The numerical solution of partial differential equations (Electrical Machine)

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Solving a differential equation analytically or by semi-numerical techniques, as demonstrated before, is only possible assuming simplifications in the differential equations that are valid in a domain that can be described in a plain mathematical way. Specific boundary conditions and various material properties in different domains make it difficult to obtain an analytical solution for problems of technical importance.

The numerical approximation methods (Table 5.1) of solving the introduced partial differential equations are not limited to such specific geometries or other rough simplifications and can handle different material characteristics with an acceptable accuracy in a single model. The two most important groups of numerical methods are the finite difference method and the projection methods.

The finite element method nowadays is the most important and most frequently used approach solving variational problems and differential equations in engineering. The most significant success of this method is founded in the possibility to develop on its base user-friendly computer programs of general application range. Due to its structured rules this is closely linked to the opportunity of the FEM to generate stable numerical schemes for considering complicated two- and three-dimensional geometries in a relatively simple way.

Overview of various numerical methods.

Fig. 5.22. Overview of various numerical methods.