Model Reduction 061904 - Faculty Server...

1 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques

MODEL REDUCTION TECHNIQUES

Peter AvitabileMechanical Engineering DepartmentUniversity of Massachusetts Lowell

[ K ]n

[ M ]n [ M ]a[ K ]a [ E ]a

[ ω ]2

Structural Dynamic Modeling Techniques & Modal Analysis Methods

Model Reduction TechniquesDynamic reduction means :

reducing a given dynamic finite element model

to one with fewer degrees of freedom

while maintaining the dynamic characteristics of the system

XA = active set of dof’sXF = full set of dof’s

• Guyan condensation

• Dynamic Condensation

• Improved Reduced System

• System Equivalent Reduction Expansion Process

• Hybrid Reduction

Model Reduction Techniques

Model Reduction

• Guyan/Irons condensation• Dynamic condensation• Improved Reduced System• System Equivalent Reduction Expansion Process• Hybrid Reduction (Kammer)

Generally, it may be necessary to reduce a finite element model to a smaller size especially when correlation studies are to be performed.Several model reduction techniques are:

General TransformationFor all model reduction/expansion techniques, there is a relationship between the master dof (adof) and the deleted dof (ddof) which can be written in general terms as

n denotes all FEM dofa denotes master or tested dofd denotes deleted or omitted dof

[ ] ad

[ ] [ ] 2121ad

an xTxorxT

General TransformationSince the energy of the system needs to be conserved, a balance can be written between the energy at state 1 and state 2 as

Substituting the transformation gives

[ ] [ ] 22T

1 xKx21xKx

21U ==

[ ] [ ] [ ] [ ] 22T

22121T

212 xKx21xTKxT

21U ==

General TransformationRearranging some terms then yields

Then the reduced stiffness is related to the original stiffness by

The mass is reduced in a similar fashion

[ ] [ ][ ] [ ] 22T

22121T

2 xKx21xTKTx

21U ==

[ ] [ ] [ ][ ] [ ] [ ] [ ][ ]TKTKorTKTK nT

122 ==

Reduction of System MatricesThe reduced mass and stiffness matrices can be written as

[M] denotes the mass matrix[K] denotes the stiffness matrix

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]TKTK

General TransformationThe transformation T will take on various forms depending on the transformation technique utilized

[ ] ad

XXAA = active set of = active set of dof’sdof’sXXFF = full set of = full set of dof’sdof’s

Eigensolution of Reduced SystemUsing the reduced mass and stiffness matrices, the eigensolution produces frequencies that are higher than those of the original system (for most of the reduction schemes).

The eigensolution of the reduced matrices

yields[ ] [ ][ ] 0xMK aaa =λ−

[ ] [ ]a2a U;ω

Expansion FormulationThe expansion of the adof from the reduced model eigensolution over all the ndof is given by

[ ] ad

XXAA = active set of = active set of dof’sdof’sXXFF = full set of = full set of dof’sdof’s

Guyan CondensationThe stiffness equation

can be partitioned into the ‘a’ active DOF and the ‘d’ deleted or omitted DOF to form two equations given as

[ ] nnn FxK =

[ ] [ ][ ] [ ]

Guyan CondensationAssuming that the forces on the deleted DOF are zero, then the second equation can be written as

which can be solved for the displacement at the deleted DOF as

[ ] [ ] 0xKxK dddada =+

[ ] [ ] ada1

ddd xKKx −−=

Guyan CondensationThe first equation can be written as

and upon substituting for the ‘d’ deleted DOF gives the equation becomes

[ ] [ ] adadaaa FxKxK =+

[ ] [ ][ ] [ ] aada1

ddadaaa FxKKKxK =+ −

Guyan CondensationThis can be manipulated to yield the desired transformation to be

[ ] [ ][ ]

= − ]K[]K[

Guyan CondensationUsing this transformation, the reduced stiffness can be written as

Guyan proposed that this same transformation be applied to the mass matrix given by

[ ] [ ] [ ][ ]snT

sGa TKTK =

[ ] [ ] [ ][ ]snT

sGa TMTM =

Summary - Guyan CondensationThe stiffness equation

can be written as

The transformation matrix can be written as

[ ] nnn FxK =

[ ] [ ][ ] [ ]

[ ] [ ][ ]

= − ]K[]K[

Guyan Condensation – MATLAB Script

Guyan Condensation• Guyan (static) condensation is only accurate for

stiffness reduction; inertial forces are not preserved

• Eigenvalues of the reduced system are always higher than those of the original system

• The quality of the eigenvalue approximation depends highly on the location of points preserved in the reduced model

• The quality of the eigenvalue approximation decreases as the mode number increases

Dynamic CondensationThe equation of motion is cast as a shifted eigenproblem. A shift value, f, is introduced into the set of equations describing the dynamic system, thus

[ ] ( )[ ][ ] 0xMfK nnn =−λ−

Dynamic CondensationThe terms are rearranged to group the constant term f times the mass matrix with the stiffness matrix to yield

Then let a new system matrix [D] be used to describe the ‘effective’ stiffness matrix as

[ ] [ ][ ] [ ][ ] 0xMMfK nnnn =λ−+

[ ] [ ] [ ][ ]nnn MfKD +=

Dynamic CondensationThis ‘effective’ stiffness equation

can be partitioned into the ‘a’ active DOF and the ‘d’ deleted or omitted DOF to form two equations given as

[ ] nnn FxD =

[ ] [ ][ ] [ ]

Dynamic CondensationAssuming that the forces on the deleted DOF are zero, then the second equation can be written as

which can be solved for the displacement at the deleted DOF as

[ ] [ ] 0xDxD dddada =+

[ ] [ ] ada1

ddd xDDx −−=

Dynamic CondensationThe first equation can be written as

and upon substituting for the ‘d’ deleted DOF this equation becomes

[ ] [ ] adadaaa FxDxD =+

[ ] [ ][ ] [ ] aada1

ddadaaa FxDDDxD =+ −

Dynamic CondensationThis can be manipulated to yield the desired transformation to be

[ ] [ ][ ]

= − ]D[]D[

Dynamic CondensationUsing this transformation, the reduced stiffness can be written as

This same transformation can be applied to the mass matrix given by

[ ] [ ] [ ][ ]fnT

ffa TKTK =

[ ] [ ] [ ][ ]fnT

ffa TMTM =

Summary - Dynamic CondensationThe equation of motion is cast as a shifted eigenproblem

and can be written as

The transformation matrix can be written as

[ ] [ ] [ ][ ] 0xMfKxD nnn =−=

[ ] [ ][ ] [ ]

[ ] [ ][ ]

= − ]D[]D[

Dynamic Condensation – MATLAB Script

Dynamic Condensation• If the shift frequency is zero, then this

reduces to Guyan reduction• The reduced model will at most contain an

eigenvalue equal one from the full model• If the shift equals as eigenvalue of the original

system, then the reduced system will also contain this eigenvalue

• All other eigenvalues of the reduced system may not be good representations of the system

• Dynamic condensation is useful when only one mode of the system is to be retained in the model

Improved Reduced SystemExtensions to the Guyan reduction process are used to account for the effects of mass inertia associated with the deleted dof

[ ] [ ][ ] [ ]is

[ ] [ ] [ ][ ] [ ] [ ][ ][ ] [ ]a1

asn1dd

KMTMK000

]K[]K[t

−−

IRS Reduction– MATLAB Script

Improved Reduced System• Adjustment terms to the Guyan reduction

process allow for the better representation of the mass associated with the deleted dof

• Improves on the accuracy of the reduced model when compared to Guyan especially for the higher modes of the system

System Equivalent Reduction Expansion Process

The modal transformation equations can be written as

and for the active set of dof

[ ] [ ][ ] pUU

[ ] pUx aa =

Least Squares Solution - ma ≥

[ ] [ ] [ ] [ ]

[ ] [ ]( ) [ ] [ ] [ ]( ) [ ] [ ]

[ ] [ ]( ) [ ] [ ] agaa

xUxUUUp

pUUUUxUUU

−−

Using a generalized inverse, this is

[ ] [ ]( ) [ ]

[ ] aga

Substituting into the modal transformation equation gives

[ ][ ] agann xUUx =

[ ][ ] [ ] ag

a xUUU

The SEREP transformation matrix can be written as

[ ] [ ][ ] [ ] [ ] [ ]( ) [ ][ ][ ] [ ] [ ]( ) [ ][ ]

== −

anuUUUUUUUUUUT

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Ua =Un =

SEREP Computational Advantages

The SEREP transformation is given by

The reduced mass and stiffness are computed as

[ ] [ ][ ]ganu UUT =

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]Un

Substituting Tu into the reduced mass computation gives

But recall from mass orthogonality that

[ ] [ ] [ ] [ ][ ][ ]gannT

aSa UUMUUM =

[ ] [ ][ ] [ ]IUMU nnT

The reduced mass is efficiently computed as

[ ] [ ] [ ][ ] [ ] [ ]gaTgaUn

Sa UUTMTM ==

NOTE: Unit modal mass scaling

Substituting Tu into the reduced stiffness computation gives

But recall from stiffness orthogonality that

[ ] [ ] [ ] [ ][ ][ ]gannT

aSa UUKUUK =

[ ] [ ][ ] [ ]2nnT

n UKU Ω=

The reduced stiffness is efficiently computed as

[ ] [ ] [ ][ ] [ ] [ ][ ]ga2TgaUn

Sa UUTKTK Ω==

NOTE: Unit modal mass scaling

SEREP Reduction – MATLAB Script

System Equivalent Reduction Expansion Process• The eigenvalues of the reduced system always

equals the eigenvalues of the full system for the modes of interest retained in the model

• The modes that are preserved in the reduced model may be arbitrarily selected from those modes of interest in the original model

• The eigensolution of the reduced system is exact and does not depend on the location or number of points preserved in the reduced model.

Hybrid Reduction

Hybrid reduction combines the advantages of the full rank nature of Guyan Reduction along with the accuracy of the SEREP process

[ ] [ ] [ ] [ ][ ][ ]PTTTT SUSH −+=

[ ] [ ][ ] [ ]SaTaa MUUP =

Hybrid Reduction

The projection operator can be rewritten as

This transformation equation can be manipulated to give

[ ] [ ][ ] [ ] [ ][ ]gaaSa

Taa UUMUUP ==

[ ] [ ][ ] [ ] [ ] [ ][ ]PITPTT SUH −+=

Hybrid Reduction

Substituting the projection operator gives

And recalling the Moore-Penrose conditions this can be rewritten as

[ ] [ ][ ] [ ][ ] [ ] [ ] [ ][ ][ ]gaaSg

anH UUITUUUUT −+=

[ ] [ ][ ] [ ] [ ] [ ][ ][ ]gaaSg

anH UUITUUT −+=

[ ] [ ] [ ] [ ] [ ][ ][ ]gaaSUH UUITTT −+=

SEREP Applications

• Exactness of the technique• arbitrary selection of modes included in the

reduced model• arbitrary selection of dof included in the

reduced model

Several simple examples are investigated in order to demonstrate the unique features of the SEREP process:

FRAME Modal DataN11

* * * * * *

N24N23N22N21N20N4

N7 N8 N9 N10 N12

Aluminum frame1-1/2 x 3-1/2 x3/1624 nodes24 planar bean elements

SEREP - Exact System Reduction

SEREP - Effect of Mode Selection

SEREP - Effect of Point Selection

SEREP Applications Summary

• Arbitrary selection of modes preserved in the reduced model

• Reduced model accuracy is not dependent on the selection of master dof

• Reduced model frequencies are identical to those of the full system model

• Expanded reduced mode shapes are identical to those of the full system model

A new modeling/mapping techniques referred to as the System Equivalent Reduction Expansion Process (SEREP) reveals the following salient features:

Comparison of Reduced Models

• Guyan• IRS• Dynamic• SEREP

Several simple examples are investigated in order to compare the different model reduction techniques - those investigated were:

Case 3 - Poor Selection of 6 DOF

Case 4 - Better Selection of 6 DOF

Case 5 - Larger Selection of DOF

Model Reduction Application Summary

• Guyan condensation always produces frequencies that are greater than those of the full model; dof selection is critical to its success

• IRS improves on Guyan by making adjustments to the inertial effects associated with the ddof

• dynamic condensation will preserve at most one of the eigenvalues of the original system

• SEREP always produces the same frequencies and mode shapes as the full system

Comparison of several different model reduction methods were presented to show distortion that results from various schemes. Main points are:

Model Reduction SummaryThe different reduction forms are:

Dynamic

[ ] [ ] [ ] [ ][ ] [ ] [ ][ ][ ] [ ]a1

asn1ddda

i KMTMK000

]K[]K[I

T −−−

[ ] [ ][ ]

= − ]B[]B[

[ ] [ ][ ]

= − ]K[]K[

[ ] [ ][ ] [ ] [ ] [ ]( ) [ ][ ][ ] [ ] [ ]( ) [ ][ ]

== −

anuUUUUUUUUUUT

Model Reduction 061904 - Faculty Server...

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