Direction-splitting implicit solvers employ the regular structure of the computational domain augmented with the splitting of the partial differential operator to deliver linear computational cost solvers for time-dependent simulations. The finite difference community originally employed this method to deliver fast solvers for PDE-based formulations. Later, this method was generalized into so-called variational splitting. The tensor product structure of basis functions over regular computational meshes allows us to employ the Kronecker product structure of the matrix and obtain linear computational cost factorization for finite element method simulations. These solvers are traditionally used for fast simulations over the structures preserving the tensor product regularity. Their applications are limited to regular problems and regular model parameters. This paper presents a generalization of the method to deal with non-regular material data in the variational splitting method. Namely, we can vary the material data with test functions to obtain a linear computational cost solver over a tensor product grid with non-regular material data. Furthermore, as described by the Maxwell equations, we show how to incorporate this method into finite element method simulations of non-stationary electromagnetic wave propagation over the human head with material data based on the three-dimensional MRI scan.

This article examines in depth the most recent thermal testing techniques for lithium-ion batteries (LIBs). Temperature estimation circuits can be divided into six divisions based on modeling and calculation methods, including electrochemical computational modeling, equivalent electric circuit modeling (EECM), machine learning (ML), digital analysis, direct impedance measurement and magnetic nanoparticles as a base. Complexity, accuracy and computational cost-based EECM circuits are feasible. The accuracy, usability and adaptability of diagrams produced using ML have the potential to be very high. However, none of them can anticipate the low-cost integrated BMS in real time due to their high computational costs. An appropriate solution might be a hybrid strategy that combines EECM and ML.