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Partitioned Formulation withLocalized Lagrange Multipliers

And its Applications**

**Carlos Felippa, Gert Rebel, Yong Hwa Park, Yasu Miyazaki, Younsik Park, Euill Jung, Damijan Markovic, Jose Gonzales

Center for Aerospace Structures (CAS),University of Colorado at Boulder

K.C. Park

Presented at EURODYNE2005, Paris, 2-5 September 2005.

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Motivations – dynamic analysis of large structures (oriented to multi-physics & multi-scale problems)

Reasons for reduced modelling

•efficiency

•physical insight

•optimal design

•coupled problems

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Objectives

Desired features for the reduced model:

•suited for parallel computing

•enables a robust mode selection criterion – adaptativity

•adapted for multi-scale and multi-physics problems

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1) Structural vibration + acoustics(low & medium frequencies)

Objectives

2) Impact problemsinvolving large structures(project at LMT-Cachan, France)

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Plan of the presentation

•Classical approach :Craig & Bampton method

•Partitioning

•Reduction of interior d.o.f

•Reduction of boundary d.o.f

•Conclusions & Prospects

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Classical approach

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Lacks of existing CMS methods

•difficult to parallelize

•there is no mode selection criterium but for local loading

•impossible to model several parts of the freq. spectrum (low + medium frequencies)

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•Partitioning localized Lagrange multiplier method

Plan of the presentation

•Classical approach :Craig & Bampton method

•Reduction of interior d.o.f

•Reduction of boundary d.o.f

•Conclusions & Prospects

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Flexibility based approach - partitioning by localized Lagrange multipliers (LLM)

Park & Felippa et al. 1997-2005

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Localized Lagrange multipliers (LLM) - advantages

Local formulation of coupling of several sub-domains• no redundancy• fewer global d.o.f.• increased modularity

More adapted to multi-physics problems than classical approach

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Localized Lagrange multipliers (LLM) - formulation

Euler-Lagrange equation•Local dynamics

•Global-local compatibility

•Global equilibrium

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Localized Lagrange multipliers (LLM) - displacement decompositions

zero energy modes (RBM)

deformation modes {,

Reason : substructures being flotting objects

K singular, K-1 non-existent

u = K-1 f, replaced by d = K+ FTf

generilized inverse

But, K+ f ¹u and a = ?

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Partitioned system to solve

global

local

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•Partitioning localized Lagrange multiplier method

Plan of the presentation

•Classical approach :Craig & Bampton method

•Reduction of interior d.o.f

•Reduction of boundary d.o.f

•Conclusions & Prospects

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Modal decomposition

retained modes residual modes

dim(Fl)† dim(Fr)

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Residual flexibility based approach

Unlike the usual approaches, the residual modesare not truncated, but approximated through l, f

dynamic interface flexibility

Park & Park, 2004

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Application to the eigenvalue problem

More practical for testing(comparison with exact eigenvalues)

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Approximation of the dynamic interface flexibility

Taylor expansion, low frequencies :

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Approximation of the dynamic interface flexibility

Diagonalized flexibility :

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Approximation of the dynamic interface flexibility

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Reduced eigenvalue problem

For which exists a very efficient resolution algorithm proposed in Park, Justino & Felippa 1997

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Mode selection criterion

Idea : find N modes Fd which renders Frb minimal

criterion

Local mode selection ispossible only with LLM

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Example – thin plate eigenvalue problem

Sub-structure 1 Sub-structure 20.4 0.2

0.3

relative error (log)of the eigenfrequency

No of eigen mode

Craig-Bampton method

residual flexibilitymethod

10-4

10-8

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•Partitioning localized Lagrange multiplier method

Plan of the presentation

•Classical approach :Craig & Bampton method

•Reduction of interior d.o.f

•Reduction of boundary d.o.f

•Conclusions & Prospects

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Reduction of interface d.o.f.

Number of interface d.o.f is smaller than interior d.o.f, but not neglegeable.

Goals : •to have a mode selection criterion•not to loose advantages of the resolution algorithm•retain a similar accuracy

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Reduction of local multipliers - separation from interior modes

ll

l

l

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Reduction of local multipliers

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Reduction of local multipliers

reduced !!

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Reduction of frame displacements

decoupled from local variables

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Complete reduction

we retain (almost) the same structure of the systemand we can use equally efficient resolution algorithm

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Completely reduced eigenvalue problem

partially reduced

fully reduced

No. of modes : Nint + Nint + 3* Nint = 5* Nint

No. of modes : Nint + 2*Nboun + Nboun = Nint + 3* Nboun

exclusively physics dependent

FE model dependent

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Complete reduction examples – thin plate

2 substructures

Craig-Bampton method

full interface

reduced interface

10-7

10-4

interior 1719 to 19LM 462 to 25frame disp. 231 to 75

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Complete reduction examples – 3D structure

Z – cross section

Y – cross section

X – cross section

Partitioned

•11 000 d.o.f. FE model

•linear tetrahedra elements

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Complete reduction – results

Craig-Bampton method

full interface

reduced interface

10-9

10-6

10-3

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•Partitioning localized Lagrange multiplier method

Plan of the presentation

•Classical approach :Craig & Bampton method

•Reduction of interior d.o.f

•Reduction of boundary d.o.f

•Conclusions & Prospects

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Conclusions

•flexibility based approach leads to the consistently reduced models •it is adapted for parallel computing

•mode selection criterion is well established

•suitable for coupled problems

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Sub-structure 1 Sub-structure 20.4 0.2

0.3

Reduction in medium frequency range

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Reduction in medium frequency range

Idea : find N modes Fd which render Frb minimal

does not work well !!!

Idea : find N modes Fd with the highest error due tothe approximation of Frb in the range of interest

modal error estimator

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Reduction in medium frequency range

low & medium freq. segment

medium freq. segment

Craig-Bampton method

10-6

10-3