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 2005. 1
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
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
Objectives 1) Structural vibration + acoustics (low & medium frequencies) 2) Impact problems involving large structures (project at LMT-Cachan, France) 4
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
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Classical approach 7
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
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
Flexibility based approach - partitioning by localized Lagrange multipliers (LLM) Park & Felippa et al. 1997-2005 10
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
Localized Lagrange multipliers (LLM) - formulation Euler-Lagrange equation Local dynamics Global-local compatibility Global equilibrium 12
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
Partitioned system to solve local global 14
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
Modal decomposition retained modes residual modes dim(f l ) dim(f r ) 16
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
Application to the eigenvalue problem More practical for testing (comparison with exact eigenvalues) 18
Approximation of the dynamic interface flexibility Taylor expansion, low frequencies : 19
Approximation of the dynamic interface flexibility Diagonalized flexibility : 20
Approximation of the dynamic interface flexibility 21
Reduced eigenvalue problem For which exists a very efficient resolution algorithm proposed in Park, Justino & Felippa 1997 22
Mode selection criterion Idea : find N modes F d which renders F rb minimal Local mode selection is possible only with LLM criterion 23
Example thin plate eigenvalue problem 0.3 Sub-structure 1 Sub-structure 2 0.4 0.2 Craig-Bampton method residual flexibility method relative error (log) of the eigenfrequency 10-4 10-8 N o of eigen mode 24
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
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
Reduction of local multipliers - separation from interior modes l l l l 27
Reduction of local multipliers 28
Reduction of local multipliers reduced!! 29
Reduction of frame displacements decoupled from local variables 30
Complete reduction we retain (almost) the same structure of the system and we can use equally efficient resolution algorithm 31
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
Complete reduction examples thin plate 2 substructures Craig-Bampton method full interface 10-4 10-7 reduced interface interior 1719 to 19 LM 462 to 25 frame disp. 231 to 75 33
Complete reduction examples 3D structure Z cross section X cross section Y cross section Partitioned 11 000 d.o.f. FE model linear tetrahedra elements 34
Complete reduction results Craig-Bampton method full interface 10-3 reduced interface 10-6 10-9 35
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
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
Reduction in medium frequency range Sub-structure 1 Sub-structure 2 0.4 0.2 0.3 38
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
Reduction in medium frequency range Craig-Bampton method low & medium freq. segment medium freq. segment 10-3 10-6 40