Specifying constant potential boundary condition

Question

I want to model a "floating" electrode on the surface of dielectric while the potential difference is applied to other electrodes and I want to calculate the potential of such "floating" electrode. So I solve the Poisson equation, I specify the Dirichlet BC on the driven electrode. Could you suggest some idea to treat the floating electrode? We know that there must be constant potential on it. But how to specify this idea to FEniCS?

The equation is
∇(ϵ∇φ)=0
, and boundary conditions φ=0 on $\Gamma_0$ φ=1 on $\Gamma_1$ and φ=const on $\Gamma_2$, we want to find the value of this constant as a result of solution of the equation. Alternatively the constant potential boundary condition can be specified as |∇φ|=0 on $\Gamma_2$, i.e. not only normal component of the gradient is zero on the boundary, but all the other components as well.

Comments

Hi, you are more likely to have your question answered if you specify mathematical formulation of your problem.

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The equation is $$\nabla ( \epsilon \nabla \varphi) = 0 $$, and boundary conditions $\varphi = 0$ on $\Gamma_0$ $\varphi = 1$ on $\Gamma_1$ and $\varphi = \mathrm{const}$ on $\Gamma_2$, we want to find the value of this constant as a result of solution of the equation. Alternatively the constant potential boundary condition can be specified as $|\nabla \varphi | = 0$ on $\Gamma_2$, i.e. not only normal component of the gradient is zero on the boundary, but all the other components as well.

Answer 1

Well, I figured out the answer myself, just posting it for the reference.

Actually, we can solve the equation twice: once we put the boundary condition $\varphi=1$ on $\Gamma_2$ (let's call this solution $\varphi_1$) and once we put the boundary condition $\varphi=0$ (let's call this solution $\varphi_0$). Notice that $\delta\varphi = \varphi_1-\varphi_0$ satisfy zero boundary condition on $\Gamma_{0}$ and $\Gamma_1$ and $\delta\varphi=1$ on $\Gamma_2$.

Now if the actual value of the floating potential on $\Gamma_2$ is $\varphi |_{\Gamma_2}=\alpha$ then $$\varphi=\varphi_0+\alpha \delta\varphi$$ is a a solution. We must take into account zero surface charge boundary condition on the floating potential surface to determine $\alpha$. This leads to $\int_{\Gamma_2} \frac{\partial \varphi}{\partial n} dS = 0$, substituting $\varphi=\varphi_0+\alpha \delta\varphi$ into this condition leaves us with a linear equation for $\alpha$, the potential of the floating electrode.