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In this article, a novel pre-processing method is proposed for magnetic field approximation based on neural networks. The process of calculating magnetic fields based on neural networks is presented first. In order to save the training time and reduce the estimation error, an excitation distance function, which is inspired by an analytical formula of magnetic potential, is proposed to integrate the input data, such as the geometry, excitations, and boundaries. This pre-processing or roughly reduces the dimension of the input layer by three quarters, while the training process is accelerated by a more integrated input layer. Preliminary experiments show that the predicted results by the proposed pre-processing method are close to the ground truth, with a significant reduction of computation time compared with the traditional finite element method (FEM). [ABSTRACT FROM AUTHOR]

In this article, an adaptive degrees-of-freedom (DoFs) finite-element method (FEM) is presented to solve the 3-D transient magnetic problems. For time-dependent problems, the error distribution changes over time, and hence mesh coarsening is needed in some regions to keep the number of DoFs small. In this novel h-type adaptive FEM, real mesh coarsening is replaced by an implicit elimination of DoFs, which avoids the following operations after mesh coarsening and the interpolation errors from fine mesh to coarse mesh. The DoFs to be eliminated are constrained by interpolation functions, which are formulated by the master DoFs. To adapt to the 3-D field, a constraint with a variable number of master DoFs and rational coefficients is adopted. The constraints are then integrated into the finite-element (FE) equation in the element level using the slave–master technique, which is followed by a global assembly process. After solving the global FE equation, the solution of constrained DoFs can be recovered from the solution of master DoFs. Several transient problems are computed to showcase the effectiveness of this proposed method. [ABSTRACT FROM AUTHOR]