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.

中文翻译：

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使用 FEniCS 计算螺旋电容器

该论文展示了如何使用强大的 Python 库 FEniCS 优化由侧面带有螺旋金属条纹的圆柱体组成的水位传感器。展示了如何使用螺旋坐标系将 3D 拉普拉斯方程简化为 2D；如何为开放区域指定正确的边界条件；如何将偏微分方程转换为 FEniCS 的变分形式；以及如何计算电容。然后显示 FEniCS 代码，用于求解拉普拉斯方程并计算电容。进一步的数值实验表明，螺旋频率和条纹宽度的最佳组合可以最大限度地提高传感器的灵敏度。给出了 Python 代码来计算最优值。