Dinamica Lineare FEM 3

7/25/2019 Dinamica Lineare FEM 3

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Dr.Dr. Ing.Ing.ValentinaValentina SalomoniSalomoni

Dipartimento di Costruzioni e TrasportiDipartimento di Costruzioni e Trasport i

UniversitUniversitdegli Studi di Padovadegli Studi di Padova

DINAMICA DELLE STRUTTUREDINAMICA DELLE STRUTTURELAUREA SPECIALISTICALAUREA SPECIALISTICA -- INGEGNERIA CIVILEINGEGNERIA CIVILE

Introduzione alla soluzione di problemi agliIntroduzione alla soluzione di problemi agli autovaloriautovalori

nellnell analisi dinamica dei continuianalisi dinamica dei continui


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Eigenvalue problems

1) Standard problem (eigenvalues of the stiffness matrix)

Eigenvalues

Eigensolution

K matrix can be positive definite of semi-definite (rigid body

motions).

1. Introduction

StructuralStructuraldynamicsdynamics

=K

n ...0 21

K =


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Eigenvalue problems

2) Generalised eigenproblem

Eigenvalues

Eigenpairs

Eigensolution

1. Introduction


MK =

),(...,),,(),,( 2

2

2

1

2

21 nn

222 ...021 n

M

K

=


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Eigenvalue problems

3) Linearised buckling analysis

Eigenvalues

Eigenpairs

Eigensolution

1. Introduction


KK =G

),(...,),,(),,( 2211 nn

K

K =G

n ...0 21


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Linear properties

1) Connection to static problems

Posing

We have to solve the static problem

2) Eigenvectors are defined as directions (not uniqueness)

2. Properties of eigenvectors


iii MK =

iwith == URUK

RM =ii

)()( iii MK =


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Linear properties

3) M-orthonormalisation

4) K-orthonormalisation (as consequence)

5) Multiple eigenvalues (and corresponding eigenvectors): with

multiplicity m

Eigenvectors are not unique, while eigenspace is unique



ijj

T

i =M

ijijTi =K

miii ++ === 11 ...


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Linear properties

6) Conditions for eigenvectors

7) In buckling analysis they are



IMT =

KT =

IKT =

K GT =


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3. Characteristic polynomials


The eigenvalues of the generalised problem are the roots of

the characteristic polynomial

or

In other terms:

If I has multeplicitym, the last m elements of D matrix arezero.

)det()( MK =p

0)( = ii MK

( ) ii

n

ii

dp1

det)(=

== TLDL


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3.1 Separation of eigenvalues


Sturm sequence property holds for the characteristic

polynomials of the constraint associated problems.

Eigenproblem of the rth associated constraint problem of the

generalised eigenproblem (obtained deleting last r rows and

columns from K and M)

Characteristic polynomial

)()()()()( rrrrr MK =

( ) ( ))()()()()( det rrrrrp MK =


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The eigenvalues of the (r+1)st associated constraint problem

separate those of the rth constraint problem, i.e.

Consequence:

If we factorise the matrix

The number of n g tiv elements in D is equal to the number

of eigenvalues smaller than

)()1(

1

)(

1

)1(

2

)(

2

)1(

1

)(

1 ... r

rn

r

rn

r

rn

rrrr

+

++

TLDLMK


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3.2 Shifting


Used to accelerate eigenvalues calculation (and eliminate

rigid body modes).

Resolve the generalised eigenproblem performing a shiftingon K-matrix by calculating

Then we consider the eigenproblem

That it can be rewritten as

Comparing this with the generalised eigenproblem

we have (due to uniqueness)

MK =

MKK =

MK )( += MK =

ii += ii =


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3.1 Effect of zero mass


If we have r zero diagonal elements in the mass (lumped)matrix M, then the generalised eigenproblem can be rewrittenas

were = -1. Hence the eigenpair

is a non-trivial solution of the generalised eigenproblem:

[ ] 0.00...010...00 == iT

i

KM =

( ) ( )kii e,, =


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3.2 Standard form of the G.E.P.


Hyp.: M positive definite (mii > 0, i=1,,n if lumped; orbanded if consistent)

Transform M using the following decomposition

and substitute in the G.E. equation

Premultiplying l.h.s. and r.h.s. by S-1, we have

with

TSSK =

TSSM=

~~~

=K

TT SSKSK == ~~ 1 and


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Choices for S - matrix

1) Cholesky factorisation

that gives

2) Spectral decomposition

that gives

T

MMLLM ~~

=

MLS ~=

DRS=

T2RDRM=


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Cholesky factorisation is more efficient than Spectraldecomposition, because less operations are involved, butthe latter can be more accurate.

If M is ill-conditioned the transformation process of theG.E.P. to S.E.P. il also ill-conditioned.

its better to use Spectral decomposition thanCholesky factorisation. (Es. 10.9)

Its also possible to use a factorisation of K-matrix, that is

usually better conditioned than M, but the transformed M-matrix becomes always full.

Other reasons to transform G.E.P. into S.E.P.

i) properties of solution of S.E. hold also for solution of G.E.

ii) Sturm sequence properties of S.E. valid for G.E.
http://lezioni%20di%20dinamica%20%28corsi%20a.a.%202004%202005%202006%202007%29/92.%20Esercizi%20cap.%2010/Es%20Bathe%2010.8%20e%2010.9.xlshttp://lezioni%20di%20dinamica%20%28corsi%20a.a.%202004%202005%202006%202007%29/92.%20Esercizi%20cap.%2010/Es%20Bathe%2010.8%20e%2010.9.xls


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3.3 Approximate solution techniques


Purpose: calculate the lowest eigenvalues and correspondingeigenvectors of the G.E.P. (with large order of system)

Techniques:

1) Static condensation2) Rayleigh-Ritz analysis

3) Component mode synthesis

4) Lanczos method

The same techniques will be used as first iteration in thesubspace iteration algorithm (used to solve large eigenvalueproblems).

MK =


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3.3.1 Static condensation


Static condensation applies if structural masses are lumpedin at some specific d.o.f. without affecting eigefrequencesand eigenvectors accuracy.

Condensed masses can be eliminated using the followingpartitioning technique:

From the second row it can be found:

0=+ cccaca KK

=

c

aaa

c

a

ccca

acaa

000M

KKKK

acac cc KK 1

=


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The following reduced problem can be found

with

A standard technique is often used to solve the above G.E.P.

An analogy exists with Gauss elimination in static analysis of

massless degrees of freedom, for which the followinginterpretation of the load vector R is given

caccacaaa KKKKK 1=

aaaa MK =

=

0

aa MR


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The equivalent static problem becomes

hence, i.e. Cholesky factorisation of Ka matrix can beperformed.

The accuracy of the technique for resolving the G.E.P. bymeans of the approximation given by the reducedeigenproblem depends on the specific mass lumping chosen.

However the bandwidth of Ka is in general increased withrespect to K, and the computational effort is higher, even if

only the smallest eigenvalues are needed to be calculated forthe reduced E.P. and this can lead to go back to the originalG.E.P.

In summary, the success of the procedure to be adopteddepends on a large extent on the experience of the analyst indistributing the masses appropriately. (Example: tower bell).

aaa RK =
http://lezioni%20di%20dinamica%20%28corsi%20a.a.%202004%202005%202006%202007%29/3.%20lezione%20n.%205,6,7/torre.ppthttp://lezioni%20di%20dinamica%20%28corsi%20a.a.%202004%202005%202006%202007%29/3.%20lezione%20n.%205,6,7/torre.ppt


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3.3.2 Rayleigh Ritz analysis


Assume that K and M are positive defined (i > 0, i=1,n) for agiven GEP.

The Rayleigh minimum principle states that

with Rayleigh quotient

and the following upper and lower bounds

M

KT

T

=)(

)(min1 =


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Ritz analysis: consider a set of vectors linearcombinations of Ritz basis vectors with a typical

vector equal to

andxi the Ritz coordinates.

In Rayleigh-Ritz analysis we aim to determine the specificvectors that best approximate the required eigenvectors.

This is obtained using Rayleigh minimum principle. Starting

from the Rayleigh quotient

m

k

mxx

kxx

q

j

ijj

q

i

i

q

j

ijj

q

i

i

~

~

~

~

)(

1 1

1 1 ==

= =

= =

i

q

i

ix =

=1

qii ,...,1, =

j

T

i

j

T

i

m

kand

M

K

=

=~

~


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The necessary condition for a minimum of is given by

in thexivariables. Being

the condition for the minimum is

2

11

~

~~2~~2

)(

m

mxkkxm

x

q

jijj

q

jijj

i

==

=

)(

qixi ,...,1,0/)( ==

qiforxmk j

q

j

ijij ,...,10)~~(

1

===


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In matrix form we have the eigenproblem

with K and M (q x q) matrices and x the vector of Ritzcoordinates

The eigensolution is given byq

eigenvalues

approximating

and q eigenvectors

used to evaluate

approximating

eigenvectors

[ ]qT

xxx ...21=x

xMxK

~~

=

[ ]

[ ]q

q

qqT

q

q

Tq

T

xxx

xxx

xxx

...

...

...

...

21

22

2

2

12

11

2

1

11

=

=

=

x

x

xq,...,1

q,...,1

q,...,1

q,...,1


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The approximation to eigenvectors are given by

and the approximated eigenvalues are upper bounds to thereal eigenvalues

The solution found is good if the vectors i span a subspaceV

q

close to the least dominant subspace of K and M spannedby .

Ritz analysis is a very general tool and also StaticCondensation can be regarded as a particular case of the

same method of analysis.

nqq ...,,, 332211

qix j

q

j

i

ji ,...,11 == =

q ,...,1


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3.3.3 Component Mode Synthesis


Also CMS is a particular case of RA (as well as SC). Again, theresults are good as far as the Ritz basis vectors are closeenough to the real eigenvectors.

The basic idea of CMS is the following: Suppose to have a large structure and to have partitioned that in

substructures

Calculate eigeinpairs separately for each substructure (withsuitable boundary conditions).

Use approximate values of eigenpairs to calculate the frequencesand modal shapes of the whole structures (Modal Synthesis).


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Matrices of component structures (substructures)

Matrices of the whole structure

Assume the lowest eigenvalues/vectors of each componentstructure have been calculaled:

....

,...

=

=

M

II

I

M

II

I

M

M

M

M

K

K

K

K

.,...,,;,...,, MIIIMIII MMMKKK


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In a component mode synthesis, approximated eigenpairs canbe obtained by perfoming a RR analysis with the followingloads at the r.h.s.

=

......0

...........

..I00

...00

...0I0

...00

R

M

IIIII,

II

III,

I

MMMMM

IIIIIIIIII

IIIII

MK

...

MK

MK

=

=

=


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3.3.4 Lanczos Method


Used for finding Eigenpairs of tridigonalized matrices.

The method starts with the transformation of the GEP in aSEP form.

An arbitrary starting vector x is used for this purpose and it isnormalised with respect to M matrixto obtain a second vectorx

1

.

Then a sequence of vectors xi

, i = 1,q is generated by means

of an algorithm, producing a matrixX for which

where Tq is tridiagonal of order q. We solve now

The eigenvalues of Tn are reciprocals of the eigenvalues ofthe given GEP and

q

1TTM)X(MKX =

~1~

=nT

~X=


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Examples

Blast doors
http://../EVO_N800v_may2005/Majorana/2004/novembre2004/singapore/keynote%20Valentina.ppthttp://../EVO_N800v_may2005/Majorana/2004/novembre2004/singapore/keynote%20Valentina.ppt

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