Application of the Finite Element Method to Some Simple Systems in One and Two Dimensions. Page: 5
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and its derivative are continuous across element boundaries. This then results in the
following
(e) [(x)p (e)/(x) w (e) / (x) ]dx + (e) (x) e) (x) I(e) (x) dx
X(e) [()( e e
(8)
- J (e) [y(x)p)) (x)T-(e) (x)]dx = 0.
Now substitute the approximate solution for the et element, Eq. (4), into Eq. (8), leading
to element equations of the form
(e) [c(x)P/ (x) ((e)/ (x) ]dx + (e)(X)pe (x) e(X) (x) dx
I (( 1(x)]dx a = 0. (9)
= - [(e) y(x)e) (x) pe) (x)]dx
Finally, the element equations in Eq. (9) are written in the matrix form
[K](e) {a}- [M](e) {a} = {0} , (lOa)
where
[K](e) = [KL](e) + [K3](e) , (10Ob)
and
Ka i (e) f(e) a((x) (Pf(e) / (X) ( (e>/) d
K0( i(e)- (e) [c(x)p (e) (x) p)e) (x)]dx,
KOj()(e) b3(X) (P (e) (X) (P(e
f i(x) ]dx
M i (e) - (e) [y(x)(P(e) (x)e) (x)dx .
A number of integrations are required to fill the system of element equations. Typically
the matrix [K] is referred to as the stiffness matrix, while the matrix [M] is referred to as
the mass matrix.
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Hunnell, Jason C. Application of the Finite Element Method to Some Simple Systems in One and Two Dimensions., thesis, May 2002; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc3087/m1/12/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .