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S P . Ladevze ,D. Nron , S.Rodriguez and R.Scanff LMT (ENS - PowerPoint PPT Presentation

Dec 03, 2022 •441 likes •1.24k views

A TIME-MULTISCALE MODEL ORDER REDUCTION METHOD IN NONLINEAR SOLID MECHANICS S P . Ladevze ,D. Nron , S.Rodriguez and R.Scanff LMT (ENS Paris-Saclay / CNRS / Universit Paris-Saclay) coll G. Nahas , P .E Charbonnel CEA , IRSN coll

  1. A TIME-MULTISCALE MODEL ORDER REDUCTION METHOD IN NONLINEAR SOLID MECHANICS S P . Ladevèze ,D. Néron , S.Rodriguez and R.Scanff LMT (ENS Paris-Saclay / CNRS / Université Paris-Saclay) coll G. Nahas , P .E Charbonnel CEA , IRSN coll U.Nackenhorst, A. Fau, M. Bhattacharyya, S.Alameddin , Hanover University

  2. Motivation fatigue loadings with large number of cycles 2

  3. Motivation initiation of a To compute time-dependent nonlinear macrocrack problems (viscoplasticity + fatigue damage) fatigue loadings with large number of cycles 3 FINAL ROM

  4. Motivation seismic loadings problems 4

  5. Motivation To compute time-dependent nonlinear damage evaluation problems (viscoplasticity + damage) seismic loadings Worst earthquake : Chili 1960 —Richter scale:9.5 —duration : 10mn 5 FINAL ROM

  6. Motivation Our answer : two levels of complexity reduction ♦ A signal theory with two time scales ♦ The time-multiscale PGD 6

  7. Another motivation MOR methods : intrusive ➡ engineering diffusion ➘➘➘➘ LATIN-PGD : version non intrusive —Very general: + —Performance: — —Applications (in progress) in SAMCEF (coll SIEMENS) and in CASTEM (coll CEA) 7

  8. Outline 1. A signal theory :a ROM for the loading 2. A new PGD approach : multiscale in time and non intrusive 3. First illustrations 4. Conclusion-Prospects 8

  9. Outline 1. A signal theory :a ROM for the loading 2. A new PGD approach : multiscale in time and non intrusive 3. First illustrations 4. Conclusion-Prospects 9

  10. A signal theory: a ROM-loading (wavelet theory: too much terms) Specific aim: minimum of time shape functions Two time-scale approximation ( micro : harmonic funct.) m µ ∂ i ( t )cos2 π τ i ( t )sin2 π τ ϕ R + ϕ I X s d ( t ) : τ i = ∞ 0 τ i τ i i = 0 | τ = t ( τ i -mode( H )) t : « macro » time τ i < 1 τ : « micro » time ≤ 4macro-time scale N ( N = 3) ( τ i , ϕ R i , ϕ I i ∈ 0,1,..., m ) | Characteristics : i | {z } macro functions 10

  11. A signal theory: a ROM-loading Two time-scale approximation (micro : harmonic funct.) m n ( τ , τ i ) T S S ( t ) A i X S s d ( t ) : | τ = t i = 0   a R a I τ   1 1 cos2 π . .   τ i . . A =   n ( τ , τ i ) :   S S S ( t ) = [ ψ 1 ,..., ψ m ]( t ) . .     τ a R a I   sin2 π FE basis m m τ i 11

  12. A signal theory: a ROM-loading Best approximation (micro : harmonic funct.) s m ( t ) : ( τ i , A i S (0, T ) S | i ∈ 0,1,..., m ) S m Z T 1 m ( t )) 2 dt ( s d ( t ) ° s 0 s m ( t ) = arg min T S (0, T ) s 0 0 m 2 S S m 12

  13. A signal theory: a ROM-loading Computational method: « greedy » (micro : harmonic funct.) n ( τ , τ ) T S New correction: S ( s m + 1 − s m ) : ( τ , A ) SA 1 τ 0 0 R T M ° 1 R τ = argmax 1 R j = FT[( s d − s m ) ψ j ] × T A = M − 1 R | τ Z T 1 ≈ ≈ ψ i ψ j dt ≈ M i j = ≈ T 0 Error : 1/ (6,6x N 2 ) for P1 ( N = 3) 1/ (165x N 3) for P2 + strict separation of computed periods 13

  14. A signal theory: a ROM-loading Computational method: « greedy » (micro : harmonic funct.) - I m Convergence proof : data = finite sum of modes (H) 14

  15. A signal theory: a ROM-loading Other writing FE shape function ( ) q n m z }| { X X θ i s m ( t ) = ψ j ( t ) j ( τ , τ i ) × j = 1 i = 0 | τ = t | {z } j − nodal contribution (harmonic functions) Theory extension harmonic function arbitrary periodic function sum of τ k- modes( H ) sum of τ k- modes( P ) 15

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