Journal of Mathematical Sciences and Modelling

2636-8692

Mahmut AKYİĞİT

10.33187/jmsm.1810588

Software Engineering (Other) Applied Mathematics (Other)

Yazılım Mühendisliği (Diğer) Uygulamalı Matematik (Diğer)

Adjoint Optimization for Poisson Problem with Applications

https://orcid.org/0000-0002-5885-790X

Ekici

Ekrem

KING FAHD UNIVERSITY OF PETROLEUM AND MINERALS,

Advanced Online Publication 77 84 10 26 2025 04 08 2026

2018

Journal of Mathematical Sciences and Modelling

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.

Adjoint method FEniCSx Optimization Poisson equation Finite element method

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