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Article History

Received: 15 September 2023

Revised: 16 January 2024

Accepted: 11 March 2024

First Online: 11 May 2024

Declarations

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: On behalf of all authors, the corresponding author states that there is no conflict of interest.

: The implementation in this work considers the commercial software COMSOL. However, all state equations and algorithm are described in the text to be implemented in other environments. In order to reproduce the results, the following steps are needed: (1) Define a mesh; (2) Define the function spaces of the state variables; (3) Define the function spaces of the design variable (); (4) Define the boundary conditions; (5) Define the state equations (NS, Species, and Current Density); (6) Define the objective function; (7) Solve the state equations, which enables COMSOL to calculate the sensitivities; (8) Calculate the initial objective function value; (9) Start the optimization loop: − Solve the state equations for current ; − Calculate the current objective function value; − Solve the adjoint model; − Pass the sensitivity to the optimizer (<i>GCMMA</i>); − Update the design variable (): If the convergence criteria is met return; Else: restart the optimization loop;