The Laplace equation calculates the potentials inside some volume of space given certain conditions on the boundary surface of that volume. Boundary conditions are typically voltages on the surfaces of electrodes, but they can also be mirror planes). SIMION’s “Refine” function solves the Laplace equation.
The Laplace equation is
The Laplace equation is a special case of the Poisson equation in which zero space-charge is assumed:
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