Computation of resistance with arbitrary local resistivity rho(x,y,z)

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SUMMARY

The discussion centers on the numerical computation of electrical resistance in a geometry defined by a tensorial resistivity, specifically for a parallelepiped with trapezoidal flanks. The user, Pierre, seeks an algorithm to compute resistance given a set of coordinates (x, y, z) and their corresponding position-dependent resistivity values. Traditional formulas like R = ρ * l / S are inadequate due to the complexity introduced by the tensorial nature of resistivity. The key equation to consider is J = σ * E, where σ(x, y, z) = 1/ρ(x, y, z).

PREREQUISITES
  • Understanding of tensor algebra and its application in electrical conductivity.
  • Familiarity with numerical methods for solving partial differential equations (PDEs).
  • Knowledge of finite element analysis (FEA) techniques for geometrical modeling.
  • Proficiency in programming languages suitable for numerical computation, such as Python or MATLAB.
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  • Research numerical methods for solving PDEs, focusing on methods applicable to electrical resistance problems.
  • Explore finite element analysis (FEA) software like COMSOL Multiphysics for modeling complex geometries with variable resistivity.
  • Learn about tensor calculus and its implications in electrical engineering, particularly in resistivity calculations.
  • Investigate existing algorithms for computing current density and electric fields in heterogeneous materials.
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pierebean
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Bonjour,

I need to numerically compute the net electrical resistance of a given geometry.

I know the shape of my object, it is relatively simple. It's close to this: http://2.imimg.com/data2/QX/UC/IMFCP-3019296/i-shape-big-1-250x250.jpg
Actually my shape is even simpler because it's a parallelepiped flanked by two trapezoids.

As data, I have many points coordinates x,y,z and the corresponding resistivity which is unusually dependent of the position.

my data table looks like that:

x1 y1 z1 rho1
x2 y2 z2 rho2
x3 y3 z3 rho3
x4 y4 z4 rho4
...
ect...
...
xn yn zn rhon

n is my number of point in my geometry.

Naturally, since the electrical current will preferentially go to the low resistivity domain. I cannot use the R=rho*l/S formula.

I probably have to solve J=sigma*E for every point with sigma(x,y,z)=1/rho(x,y,z)

Does someone have any idea of algorithm that can compute the resistance?

Thank you very much

Pierre
 
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I forgot to mention that my resistivity was a tensor and not a mere scalar.
 
Any leads maybe?
 

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