A unified library of nonlinear solution schemes

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1 A unfed lbrary of nonlnear soluton schemes Sofe E. Leon, Glauco H. Paulno, Anderson Perera, Ivan F. M. Menezes, Eduardo N. Lages 7/27/2011

2 Motvaton Nonlnear problems are prevalent n structural, flud, contnuum, etc. mechancs Nonlnear wave nteracton smulated wth lbmesh. lbmesh wk. Smulaton of the electronc behavor of the heart performed wth PETSc. Antak, J., et al., Procceedngs of SC2000, Maxwell model used for vscoelastcty n asphalt pavements. Dave et al. IJNME, 2011 Nonlnear fracture process zone n dynamc fracture processes. Park et al. IJNME, /27/2011 USNCCM-11 A unfed lbrary of nonlnear soluton schemes 2

3 Motvaton Many algorthms have been developed to solve such problems, but no sngle algorthm s capable of solvng any and all nonlnear problems u 2 A lbrary of nonlnear soluton schemes, defned by unque constrant equatons, s unfed nto a sngle space P u 1 Bergan, P. G., et al. IJNME Yang, Y.B. and Sheh, M.S. AIAA Journal Rezaee-Pajand, M., et al. Internatonal Journal of Engneerng, Transactons B: Applcatons Salnger, A. G. et. al., "LOCA 1.0 Lbrary of contnuaton algorthms", Sanda Natonal Laboratores (2002). Balay, S. et al Modern Software Tools n Scentfc Computng (Brkhäuser Press, 1997) 7/27/2011 USNCCM-11 A unfed lbrary of nonlnear soluton schemes Sofe E. Leon 3

4 Presentaton Outlne Nonlnearty overvew N+1 dmensonal space formulaton Soluton schemes Numercal results Evaluaton and conclusons 7/27/2011 USNCCM-11 A unfed lbrary of nonlnear soluton schemes Sofe E. Leon 4

5 Nonlnearty overvew Geometrc Large strans and/or rotatons Lnear or nonlnear consttutve relaton Loads have an effect on deformed confguraton or confguraton can have an effect on the load Load Load Lmt Pont A B Dsplacement Lmt Ponts E Materal.e. nelastcty, stran-softenng C D Load Lmt Pont Crtcal ponts Structure loses stablty or bfurcaton occurs O Stable Unstable Stable Dsplacement 7/27/2011 USNCCM-11 A unfed lbrary of nonlnear soluton schemes Sofe E. Leon 5

6 N+1 dmensonal space formulaton Load p j δp j p j 1 p 1 prev p prev u u1 u j 1 δu j Dsplacement u j Decomposton Load parameter Batoz JJ and Dhatt G. IJNME, /27/2011 USNCCM-11 A unfed lbrary of nonlnear soluton schemes Sofe E. Leon 6

7 Unfed scheme NLS++ Incrementalteratve scheme s the same for any algorthm mplemented n the N+1 space K 0 Step 1 K( u ) j 1 j ==1 Yes δu II j 0 λ λ r p q( u ) j + δλ j j j j u u + ( δλ δu I + δu II ) p λ p Compute q( u ) Only the computaton of the load factor changes for each algorthm No ( ) 1 δu II K j j 1 rj 1 ) j j 1 u ( 1 δ I K p Computeδλ j r j 1 λ p < TOL No Compute Κ j Yes +1 j j +1 7/27/2011 USNCCM-11 A unfed lbrary of nonlnear soluton schemes Sofe E. Leon 7

8 Nonlnear soluton schemes Crsfeld MA. Non-lnear Fnte Element Analyss of Solds and Structures. Volume 1: Essentals Ramm E. Nonlnear Fnte Element Analyss n Structural Mechancs Bergan PG, et al. IJNME Rks E. Internatonal Journal of Solds and Structures Wempner GA. Internatonal Journal of Solds and Structures Yang Y-B, McGure W. Proceedngs of the 1985 Internatonal Conference on Numercal Methods n Engneerng: Theory and Applcatons. Yang, Y.B. and Sheh, M.S. AIAA Journal Krenk S. IJNME Kouha R. CMAME /27/2011 USNCCM-11 A unfed lbrary of nonlnear soluton schemes Sofe E. Leon 8

9 Common nonlnear soluton schemes Load control method Arc-length control method λp Constrant Surface λp Constrant Surface j =1 j = 2 j = 3 λp A B λ 1 p λ 2 p λ 3 p s B A u u u 1 u 2 u u 3 7/27/2011 USNCCM-11 A unfed lbrary of nonlnear soluton schemes Sofe E. Leon 9

10 Generalzed dsplacement control method Load parameter s based on a physcal quantty called the Generalzed Stffness Parameter (GSP) The sgn of the GSP changes only at load lmt ponts Load - + Sgn of δ : 1 u δu I I Dsplacement Yang, Y.B. and Sheh, M.S. AIAA Journal /27/2011 USNCCM-11 A unfed lbrary of nonlnear soluton schemes Sofe E. Leon 10

11 Two-dmensonal functon Internal force gven by Tangent matrx LCM = load control method DCM = dsplacement control method ALCM = arc-length control method WCM = work control method GDCM = generalzed dsplacement control method ORP = orthogonal resdual procedure var = varable u1,u2 = control degrees of freedom Ilnca Stanculescu. Personal communcatons /27/2011 USNCCM-11 A unfed lbrary of nonlnear soluton schemes Sofe E. Leon 11

12 Two-dmensonal functon and the varable DCM Systematcally select the best control parameter Varable dsplacement control method Step Control Snap back Fuj F, et al. Computers & Structures /27/2011 USNCCM-11 A unfed lbrary of nonlnear soluton schemes Sofe E. Leon 12

13 Twelve bar truss 1.5P 0.75P 0.25P 0.25P P 2 EA/ 2 u 2 u 1 1.5P u 2 u 1 u 3 u 3 EA 4 EA 3 1 EA/ z y x x-z vew y-z vew Yang Y-B and Leu LJ. CMAME Krenk S and Hededal O. CMAME NRM = newton raphson method, DCM = dsplacement control method, ALCM = arc-length control method, WCM = work control method, GDCM = generalzed dsplacement control method, ORP = orthogonal resdual procedure 7/27/2011 USNCCM-11 A unfed lbrary of nonlnear soluton schemes Sofe E. Leon 13

14 Lee frame x P y u 2 u 1 E, A I, G Lee SL et al. ASCE Journal of Engneerng Mechancs Schwezerhof KH and Wrggers P. CMAME Parente E and Vaz LE. IJNME 2001 LCM = load control method DCM = dsplacement control method ALCM = arc-length control method WCM = work control method GDCM = generalzed dsplacement control method ORP = orthogonal resdual procedure 7/27/2011 USNCCM-11 A unfed lbrary of nonlnear soluton schemes Sofe E. Leon 14

15 Evaluaton of soluton schemes In general the most robust and powerful soluton schemes n the context of the N+1 space are Varable dsplacement control method Arc-length control method Generalzed dsplacement control method Standard dsplacement control method and work control method have predctable dffcultes at dsplacement lmt ponts Orthogonal resdual procedure s senstve to nput parameters and behavor s sometmes unpredctable near lmt ponts 7/27/2011 USNCCM-11 A unfed lbrary of nonlnear soluton schemes Sofe E. Leon 15

16 Conclusons and future work NLS++ s a powerful computatonal framework that provdes seamless mplementaton of soluton schemes of nterest Gves user flexblty to select the most approprate scheme for a partcular problem Portable, effectve and extendable Plan to ntegrate NLS++ wth the new fnte element code based on topologcal data representaton, called TopFEM Detals and code avalable n M.S. thess 7/27/2011 USNCCM-11 A unfed lbrary of nonlnear soluton schemes Sofe E. Leon 16

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