Electrical Flow Field

Module: devtests.bidomain.electric_flow_field.run

Section author: Anton Prassl <anton.prassl@medunigraz.at> and Gernot Plank <gernot.plank@medunigraz.at>

../../_images/electric_flow_field.png

Fig. 13 Experimental setup to validate the electrical flow field.

Initial Conditions

\sigma &= 1 S/m \\
\phi_{x=0} &= 1V \\
\phi_{x=1mm} &= 0V \\
l_{cube} &= 1mm

Electric Conductivity and Resistance

Conductivity\, G &= \sigma \cdot \frac{A}{l} = 1[\frac{S}{m}] \cdot \frac{1[mm^2]}{1[mm]} = 1\left[ \frac{S}{m} \right] \cdot 1[mm] = 1mS\\
Resistance\, R &= \frac{1}{G} = 1 k\Omega

Electric Field E

E_x &= \frac{1[V]}{1[mm]} \vec{e_x} = \frac{1000[mV]}{10^3[\mu m]} \vec{e_x} = 1\left[ \frac{mV}{\mu m}\right] \vec{e_x} \\
\vec{E} &= \begin{bmatrix} 1 \\
                           0 \\
                           0
           \end{bmatrix} \left[\frac{mV}{\mu m} \right] \\

Electric Current Density J

\vec{J} &= \begin{bmatrix} \sigma\, 0\, 0 \\
                           0\, \sigma\, 0\\
                           0\, 0\, \sigma
           \end{bmatrix}
           \begin{bmatrix} E_x \\
                           E_y \\
                           E_z
           \end{bmatrix} =
           \begin{bmatrix} E_x \\
                           0\\
                           0
           \end{bmatrix} 
           \left[\frac{S}{m} \cdot \frac{mV}{\mu m}  \right] \\

J_x &= 1  \left[ \frac{S}{m} \right] \cdot 1 \left[ \frac{mV}{\mu m} \right] = 1 \left[ \frac{10^3 mS}{10^6 \mu m} \right] \cdot 1 \left[ \frac{mV}{\mu m} \right] = 10^{-3} \left[ \frac{\mu A}{\mu m^2} \right]

Electric Current

I = U \cdot G = 1[V] \cdot 1[mS] = 1 mA

Electric Power P

P = U \cdot I = 1[V] \cdot 1[mA] = 1 mW

Electric Power Density

Power Density = P / V = \frac{1 [mW]}{1 [mm^3]} = 1 \left[ \frac{mW}{mm^3} \right]

Tests

serial

Solve Laplace’s equation on a 1mm cube and compare \phi, \vec{J} and \vec{E} against reference.

Tags: FAST SERIAL

Checks:

  • Compare against stored reference: max_error(phie.igb)
  • Compare against stored reference: max_error(PostProcess/E_field.igb)
  • Compare against stored reference: max_error(PostProcess/J_field.igb)

Last run: 2023-12-06 00:12:38.245428, revision {‘base’: ‘f172baea’}, dependency revisions {PT_C: 31642c1e,cvsys: 69164767,eikonal: 5fbbfda3,elasticity: edfdcbb3}

Runtime: 0:00:00.768151
ALL PASSED
  PASS max_error(phie.igb): 0.0
  PASS max_error(PostProcess/E_field.igb): 0.0
  PASS max_error(PostProcess/J_field.igb): 0.0

parallel

Solve Laplace’s equation on a 1mm cube and compare \phi, \vec{J} and \vec{E} against reference.

Tags: FAST PARALLEL

Checks:

  • Compare against stored reference: max_error(phie.igb)
  • Compare against stored reference: max_error(PostProcess/E_field.igb)
  • Compare against stored reference: max_error(PostProcess/J_field.igb)

Last run: 2023-12-06 00:12:39.047510, revision {‘base’: ‘f172baea’}, dependency revisions {PT_C: 31642c1e,cvsys: 69164767,eikonal: 5fbbfda3,elasticity: edfdcbb3}

Runtime: 0:00:00.694379
ALL PASSED
  PASS max_error(phie.igb): 0.0
  PASS max_error(PostProcess/E_field.igb): 0.0
  PASS max_error(PostProcess/J_field.igb): 0.0