DOI: https://doi.org/10.15588/1607-6761-2018-4-4

### EDGE-BASED VECTOR FUNCTIONS FOR APPROXIMATION OF ELECTROMAGNETIC FIELD IN VOLUME AND SURFACE FINITE ELEMENTS

M.V. Ostrenko

#### Abstract

Purpose. Analysis of the properties of Edge-Based vector of basis functions for the volume elements and working surface Edge-Based FEM vector of basis functions for approximating the vector values of the electromagnetic field in the surface integrals.

Methodology. Based on the properties of simplex coordinates, the mathematical model of Edge-Based vector of basic functions for volume, and for the surface elements is proposed.

Findings. The analysis of Edge-Based vector basis functions for the finite element method (FEM) is performed. Based on the analysis, Edge-Based functions for surface elements (triangles) are proposed. The properties of Edge-Based basis functions are obtained, which make it possible to more accurately simulate a magnetic field, eddy currents and losses in comparison with Nodal-Based vector basis functions. The proposed mathematical model shows the orthogonality of the proposed system Edge-Based vector basis functions, which allows their use in FEM. Moreover, it is shown that the tangent component of a vector field, represented as a linear combination of a system of both volume and surface Edge-Based basis functions on the edge of an element, is equal to the coefficient of the linear combination corresponding to the given edge. This property allows us to combine the volume and surface integrals of the FEM into one system of linear equations.

Originality.  The originality of the proposed mathematical model is the form, which makes it possible to go from volume integrals to surface integrals and merge volume and surface integrals of FEM into one system of linear equations. This allows you to significantly reduce the dimension of the problem and, as a result, reduce the resource intensity of the method and increase its speed without significant loss of accuracy.

Practical value. The Edge-Based vector basis functions for volume and surface finite elements were used in the developed software package ELMAD-3D, designed to calculate losses and overheating from the leakage fields of power transformers and reactors. The use of surface Edge-Based vector basis functions allowed us to significantly increase the speed of calculation methods and reduce their resource intensity.

#### Keywords

finite element method; volume and surface vector basis functions

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#### References

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