5.2 THE WHITNEY COMPLEX 135
to account for time-dependence. The four pillars are connected by
horizontal beams, which link entities related by constitutive laws. This is
like a building, in which as we'll see Maxwell's equations are well at
home: "Maxwell's house", let's say.
Joints between pillars and beams make as many niches for
electromagnetic-related entities. For instance, magnetic field, being
associated with lines (dimension 1) is at level 1 on the right, whereas b,
associated with surfaces (dimension 2), is at level 2 on the left, at the
right position to be in front of h. Note how the equations can be read off
the diagram. Amp6re's relation, for instance, is obtained by gathering at
level 2, right, back, the outcomes of the arrow actions on nearby fields: -
3td comes from the front and rot h from downstairs, and they add up to
j. All aspects of the diagram shoud be as easy to understand, except the
leftmost and rightmost columns. These concern the finite dimensional
spaces W p of
Whitney elements
announced in the introduction, which we
now address.
5.2 THE WHITNEY COMPLEX
Let us start back from the notion of finite element mesh of Chapter 3:
Given a regular bounded domain D
c E3,
with a piecewise smooth
boundary S, a
simplicial mesh
is a tessellation of D by tetrahedra, subject
to the condition that any two of them may intersect along a common
face, edge or node, but in no other way. We denote by YG E, Y, T (nodes,
edges, faces, and tetrahedra, respectively) the sets of simplices of dimension
0 to 3 thus obtained, 9 and by m the mesh itself. (The possibility of
having curved tetrahedra is recalled, but will not be used explicitly in
this section, which means that D is assumed to be a polyhedron.)
Besides the list of nodes and of their positions, the mesh data structure
also contains
incidence matrices,
saying which node belongs to which edge,
which edge bounds which face, and so on. Moreover, there is a notion of
orientation of the simplices, which was downplayed up to now. In short,
an edge, face, etc., is not only a two-node, three-node, etc., subset of N,
but such a set
plus
an
orientation
of the simplex it subtends. Let's define
these concepts (cf. A.2.5 for more details).
9Note that if a simplex s belongs to the mesh, all simplices that form the boundary
3s also belong. Moreover, each simplex appears only once. (This restriction may be lifted
to advantage in some circumstances, for instance when "doubling" nodes or edges, as we'll
do without formality in Chapter 6.) The structure thus defined is called a
simplicial complex.
136
CHAPTER 5 Whitney Elements
5.2.1 Oriented simplices
An edge {m, n} of the mesh is oriented when, standing at a point of e,
one knows which way is "forward" and which way is "backward". This
amounts to distinguishing two classes of vectors along the line that
supports e, and to select one of these classes as the
"forward" (or positively oriented) one. To denote ~ n
the orientation without too much fuss, we'll make
the convention that edge e = {m, n} is oriented m
from m to n. All edges of the mesh are oriented,
and the opposite edge {n, m} is not supposed to belong to E if e does.
Now we define the so-called
incidence numbers
G e
n -- 1, G e m "- -1,
and
Gek-
0 for nodes k other than n and m. They form a rectangular
matrix G, with N and E as column set and row set, which describes
how edges connect to nodes. (See A.2.2 for the use of boldface.)
Faces also are oriented, and we shall adopt a similar convention to
give the list of nodes that define one and its orientation, all in one stroke:
A face f = {~, m, n} has three vertices, which are nodes g, m, and n; we
regard even permutations of nodes, {m, n, ~} and {n, g, m}, as being the
same face, and odd permutations as defining the oppositely oriented
face, which is not supposed to belong to F if f does. This does orient
the face, for when sitting at a point of f, one knows what it means to
"turn left" or to "turn right" (clockwise). In more precise terms, vectors
~m and gn, for instance, form a reference frame in the plane supporting
f. Given two independent vectors v 1 and v 2 at a point of the face, lying
in its plane, one may form the determinant of their coordinates with
respect to this basis. Its sign, + or -, tells whether v 2 is to the left or to
the right with respect to vl. Observe that v 1 and v 2 also form a frame,
so this sign comparison defines an equivalence relation with two classes,
positively oriented and negatively oriented
frames. The n
positive ones include {gm, gn}, and also of course
{mn, m~} and {n~, nm}.
An orientation of f induces an orientation of
its boundary: A tangent vector ~ along the
boundary is positively oriented if {v, ~} is a direct
frame, where v is any outgoing vector 1° in the m
plane of f, originating from the same point as
(inset). Thus, with respect to the orientation of the face, an edge may
1°No ambiguity on that: In the plane of f, the boundary is a closed curve that separates
two regions of the plane, so "outwards" is well defined. Same remark for the surface of a
tetrahedron (Fig. 5.2).
5.2 THE WHITNEY COMPLEX 137
"run along", like e = {m, n}, when its orientation matches the orientation
of the boundary, or "run counter" when it doesn't.
We can now define the incidence number Rfe" it's + 1 if e runs
along the boundary, -1 otherwise, and of course 0 if e is not one of
the edges of f. Hence a matrix R, indexed over E and F.
A matrix D, indexed over F and % is similarly defined"
DTf- 4-
1
if face f bounds tetrahedron T, the sign depending on whether the
orientations of f and of the boundary of T match or not. This makes
sense only after the tetrahedron T itself has been oriented, and our
convention will be that if T
--
{k, ~, m, n}, the vectors k~, km, and kn, in
this order, define a positive frame. (Beware: {~, m, n, k} has the opposite
orientation, so it does not belong to T if T does.) The orientation of T
may or may not match the usual orientation of space (as given by the
corkscrew rule): these are independent things (Fig. 5.2).
i D{k, ~, m, n}, {~, k, n}
I
i
~J
X
n n
=-1
_._ V
k k
Y m g
FIGURE 5.2. Left: Standard orientation of space. Right: The tetrahedron T -
{k, g, m, n}, "placed" this way in E3, has "counter-corkscrew" orientation. See
how, thanks to the existence of a canonical "crossing direction" (here inside-out,
materialized by the outgoing vector v), this orientation induces one on the
boundary of the tetrahedron, which here happens to be opposite to the orientation
of f- {k, n, ~}. Concepts and graphic conventions come from [VW] and [Sc],
via [Bu].
Remark 5.2. The orientation of faces is often casually defined by providing
each face with its own normal vector, which is what we did earlier when
we had to consider crossing directions. This is all right if the ambient
space E 3 has been oriented, which is what we assume as a rule (the
standard orientation is that of Fig. 5.2, left). In that case, the normal
vector and the ambient orientation join forces to orient the face. But
there are two distinct concepts of orientation here. What we have described
above is inner orientation, which is intrinsic and does not depend on the
simplex being embedded in a larger space. In contrast, giving a crossing
direction 11 for a surface is outer, or external orientation. More generally,
when a manifold (line, surface,... ) is immersed in a space of higher
138 CHAPTER 5 Whitney Elements
dimension, an outer orientation of the tangent /
space at a point is by definition an inner ~
orientation of its complement. (Outer orienting ~J ~ ~--~------_~
a line is thus the same as giving a way to turn
around it, cf. the inset.) So
if
the encompassing
space is oriented, outer orientation of the tangent
space at a point of the manifold determines its
inner orientation, and the other way around. (Cf. A.2.5 for more detail.)
It's better not to depend on the orientation of E3, however, so let it be
clear that faces have inner orientations, like edges and tetrahedra.
Remark 5.3. For consistency, one is now tempted to attribute an orientation
to nodes as well, which is easy to do: Just attribute a sign, + 1 or- 1, to
each node, and for each node n with "orientation" - 1, change the sign
of all entries of column n in the above G. Implicitly, we have been
orienting all nodes the same way (+ 1) up to now, and we'll continue to
do so, but all proofs below are easily adapted to the general case.
DTf= - DTg e DT f = DTg
R fe - R fe =- R ge
FIGURE 5.3. Opposition of incidence numbers, leading to DR = 0, whatever
the orientations.
Next point:
Proposition 5.3.
One has
DR = 0, RG = 0. (Does that ring a bell?)
Proof.
For e~ E and TE %the {T, e}-entry of DR
is ~f~FDwfRfe.
The
only nonzero terms are for faces that both contain e and bound T,
which means that e is an edge of T, and there are exactly two faces f
and g of T hinging on e (Fig. 5.3). If Dig =DTf,theirboundariesare
oriented in such a way that e must run along one and counter the other,
SO I~ g e -- -- I~ f
el ~ and the sum is zero. If D
T g -- --
D
W f'
the opposite
happens,
Rg
e --
f e' with the same final result. The proof of RG = 0 is
similar.
11Which doesn't require a
normal
vector, for any outgoing vector will do; cf. Fig. 5.2,
middle.
5.2 THE WHITNEY COMPLEX 139
Let us finally mention some facts about the dual mesh m* of 4.1.2,
obtained by barycentric subdivision and reassembly (although we shall
not make use of it as such). Each dual cell s* inherits from s an
outer
orientation (and hence, an inner one if space
E 3
is oriented). Incidence
relations between dual cells are described by the same matrices G, R,
D, only transposed"
R fe4: 0,
for instance, means that the bent edge f* is
part of the boundary of the skew face e* (cf. Fig. 4.4), and so on.
5.2.2 Whitney elements
Now, we assign a function or a vector field to all simplices of the mesh.
For definiteness, assume the ususal orientation of space, although concepts
and results do not actually depend on it.
For notational consistency, we make a change with respect to Chapter
3, which consists of denoting by w n the continuous, piecewise affine
function, equal to 1 at n and to 0 at other nodes, that was there called
Kn. The w stands for "Whitney", and as we shall see, the hat function
Kn, now w n, is the Whitney element of lowest "degree", this word referring
not to the degree of w n as a polynomial, but to the dimension of the
simplices it is associated with (the nodes). We shall have Whitney elements
associated with edges, faces, and tetrahedra as well, and the notation for
them, as uniform as we can manage, will be w e , wf, and w w. Recall the
identity
(8) ~n~NW =1
over D. We shall denote by W ° the span of the WnS (that was, in
Chapter 3, space Ore). Finite-dimensional spaces W p will presently be
defined also, for p = 1, 2, 3. They all depend on m, and should therefore
rather be denoted by W°(m), or W°m, but the index m can safely be
understood and is omitted in what follows.
Next, degree, 1. To edge e = {m, n}, let us associate the vector field
(9)
We-- Wm VWn -- Wn Vw m
(cf. Fig. 5.4, left), and denote by W 1 the finite-dimensional space generated
by the WeS. Similarly, W 2 will be the span of the wfs, one per face f =
{~, m, n}, with
(10) wf = 2(w~
VW m X VW n q- WmVW n X VW~ if- Wn VW~. X VWm)
(cf. Fig. 5.4, right). Last,
W 3
is generated by functions
WT, one
for each
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