Edge Elements

The unsatisfactory performance of elements using node based expansion functions in many electromagnetic applications has prompted the search for an alternative repre- sentation of the field within an element. Such elements exist and are generally termed

i

j

k l

i

j

k l

Figure 2.2: Degrees of freedom for Whitney and linear edge elements

vector elements or edge elements because the degrees of freedom are associated with the circulations of the field along the edges of the element rather than with the nodes. Vec- tor finite elements were first proposed by Raviart & Thomas [1977] for the solution of two dimensional fluid flow problems, and were subsequently extend to three dimensional Maxwell’s equations by N´ed´elec [1980]. Bossavit [1988b] related N´ed´elec’s elements to the differential forms proposed by Whitney [1957] which have a structure closely resem- bling that of Maxwell’s equations when the field quantities are expressed as differential forms. It has been argued that differential forms provide a more natural representation of Maxwell’s equations than the more common vector field approach [Deschamps, 1981;

Baldomir, 1986; Hammond & Baldomir, 1988]. These elements are therefore well suited to the solution of Maxwell’s equations.

Edge elements differ from nodal elements in that the degrees of freedom are associ- ated with the edges of the element rather than with the nodes, a consequence of which is that they only impose tangential continuity at the element boundaries leaving the normal component free to jump. This allows edge elements to model either E or H when both µ and ²^∗ are discontinuous throughout the domain. At sharp corners the normal component of the field is not explicitly prescribed. This extra freedom allows edge elements to correctly model the behaviour surrounding a corner Webb [1993].

2.2 Finite Element Discretisation 31

For first order tetrahedral elements the basis function associated with an edge having verticesi and j is given by,

w_i,j =λ_i∇λ_j−λ_j∇λ_i (2.12)

whereλ_i is the barycentric function associated with node i [Bossavit, 1988b], as shown in Figure 2.2. The elements proposed by N´ed´elec have basis functions that vary linearly within the element, however, they are constant along an edge. This lack of first order completeness has caused suspicions about the accuracy of edge elements among some workers [Mur, 1994] and has prompted the design of consistently linear edge elements [Mur & Dehoop, 1985; N´ed´elec, 1986] which have two degrees of freedom per edge,

w_i,j =λ_i∇λ_j and w_j,i =λ_j∇λ_i. (2.13) While the interpolation function is now linear along each edge as well as in the body of the element, it is at the expense of doubling the number of degrees of freedom.

Furthermore, linear edge elements approximate thecurl in an identical way to Whitney elements. This can be seen by rewriting the two basis functions for the edgei, j as,

w_i,j^a =w_i,j−w_j,i =λ_i∇λ_j−λ_j∇λ_i,

w_i,j^b =w_i,j +w_j,i =λ_i∇λ_j+λ_j∇λ_i =∇(λ_iλ_j). (2.14) w^a_i,j is identical to the Whitney element basis function, while w_i,j^b has a curl which is identically zero. Linear edge elements are claimed to be more accurate than the Whitney elements, and to have a local approximation error ofO(h²) compared toO(h) for Whitney elements [Mur, 1994], a claim which has recently been challenged [Bossavit, 1994]. Monk [1992] shows that Whitney elements do have a linear dependence uponh compared to a quadratic dependence for linear edge elements when measured with the`₂- norm, however, when the discrete maximum norm is used instead both types of elements produce O(h²) convergence with Whitney elements being consistently more accurate for a given number of unknowns. Results comparing the accuracy of the two types of elements are given in§2.7, where Monk’s observations are confirmed. It has recently been shown [Bossavit, 1994] that while first order completeness gives nice smooth solutions it is not a prerequisite for achieving accurate answers.

The form (2.14) is that given by Webb & Forghani [1993] who proposed a family of hierarchal edge elements. This allows Whitney elements to be used in some regions and higher order edge elements in others where a greater accuracy is required. The family proposed by Webb extends to elements that are complete to degree two. This representation, which for the first order element is the same as (2.14), is therefore more convenient than (2.13) in certain situations. Alternative implementations of edge ele- ments have been proposed [Barton & Cendes, 1987], which use a different representation of the interpolation function, namely

w_i,j =α×r+β, (2.15)

where α and β are constant vectors chosen so that the tangential component of w_i,j vanishes on all edges other than i, j while r is the position vector inside the element.

The basis function (2.15) is mathematically equivalent to Whitney elements, however, it is not as convenient to programme.

It is often convenient to use shapes other than tetrahedrons for discretising the problem. This is often due to constraints imposed by the mesh generating software which is often not capable of generating tetrahedral meshes directly. One answer, of course, would be to use software that has this facility but when this is not available other methods must be found. A mesh consisting of hexahedra can be broken up so that each hexahedron is replaced by five tetrahedra [Webb, 1981]. However, this requires that the hexahedral mesh is fairly regular, otherwise conflicts at element faces are encountered where elements on either side do not match. An alternative is to break each hexahedron into twelve tetrahedra by inserting an extra node in the centre of the hexahedron.

This has the disadvantage of creating considerably more elements. It can therefore be desirable to use hexahedral edge elements so that the hexahedral mesh can be used unmodified. When the geometry is suitable it is possible to extrude a two dimensional triangular mesh into three dimensions so producing prism elements. This technique is used for the TM₀₁₀ cavity analysed in Chapter 5. The method for creating hexahedral elements was outlined by van Welij [1985], however, this author has not seen a complete listing of the shape functions for hexahedral and prismatic edge elements so these are given in Appendix A.