basil - numbering, banded matrix, mandwidth minimisation

8/12/2019 BASIL - Numbering, Banded Matrix, Mandwidth Minimisation

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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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4

Different methods


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5

Different methods


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6

Different methods


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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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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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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 . .

basil - numbering, banded matrix, mandwidth minimisation

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