The direct current conduction problems, such as electrolysis and computation of resistances of grounding plates, involve a steady current passing through a conductive medium. The current density J is related to the electric field E as J = σ E, where σ is the conductivity of the medium. The electric field E is the gradient of the electric potential V, E = –∇V. Thus, the continuity equation ∇ · J = Q, where Q is the current source, yields the elliptic Poisson's equation:
–∇ · (σ ∇V) = Q.
The toolbox supports the following boundary conditions for DC conduction problems:
Dirichlet boundary condition assigning values of V at the boundaries, which are typically metallic conductors.
Neumann boundary condition assigning the value of the normal component of the current density (n · (σ ∇V)).
Generalized Neumann condition n · (σ ∇V) + qV = g, where q is film conductance for thin plates.