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Chapter 2, Mathematical preliminaries47
dimension problems it is the flux vector, q. On a boundary we
may be interested in a
related scalar term, qn=q . n, which is the flux normal to the
boundary defined by theunit normal vector n. For the special case
of the one-dimensional form being considered
here we need to note that at the left limit of the domainn = 1 i
while at the right limit itisn = + 1 i. In this case, the Galerkin
method states that the function, w, that satisfies the
boundary conditions and the integral form:(2.36)I =
L
0[dw
dx
du
dx w u w Q ] dx + u ( 0 ) w ( 0 ) u (L ) w (L ) = 0,
also satisfies Eq. 2.15. For a finite element model we must
generate a mesh that
subdivides the domain and (usually) its boundary. The unknown
coefficients in the finite
element model,D, will be assigned to the node points of the
mesh. Within each element
the solution will be approximated by an assumed local spatial
behavior. That in turn
defines the assumptions for spatial derivatives in an element
domain. To illustrate this in
one-dimension consider Fig. 2.10 which compares an exact
solution (dashed) and a
piecewise linear finite element model. The domains of influence
of a typical element and
a typical node are sketched there. In a finite element model,I
is assumed to be the sum
of the ne element and nb boundary segment contributions so
that
(2.37)I =ne
e=1 Ie +
nb
b=1 Ib,
where here nb = 2 and consists of the last two terms given in
Eq. 2.36. A typical elementterm is
Ie =Le ( du
e/dx )2 (ue)2 Qe ue
dx,
where Le is the length of the element. To evaluate such a
typical element contribution, it
is necessary to introduce a set of interpolation functions, H,
so ue(x)= He(x)De, and
(2.38)due/dx = dHe/dx De = DeT
dHeT
/dx,
where De denotes the nodal values ofu for element e. One of the
few standard notations
in finite element analysis is to denote the result of the
differential operator acting on the
interpolation functions, H, by the symbol B. That is, Be dHe/dx.
Thus, a typicalelement contribution is
(2.39)Ie = DeT
Se De DeT
Ce,
withSe = (Se1 Se
2) and where the first contribution to the square matrix is
Se1 Le
dHeT
dx
dHe
dxdx =
LeBe
T
Be dx,
which, for this linear element, has a constant integrand and can
be integrated by
inspection. The second square matrix contribution and the
resultant source vector are:
Se2 LeHe
T
He dx, Ce LeQe He
T
dx.
Clearly, both the element degrees of freedom,De, and the
boundary degrees of freedom,
Db, are subsets of the total vector of unknown parameters, D.
That is, De D andDb D. Of course, the Db are usually a subset of
the De ( i.e., Db De and in higher
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