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02nd June 2014 F. E. A. TERM PAPER PRESENTATION Basil Mathai
131043
M2 G.E., C.E.T.
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Numbering of Nodes
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Once domain is discretised, unique element number
& node number to identify nodes
Global & Local numberings
Plays an important role in shape and size of the
profile of the FE stiffness matrix
Generally done in a manner that minimizes the
half-bandwidthof stiffness matrix
Several algorithmsfor nearly optimal node
numbering
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Numbering of Nodes (cntd.)
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Different methods
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Numbering of Nodes (cntd.)
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Different methods
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Numbering of Nodes (cntd.)
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Different methods
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Numbering of Nodes (cntd.)
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Different methods
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Numbering of Nodes (cntd.)
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Generally, for a 2D continuum,
Start numbering from bottom-left node & proceed in anti-
clockwise direction
Numberprimary nodes first
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Banded Matrix
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It is a sparse matrix whose non-zero entries are
confined to a diagonal band
Comprises of main diagonal and zero or more
diagonals on either side
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Banded Matrix (cntd.)
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Banded Matrix (cntd.)
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Bandwidth is described by, Sub-diagonals
Super-diagonals
Semi/Half bandwidth
Instead of storing 8x8=64elements,
in a band matrix only 8x5=40elements are stored
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Banded Matrix (cntd.)
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Half-bandwidth, NB= (D+1) x f
(max nodal difference + 1) x dof
D: maximum largest difference of node numbers
occurring for all elements of the assemblage f : number of dof at each node
Example:
NB= (8 + 1) x 2 = 18
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Minimisation of Bandwidth
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Computer time is proportional to square of
bandwidth
Helps in reducing both solution time and storage
requirements Nodal numbering is given such that maximum nodal
difference is always minimum
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Minimisation of Bandwidth (cntd.)
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Steps
Avoiding secondary external nodes
By choosing displacement derivatives as additional dof
at primary nodes
Systematic subdivision, adopt appropriate node-
numbering
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Minimisation of Bandwidth (cntd.)
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Example Numbering A
D = 91 = 8
Assume f = 2
So, NB = (8+1)
x 2 =
18
Numbering BD = 41 = 3
Assume f = 2
So, NB = (3+1)
x 2 =
NB= (D+1) x f
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References
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Finite Element Method with Applications in EngineeringY. M. Desai, T. I. Eldho & A. H.
Shah
Numerical Methods in Science and Engineering
S. Rajasekaran
The Finite Element Method and Applications in
Engineering Using ANSYS
Erdogan Madenci, Ibrahim
Guven
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t h a n k s . .