Spiral capacitor calculation using FEniCS

Abstract: The paper shows how to optimize a water level sensor consisting of a cylinder with spiraling metal stripes on the side, using a powerful Python library FEniCS. It is shown how to reduce a 3D Laplace equation to a 2D, using a spiraling coordinate system; how to specify the correct boundary conditions for an open region; how to convert the partial differential equation to a variational form for FEniCS; and how to calculate the capacitance. Then the FEniCS code is shown that solves the Laplace equation and calculates the capacitance. The further numeric experiments show that there is an optimal combination of the spiral frequency and the width of the stripes that maximizes the sensitivity of the sensor. The Python code is given to calculate the optimum.