A unified library of nonlinear solution schemes Sofie E. Leon, - PowerPoint PPT Presentation

A unified library of nonlinear solution schemes Sofie E. Leon, Glaucio H. Paulino, Anderson Pereira, Ivan F. M. Menezes, Eduardo N. Lages 7/27/2011

Motivation • Nonlinear problems are prevalent in structural, fluid, continuum, etc. mechanics Simulation of the electronic behavior of the heart performed with PETSc. Antaki, J., et al., Procceedings of SC2000, 2000. Nonlinear wave interaction simulated with libMesh. libMesh wiki. Nonlinear fracture process zone in dynamic Maxwell model used for viscoelasticity in fracture processes. Park et al. IJNME, 2010 asphalt pavements. Dave et al. IJNME , 2011 7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes 2

Motivation Many algorithms have been developed to solve such problems, but no single • algorithm is capable of solving any and all nonlinear problems u 2 u 1 P • A library of nonlinear solution schemes, defined by unique constraint equations, is unified into a single space Bergan, P . G., et al. IJNME 1978. | Yang, Y.B. and Sheih, M.S. AIAA Journal 1990. | Rezaiee-Pajand, M., et al. International Journal of Engineering, Transactions B: Applications . 2009. | Salinger, A. G. et. al., "LOCA 1.0 Library of continuation algorithms", Sandia National Laboratories (2002). | Balay, S. et al Modern Software Tools in Scientific Computing (Birkhäuser Press, 1997) 7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon 3

Presentation Outline • Nonlinearity overview • N+1 dimensional space formulation • Solution schemes • Numerical results • Evaluation and conclusions 7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon 4

Nonlinearity overview • Geometric – Large strains and/or rotations – Linear or nonlinear constitutive relation Load Load Limit Point – Loads have an effect on deformed A Displaceme nt E Limit Points configuration or configuration can have an effect on the load B C D • Material Load Limit – i.e. inelasticity, strain-softening Point O Displaceme nt Stable Unstable Stable Critical points • – Structure loses stability or bifurcation occurs 7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon 5

N+1 dimensional space formulation Load δ p j ∆ ∆ p p − ∆ j 1 1 p j prev p δ u ∆ Displaceme nt u j prev u 1 ∆ u − j 1 ∆ u j • Decomposition Load parameter • Batoz JJ and Dhatt G. IJNME , 1979. 7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon 6

Unified scheme – NLS++ • Incremental- λ ← λ + δλ i i i Step i j ← + δλ δ + δ iterative scheme i i i i i u u ( u u ) j I j II j ← − i i 1 is the same for ( ) K K u 0 ← λ i i p p any algorithm ← j 1 implemented in i Compute ( ) q u the N+1 space Yes == δ ← i 0 j 1 u II j ← − i i i r p q ( u ) j No • Only the δ ← − i i 1 i u II ( K 1 ) r − − j j 1 j computation of i r Yes j < ← i + TOL 1 i ∆ λ i p 1 the load factor δ ← − i 1 i u I ( K 1 ) p − j j changes for each No δλ algorithm i Compute Compute Κ i j j ← j + j 1 7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon 7

Nonlinear solution schemes • Crisfield MA. Non-linear Finite Element Analysis of Solids and Structures. Volume 1: Essentials . 1991. | Ramm E. Nonlinear Finite Element Analysis in Structural Mechanics . 1981. | Bergan PG, et al. IJNME . 1978. | Riks E. International Journal of Solids and Structures . 1979. | Wempner GA. International Journal of Solids and Structures . 1971. | Yang Y-B, McGuire W. Proceedings of the 1985 International Conference on Numerical Methods in Engineering: Theory and Applications . | Yang, Y.B. and Sheih, M.S. AIAA Journal 1990. |Krenk S . IJNME 1995. | Kouhia R. CMAME 2008. 7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon 8

Common nonlinear solution schemes Load control method Arc-length control method Constraint = Surface λ λ Constraint j 1 p p = Surface j 2 = 3 j ∆ λ B B p 1 ∆ λ ∆ λ p p A ∆ λ 2 p 3 ∆ i s A u ∆ ∆ u u u 1 ∆ u 2 ∆ u 3 7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon 9

Generalized displacement control method Load parameter is based on a physical quantity called the Generalized Stiffness • Parameter (GSP) • The sign of the GSP changes only at load limit points Load Sign of : - − δ ⋅ δ 1 i i u u I I + 1 1 + + + - Yang, Y.B. and Sheih, M.S. AIAA Displaceme nt Journal 1990. 7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon 10

Two-dimensional function Internal force given by • • Tangent matrix LCM = load control method DCM = displacement control method ALCM = arc-length control method WCM = work control method GDCM = generalized displacement control method ORP = orthogonal residual procedure var = variable u1,u2 = control degrees of freedom Ilinca Stanciulescu. Personal communications. 2009. 7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon 11

Two-dimensional function and the variable DCM • Systematically select the best control parameter Variable displacement control method Step Control Snap back 1-73 74-216 217-445 Fujii F, et al. Computers & Structures. 1992. 7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon 12

Twelve bar truss 1 . 5 P 0 . 75 0 . 25 P P 0 . 25 P P 2 EA / 2 u u u 1 . 5 P u 1 2 1 u 2 3 4 1 . 0 1 3 EA u 3 EA EA / 2 1 . 414 1 . 0 z 1 . 697 x x-z view y-z view y NRM = newton raphson method, DCM = displacement control method, ALCM = Yang Y-B and Leu LJ. CMAME 1991. arc-length control method, WCM = work control method, GDCM = generalized Krenk S and Hededal O. CMAME 1995. displacement control method, ORP = orthogonal residual procedure 7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon 13

Lee frame P x y u 2 u 1 120 E , A I , G 120 LCM = load control method DCM = displacement control method ALCM = arc-length control method Lee SL et al. ASCE Journal of Engineering Mechanics . 1968. WCM = work control method Schweizerhof KH and Wriggers P . CMAME 1986. GDCM = generalized displacement control method Parente E and Vaz LE. IJNME 2001 ORP = orthogonal residual procedure 7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon 14

Evaluation of solution schemes • In general the most robust and powerful solution schemes in the context of the N+1 space are – Variable displacement control method – Arc-length control method – Generalized displacement control method • Standard displacement control method and work control method have predictable difficulties at displacement limit points • Orthogonal residual procedure is sensitive to input parameters and behavior is sometimes unpredictable near limit points 7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon 15

Conclusions and future work • NLS++ is a powerful computational framework that provides seamless implementation of solution schemes of interest – Gives user flexibility to select the most appropriate scheme for a particular problem – Portable, effective and extendable • Plan to integrate NLS++ with the new finite element code based on topological data representation, called TopFEM • Details and code available in M.S. thesis www.ideals.illinois.edu www.sofieleon.com 7/27/2011 USNCCM-11 | A unified library of nonlinear solution schemes | Sofie E. Leon 16