Numerical solution of differential and integral equations
The variational statements to simulate electromagnetic fields using edge elements are considered in the report. For the problems with significant displacement currents the variational statement using only vector potential is considered. And when the displacement currents are negligible, it is proposed to use the authorsТ developed approach with joint use of vector and scalar potentials, and, respectively, with joint use of edge and nodal finite elements. In the last statement the magnetic field is described by the scalar potential in subareas with zero conductivity (air, dielectrics). In the subareas with non-zero conductivity the electromagnetic field is described by the vector potential. The main advantage of using scalar potential in subareas with zero conductivity is to avoid many difficulties of using vector potential to describe magnetic field in non-conducting media, where vector potential is not unique. Both variational statements allow the use of the idea of extracting of main field part.
For both variational statements the finite element formulations are described. It is shown what local matrixes must be calculated for edge elements and nodal elements and what local matrixes are necessary for mixed type elements. Besides the local matrixes necessary for recalculate main field part and the use of it as field source are described for each elements type.
The report contains finite element approximation on tetrahedron edge and nodal elements. Using an example of model problem solving, the one of disadvantages of finite element approximation of vector potential on tetrahedron elements is shown. This disadvantage became apparent when one of the vector potential components (for example, z-component) is sufficiently greater than one another (for example, x-component), and the less component (i.e. x) is non-zero. This disadvantage is absent when the parallelepiped elements are used.
Moreover, the algorithm to build irregular parallelepiped meshes with terminal nodes and edge and nodal finite elements for these meshes are considered. The local matrixes for irregular parallelepipeds with terminal nodes are shown for both variational statements.
In conclusion, the examples of model problems solving are shown.
Note. Abstracts are published in author's edition
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