region attributes.

How to use

The specific parameters to set are conductivity, anode attributes, cathode attributes and robin coefficients. Default conductivity is 1, default Robin coefficient is 0.

problem type : flux anode cathode
 
output file name : ./results/poisson
 
3d mesh : ./meshes/catheterBloodTissue.vtk
 
conductivity : CONSTANT 1.2 0. 0. 0. 0.4 0. 0. 0. 0.4
anode attributes : 2
cathode attributes : 3
 
# Optional
# robin coefficients : 1 30
 
spatial discretization : FE 1
linear solver absolute tolerance : 1.E-12
linear solver relative tolerance : 1.E-12
linear solver type               : GMRES