Optimization using finite element analysis and the adjoint variable method to solve engineering problems appears in various application areas. However, to the best of the authors’ knowledge, there is a lack of detailed explanation on the implementation of the adjoint variable method in the context of electromagnetic modeling. This paper aimed to provide a detailed explanation of the method in the simplest possible general framework. Then, an extended explanation is offered in the context of electromagnetism. A discrete design methodology based on adjoint variables for magnetostatics was formulated, implemented, and verified. This comprehensive methodology supports both linear and nonlinear problems. The framework provides a general approach for performing a very efficient and discretely consistent sensitivity analysis for problems involving geometric and physical variables or any combination of the two. The accuracy of the implementation is demonstrated by independent verification based on an analytical test case and using the finite-difference method. The methodology was used to optimize the parameters of a superconducting energy storage device and a magnet press and the optimization of the topology of an electromagnet. The objective function of each problem was successfully decreased, and all constraints stipulated were met.

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