Method of solving potential field problems with complicated geometries

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Abstract
The method of solving potential field problems by dividing the potential field into an appropriate number of regions, expressing the solution in each region by eigenfunction expansion, and determining the expansion coefficients from the matching conditions at the dividing surfaces is commonly used. However, the existing solution is limited to the cases where the eigenfunctions are the same for all regions. This paper presents a method of determining the coefficients of expansion when the eigenfunctions are different for each region, which is the case for problems with complicated geometries. The method greatly increases the analytical power of solving potential fields such as electric or magnetic field, potential flow, heat conduction, and neutron diffusion, etc. The method is illustrated by solving (i) potential flow in a channel with a sudden change in cross‐section area; (ii) conduction of heat in a cylinder with a sudden change in cross‐section area; (iii) potential flow in a rectangular container with an inlet pipe and an outlet pipe.

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