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# fieldmap

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# fieldmap

{  FIELDMAP.PDE

This example shows the use of the adjoint equation to display Electric field

lines and to compare these to the vector plot of E.

The problem shows the electrostatic potential and the electric field

in a rectangular domain with an internal region in which the dielectric

constant is five times that of the surrounding material.

The electric field E is -grad(V), where V is the electrostatic potential.

}

 title 'Electrostatic Potential and Electric Field'   variables  V  Q   definitions  eps = 1   equations { Potential equation }  V:    div(eps*grad(V)) = 0   { adjoint equation }  Q:    div(grad(Q)/eps) = 0   boundaries region 1   start (0,0)     value(V) = 0        natural(Q) = tangential(grad(V))   line to (1,0)     natural(V) = 0      natural(Q) = tangential(grad(V))   line to (1,1)     value(V) = 100      natural(Q) = tangential(grad(V))   line to (0,1)     natural(V) = 0      natural(Q) = tangential(grad(V))   line to close

region 2

eps = 5

start (0.4,0.4)

line to (0.8,0.4) to (0.8,0.8) to (0.6,0.8)

to (0.6,0.6) to (0.4,0.6) to close

monitors

contour(V) as 'Potential'

contour(Q) as 'Field'

plots

grid(x,y)

contour(V) as 'Potential'

contour(Q) as 'Field Lines'

contour(V,Q) as 'Potential and Field Lines'

vector(-dx(V),-dy(V)) as 'Electric Field'

vector(-dx(V),-dy(V)) norm notips as 'Electric Field'

end