Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

Appendix C
Edge-based ®nite element
formulation
The edge-based data structure has been used in many recent ®nite element formulations
for ¯ow problems. As mentioned in Sec. 6.8, Chapter 6, this formulation has
many advantages such as smaller storage, etc. To explain the formulation we shall
consider the Euler equations and a few assembled linear triangular elements on a
two-dimensional ®nite element mesh as shown in Fig. C.1. From Eq. (1.24) we rewrite
the following Euler equations
@
@t ‡ @Fi ˆ 0 …C:1†
@xi where are the conservative variables. If the element-based formulation for the
above equation omits the stabilization terms, the weak form can be written as
…
N k
…
d ˆÿ …N
t k † T @Fi d …C:2†
@xi In a fully explicit form of solution procedure, the left-hand side becomes M… = t†
and here M is the consistent mass matrix (see Chapter 3). We can write the RHS of
the above equation for an interior node I (Fig. C.1(a)) by interpolating Fi in each
element and after applying Green's theorem as
X
…
@N
E 2 I
AE I
…N
@xi k F k i † d ˆ X AE @NI …F
3 @x E 2 I
i E
I i ‡ F J i ‡ F K i † …C:3†
where A E is the area and I, J and K are the three nodes of the element (triangle) E.
This is an acceptable added approximation which is frequently used in the Taylor±
Galerkin method (see Chapter 2). In another form, the above RHS can be written
as (Fig. C.1(a))
A 1
3
@NI …FI ‡ F1 ‡ F
@x
2†‡
i
A2 @NI …FI ‡ F2 ‡ F
3 @x
3†‡
i
A3 @NI …FI ‡ F3 ‡ F
3 @x
1† …C:4†
i
where A1, A2 and A3 are the areas of elements 1, 2 and 3 respectively. For integration
over the boundary on the RHS, we can write the following in the element
formulation
X
…
B 2 I
ÿ B
N I …N k F k i † dÿn B ˆ X
B 2 I
ÿ B
6 …2FI i ‡ F J i †n B
…C:5†