Partitioned Formulation with Localized Lagrange Multipliers And its Applications **

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Eleanore Wilkerson
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1 Partitioned Formulation with Localized Lagrange Multipliers And its Applications ** K.C. Park Center for Aerospace Structures (CAS), University of Colorado at Boulder ** Carlos Felippa, Gert Rebel, Yong Hwa Park, Yasu Miyazaki, Younsik Park, Euill Jung, Damijan Markovic, Jose Gonzales Presented at EURODYNE2005, Paris, 2-5 September

2 Motivations dynamic analysis of large structures (oriented to multi-physics & multi-scale problems) Reasons for reduced modelling efficiency physical insight optimal design coupled problems 2

3 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 3

4 Objectives 1) Structural vibration + acoustics (low & medium frequencies) 2) Impact problems involving large structures (project at LMT-Cachan, France) 4

5 Classical approach : Plan of the presentation Craig & Bampton method Partitioning Reduction of interior d.o.f Reduction of boundary d.o.f Conclusions & Prospects 5

6 6

7 Classical approach 7

8 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) 8

9 Classical approach : Plan of the presentation Craig & Bampton method Partitioning localized Lagrange multiplier method Reduction of interior d.o.f Reduction of boundary d.o.f Conclusions & Prospects 9

10 Flexibility based approach - partitioning by localized Lagrange multipliers (LLM) Park & Felippa et al

11 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 11

12 Localized Lagrange multipliers (LLM) - formulation Euler-Lagrange equation Local dynamics Global-local compatibility Global equilibrium 12

13 Localized Lagrange multipliers (LLM) - displacement decompositions { deformation modes zero energy modes (RBM), Reason : substructures being flotting objects K singular, K -1 non-existent generilized inverse u = K -1 f, replaced by d = K + F T f But, K + f ¹u and a =? 13

14 Partitioned system to solve local global 14

15 Classical approach : Plan of the presentation Craig & Bampton method Partitioning localized Lagrange multiplier method Reduction of interior d.o.f Reduction of boundary d.o.f Conclusions & Prospects 15

16 Modal decomposition retained modes residual modes dim(f l ) dim(f r ) 16

17 Residual flexibility based approach Unlike the usual approaches, the residual modes are not truncated, but approximated through l, f Park & Park, 2004 dynamic interface flexibility 17

18 Application to the eigenvalue problem More practical for testing (comparison with exact eigenvalues) 18

19 Approximation of the dynamic interface flexibility Taylor expansion, low frequencies : 19

20 Approximation of the dynamic interface flexibility Diagonalized flexibility : 20

21 Approximation of the dynamic interface flexibility 21

22 Reduced eigenvalue problem For which exists a very efficient resolution algorithm proposed in Park, Justino & Felippa

23 Mode selection criterion Idea : find N modes F d which renders F rb minimal Local mode selection is possible only with LLM criterion 23

24 Example thin plate eigenvalue problem 0.3 Sub-structure 1 Sub-structure Craig-Bampton method residual flexibility method relative error (log) of the eigenfrequency N o of eigen mode 24

25 Classical approach : Plan of the presentation Craig & Bampton method Partitioning localized Lagrange multiplier method Reduction of interior d.o.f Reduction of boundary d.o.f Conclusions & Prospects 25

26 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 26

27 Reduction of local multipliers - separation from interior modes l l l l 27

28 Reduction of local multipliers 28

29 Reduction of local multipliers reduced!! 29

30 Reduction of frame displacements decoupled from local variables 30

31 Complete reduction we retain (almost) the same structure of the system and we can use equally efficient resolution algorithm 31

32 Completely reduced eigenvalue problem partially reduced No. of modes : N int + 2*N boun + N boun = N int + 3* N boun FE model dependent fully reduced No. of modes : N int + N int + 3* N int = 5* N int exclusively physics dependent 32

33 Complete reduction examples thin plate 2 substructures Craig-Bampton method full interface reduced interface interior 1719 to 19 LM 462 to 25 frame disp. 231 to 75 33

34 Complete reduction examples 3D structure Z cross section X cross section Y cross section Partitioned d.o.f. FE model linear tetrahedra elements 34

35 Complete reduction results Craig-Bampton method full interface 10-3 reduced interface

36 Classical approach : Plan of the presentation Craig & Bampton method Partitioning localized Lagrange multiplier method Reduction of interior d.o.f Reduction of boundary d.o.f Conclusions & Prospects 36

37 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 37

38 Reduction in medium frequency range Sub-structure 1 Sub-structure

39 Reduction in medium frequency range Idea : find N modes F d which render F rb minimal does not work well!!! Idea : find N modes F d with the highest error due to the approximation of F rb in the range of interest modal error estimator 39

40 Reduction in medium frequency range Craig-Bampton method low & medium freq. segment medium freq. segment

Partitioned Modeling of Coupled Problems and Applications: New Interpretations of Lagrange Multipliers

Partitioned Modeling of Coupled Problems and Applications: New Interpretations of Lagrange Multipliers Carlos A. Felippa and K. C. Park Center for Aerospace Structures University of Colorado at Boulder