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46 Finite Element Analysis with Error Estimators
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0.08Cubic Global Galerkin, by parts, u"+u+x=0, u(0)=0=u(1)
X
u,
*=FEAapproximation
Figure 2.8Exact (-) and cubic global Galerkin (*) solutions
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0.08Quadratic Global Galerkin, by parts, u"+u+x=0, u(0)=0=u(1)
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u,o=FEAapproximation
Figure 2.9Exact (-) and quadratic global Galerkin (o) solutions
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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