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Unformatted text preview: Additional Features of TwoDimensional
Finite Element Analysis Assembly
Bandwidth
Boundary Conditions
Higher Order Elements Assembly of Element Matrices Global E» Local
Kn K 1( l )
K12 K12”
K22 K55) + K1?
K l 4 0
K15 O
K 23 K31” K12)
K14 K11),
K34 K31) + K11)
K17 0
K44 K19 + me + K9) + Kw
K 45 K 1%) + K8)
K 56 0
‘ K47 K13.) + Kg)
2 (b)
(1,1) (1, l) of elementl
(1,2) (1,2) of element 1
(1,3) (1,3) of element 1
(1,4) No correspondence
(1, 5) No correspondence
(2,2) (2,2) of element 1 and
(1, 1) of element 2
(2,3) (2,3) of element 1 and
(1,4) of element 2
(2, 4) (1,2) of element 2
(2,5) (1,3) of element 2
(3, 3) (3,3) of element 1 (and
(4,4) of element 2
(3,4) (4,2) of element 2
(3,5) (4, 3) of element 2
(4,4) (2,2) of element 2
(4,5) (2,3) of element 2 (5, 5) (3,3) of element 2 Labeling of the Nodes
Bandwidth Consideration The nature of the stiffness matrix will change depending on the numbering scheme used
for the nodes. In speciﬁc, the so—called bandwidth, B will be effect by such numbering as follows ‘ B=(R+l)NDOF C C C o C \0 0 o 0
C C C C C C\\\\O o o
C C C C 0 C d\\o 0
0 C C C C C C C\\o
C C o C C C C o 2*
\0\\\C C C C C C C C
0 0\\\ C C C C C C o
o 0 \o\\c 0 C C C C
O O 0 B\ C C 0 C C In the formula for B, R is the largest difference between the node numbers in any given element, and NDOF is the number of degrees of freedom at each node.
As an example, consider for a scalar valued problem, the two meshes and their respective stiffness matrices shown below X
XXX XXX
X I><><><><
xxxx
><><><><>V<><t
><><><><'><><
><><><><><><
><><>4><><><
><
><
><
><
><
>< ><><><><
><
_><
><
><
><
x ><><><><><><
><><><><><><
><><><>< Imposition of Boundary Conditions specification of two different values of a primary variable at the
same boundary point (l1) 4 X mesh (mesh 1) (1)) (19:1? ] (19:0 9 81 9 W ,7 T7 if: (a O O... l .— (c) 8 X 8 mesh (mesh 2) (d) Figure 4.20 Effect of specifying (either of the) two values of a primary variable at a boundary node
[node 5 in (a) and (b) and node 9 in (c) and (a’)]. INTERPOLATION FUNCTIONS Triangular Elements Pascal's triangle Degree of the Number of Element with nodes
polynomial terms in the
polynomial
1 0 1
/ \
x=——= y 1 3
/ \
x2 w— xy ——y2 2 6 ———«~——=——>=
’ \
x3—x2y——xy2——ty3 3 10 ________,,,
/ \
x4 —x3y—.vc2)22wc_v3~,V4 4 15
/ \
lx5——x“y——~x3y2~x2y3~xy4—yi 5 21
féhxsyﬂxWZPW «WtWt” 6 28 Pascal’s triangle for the generationof the Lagrange family of triangular elements. Rectangular Elements ‘ Pascal's triangle Rectangular array Rectangular elements  I I y W x2y x3y x4y I I
xzyz x3y2 x4y2m 13
l ._____..~..._————»
3 3 xya—x2y3—x y 3(4))3 one o o no. Lagrange family of rectangular elements of various order. ' Condensation of Internal Nodes Internal nodes of the hi gher—order elements of the Lagrange family do not contribute to the
interelement connectivity. Hence they can becondensed out of the problem at the element level so
that the size of the element matices does not become excessively large. Consider, as an example,
the triangluar element with one internal node as shown. The element equation for this case would read K11 X12 K13 X14 “1 F1
K22 K23 K24 “2 F2
X33 K34 “3 F3 K44 “4 F4 This equation can be written in the equivalent form
[K7] {Km} [Mi] {117/}
T _
{m [(44 F4 “4 This system of equations can thus be separated, and solving the last equation for u4 yields : _1._ _ // T
U, {(44 (E, {K } {u}) Using this result back into the ﬁrst partioned system, yields {K/ll} [K’Hul +
K44 (F4 — {](”}T{u}) = {F’} which can be rewritten in the standard form A [a] {u} = {n with appropriate deﬁnitions for the new stiffness and column matrices. Thus this new element equation now contains only the boundary nodal unknowns. This procedure can be carried out for
any number of internal nodes. ...
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This note was uploaded on 10/03/2011 for the course MCE 561 taught by Professor Sadd during the Spring '11 term at Rhode Island.
 Spring '11
 Sadd

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