New vector finite elements for three-dimensional magnetic field computation

Abstract

Finite-element vector potential solutions of three-dimensional magnetic field problems are usually obtained by approximating each component of the vector potential by a separate set of scalar finite-element approximation functions and by imposing continuity conditions between elements on all three components. This procedure is equivalent to imposing continuity of both the normal and the tangential components of the vector potential. We show in this paper that this procedure is too restrictive: While continuity of the tangential component of the vector potential is required, continuity of the normal components is not essential in the variational formulation. We introduce a new type of vector finite-element approximation function that has the property that it interpolates not to point values of each component of vector potential, but rather to the tangential projection of the vector potential on each edge of tetrahedral finite elements. With the new basis functions, continuity of the normal component of the vector potential is provided only approximately by means of the natural interface conditions inherent in the variational procedure. This results in a more efficient procedure for the solution of three-dimensional magnetostatic field problems than is obtained by enforcing normal component continuity exactly.

Numerical Analysis of Thermo-electrically Conducting Fluids in a Cubic Cavity Using Vector Finite Element Method for Induction Equations

“…The generalization of the FDTD method to arbitral elements is the vector finite element method. [8][9][10][11][12] The vector finite element has two types of element just like variables of FDTD method are defined at the edge and the surface center on the element. One is facet element and the other is edge element.…”

Application of edge-based finite elements and vector ABCs in 3-D scattering

“…They do, however, lead to a higher unknown count but this is balanced by the greater sparsity of the resulting finite element matrix. Thus the computation time required to solve such a system iteratively with a given degree of accuracy is less than the traditional node-based approach [2].…”

“…However, for 5 = 2, the functional yields a sparse, symmetric matrix. Thus, the second order boundary operator can be expressed as B 2 (E) = -(r x V x E) + P 2 (E) 6 where P 2 …”

Spurious-free approximations of electromagnetic eigenproblems by means of Nedelec-type elements

“…Some years later, Nedelec introduced a family of elements [29] well suited to the study of 3D vector problems involving electromagnetic fields. These new elements (mostly the lowest order one on tetrahedra) became quite popular, usually under the name of edge elements, among researchers developing numerical simulators for the calculations of electromagnetic fields [2,7,8]. Very soon they realized that edge elements had solved the annoying problem of spurious modes, in all cases in which they had been used, and their popularity increased even further [9].…”