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The range of problems in high voltage and materials engineering which can be addressed through mathematical and numerical modeling has expanded with the reduction in the price of computing power. Presently, very complex transient nonlinear finite element problems can be solved in both 2D and 3D. In addition, density functional theory (DFT), the most useful implementation of computational quantum mechanics, allows material properties, such as dielectric constant, band offsets, band gap and impurity states therein, etc. to be computed at a molecular level, including the effects of compatibilizing agents between inorganic fillers and polymer matrices. These effects become especially important at the nano level, where the oxidation of the surface of a nanoparticle can, in some cases, change its dielectric properties completely. The role of the more sophisticated forms of numerical modeling presently available will be explored through applications to lightning arresters, materials design, etc.