Adjoint Optimization for Poisson Problem with Applications

In this paper, we aim to introduce a detailed, explanatory adjoint-based optimization procedure for the Poisson problem with possible extension to industrial problems. We present a rigorous derivation for direct and adjoint Poisson problems and derive their weak formulations. We solve these systems using the open-source finite element framework FEniCSx to employ numerical optimization. We provide two example problems; the first is to optimize the forcing term to match the numerical solution with the analytical relation for the model Poisson problem. The second is to control the power of the heat source on the PCB to prevent the maximum temperature from exceeding a certain temperature limit of the PCB material. By combining detailed mathematical theory with open-source tools, this work provides an extendable framework for adjoint-based optimization in real-world industrial Poisson applications.